hsn€¦ · Part Marks Level Calc. Content Answer U3 OC2 1 C NC C21, C19 1994 P1 Q17 3 A/B NC C21,...

Higher Mathematics

A/B Integration

Paper 1 Section B

1.[SQA] Find∫ √1+ 3x dx and hence find the exact value of

∫ 1

0

√1+ 3x dx . 4

Part Marks Level Calc. Content Answer U3 OC2

4 A/B NC C22 1993 P1 Q16

2.[SQA] The graph of y = f (x) passes through the point(

π

9 , 1)

.

If f ′(x) = sin(3x) express y in terms of x . 4


4 A/B NC C18, C23 y = − 13 cos(3x) + 76 2000 P1 Q8

•1 ss: know to integrate•2 pd: integrate•3 ic: interpret ( π

9 , 1)•4 pd: process

•1 y =∫

sin(3x) dx stated or implied by•2

•2 − 13 cos(3x)•3 1 = − 13 cos( 3π9 ) + c or equiv.

•4 c = 76

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Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes

Higher Mathematics

3.[SQA]


(a) 1 C NC T1 1998 P1 Q15

(b) 1 C NC C16

(b) 3 A/B NC C23

4.[SQA]

(a) Evaluate∫ π

2

0cos 2x dx . 3

(b) Draw a sketch and explain your answer. 2


(a) 3 A/B NC C23 1992 P1 Q14

(b) 1 C NC T1, C16

(b) 1 A/B NC T1, C16

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Higher Mathematics

5.[SQA]

(a) Show that (cos x+ sin x)2 = 1+ sin 2x . 1

(b) Hence find∫

(cos x+ sin x)2 dx . 3


(a) 1 C NC T8 1993 P1 Q19

(b) 3 A/B NC C23

6.[SQA] The curve y = f (x) passes through the point ( π

12 , 1) and f′(x) = cos 2x .

Find f (x) . 3


3 A/B NC C23 1997 P1 Q15

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Higher Mathematics

7.[SQA]

(a) By writing sin 3x as sin(2x+ x) , show that sin 3x = 3 sin x− 4 sin3 x . 4

(b) Hence find∫

sin3 x dx . 4


(a) 2 C NC T8, T8 1995 P2 Q9

(a) 2 A/B NC T8, T8

(b) 4 A/B NC C23

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Higher Mathematics

8.[SQA]


(a) 4 C NC CGD 1989 P2 Q8

(b) 2 C NC C23, C15

(b) 3 A/B NC C23, C15

(c) 2 C NC T1

(d) 2 A/B NC CGD

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Higher Mathematics

9.[SQA]


2 C NC C23, C16 1996 P2 Q5

4 A/B NC C23, C16

10.[SQA] Find∫

x2 − 5x√xdx . 4


2 C NC C14 1999 P1 Q20

2 A/B NC C13

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Higher Mathematics

11.[SQA] Find the value of∫ 2

1

u2 + 2

2u2du . 5


4 C NC C15 1989 P1 Q16

1 A/B NC C15

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Higher Mathematics

12.[SQA]


(a) 5 C NC C17 1995 P2 Q10

(b) 3 A/B NC C15

(c) 2 C NC A21, A22

(c) 3 A/B NC A21

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Higher Mathematics

13.[SQA]


(a) 3 C NC G2, G8 1989 P2 Q10

(a) 2 A/B NC G2, G8

(b) 1 C NC G3, A6

(b) 3 A/B NC G3, A6

(c) 6 A/B NC C16

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Higher Mathematics

14.[SQA]


(a) 2 C NC A21 1997 P2 Q4

(b) 6 C NC C16

(b) 1 A/B NC C16

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Higher Mathematics

15.[SQA]


(a) 3 C NC C17 1990 P2 Q7

(a) 3 A/B NC C17

(b) 2 C NC C17

(b) 2 A/B NC C17

(c) 4 A/B NC CGD

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Higher Mathematics

16.[SQA]


(a) 2 C NC A7 1998 P2 Q4

(b) 4 C NC C16

(ci) 2 C NC A23

(cii) 3 A/B NC C17

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Higher Mathematics

17.[SQA]


(a) 2 C NC A6 1999 P2 Q10

(b) 7 A/B NC C17

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Higher Mathematics

18.[SQA] Differentiate sin3 x with respect to x .

