Combined QP - C3 Edexcel - Physics & Maths...

60
PhysicsAndMathsTutor.com

Transcript of Combined QP - C3 Edexcel - Physics & Maths...

Page 1: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

PhysicsAndMathsTutor.com

Physics & Maths Tutor
Typewritten Text
Edexcel Maths C3
Physics & Maths Tutor
Typewritten Text
Topic Questions from Papers
Physics & Maths Tutor
Typewritten Text
Trigonometry
Page 2: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leaveblank

1. (a) Given that sin2θ + cos2θ ≡ 1, show that 1 + tan2θ ≡ sec2θ.(2)

(b) Solve, for 0 !"θ < 360°, the equation

2 tan2θ + secθ = 1,

giving your answers to 1 decimal place.(6)

__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

2 *n23494B0220*

physicsandmathstutor.com June 2005

Page 3: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leaveblank

5. (a) Using the identity cos(A + B) ≡ cosA cosB – sin A sinB, prove that

cos 2A ≡ 1 – 2 sin2A.(2)

(b) Show that

2 sin 2θ – 3 cos 2θ – 3 sin θ + 3 ≡ sin θ (4 cos θ + 6 sin θ – 3).(4)

(c) Express 4 cos θ + 6 sin θ in the form R sin(θ + α), where R > 0 and 0 < α < π.(4)

(d) Hence, for 0 ! θ < π, solve

2 sin 2θ = 3(cos 2θ + sin θ – 1),

giving your answers in radians to 3 significant figures, where appropriate.(5)

____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

12

10 *n23494B01020*

physicsandmathstutor.com June 2005

Page 4: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leaveblank

______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

11

Turn over*n23494B01120*

Question 5 continued

physicsandmathstutor.com June 2005

Page 5: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leaveblank

6. f(x) = 12 cos x – 4 sin x.

Given that f(x) = R cos(x + α), where R 0 and 0 α 90°,

(a) find the value of R and the value of α.(4)

(b) Hence solve the equation

12 cos x – 4 sin x = 7

for 0 x < 360°, giving your answers to one decimal place.(5)

(c) (i) Write down the minimum value of 12 cos x – 4 sin x.(1)

(ii) Find, to 2 decimal places, the smallest positive value of x for which thisminimum value occurs.

(2)

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

12 *N23495A01220*

physicsandmathstutor.com January 2006

Page 6: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leaveblank

Question 6 continued

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________ Q6

(Total 12 marks)

13

Turn over*N23495A01320*

physicsandmathstutor.com January 2006

Page 7: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leaveblank

7. (a) Show that

(i) x ≠ (n – )π, n ∈ Z,(2)

(ii) (cos 2x – sin 2x) ≡ cos2 x – cos x sin x – .(3)

(b) Hence, or otherwise, show that the equation

can be written as

sin 2θ = cos 2θ.(3)

(c) Solve, for 0 θ < 2π,

sin 2θ = cos 2θ,

giving your answers in terms of π.(4)

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

cos 2 1coscos sin 2

θθθ θ

⎛ ⎞ =⎜ ⎟+⎝ ⎠

12

12

14

cos 2 cos sin ,cos sin

x x xx x

≡ −+

14 *N23495A01420*

physicsandmathstutor.com January 2006

Page 8: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leaveblank

Question 7 continued

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

15

Turn over*N23495A01520*

physicsandmathstutor.com January 2006

Page 9: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leaveblank

14

6. (a) Using sin2θ + cos2θ ≡ 1, show that cosec2θ – cot2θ ≡ 1.(2)

(b) Hence, or otherwise, prove that

cosec4θ – cot4θ ≡ cosec2θ + cot2θ.(2)

(c) Solve, for 90° < θ < 180°,

cosec4θ – cot4θ = 2 – cot θ.(6)

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

*N23581A01424*

physicsandmathstutor.com June 2006

Page 10: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leaveblank

22

8. (a) Given that where 270° < A < 360°, find the exact value of sin 2A.(5)

