SPM Past Year Questions : AM Form 5 Chapter 3 - Integration

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SPM PAST YEAR QUESTIONS ADDITIONAL MATHEMATICS

FORM 5 CHAPTER 3 : INTEGRATION

marks) (3 axis.-y about the 360 through rotated is

diagram in theregion shaded when thegenerated solid theof volume theCalculate a)2

marks) (2 .2

2)(for value thefind ,8)(Given b)

marks) (4 5)-(3x

18 ii)

,x)x)(4-(4

i)

Find 1a)

1993 SPM

0

3

0

3

0

3

2

dxxf

dxxf

dx

dxx

marks) (7 door. new theof surfacefront he t

of area thefind ,maintained are AKD arc theof shape theand BC of

width the whereas4m, toincreased are DC and AB ofheight theIf (iii)

surface.front sdoor' theof area theCalculate (ii)

BC. of width theCalculate (i)

BC. of level thefrompoint highest theisK and 3mDCABGiven

.4

x-4ygraph theofpart with shape same thehas

that arc lsymmetrica a is AKD ly.horizontal lying is BC andcally verti

standing are DC and AB door. a of surfacefront theshows diagram The b)

2

SPM Past Year Questions : AM Form 5 Chapter 3 - Integration

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marks) (3 1. when x6y if x of in termsy find ,14dx

dyGiven 4.

marks) (2 .7)(2x Find 3.

1994 SPM

3

x

dx

5.

marks) (5

region. shaded theof area theCalculate x.-7y and

74xy of graphs theshows diagram The 2

x

marks) (4

axis.- xabout the 360rough th

revolved isregion shaded when thegenerated solid of lume vo

theCalculate R. point, maximum at the curve thengent to ta

a is PR .3x-6xygraph theofpart shows diagram The a) 6. 2

1

1-

2

2

1

2

marks) (3 g(x)dx.for

value theFind in x.function a is g(x) where2g(x)dx

dy and

x

1-2xyGiven b)

marks) (3 .3x

3)3)(x-2(x Evaluate a) 7.

1995 SPM

marks) (4

curve.each ofequation thefind (2,4),point at intersect curves theIf ly.respective 2

9-2x

and2x -6by given are curves two the to tangents two theoffunction gradien The b)

dx

SPM Past Year Questions : AM Form 5 Chapter 3 - Integration

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marks) (4 region. shaded

theof area theCalculate axis.- with xparallel is PQ linestraight theGiven that Q.point at

1 xlinestraight with theintersects that )1(4y curve theofpart shows diagram The a) 8. 2 x

marks) (6

axis.-yat 180 angle through revolved is OKMNsector when thegenerated e volum

theCalculate .90KON and Oat centre with thecircle a ofsector a is OKMNGiven

axis.-yat lsymmetrica is which 2 xcurve theofpart is KMN diagram, In the b) 22

y

marks) (2 .14kx-g(x) ifk b)

marks) (2 g(x)dx, 2f(x)dx a)

of value theFind 6.g(x)dx and 5f(x)dxGiven 9.

1996 SPM

3

1

4

0

1

3

4

0

3

1

dx

SPM Past Year Questions : AM Form 5 Chapter 3 - Integration

Compiled by Miss Ng @ www.epitomeofsuccess.com Page 4

marks) (6 1. xline and -2 xline axis,- xcurve, by the

boundedregion theof area theCalculate 3).1)(x-x(xy curve theshows diagram The a) 10.

marks) (4 k. of value theFind .unit 18

is generated volume theaxis,- xabout the 360 rotated isregion When thek. xline

and 1 xline ,12xy curve by the boundedregion shaded theshows diagram The b)

3

marks) (5 region. shaded of area the(ii)

h, of value the(i)

Find (h,3).point at

intersect which 4)-2)(x-(xy curve a and 12xy linestraight a shows diagram The a) 12.

marks) (2 dx. 5-2f(x)for value thefind 6,f(x)dxGiven 11.