Hence find∫

sin2 x cos x dx . 4


1 C NC C21, C19 1994 P1 Q17

3 A/B NC C21, C19

[END OF PAPER 1 SECTION B]

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All others c© Higher Still Notes

Higher Mathematics

Paper 2

1. (a) A curve has equation y = (2x− 9) 12 .Show that the equation of the tangent to this curve at the point where x = 9is y = 1

3x . 5

(b) Diagram 1 shows part of the curve and the tangent.

The curve cuts the x-axis at the point A.

A 9

13

=

12(2 9)= −

Diagram 1

O

x

x

x

y

y

y

Find the coordinates of point A. 1

(c) Calculate the shaded area shown in diagram 2. 7

9

13

=

12(2 9)= −

A

Diagram 2

O

x

x

x

y

y

y


(a) 5 B CN C21, C24 proof 2010 P2 Q6

(b) 1 C CN A6 ( 92 , 0)

(c) 7 A CN C17, C22 92 = 412 = 4·5

•1 ss: know to and start to differentiate•2 pd: complete chain rule derivative•3 pd: gradient via differentiation•4 pd: obtain ycurve at x = 9•5 ic: state equation and complete

•6 ic: obtain coordinates of A

•7 ss: strategy for finding shaded area•8 ss: know to integrate (2x− 9) 12•9 pd: start integration•10 pd: complete integration•11 ic: limits xA and 9•12 pd: substitute limits•13 pd: evaluate area and complete

•1 12(2x− 9)−12 · · ·

•2 · · · × 2•3 13•4 3•5 y− 3 = 1

3(x− 9) and complete

•6 ( 92 , 0)

•7 Shaded area = area of large△− area under curve•8

∫

(2x− 9) 12 dx

•9 (2x− 9) 3232

· · ·

•10 · · · × 12

•11 92 and 9

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Higher Mathematics

2.[SQA] Find∫

1

(7− 3x)2 dx . 2


2 A/B CN C22, C141

3(7− 3x) + c 2000 P2 Q10

•1 pd: integrate function•2 pd: deal with function of function

•1 1−1(7− 3x)−1

•2 × 1−3

3.[SQA] The graphs of y = f (x) and y = g(x) areshown in the diagram.

f (x) = −4 cos(2x) + 3 and g(x) is of theform g(x) = m cos(nx) .

(a) Write down the values of m and n . 1

(b) Find, correct to one decimal place,the coordinates of the points ofintersection of the two graphs in theinterval 0 ≤ x ≤ π . 5

(c) Calculate the shaded area. 6

π

= f( )

= g( )

7

3

0

–1

–3x

x

x

y

y

y


(a) 1 C CN T4 m = 3 and n = 2 2009 P2 Q5

(b) 5 C CR T6 (0·6, 1·3), (2·6, 1·3)(c) 6 B CR C17, C23 12·4

•1 ic: interprets graph

•2 ss: knows how to find intersection•3 pd: starts to solve•4 pd: finds x-coord in 1st quadrant•5 pd: finds x-coord in 2nd quadrant•6 pd: finds y-coordinates

•7 ss: knows how to find area•8 ic: states limits•9 pd: integrate•10 pd: integrate•11 ic: substitute limits•12 pd: evaluate area

•1 m = 3 and n = 2

•2 3 cos 2x = −4 cos 2x+ 3•3 cos 2x = 3

7•4 x = 0·6•5 x = 2·6•6 y = 1·3, 1·3

•7∫ (

−4 cos 2x+ 3− 3 cos 2x)

dx

•8∫ 2·60·6 · · ·

•9 “−7 sin 2x”•10 3x− 7

2 sin 2x

•11 (3× 2·6− 72 sin 5·2)− (3× 0·6− 7

2 sin 1·2)•12 12·4

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Higher Mathematics

4.[SQA] A curve for whichdy

dx= 3 sin(2x) passes through the point

(

5π12 ,

√3)

.