(b) (i) Show that (3)

Given that

(ii) show that (4)

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

d sin 2 .dy xx

=

23sin cos 2 cos 2 ,3 3

y x x xπ π= + + + −

cos 2 cos 2 cos 2 .3 3

x x xπ π+ + − ≡

3cos ,4

A =

*N23581A02224*

physicsandmathstutor.com June 2006

Page 11: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leaveblank

24

Question 8 continued

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

TOTAL FOR PAPER: 75 MARKS

END

*N23581A02424*

Q8

(Total 12 marks)

physicsandmathstutor.com June 2006

Page 12: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leaveblank

2

1. (a) By writing sin 3θ as sin (2θ +θ , show that

(5)

(b) Given that sin , find the exact value of sin3θ .(2)

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

34

θ =

3sin 3 3sin 4sin .θ θ θ= −

*N23583A0224*

physicsandmathstutor.com January 2007

Page 13: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leaveblank

12

5. Figure 1

Figure 1 shows an oscilloscope screen.

The curve shown on the screen satisfies the equation

(a) Express the equation of the curve in the form y = Rsin(x + ), where R and areconstants,

(4)

(b) Find the values of x, , for which y = 1.(4)

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

0 2x π<-

0 and 0 .2

R π< <

cos sin .y x x= +

*N23583A01224*

physicsandmathstutor.com January 2007

Page 14: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leaveblank

13

Question 5 continued

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

Turn over

Q5

(Total 8 marks)

*N23583A01324*

physicsandmathstutor.com January 2007

Page 15: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leaveblank

22

8. (i) Prove that

(3)

(ii) Given that

(a) express arcsin x in terms of y.(2)

(b) Hence evaluate arccos x + arcsin x. Give your answer in terms of π.(1)

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

arccos , 1 1 and 0 ,y x x y π= − - -� -�-

2 2 2 2sec cosec tan cot .x x x x− ≡ −�

*N23583A02224*

physicsandmathstutor.com January 2007

Page 16: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leaveblank

24

Question 8 continued________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________________

TOTAL FOR PAPER: 75 MARKS

END

*N23583A02424*

Q8

(Total 6 marks)

physicsandmathstutor.com January 2007

Page 17: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leave blank

14

*N26109A01424*

6. (a) Express 3 sin x + 2 cos x in the form R sin(x + α) where R > 0 and 0 < α < π2

.(4)

(b) Hence find the greatest value of (3 sin x + 2 cos x)4.(2)

(c) Solve, for 0 < x < 2π, the equation

3 sin x + 2 cos x = 1,

giving your answers to 3 decimal places.(5)

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

physicsandmathstutor.com June 2007

Page 18: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leave blank

15

Turn over*N26109A01524*

Question 6 continued

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

physicsandmathstutor.com June 2007

Page 19: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leave blank

18

*N26109A01824*

7. (a) Prove that

sin cos 2cosec 2 , 90 .cos sin

θ θ θ θ nθ θ+ = ≠ °

(4)

(b) On the axes on page 20, sketch the graph of y = 2 cosec 2θ for 0° < θ < 360°.(2)

(c) Solve, for 0° < θ < 360°, the equation

sincos

cossin

,θθ

θθ

+ = 3

giving your answers to 1 decimal place.(6)

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

physicsandmathstutor.com June 2007

Page 20: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leave blank

20

*N26109A02024*

Question 7 continued

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

_________________________________________________________________________________________________________________________________

y

O 90° 180° 270° 360° θ

physicsandmathstutor.com June 2007

Page 21: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leave blank

14

*H26315RB01424*

6. (a) Use the double angle formulae and the identity

cos( ) cos cos sin sinA B A B A B+ −

to obtain an expression for cos 3x in terms of powers of cos x only.(4)

(b) (i) Prove that

cossin

sincos

sec , ( )xx

xx

x x n1

1 2 2 12+

+ + ≡ ≠ + π .

