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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)

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2 *n23494B0220*

physicsandmathstutor.com June 2005C3 past papers C3

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2. (a) Differentiate with respect to x

(i) 3 sin2x + sec2x,(3)

(ii) {x + ln(2x)}3.

(3)

Given that

(b) show that

(6)

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3

d 8.

d ( 1)

yx x

= −−

2

2

5 10 9, 1,

( 1)

x xy xx− +

= ≠−

4 *n23494B0420*

physicsandmathstutor.com June 2005

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3. The function f is defined by

(a) Show that

(4)

(b) Find f –1(x).(3)

The function g is defined by

g : x → x2 + 5, x ∈ R.

(c) Solve fg(x) = .

(3)

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14

2f ( ) , 1.

1x x

x= >

−

2

5 1 3f : , 1.

2 2

xx xx x x

+→ − >

+ − +

6 *n23494B0620*

physicsandmathstutor.com June 2005

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4. f(x) = 3ex – ln x – 2, x > 0.

(a) Differentiate to find f ´(x).(3)

The curve with equation y = f(x) has a turning point at P. The x-coordinate of P is α.

(b) Show that α = e–α.

(2)

The iterative formula

is used to find an approximate value for α.

(c) Calculate the values of x1, x2, x3 and x4, giving your answers to 4 decimal places.

(2)

(d) By considering the change of sign of f ´(x) in a suitable interval, prove that α = 0.1443

correct to 4 decimal places.

(2)

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11 06

e , 1,nxnx x−

+ = =

16

12

8 *n23494B0820*

physicsandmathstutor.com June 2005

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5. (a) Using the identity cos(A + B) ≡ cosA cosB – sinA 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)

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12

10 *n23494B01020*

physicsandmathstutor.com June 2005

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6. Figure 1

Figure 1 shows part of the graph of y = f(x), x ∈ R. The graph consists of two line

segments that meet at the point (1, a), a < 0. One line meets the x-axis at (3, 0). The other

line meets the x-axis at (–1, 0) and the y-axis at (0, b), b < 0.

In separate diagrams, sketch the graph with equation

(a) y = f(x + 1),

(2)

(b) y = f( |x | ).

(3)

Indicate clearly on each sketch the coordinates of any points of intersection with the axes.

Given that f(x) = |x – 1| – 2, find

(c) the value of a and the value of b,(2)

(d) the value of x for which f(x) = 5x.(4)

14 *n23494B01420*

y

xb

–1 O 3

(1, a)

physicsandmathstutor.com June 2005

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7. A particular species of orchid is being studied. The population p at time t years after the

study started is assumed to be

where a is a constant.

Given that there were 300 orchids when the study started,

(a) show that a = 0.12,

(3)

(b) use the equation with a = 0.12 to predict the number of years before the population

of orchids reaches 1850.

(4)

(c) Show that

(1)

(d) Hence show that the population cannot exceed 2800.

(2)

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0.2

336.

0.12 e tp −=+

0.2

0.2

2800 e,

1 e

t

t

apa

=+

18 *n23494B01820*

physicsandmathstutor.com June 2005

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1. Figure 1

Figure 1 shows the graph of y = f(x), –5 x 5.The point M (2, 4) is the maximum turning point of the graph.

Sketch, on separate diagrams, the graphs of

(a) y = f(x) + 3,(2)

(b) y = | f(x) | ,(2)

(c) y = f( |x | ).(3)

Show on each graph the coordinates of any maximum turning points.

2 *N23495A0220*

y

x–5 O 5

M (2, 4)

physicsandmathstutor.com January 2006C3

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2. Express

as a single fraction in its simplest form.(7)

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2

2

2 3 6

(2 3)( 2) 2

x x

x x x x

+ −+ − − −

4 *N23495A0420*

physicsandmathstutor.com January 2006

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3. The point P lies on the curve with equation The x-coordinate of P is 3.

Find an equation of the normal to the curve at the point P in the form y = ax + b, wherea and b are constants.

(5)

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1ln .

