HSC Papers, Sorted by Topic (1984-1997), Part 2 of 4

8/11/2019 HSC Papers, Sorted by Topic (1984-1997), Part 2 of 4

1/20

4UNIT MATHEMATICSINTEGRATION HSC

BOARD OF STUDIESNSW1984-1997

EDUDATA:DATAVER1.01995

Integration4U97-1a)!

Evaluate2

0

5

x + 4dx .

4 4U97-1b)!

Evaluatesin

cosd

4

0

4

.

13

2 2 1

4U97-1c)!

Find1

2x x + 3dx

2 .

1

2

tan x +1

2

C-1

4U97-1d)!

Find4t -6

(t +1)(2tdt

2 3).

Lnc (2t

t +1)

2

2

3)

(

4U97-1e)!

Evaluate x sec x dx2

0

3

.

3

3 Ln2

4U96-1a)!

Evaluate4

(2 x)dx

21

3

.

8

15

4U96-1b)!

Find sec tan2

d .

1

2tan2+ c

4U96-1c)!

Find5t 3

t(t 1)dt

2

2

.

3 Ln t+ Ln (t2+ 1) + c 4U96-1d)!

Using integration by parts, or otherwise, find x x dxtan 1 .


2/20




1

2(x2tan-1x + tan-1xx) + c

4U96-1e)!

Using the substitution x 2sin , or otherwise, calculatex

4 xdx

2

21

3

.

3

4U96-3b)!

i. Show that (sin ) cosx xdxk

k2

0

2 1

2 1

, where k is a positive integer.

ii. By writing (cos ) ( sin )x xn n2 21 , show that (cos ) ( )

x dxk

n

kn

k

k

n2 1

00

21

2 1

.

iii. Hence, or otherwise, evaluate cos5

0

2

xdx

.

i) Proof ii) Proof iii)8

15

4U96-7b)!

i. Let f(x) = Lnx - ax + b, for x > 0, where a and b are real numbers and a > 0. Show that

y = f(x) has a single turning point which is a maximum.

ii. The graphs of y = Lnx and y = ax - b intersect at points A and B. Using the result of part (i),

or otherwise, show that the chord AB lies below the curve y = Lnx.

iii. Using integration by parts, or otherwise, show that1

k

Lnx dx = k Lnk - k + 1.

iv. Use the trapezoidal rule on the intervals with integer endpoints 1, 2, 3,..., k to show that

Lnx dx

k

1

is approximately equal to 1

2 1Lnk Ln k ( )! .

v. Hence deduce that k e k k

e

k

!

.

Proof 4U95-1a)!

Find dx

x(lnx)2

.

1

ln xc

4U95-1b)!

Find xedxx .ex- xex+ c

4U95-1c)!

Show that6t 23

(2t 1)(t 6)dt ln 70

1

4

.

Proof

4U95-1d)!


3/20




Findd

dx(x sin x)1 , and hence find sin x dx1 .

x

1 xsin x

2

1

, x sin x 1 x c1 2

4U95-1e)!

Using the substitution t = tanx

2, or otherwise, calculate

dx

5 3sin x 4cos x02

.

1

6

4U95-4c)!

i. Show that, if 0 < x Ik + 1.

iv. Hence, using the fact that I2n - 1> I2nand I2n> I2n + 1, show that

2

2n2n 1

2 .4 ...(2n)

1.3 .5 ...(2n 1) (2n 1)

2 2 2

2 2 2

2.

Proof 4U94-1a)!

Find

i.4 12

6 132x

x x dx

ii.1

6 132x x dx

.

(i) 2log ce x x2 6 13 (ii) 12 1tan x 3

2c

4U94-1b)!

Evaluate 9 232

3

u du .


4/20


5/20




i. Show that f(x) f 2 x 1 for 0 x 2

.

ii. Sketch y = f(x) for 0 x2

.

[There is no need to find f xbut assume y = f(x) has a horizontal tangent at x = 0. Yourgraph should exhibit the property of (b)(i).]

iii. Hence, or otherwise, evaluate

dx

1 (tan x)k02

.

(i) Proof (ii)2

1

y

x (iii)

4

4U92-1a)!