(4)

(ii) Hence find, for 0 x< < 2π , all the solutions of

cossin

sincos

xx

xx1

1 4+

++

= .

(3)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

physicsandmathstutor.com January 2008

Page 22: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leave blank

15

Turn over*H26315RB01524*

Question 6 continued

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

physicsandmathstutor.com January 2008

Page 23: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leave blank

4

*N30745A0424*

2.

Given that where R > 0 and

(a) find the value of R and the value of α to 3 decimal places.(4)

(b) Hence solve the equation

for 0 x < 2π.(5)

(c) (i) Write down the maximum value of (1)

(ii) Find the smallest positive value of x for which this maximum value occurs.(2)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

f ( ) cos sinx x x= +5 12

f ( ) cos( ),x R x= −α 0 α π ,

5 12 6cos sinx x+ =

5 12cos sin .x x+

physicsandmathstutor.com June 2008

Page 24: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leave blank

5

*N30745A0524* Turn over

Question 2 continued

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

physicsandmathstutor.com June 2008

Page 25: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leave blank

16

*N30745A01624*

5. (a) Given that sin2 θ + cos2 θ ≡ 1, show that 1 + cot2 θ ≡ cosec2 θ .(2)

(b) Solve, for 0 θ < 180°, the equation

2 cot2 θ – 9 cosec θ = 3,

giving your answers to 1 decimal place.(6)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

physicsandmathstutor.com June 2008

Page 26: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leave blank

16

*H31123A01628*

6. (a) (i) By writing 3θ = (2θ + θ), show thatsin 3θ = 3 sin θ – 4 sin3θ.

(4)

(ii) Hence, or otherwise, for solve

8 sin3θ – 6 sin θ + 1 = 0.

Give your answers in terms of .(5)

(b) Using or otherwise, show that

(4)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

0 θ π3

,

sin( ) sin cos cos sin ,θ α θ α θ α− = −

sin15 14

= − .

physicsandmathstutor.com January 2009

Page 27: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leave blank

17

*H31123A01728* Turn over

Question 6 continued

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

physicsandmathstutor.com January 2009

Page 28: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leave blank

24

*H31123A02428*

8. (a) Express 3 cos θ + 4 sin θ in the form R cos(θ – α), where R and α are constants, R > 0 and 0 < α < 90°.

(4)

(b) Hence find the maximum value of 3 cos θ + 4 sin θ and the smallest positive value of θ for which this maximum occurs.

(3)

The temperature, f (t), of a warehouse is modelled using the equation

f (t) = 10 + 3 cos(15t)° + 4 sin(15t)°,

where t is the time in hours from midday and 0 t < 24.

(c) Calculate the minimum temperature of the warehouse as given by this model.(2)

(d) Find the value of t when this minimum temperature occurs.(3)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

physicsandmathstutor.com January 2009

Page 29: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leave blank

27

*H31123A02728*

Question 8 continued

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

TOTAL FOR PAPER: 75 MARKSEND

Q8

(Total 12 marks)

physicsandmathstutor.com January 2009

Page 30: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leave blank

18

*H34264A01828*

6. (a) Use the identity cos( ) cos cos sin sin ,A B A B A B+ = − to show that

cos sin2 1 2 2A A= −(2)

The curves C1 and C2 have equations

C1: 3sin 2y x=

C2: y x= − 2 cos 2x4 2sin

(b) Show that the x-coordinates of the points where C1 and C2 intersect satisfy the equation

4cos 2 3sin 2 2x x+ =(3)

(c) Express 4cos 2 3sin 2x x+ in the form R cos (2x – α), where R > 0 and 0 < α < 90°, giving the value of α to 2 decimal places.