Find y in terms of x . 4


4 A/B CN C18, C23 y = − 32 cos(2x) + 14

√3 2001 P2 Q10

•1 pd: integrate trig function•2 pd: integrate composite function•3 ss: use given point to find “c”•4 pd: evaluate “c”

•1∫

3 sin(2x) dx stated or implied by •2•2 − 32 cos(2x)•3

√3 = − 32 cos(2× 5

12π) + c

•4 c = 14

√3 (≈ 0·4)

5. (a) The expression 3 sin x− 5 cos x can be written in the form R sin(x+ a) whereR > 0 and 0 ≤ a < 2π .

Calculate the values of R and a . 4

(b) Hence find the value of t , where 0 ≤ t ≤ 2, for which

t∫

0

(3 cos x+ 5 sin x) dx = 3.

7


(a) 4 C CN T13 R =√34, a = 5·253 2011 P2 Q6

(b) 7 B CN C23, T3, T16 t = 0·6

•1 ss: use compound angle formula•2 ic: compare coefficients•3 pd: process R•4 pd: process a

•5 pd: integrate given expression•6 ic: substitute limits•7 pd: process limits•8 ss: know to use wave equation•9 ic: write in standard format•10 ss: start to solve equation•11 pd: complete and state solution

•1 R sin x cos a+ R cos x sin a•2 R cos a = 3 and R sin a = −5•3

√34 (accept 5·8)

•4 5·253 (accept 5·3)

•5 3 sin x− 5 cos x•6 (3 sin t− 5 cos t) − (3 sin 0− 5 cos 0)•7 3 sin t− 5 cos t+ 5•8

√34 sin(t+ 5·3) + 5

•9 sin(t+ 5·3) = − 2√34

•10 t+ 5·3 = 3·5, 5·9•11 t = 0·6

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Higher Mathematics

6.[SQA]


(a) 4 C CR C16 1992 P2 Q10

(b) 2 C CR A6

(c) 2 C CR T16, C23, C17

(c) 8 A/B CR T16, C23, C17

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Higher Mathematics

7. The diagram shows the curve with equation y = x3− x2− 4x+ 4 and the line withequation y = 2x+ 4. The curve and the line intersect at the points (−2, 0) , (0, 4)and (3, 10) .

= 2 + 4

=3 – 2 – 4 + 4

–2

3O x

xxx

x

y

yy

Calculate the total shaded area. 10


10 B CN C17 21 112 2011 P2 Q4

•1 ss: know to integrate•2 ic: know to deal with areas on eachside of y-axis

•3 ic: interpret limits of one area•4 ic: use “upper− lower”•5 pd: integrate•6 ic: substitute in limits•7 pd: evaluate the area on one side•8 ss: interpret integrand with limitsof the other area

•9 pd: evaluate the area on the otherside

•10 ic: state total area

•1∫

· · · or attempt integration•2 evidence of treating areas separately•3 e.g.

∫ 30

•4 (2x+ 4) − (x3 − x2 − 4x+ 4)•5 3x2 + 1

3x3 − 1

4 x4

•6(

3(3)2 + 13(3)

3 − 14(3)

4)

•7 634•8

∫ 0−2(x

3 − x2 − 4x+ 4) − (2x+ 4) dx

•9 163•10 21 112 or

25312 or 21·1

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Higher Mathematics

8.[SQA]


(a) 5 C CN C4, G3, A23 1996 P2 Q8

(a) 3 A/B CN C4, G3, A23

(b) 3 A/B CN C17

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Higher Mathematics

9.[SQA]


3 C CN C17, CGD 1994 P2 Q10

6 A/B CN C17, CGD

[END OF PAPER 2]

hsn.uk.net Questions marked ‘[SQA]’ c© SQA

All others c© Higher Still Notes

hsn€¦ · Part Marks Level Calc. Content Answer U3 OC2 1 C NC C21, C19 1994 P1 Q17 3 A/B NC C21,...

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