Find:

i. tan sec 2 d .

ii.2 6

6 12x

x x dx

.(i) 12

2tan c (ii) logx 6x 1 c2

4U92-1b)!

Evaluatedx

(x 1)(3 x)32

5

2

by using the substitution u = x - 2.

3

4U92-1c)!

Evaluate5 1

1 3 20

1 ( )

( )( )

t

t t dt .

log 4

3e

4U92-1d)!

i. Find xe dxx2

.

ii. Evaluate 2x e dx3 x0

1 2

.(i) 12

xe c2

(ii) 1 4U92-4a)!

Each of the following statements is either true or false. Write TRUE or FALSE for each statement and

give brief reasons for your answers. (You are not asked to find the primitive functions).


6/20




i. sin7

2

2

0

d

ii. sin70

0

d

iii. e x

2

1

10dx

iv. (sin80

2 0

- cos ) d8

v. For n = 1, 2, 3, ...., dt

tdttn n1 10

1

10

1

.(i) True (ii) False (iii) False (iv) True (v) True

4U91-1a)!

Find:

i.t1

t3

dt ;

ii.e

x

e2x 9 dx , using the substitution u = ex.

(i) 1

t

1

2t c2 (ii)

13tan

e

3 c1

x

4U91-1b)!

i. Evaluate5

(2t 1)(2t)dt

0

1

.

ii. By using the substitution t = tan

2 and (i), evaluate

d

3 40

2

sin cos

.

(i) ln 6 (ii)1

5ln 6

4U91-1c)!

Let I sin x dxnn

0

2

, where n is a non-negative integer.

i. Show that I (n 1) sin xcos x dxnn 2 2

0

2

when n 2.

ii. Deduce that In n 1

nIn2 when n 2.

iii. Evaluate I4.

(i) Proof (ii) Proof (iii) 316

4U90-2a)!

Find the exact values of:

i.x 1

x 2x 5dx

22

3


7/20




ii. 4 x dx2

0

2

(i) 2 5 13 (ii)

21

4U90-2b)!

Find:

i.dx

(x 1)(x2 2) ;ii. cos x dx3 , by writing cos

3x(1sin2 x)cosx , or otherwise.

(i) 13 162 1ln x 1 ln(x 2)

1

3 2tan

x

2c (ii) sin x sin x c13

3

4U90-2c)!

Let I (1 t ) dt, n 1,2,3, .. .n2

0

xn

Use integration by parts to show that I 1

2n 1

(1 x ) x 2n

2n 1

I 1n2 n

n

.

Hint: Observe that (1 t2)n1 t2(1 t2)n 1 (1 t2)n .Proof

4U89-2a)!

Evaluate:

i.2

920

3

x dx ;

ii. x lnx dx2

1

3

;

iii. 4 x dx2

3

2

.

(i) 9

(ii) 9 3 26

9ln (iii)

33

2

4U89-2b)!

i. Write4x2 5x 7

(x 1)(x2 x 2)in the form

A

x 1

Bx C

x2 x 2

.

ii. Hence evaluate4x2 5x 7

(x1)(x2 x 2)dx

1 .

(i) 2

x 16x 3

x x 22 (ii) 2 ln 2

4U88-1a)!

Find the exact value of:

i.2x

1 2xdx

0

1

;

ii.(logex)

2

xdx

1

e

;


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iii. cos x dx1

0

1

2

.

(i) 1 312 loge (ii)1

3 (iii)

1

3

2

4U88-1b)!

Use the substitution t tan2

to find the exact value ofd

sin 20

2

.

3 3

4U87-1i)Prove that:

a.t 1

t2 dt

1

2

1

2loge2;

b.

4 dx

(x 1)(x 3) 2log

9

5e4

6

.Proof

4U87-1ii)

a. Use the substitution x2

3sin to prove that 4 9x dx

32

0

2

3

.

b. Hence, or otherwise, find the area enclosed by the ellipse 9x2+ y2= 4.

(a) Proof (b) 43

units2

4U87-1iii)

a. Given that I x dxnn

cos0

2

, prove that I n 1

nIn n 2

where n is and integer and n 2.

b. Hence evaluate cos x dx5

0

2

.