(3)

(d) Hence find, for 0 180x < °, all the solutions of

4cos 2 3sin 2 2x x+ =

giving your answers to 1 decimal place.(4)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

physicsandmathstutor.com June 2009

Page 31: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leave blank

19

*H34264A01928* Turn over

Question 6 continued

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

physicsandmathstutor.com June 2009

Page 32: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leave blank

22

*N35381A02228*

8. Solvecosec2 2x – cot 2x = 1

for 0 - x - 180°.(7)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

physicsandmathstutor.com January 2010

Page 33: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leave blank

2

*H35385A0228*

1. (a) Show that

sincos

tan21 2

θθ

θ+

=

(2)

(b) Hence find, for –180° - θ < 180°, all the solutions of

sincos

21 2

1θθ+

=2

Give your answers to 1 decimal place.(3)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

physicsandmathstutor.com June 2010

Page 34: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leave blank

22

*H35385A02228*

7. (a) Express 2 sin θ – 1.5 cos θ in the form R sin (θ – α), where R > 0 and 0 < <α π2

.

Give the value of α to 4 decimal places.(3)

(b) (i) Find the maximum value of 2 sin θ – 1.5 cos θ.

(ii) Find the value of θ, for 0 - θ < , at which this maximum occurs. (3)

Tom models the height of sea water, H metres, on a particular day by the equation

H t t= + −6 2 425

1 5 425

sin . cosπ π⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

, 0 12,t- <

where t hours is the number of hours after midday.

(c) Calculate the maximum value of H predicted by this model and the value of t, to 2 decimal places, when this maximum occurs.

(3)

(d) Calculate, to the nearest minute, the times when the height of sea water is predicted, by this model, to be 7 metres.

(6)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

physicsandmathstutor.com June 2010

Page 35: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leave blank

23

*H35385A02328* Turn over

Question 7 continued

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

physicsandmathstutor.com June 2010

Page 36: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leave blank

8

*H35404RA0828*

3. Find all the solutions of

2 cos 2 = 1 – 2 sin

in the interval 0 360°.(6)

________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

physicsandmathstutor.com January 2011

Page 37: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leave blank

12

*P38159A01224*

6. (a) Prove that

12

22

90sin

cossin

tan , ,θ

θθ

θ θ− = ≠ °n n∈!(4)

(b) Hence, or otherwise,

(i) show that tan 15° = 2 – 3,(3)

(ii) solve, for 0 x 360°,

cosec 4x – cot 4x = 1(5)

______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

physicsandmathstutor.com June 2011

Page 38: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leave blank

13

Turn over*P38159A01324*

Question 6 continued_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

physicsandmathstutor.com June 2011

Page 39: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leave blank

10

*P40084A01024*

5. Solve, for 0 180°,

2cot2 3 7 cosec 3 – 5

Give your answers in degrees to 1 decimal place.(10)

________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

physicsandmathstutor.com January 2012

Page 40: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leave blank

12

*P40084A01224*

6. f cosx x x x( ) = − + ( )2 123 2 , 0 x

(a) Show that the equation f (x) = 0 has a solution in the interval 0.8 < x < 0.9(2)

The curve with equation y = f (x) has a minimum point P.

(b) Show that the x-coordinate of P is the solution of the equation

xx

=+ ( )3

2

12sin

(4)

(c) Using the iteration formula

xx

xnn

+ =+ ( )

=1

12

0

32

2sin

,

find the values of x1, x2 and x3 , giving your answers to 3 decimal places.(3)

(d) By choosing a suitable interval, show that the x-coordinate of P is 1.9078 correct to 4 decimal places.