3y x

⎛ ⎞= ⎜ ⎟⎝ ⎠

6 *N23495A0620*

physicsandmathstutor.com January 2006

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4. (a) Differentiate with respect to x

(i) x2e3x + 2,(4)

(ii) .(4)

(b) Given that x = 4 sin(2y + 6), find in terms of x.

(5)

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d

d

y

x

3cos(2 )

3

x

x

8 *N23495A0820*

physicsandmathstutor.com January 2006

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5. f(x) = 2x3 – x – 4.

(a) Show that the equation f(x) = 0 can be written as

x = √(3)

The equation 2x3 – x – 4 = 0 has a root between 1.35 and 1.4.

(b) Use the iteration formula

xn + 1 = √with x0 = 1.35, to find, to 2 decimal places, the values of x1, x2 and x3.

(3)

The only real root of f(x) = 0 is α.

(c) By choosing a suitable interval, prove that α = 1.392, to 3 decimal places.(3)

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2 1,

2nx

⎛ ⎞+⎜ ⎟

⎝ ⎠

2 1.

2x⎛ ⎞+⎜ ⎟⎝ ⎠

10 *N23495A01020*

physicsandmathstutor.com January 2006

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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)

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12 *N23495A01220*

physicsandmathstutor.com January 2006

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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)

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cos 2 1cos

cos sin 2

θθθ θ

⎛ ⎞ =⎜ ⎟+⎝ ⎠

12

12

14

cos 2cos sin ,

cos sin

xx x

x x≡ −

+

14 *N23495A01420*

physicsandmathstutor.com January 2006

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8. The functions f and g are defined by

f:x → 2x + ln 2, x ∈ R,

g :x → e2x, x ∈ R.

(a) Prove that the composite function gf is

gf :x → 4e4x, x ∈ R.(4)

(b) In the space provided on page 19, sketch the curve with equation y = gf(x), and showthe coordinates of the point where the curve cuts the y-axis.

(1)

(c) Write down the range of gf.(1)

(d) Find the value of x for which , giving your answer to 3 significant

figures.(4)

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d[gf ( )] 3

dx

x=

18 *N23495A01820*

physicsandmathstutor.com January 2006

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2

1. (a) Simplify .(3)

(b) Hence, or otherwise, express as a single fraction in its simplest

form.(3)

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2

2

3 2 1

1 ( 1)

x x

x x x

− −−

− +

2

2

3 2

1

x x

x

− −−

*N23581A0224*

physicsandmathstutor.com June 2006C3

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4

2. Differentiate, with respect to x,

(a) e3x + ln 2x,(3)

(b) .(3)

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322(5 )x+

Q2

(Total 6 marks)

*N23581A0424*

physicsandmathstutor.com June 2006

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5

3. Figure 1

Figure 1 shows part of the curve with equation y = f(x), x ∈R, where f is an increasingfunction of x. The curve passes through the points P(0, –2) and Q(3, 0) as shown.

In separate diagrams, sketch the curve with equation

(a) y = |f (x) | ,(3)

(b) y = f –1(x),(3)

(c) y = f(3x).(3)

Indicate clearly on each sketch the coordinates of the points at which the curve crosses ormeets the axes.

12

Turn over*N23581A0524*

y

xO

P

Q (3, 0)

y = f (x)

(0, –2)

physicsandmathstutor.com June 2006

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8

4. A heated metal ball is dropped into a liquid. As the ball cools, its temperature, T °C, t minutes after it enters the liquid, is given by

T = 400 e–0.05t + 25, t 0.

(a) Find the temperature of the ball as it enters the liquid.(1)

(b) Find the value of t for which T = 300, giving your answer to 3 significant figures.(4)

(c) Find the rate at which the temperature of the ball is decreasing at the instant when t = 50. Give your answer in °C per minute to 3 significant figures.

(3)

(d) From the equation for temperature T in terms of t, given above, explain why thetemperature of the ball can never fall to 20 °C.

(1)

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*N23581A0824*

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10

5. Figure 2

Figure 2 shows part of the curve with equation

The curve has a minimum at the point P. The x-coordinate of P is k.