(a) Proof (b)8

15

4U86-1i)

Evaluatex

xdx

( )20

1

.

23

4 2 - 5

4U86-1ii)

Use integration by parts to show that tan x dx4

12

log 21

0

1

e .

Proof 4U86-1iii)


9/20




Find numbers A, B, C such thatx

4x A

B

2x 3

C

2x 3

2

2 9

.

Hence evaluatex

2

4x2 9dx

0

.

A 1

4 , B

3

8 , C = -

3

8;

1

4

3

16log 5e

4U86-1iv)

Using the substitution t = tan(1

2 ), or otherwise, show that

1

1 3 1

0

3

sin d .

Proof 4U85-1i)

Find:

a. x cos x dx0

3

;

b. 1x2 4x 5

dx0

.

(a) -2 (b) 0.1419 (to 4 decimal places) 4U85-1ii)

Find real numbers A, B, C such thatx

(x 1)2(x 2)

A

x 1

B

(x 1)2

C

x 2.

Hence show thatx

(x 1) (x 2)dx 2log

3

21

2

0

1

2

e

.

A = -2, B = -1, C = 2, Proof 4U85-1iii)!

Use the substitution x = a - t, where a is a constant, to prove that f(x)dx f(a t)dt0

a

0

a

. Hence, or

otherwise, show that x x dx( )1 1

10 10099

0

1

.

Proof 4U84-1i)

Show that:

a. 4 xlog x dx 7e ee8 2

e

e4

;

b.dx

xloge x2 loge

e

e4

2 .

(a) Proof (b) Proof 4U84-1ii)

Evaluate:

a.x 1

x 1dx

2

0

3


10/20




b.x 6

x 5dx

4

4

c. sin 4x cos 2x dx

6

6

.

(a) log 2 3e

(b) 2113 (c) 0


11/20

4UNIT MATHEMATICSVOLUMESHSC



Volumes4U97-3a)!

y

xO

(3, 2)C(0, 2)

In the diagram, the shaded region is bounded by the x axis, the line x = 3, and the circle with centre

C(0, 2) and radius 3. Find the volume of the solid formed when the region is rotated about the y

axis.

8

3units3

4U97-5a)!

i. Sketch the graph of y = -x2for 2 x 2.

ii. Hence, without using calculus, sketch the graph of y = e-x2

for 2 x 2.

iii. The region between the curve y = e-x2

, the x axis, the y axis, and the line x = 2 is rotated

around the y axis to form a solid. Using the method of cylindrical shells, find the volume of

the solid.

i)

Y

-4

-2 2X

ii)

1

y

xO-2 2

iii) 1 1

eunits

4

3

4U96-3a)!


12/20




x + y = 0

0

-1

1

41 x

S

The shaded region is bounded by the lines x = 1, y = 1, and y = -1 and by the curve x y 2 0 . The

region is rotated through 360about the line x = 4 to form a solid. When the region is rotated, the line

segment S at height y sweeps out an annulus.

i. Show that the area of the annulus at height y is equal to ( )y y4 28 7 .

ii. Hence find the volume of the solid. i) Proof ii)

296

15 units3

4U95-3b)!

The circle x + y is rotated about the line x = 9 to form a ring.

When the circle is rotated, the line segment S at height y sweeps out an annulus.

0 4

s

4

-4

-4 x

y

x = 9

The x coordinates of the end-points are x1and - x1, where x1= 16 2 y Error! Switch argument

not specified..

i. Show that the area of the annulus is equal to 36 16 2 y Error! Switch argument not

specified.

ii. Hence find the volume of the ring.

(i) Proof (ii) 2882units3

4U94-3b)!


13/20




-2 x

y

2 4 6

0

E

t = 0

The graph shows part of the curve whose parametric equations are

x t t 2 2 , y t t 3 , t0 .. i. Find the values of t corresponding to the points O and E on the curve.

ii. The volume V of the solid formed by rotating the shaded area about the y axis is to be

calculated using cylindrical shells.

Express V in the form V f ta

b 2 ( ) dt .

Specify the limits of integration a and b and the function f(t). You may leave f(t) in

unexpanded form.

Do NOT evaluate this integral.