1997 SPM

3

1

3

1

SPM Past Year Questions : AM Form 5 Chapter 3 - Integration

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marks) (5 k. of value the

Determine .unit 6 is generated volume theaxis,-yabout 360 revolved isregion shaded

When theaxis.-y andk y ,x-4yby boundedregion shaded theshows diagram The b)

3

2

marks) (5 axis.-yabout 360 through revolved isregion shaded when thegenerated

volume theFind .153

x linestraight and 2xygraph theshows diagram The b)

marks) (5 6. xline and axis- x,4xyby bounded

region theof area thefind Hence 6.x0 of range in the 4xygraph Sketch the a) 14.

(3marks)

.3,2

1point through and 1)(2x ofgradient a has that curve theofequation theFind b)

marks) (2 .1

2x Find a) 13.

1998 SPM

2

2

2

3

4

2

y

x

x

dxx

SPM Past Year Questions : AM Form 5 Chapter 3 - Integration

Compiled by Miss Ng @ www.epitomeofsuccess.com Page 6

marks) (5 region. shaded theof area thecalculate Hence,

mark) (1 B.Point of coordinate the(iii)

marks) (3 , f(x) (ii)

mark) (1 A,point of coordinate the(i)

Find a)

.4x(x)f'Given

B.point at curve theo tangent t theis axis,- with xparallel is which BC, linestraight The

B.at axis-y cuts andA point at axis- xches which touf(x),ygraph theshows diagram The 16.

marks) (4 x.of in termsy Find .3dx

dy and

2

1y -1, When x.14Given 15.

1999 SPM

2

3

2

2

x

xdx

yd

marks) (6 a. of value thefind

,unit 2

1 is produced volume theIf axis.-yabout 180 through revolved is ay line

and 1xy curve by the boundedregion when theproduced ish block whic a is Q b)

region. shaded of area theCalculate a)

questions. following answer the diagram, the toreference With 18.

marks) (4 .11kxf(x)dx f(x)dx ifconstant k b)

mark) (1 2f(x)dx, a)

of value theFind 4.f(x)dxGiven 17.

2000 SPM

3

2

2

1

5

2

5

1

5

1

dx

SPM Past Year Questions : AM Form 5 Chapter 3 - Integration

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marks) (2 xdy. of value thefind Hence, (ii)

xdy.by drepresenteregion theShade (i)

graph. curve a ofpart ofsketch a shows diagram The b)

marks) (3 .6x

2 of value theFind a) 19.

2001 SPM

5

0

8

2

8

2

2

1

3

ydx

xdx

marks) (3 .g(x)-xfor value theFind ).(1dx

dGiven .20

2

0

2

dxxgx

x

marks) (6 axis.-y and axis-x

,5)-(xy curve theg(x),y linestraight by the bounded area theFind (2,9).point at

5)-(xy curve theintersects which g(x)y linestraight a shows diagram The (ii)

2g(x)dx. of value theFind (i)

function.linear a is g(x) where14g(x)dxGiven a) .21

2

2

0

2

2

0

SPM Past Year Questions : AM Form 5 Chapter 3 - Integration

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marks) (4 k. of value thefind ,unit 8 is axis-y about the 360 revolved

isregion when thegenerated volume theIf constant. a isk where2-k isA point for x of

coordinate theand axis- with xparallel is AB linestraight theknown that isIt axis.-y and AB

linestraight the,1xy curve theofpart by boundedregion shaded a shows diagram The b)

3

2

marks) (3 12. t

when of valuemaximum thefind 6, when t4 Given that ).12(2

1

dt

dby given

is s, t time, the tocompared , re, temperatuin the increase of rate theday, specific aOn 22.