(3)_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

physicsandmathstutor.com January 2012

Page 41: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leave blank

13

Turn over*P40084A01324*

Question 6 continued_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

physicsandmathstutor.com January 2012

Page 42: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leave blank

20

*P40084A02024*

8. (a) Starting from the formulae for sin ( A + B ) and cos ( A + B ), prove that

tan tan tantan tan

A B A BA B

+( ) = +−1 (4)

(b) Deduce that1 3tantan

6 3 tanπ θθ

θ+⎛ ⎞+ =⎜ ⎟⎝ ⎠ − (3)

(c) Hence, or otherwise, solve, for 0! !θ π ,

1 3 3+ = −( ) −( )tan tan tanθ θ π θ√ √

Give your answers as multiples of (6)

________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

physicsandmathstutor.com January 2012

Page 43: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leave blank

24

*P40084A02424*

Question 8 continued________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

TOTAL FOR PAPER: 75 MARKS

END

Q8

(Total 13 marks)

physicsandmathstutor.com January 2012

Page 44: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leave blank

16

*P40686RA01632*

5. (a) Express 4 2 2cosec cosec2 θ θ− in terms of sin ! and cos !.(2)

(b) Hence show that

4 2 2cosec cosec2 θ θ− 2sec= θ

(4)

(c) Hence or otherwise solve, for 0 < !"<!" ,

4 2 2cosec cosec2 θ θ− 4=

giving your answers in terms of ". (3)

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

physicsandmathstutor.com June 2012

Page 45: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leave blank

28

*P40686RA02832*

8. f ( ) cos sinx x x= −7 2 24 2

Given that f ( ) cos( )x R x= +2 α , where R" 0 and 0 90< < !α ,

(a) find the value of R and the value of $ . (3)

(b) Hence solve the equation

7 2 24 2 12 5cos sin .x x− =

for 0 180! x# " , giving your answers to 1 decimal place.(5)

(c) Express 14 482cos sin cosx x x− in the form a x b x ccos sin2 2+ + , where a, b, and c are constants to be found.

(2)

(d) Hence, using your answers to parts (a) and (c), deduce the maximum value of

14 482cos sin cosx x x−(2)

____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

physicsandmathstutor.com June 2012

Page 46: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leave blank

32

*P40686RA03232*

Question 8 continued________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

TOTAL FOR PAPER: 75 MARKS

END

Q8

(Total 12 marks)

physicsandmathstutor.com June 2012

Page 47: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leave blank

10

*P41486A01028*

4. (a) Express 6 cos!"!+!8!sin!" in the form R!cos("!!"!), where R > 0 and 0 2< < πα .

Give the value of !#to 3 decimal places.(4)

(b) p( ) cos sin" " "= + +4

12 6 8 , 0 2! !θ π

Calculate

(i) the maximum value of p("),

(ii) the value of " at which the maximum occurs.(4)

______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

physicsandmathstutor.com January 2013

Page 48: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leave blank

18

*P41486A01828*

6. (i) Without using a calculator, find the exact value of

(sin 22.5° + cos 22.5°)2

You must show each stage of your working.(5)

(ii) (a) Show that cos 2$ + sin $#= 1 may be written in the form

k sin2 $ – sin $#= 0, stating the value of k.(2)

(b) Hence solve, for 0 "#$#< 360°, the equation

cos 2$ + sin $#= 1(4)

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

physicsandmathstutor.com January 2013

Page 49: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leave blank

19

*P41486A01928* Turn over

Question 6 continued_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

physicsandmathstutor.com January 2013

Page 50: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

physicsandmathstutor.com June 2013 (R)

Page 51: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

physicsandmathstutor.com June 2013 (R)

Page 52: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

physicsandmathstutor.com June 2013 (R)

Page 53: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

physicsandmathstutor.com June 2013 (R)

Page 54: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

physicsandmathstutor.com June 2013 (R)

Page 55: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leave blank

8

*P43016A0832*

3. Given that

2cos(x + 50)! = sin(x + 40)!

(a) Show, without using a calculator, that

tan x! = 13

tan "#!(4)

(b) Hence solve, for 0 ! ! < 360,

2cos(2! + 50)! = sin(2! + 40)!

giving your answers to 1 decimal place.(4)

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

physicsandmathstutor.com June 2013

Page 56: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leave blank

28

*P43016A02832*

8.

Figure 2

Kate crosses a road, of constant width 7 m, in order to take a photograph of a marathon runner, John, approaching at 3 m s–1.

