AJC H2 Math 2013 Prelim P1

8/11/2019 AJC H2 Math 2013 Prelim P1

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ANDERSON JUNIOR COLLEGE2013 Preliminary Examinations

H2 MATHEMATICS (JC2)PAPER 1

Duration : 3 hr

Answer ALL questions.

1. It is given that 21

5 2cos ( ) y

ax , where a is a positive constant. If x is

sufficiently small such that the series expansion for 21

5 2cos ( )ax in ascending

powers up to and including the term in x2 is 21 187 49

x , find the value of a . [4]

2. By means of the substitution 2 cosu x , show that1

2 2 22 2

0 0

14 1 4 d sin 2 d

8u u u x x

,and evaluate this definite integral exactly. [3]

Hence or otherwise, evaluate 1 3

220

1 4 d u u exactly. [2]

3. The function f is a strictly increasing function such that f ( ) y x for x 0. The

coordinates of certain points on the curve of f ( ) y x are as follows:

x 0

1

3

1 2

72

4 6 8 11 14

y 112

172

8 14 19 24 26

(i) State the value of ff (6) and the value of 1f (8) . [2]

Another function g is defined by g: x tan 2 x for 0 .8

x

(ii) Explain why the composite function fg exists. [1]

(iii) Find the range of fg. Hence find the set of values of x such that the composite

function fg satisfies the inequality fg 1 x . Leave your answer in exact form. [3]


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4. Given that 1sin (2 ) y x , show that

22 d (1 4 ) 4

d y

x x

. [1]

(i) By further differentiation of this result, find the Maclaurins series of y up to and

including the term in 3 x . [4]

(ii) Region R is bounded by the curve 1sin (2 ) y x , the x-axis, the lines 12

x and

12

x . Using your answer to part (i) , write down a definite integral that will give an

approximate value for the volume of the solid generated when R is rotated through

2 radians about the x-axis. [1]

By referring to the graph of 1sin (2 ) y x and the Maclaurins series of y in part ( i),

determine whether this approximate value is an under-estimation or over-estimation

to the actual volume of the solid generated. [2]

5. A trough is 8 metres long and its cross sectional ends are in the shape of an isosceles

triangle whose width is 5 metres and height is 2 metres. It is held in the position as

shown in the figure by stands of the same height. At time t seconds, the height of thewater in the trough is h metres and the width of the water surface is w metres. The

trough is initially empty and water is being pumped in at a constant rate of 5 m 3/s.

(i) Show that 52

w h . Hence find the volume of the water in the trough in terms of h. [2]

(ii) Find the rate of change of h at t = 2. [3]

From t = 2, the rate at which water is being pumped into the trough is changed to

2h m3/s.

(iii) Find the total time taken for the trough to be completely filled. [3]

58

2hw


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6. (a) The first term of an increasing arithmetic progression is 1. S n is the sum of the first

n terms of the arithmetic progression and S 5, S 10 and S 20 form a geometric

progression. Show that the common difference of the arithmetic progression is 2. [3]

Without the use of a graphing calculator, find the least value of n such that

1

2 1100n

n

S S

. [3]

(b) Mr Wee borrowed $3400 from an unlicensed money lender on 1 st June 2013. The

amount he owes the lender at the beginning of each month is twice the outstanding

amount at the end of the previous month.

From the month of August 2013, Mr Wee decided to repay $7000 in the middle of

each month. Find, in terms of n, the outstanding amount owed at the end of the nth

month.Hence find the earliest month when he has fully repaid his loan. [5]

7. The diagram shows the graph of . The curve has a minimum point at (-1, 0)

and a stationary point of inflexion at (1, -1). The asymptotes are 4 y , 0 y , 0 x and

2 x .

Sketch , on separate clearly labelled diagrams, the graphs of(i) 2 f y x , [3]

(ii) . [3]

In each case, indicate clearly the coordinates of any points where the curve crosses the

axes, turning points and equations of asymptotes whenever possible.

Find the number of points of intersection between the curves = f 1 y x and f ' y x . [3]

f y x

f y x


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8. The plane p with equation 154

a

b

r contains point Q with coordinates (0,5,0).

Plane p makes an acute angle of with the x-axis.

(i) Find b. [1]

(ii) Write down an equation relating , a and b. [1]

It is known that the angle between

4

a

b

and the positive x-axis direction is obtuse and

45 o .

(iii) Show that 5a . [2]

Plane p , the x-z plane and the x-y plane all meet at a common point W .

(iv) Find the position vector of W . [2]

The point M with coordinates (1,0,5) is a point on plane p . The line l is the

intersection line between p and the x-z plane.

(v) Explain why M is on l . Hence or otherwise, find the shortest distance of Q to

the line l . [3]

9. (a) The sequence of real numbers 1 2 3, , , . . .u u u is defined by

1 12

2and 2 , wheren n

nu u u a a

n is a positive real constant.

(i) Prove by mathematical induction that for .( 1)( 1)!n

na n n

un

[4]

(ii) Show that 1n nu u for 3n . Hence show that16

384r

r u a . [5]

(b) Show that1

222

2 ( )d ln

1 ( 1)

nn r

r

p q x

x n n

where p and q are constants to bedetermined.

Hence find the exact value of12 2

226

2d

1

r

r

x x

. [6]


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10. (a) The general solution of the differential equation 2d 2 2d y

y x x

is 214

y x x C

,

where C is an arbitrary constant.

(i) When C = 4, show that there is no stationary point for this particular solution. [2]

(ii) Sketch, on separate diagrams, the graph of the solution in part ( i), and 2 other

typical members of the family of solution curves.

[You need not work out the values of the axial intercepts and the equations of the

asymptotes (if any) in your diagrams.] [3]

(b)A certain circular-shaped leaf has radius r cm that is proportional to the amount of

water, w, it contains during a period of its growth at time t . The leaf absorbs water

from the plant at a rate equals to 8 times the radius of the leaf and loses water by

evaporation at a rate equals to1 times the area of the leaf. During the period of

growth, it may be assumed that the shape of the leaf will be the same.

Given thatd

6d r t

when r = 2, show that the growth of the leaf can be represented by

the differential equation 2d 1 8d 2

r r r

t

[3]

Given that, when t = 0, r = 4, find r in terms of t . What happens to the radius of the

leaf for large values of t . [4]

11. (a) The complex numbers s and w satisfy the equations

6s w i and 10sw .Given that Re( s) > 0, solve the equations for s and w , giving all answers in the

form x iy , where x and y are real. [4]

Hence find the solution to the following equations

6u v and 10uv .

Give your answers for u and v in the form x iy , where x and y are real. [2]


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(b) Find, in the form i z re , the three roots 1 z , 2 z and 3 z of the equation3 3 3 z i

where 1 2 3arg( ) arg( ) arg( ) z z z . Give your answers in exact form. [3]

The points Z 1, Z

2 and Z

3 represent

1 z ,

2 z and

3 z respectively. Find the area of the

triangle formed by Z 1 Z 2 Z 3. [2]

The constant c is a complex number such that the points representing 1cz , 2cz and

3cz forms another equilateral triangle which is congruent to triangle Z 1 Z 2 Z 3, and

one of its vertices lies on the positive real axis. Find a suitable value for the

complex constant c in the exponential form. [2]

END OF PAPER

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AJC H2 Math 2013 Prelim P1