2002 SPM

t

marks) (3

k. of valuespossible theFind constant. a isk where,3

7816xGiven b)

marks) (2 x.of in termsy Express .4dx

dy23xGiven a) .23

2

1

0

2

dxkkx

SPM Past Year Questions : AM Form 5 Chapter 3 - Integration

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marks) (5

t.of value thecalculate Q,region

of area the times threeis Pregion of area at the th

Given Q. and P regions, twointo axis- xand 5 x

1, xlinestraight the,1

y curve by the bounded

region shaded thedividest xlinestraight The

.x

1y curve theofpart shows diagram The a) .24

2

2

x

marks) (5

generated. ish block whic theof volume theFind axis.-y

about 360 through revolved is 16y linestraight and

-2 xlinestraight curve, by the boundedregion shaded

The .xy curve a ofgraph theshows diagram The b) 2

marks) (3 n. of value theandk of value thefind ,)1(x)(1

5 25.

2003 SPM

4cxkdx n

marks) (3

k. of value thefind ,units 64 isregion shaded theof area theIf

k. xlinestraight a and 3xy curve a shows diagram The .26

2

2

marks) (3 x.of in termsy find -1, when x6y and 22dy

Given .27 xdx

SPM Past Year Questions : AM Form 5 Chapter 3 - Integration

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marks) (6

axis.-yabout 360

throughrevolved isregion shaded when thegenerated

volume theCalculate A.point at 2x 3y linestraight

aintersect which 1y xcurve a shows diagram The .28 2

marks) (3 curve. theofequation theFind

.63x is A(1,-12) through passes that curve a offunction gradient The .30

marks) (4 k. of value thefind -1,k where,6)32(Given 29.

2004 SPM

2

k

1-

x

dxx

marks) (6 . of in terms generated, volume theFind

axis.-about x 360 through revolved isregion The (b)

region. theof area theFind (a)

3. xlinestraight and 2, xlinestraight axis,- xcurve,

by the bounded isregion A A(1,3).point through passes

which 1)-(2x

3y curve a ofpart shows diagram The .31

2

marks) (3

curve. theofequation theFind (1,3).point through 43x offunction gradien a has curveA 33.

marks) (4 k. of value theFind .10kx-2f(x) and 7)(Given 32.

2005 SPM

2

6

2

6

2

x

dxdxxf

SPM Past Year Questions : AM Form 5 Chapter 3 - Integration

Compiled by Miss Ng @ www.epitomeofsuccess.com Page 11

marks) (3

axis.-y about the 360 through revolved is

3y linestraight and axis-y curve, by the bounded

region when the, of in terms generated, volume the(b)

marks) (4 region. shaded theof area The (a)

Find A(2,3).at 12

xy curve

the tonormal a is PQ linestraight diagram, In the .34

2

b

a

2

marks) (2 2f(x)dx.

of value thefind ,units 5 isregion shaded the

of area theGiven that b. xand aat x axis- x

thecutting f(x)y curve theshows diagram The 35.

2006 SPM

marks) (4 .10g(x)-kx ifk of value the(b)

g(x)dx, of value the(a)

find 8,g(x)dx Given that .36

5

1

1

5

5

1

dx

marks) (3 axis.- xabout the 360 revolved is Qregion

shaded when the, of in terms generated, volume the(c)

marks) (5 P,region shaded theof area the(b)

marks) (2 k, of value the(a)

Find

B. andA points at the 2)-(xy curve thengintersecti

4xy linestraight theshows diagram The .37

2

SPM Past Year Questions : AM Form 5 Chapter 3 - Integration

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marks) (3 curve. theofequation theFind

(1,8).at point turninga has ,x

2-2xfunction gradient with curveA 39.

marks) (4 h(x)-5 (b)

h(x)dx, (a)

find ,3h(x)dx Given that 38.

2007 SPM

2

7

2

2

7

7

2

dx

marks) (7

axis.- xabout the 360 through revolved is axis-y he t

and axis- x thecurve, by the bounded is which R

region when the, of in term generated, volume the(b)

P,region shaded theof area the(a)

Calculate

A.point at 3 xlinestraight theintersects curve The

.1)-2(xy curve theofpart shows diagramgiven The 40. 3