Kate is 24 m ahead of John when she starts to cross the road from the fixed point A. John passes her as she reaches the other side of the road at a variable point B, as shown in

Figure 2. Kate’s speed is V m s–1 and she moves in a straight line, which makes an angle !,

0 < ! < 150!, with the edge of the road, as shown in Figure 2.

You may assume that V is given by the formula

V = 21

24sin 7cos+θ θ , 0 < ! < 150!

(a) Express 24sin! + 7cos! in the form Rcos(! – #), where R and # are constants and where R > 0 and 0 < # < 90!, giving the value of # to 2 decimal places.

(3)

Given that !$varies,

(b) find the minimum value of V.(2)

Given that Kate’s speed has the value found in part (b),

(c) find the distance AB.(3)

Given instead that Kate’s speed is 1.68 m s–1,

(d) find the two possible values of the angle !, given that 0 < !$< 150!.(6)

__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

3 m s–1

V m s–1

B

!A

24 m

7 m

physicsandmathstutor.com June 2013

Page 57: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Leave blank

32

*P43016A03232*

Question 8 continued________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

TOTAL FOR PAPER: 75 MARKSEND

Q8

(Total 14 marks)

physicsandmathstutor.com June 2013

Page 58: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

6 Edexcel AS/A level Mathematics Formulae List: Core Mathematics C3 – Issue 1 – September 2009

Core Mathematics C3 Candidates sitting C3 may also require those formulae listed under Core Mathematics C1 and C2.

Logarithms and exponentials

xax a=lne

Trigonometric identities

BABABA sincoscossin)(sin ±=± BABABA sinsincoscos)(cos m=±

))(( tantan1tantan)(tan 2

1 π+≠±±=± kBABABABA

m

2cos

2sin2sinsin BABABA −+=+

2sin

2cos2sinsin BABABA −+=−

2cos

2cos2coscos BABABA −+=+

2sin

2sin2coscos BABABA −+−=−

Differentiation

f(x) f ′(x)

tan kx k sec2 kx

sec x sec x tan x

cot x –cosec2 x

cosec x –cosec x cot x

)g()f(

xx

))(g(

)(g)f( )g()(f2x

xxxx ′−′

Page 59: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

Edexcel AS/A level Mathematics Formulae List: Core Mathematics C2 – Issue 1 – September 2009 5

Core Mathematics C2 Candidates sitting C2 may also require those formulae listed under Core Mathematics C1.

Cosine rule

a2 = b2 + c2 – 2bc cos A

Binomial series

2

1

)( 221 nrrnnnnn bbarn

ban

ban

aba ++++++=+ −−− KK (n ∈ ℕ)

where )!(!

!C rnr

nrn

rn

−==

∈<+×××

+−−++×−++=+ nxx

rrnnnxnnnxx rn ,1(

21)1()1(

21)1(1)1( 2 K

K

KK ℝ)

Logarithms and exponentials

ax

xb

ba log

loglog =

Geometric series un = arn − 1

Sn = r ra n

−−

1)1(

S∞ = r

a−1

for ⏐r⏐ < 1

Numerical integration

The trapezium rule: b

a

xy d ≈ 21 h{(y0 + yn) + 2(y1 + y2 + ... + yn – 1)}, where

nabh −=

Page 60: Combined QP - C3 Edexcel - Physics & Maths Tutorpmt.physicsandmathstutor.com/download/Maths/A-level/C3/Topic-Qs... · Leave blank. 24 *H31123A02428* 8. (a) Express 3 cos θ + 4 sin

4 Edexcel AS/A level Mathematics Formulae List: Core Mathematics C1 – Issue 1 – September 2009

Core Mathematics C1

Mensuration

Surface area of sphere = 4π r 2

Area of curved surface of cone = π r × slant height

Arithmetic series

un = a + (n – 1)d

Sn = 21 n(a + l) =

21 n[2a + (n − 1)d]