(a) Show that k satisfies the equation

4k + sin 4k – 2 = 0.(6)

The iterative formula

is used to find an approximate value for k.

(b) Calculate the values of x1, x2, x3 and x4, giving your answers to 4 decimal places.(3)

(c) Show that k = 0.277, correct to 3 significant figures.(2)

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1 0

1(2 sin 4 ), 0.3,

4n nx x x+ = − =

(2 1) tan 2 , 0 .4

y x x xπ

= − <

*N23581A01024*

y

O

P

x4

π

physicsandmathstutor.com June 2006

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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)

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*N23581A01424*

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18

7. For the constant k, where k > 1, the functions f and g are defined by

f: x ln (x + k), x > –k,g: x |2x – k |, x ∈ R.

(a) On separate axes, sketch the graph of f and the graph of g.

On each sketch state, in terms of k, the coordinates of points where the graph meets thecoordinate axes.

(5)

(b) Write down the range of f.(1)

(c) Find in terms of k, giving your answer in its simplest form.(2)

The curve C has equation y = f(x). The tangent to C at the point with x-coordinate 3 isparallel to the line with equation 9y = 2x + 1.

(d) Find the value of k.(4)

fg4

k

*N23581A01824*

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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)

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dsin 2 .

d

yx

x=

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 ,

4A =

*N23581A02224*

physicsandmathstutor.com June 2006

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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)

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3

4θ

√=

3sin 3 3sin 4sin .θ θ θ= −

*N23583A0224*

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4

2.

(a) Show that (4)

(b) Show that x2 + x + 1 > 0 for all values of x.(3)

(c) Show that f (x) > 0 for all values of x, x ≠ –2.(1)

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2

2

1f ( ) , 2.

( 2)

x xx x

x

+ += ≠ −

+

2

3 3f ( ) 1 , 2.

2 ( 2)x x

x x= − + ≠ −

+ +

*N23583A0424*

physicsandmathstutor.com January 2007

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6

3. The curve C has equation

x = 2 sin y.

(a) Show that the point lies on C.(1)

(b) Show that at P.(4)

(c) Find an equation of the normal to C at P. Give your answer in the form y = mx + c,where m and c are exact constants.

(4)

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d 1

d

y

x=

√ 2

,4

Pπ √ 2

*N23583A0624*

physicsandmathstutor.com January 2007

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8

4. (i) The curve C has equation

Use calculus to find the coordinates of the turning points of C.(6)

(ii) Given that

find the value of .(5)

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d 1at ln 3

d 2

yx

x=

32 2(1 e ) ,xy = +

2.

9

xy

x=

+

*N23583A0824*

physicsandmathstutor.com January 2007

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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 + α), where R and α are

constants, (4)

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

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0 2x π<

0 and 0 .2

Rπ

α> < <

cos sin .y x x= √ 3 +

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6. The function f is defined by

(a) Show that the inverse function of f is defined by

and write down the domain of f –1.(4)

(b) Write down the range of f –1.(1)

(c) In the space provided on page 16, sketch the graph of y = f –1(x). State the coordinatesof the points of intersection with the x and y axes.

(4)

The graph of y = x + 2 crosses the graph of y = f –1(x) at x = k.

The iterative formula

is used to find an approximate value for k.

(d) Calculate the values of x1 and x2, giving your answers to 4 decimal places.(2)

(e) Find the value of k to 3 decimal places.(2)

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1 0

1e , 0.3

2nx

nx x+ = − = −

1 1f : 2 e

2xx− −

f : ln (4 2 ), 2 and .x x x x− < ∈

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18

7.

(a) Show that there is a root of f (x) = 0 in the interval [–2, –1].(3)

(b) Find the coordinates of the turning point on the graph of y = f (x).(3)

(c) Given that f (x) = (x – 2)(x3 + ax2 + bx + c), find the values of the constants, a, b and c.(3)

(d) In the space provided on page 21, sketch the graph of y = f (x).(3)

(e) Hence sketch the graph of y = |f (x)|.(1)

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4f ( ) 4 8.x x x= − −

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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)

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arccos , 1 1 and 0 ,y x x y π= −

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

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