(i) t = 1, 2 (ii) V 2 (t t 2)(t t)(2t 1) dt2 31

2


14/20




4U93-6a)!

The solid S is generated by moving a straight edge so that it is always parallel to a fixed plane P. It is

constrained to pass through a circle C and a line segment L. The circle C, which forms a base for S,

has radius a and the line segment L is distance d from C. Both C and L are perpendicular to P and sit

on P in such a way that the perpendicular C at its centre O bisects L.

P

O

lS

F

G

E

C

a

d

x

i. Calculate the area of the triangular cross-section EFG which is parallel to P and distance x

from the centre O of C.

ii. Calculate the volume of S.

(i) a x d2 2 units2 (ii) 12 a2d units3

4U92-5a)!

2a

axa

a The solid S is a rectangular prism of dimensions a a 2a from which right square pyramids of basea a and height a have been removed from each end. The solid T is a wedge that has been obtained

by slicing a right circular cylinder of radius a at 45through a diameter AB of its base.Consider a cross-section of S which is parallel to its square base at distance x from its centre, and a

corresponding cross-section of T which is perpendicular to AB and at distance x from its centre.

i. The triangular cross-section of T is shown on the diagram.

Show that its area is

1

2

2 2

( )a x

.


15/20




ii. Draw the cross-section of S and calculate its area.

iii. Express the volumes of S and T as definite integrals.

iv. What is the relationship between the volumes of S and T? (There is no need to evaluate

either integral).

(i) Proof (ii)

x

a

A B

C

A = a2- x2 (iii) V 2 (a x ) dxS2 2

0

a

,

V (a x ) dxT2 2

0

a

. (iv) V VT 12 S


16/20




4U91-5b)!

E

B

O

Figure 1

Water surface

D

C

A F

G

H

H

E

a

C'

FG

Water

Figure 2

Figure 3

ha

F D

O

C

H

A drinking glass having the form of a right circular cylinder of radius a and height h, is filled withwater. The glass is slowly tilted over, spilling water out of it, until it reaches the position where the

waters surface bisects the base of the glass. Figure 1 shows this position.

In Figure 1, AB is a diameter of the circular base with centre C, O is the lowest point on the base, and

D is the point where the waters surface touches the rim of the glass.

Figure 2 shows a cross-section of the tilted glass parallel to its base. The centre of this circular section

is Cand EFG shows the water level. The section cuts the lines CD and OD of Figure 1 in F and H

respectively.

Figure 3 shows the section COD of the tilted glass.

i. Use Figure 3 to show that FH =a

h(h x) , where OH = x.

ii. Use Figure 2 to show that C'F = axh

and HC'G = cos-1 x

h .

iii. Use (ii) to show that the area of the shaded segment EGH is

a cos x

h

x

h1

x

h

2 12

.

iv. Given that cos d cos 11 1 2 , find the volume of water in the tilted glass ofFigure 1.

(i) Proof (ii) Proof (iii) Proof (iv)2a h

3

2

units3


17/20




4U90-3a)!

x

y

y = e - x

- a a

S

t

The region under the curve yex

2

and above the x axis for a x a is rotated about the y axis toform a solid.

i. Divide the resulting solid into cylindrical shells S of radius t as in the diagram. Show that

each shell S has approximate volume V 2tet2

t where t is the thickness of the shell.

ii. Hence calculate the volume of the solid.iii. What is the limiting value of the volume of the solid as a approaches infinity?

(i) Proof (ii) (1 e )a2

units3 (iii) units34U89-5b)!

P

OZ

XY

B

T

a

A Let ABO be and isosceles triangle, AO = BO = r, AB = b.

Let PABO be a triangular pyramid with height OP = h and OP perpendicular to the plane of ABO as in

the diagram. Consider a slice S of the pyramid of width a as in the diagram. The slice s is

perpendicular to the plane of ABO at XY with XY || AB and XB = a. Note that XT || OP.

i. Show that the volume of S isr a

r b

ahr

a

, when a is small. (You may assume that the

slice is approximately a rectangular prism of base XYZT and height a.)

ii. Hence show that the pyramid PABO has volume1

6hbr.


18/20




iii. Suppose now that AOB =2

nand that n identical pyramids PABO are arranged about O as

centre with common vertical axis OP to form a solid C. Show that the volume Vnof C is

given by Vn1

3r

2hnsin

n.

iv. Note that when n is large, the solid C approximates a right circular cone. Using the fact thatsinx

x

approaches 1 as x approaches 0, find limn

Vn. Hence verify that a right circular cone of

radius r and height h has volume1

3r

2h .

(i) Proof (ii) Proof (iii) Proof (iv) 13r h2

4U88-7a)!

K

L

H

J

M

I20cm

xcm

16cm

4cm

hcm

FIGURE NOT TO SCALE

i. A trapezium HIJK has parallel sides KJ = 16 and HI = 20cm. The distance between these

sides is 4cm. L lies on HK and M lies on IJ such that LM is parallel to KJ. The shortest

distance from K to LM is h cm and LM has length x cm. Prove that x = 16 + h.

ii.

20cm

16cm

12cm

10cm

FIGURE NOT TO SCALE

The diagram above is of a cake tin with a rectangular base with sides of 16cm and 10cm. Its

top is also rectangular with dimensions 20cm and 12cm. The tin has depth 4cm and each of

its four side faces is a trapezium. Find its volume.

(i) Proof (ii) 794 cm233

4U87-5i)

a. On a number plane shade the region representing the inequality (x - 2R)2+ y2R2.

b. Show that the volume of a right cylindrical shell of height h with inner and outer radii x and

x + x respectively is 2xhx when x is sufficiently small for terms involving (x)2to be

neglected.

c. The region (x - 2R)2+ y2R2is rotated about the y axis forming a solid of revolution called

a torus. By summing volumes of cylindrical shells show that the volume of the torus is given

by: V = 42R3.


19/20




(a)

2R

R 3R x

y

(x - 2R)2+ y2R2 (b) Proof (c) Proof

4U86-5i)

a. The coordinates of the vertices of a triangle ABC are (0, 2), (1, 1), (-1, 1) respectively and His the point (0, h). The line through H, parallel to the x-axis meets AB and AC at X and Y.

Find the length of XY.

b. The region enclosed by the triangle ABC, defined in (a) above, is rotated about the x axis.

Find the volume of the solid of revolution so formed.

(a) (4 - 2h) units (b)8

3

units3

4U85-5ii)

a. Using the substitution x = asin, or otherwise, verify that (a2 x2)

1

2 dx1

4a2

0

a

.

b. Deduce that the area enclosed by the ellipsex

a

y

b1

2

2

2

2 is ab.

c.

h

H

The diagram shows a mound of height H. At height h above the horizontal base, the

horizontal cross-section of the mound is elliptical in shape, with equationx

a

y

b

2

2

2

2

2 ,

where 1h

2

H2, and x, y are appropriate coordinates in the plane of the cross-section.

Show that the volume of the mound is8abH

15.

Proof

4U84-5)


20/20


x

y

- a 0 a

t t

S

i. The diagram shows the area A between the smooth curve y = f(x), -a x a, and the x axis.

(Note that f(x) 0 for -a x a and f(-a) = f(a) = 0). The area A is rotated about the line

x = -s (where s > a) to generate the volume V. This volume is to be found by slicing A into

thin vertical strips, rotating these to obtain cylindrical shells, and adding the shells. Two

typical strips of width t whose centre lines are distance t from the y axis are shown.

a. Show that the indicated strips generate shells of approximate volume 2f(-t)(s -

t)t, 2f(t)(s + t)t, respectively.

b. Assuming that the graph of f is symmetrical about the y axis, show that V = 2sA.ii. Assuming the results of part (i), solve the following problems.

a. A doughnut shape is formed by rotating a circular disc of radius r about and axis in

its own plane at a distance of s(s > r) from the centre of the disc. Find the volume

of the doughnut.

b. The shape of a certain party jelly can be represented by rotating the area between

the curve y = sinx, 0 x , and the x axis about the line x =

4. Find the

volume generated.

(i)(a) Proof (b) Proof (ii)(a) 22r2s units3 (b) 3

2units3

Documents

HSC Papers, Sorted by Topic (1984-1997), Part 2 of 4