1

2016 IJC H2 Math 9758 Promotional Examination

1 One root of the equation 3 2 12 0z az bz+ + + = , where a and b are real numbers, is

1 i 3z = − √ . Without using a calculator, find the values of a and b and the other roots.

[4]

2 Without using a calculator, solve the inequality( )2

2

3 21

3

x

x

−≥ −

−. [4]

3 At the beginning of an experiment, an inverted circular cone is filled with sand. The

base radius of the cone is 5 cm and the slant height is 13 cm. Sand is leaking through

a small hole at the bottom of the cone at the rate of 7 cm3s−1 at the instant when the

depth of the sand in the cone is 2.5 cm. Find the rate of decrease of the depth of the

sand at this instant. [5]

[It is given that the volume of a circular cone with base radius r and vertical height h

is 21

3r hπ .]

4 The diagram shows the graph of ( )fy x= with a maximum point at ( )3 ,a b− and a

horizontal asymptote y b= − where 0a > and 0b > . The curve intersects the axes at

( ) ( )5 ,0 , ,0a a− − and ( )0, a− .

Sketch, on separate diagrams, the graphs of

(i) ( )f 2y x b= + , [3]

(ii) 1

f ( )y

x= , [3]

(iii) f ( )y x′= , [3]

showing in each case the equations of the asymptote(s) and the coordinates of any

turning point(s) and point(s) of intersection with the axes whenever possible.

2

5 (a) Given that 2 2

3

1

( 1)

4

n

r

n nr

=

+=∑ , find ( )

234 5 1

n

r n

r n

=

+ +∑ in terms of n. Give your

answer in fully factorised form. [4]

(b) (i) Verify that 2 2 2 2

1 1 4

( 1) ( 1) ( 1)

n

n n n− =

− + − . [1]

(ii) Find 2

2 22 ( 1)

n

r

r

r= −∑ . (There is no need to express your answer as a single

algebraic fraction.) [3]

(iii) Hence find the exact value of 2 2

3 ( 1)r

r

r

∞

= −∑ . [2]

6

Fig. 1 shows a piece of square card, ABCD, with sides n cm, where n is a positive constant. A rectangle EFGH, as shown in Fig. 2, is being cut from the card such that

EF = HG = 2x cm and BF = GC = x cm. The rectangle is rolled up to form a cylinder

of height 2x cm, where EH and FG are the circumference of the circular ends of the cylinder, as shown in Fig. 3. The cylinder is placed on a horizontal table top and the volume of this open cylinder is V cm3.

(i) Show that ( )3 2 214 4

2V x nx n x

π= − + . [3]

(ii) Use differentiation to find, in terms of n, the stationary value of V as x varies.

Determine whether the volume is a maximum or a minimum. [6]

E

C

H

G F B n

Fig.

1

E H

G F

Fig.

2

Fig.

3

A D

3

7 The function f is defined by

3 5 3f : , for , .

4 3 4

xx x x

x

+∈ ≠

−֏ ℝ

(i) Define 1f ( )x

−. [3]

(ii) Deduce the rule of 2f ( )x and state the range of 2f . [2]

(iii) Using part (i), describe the symmetry of the graph of f . [1]

The function g is defined by

3 2g : 2 , for , 1.x x x x x+ ∈ ≥֏ ℝ

(iv) Explain why the composite function fg exists and find the exact value of

fg(2) . [4]

8 (a) Show that, when x is sufficiently small for 3x and higher powers of x to be

neglected,

2cos

1 sin2

xa bx cx

x≈ + +

−,

where the values of a, b and c are to be determined. [3]

(b) (i) Given that 3 6e )xy

−= √( + , show that d

3ed

xyy

x

−= − .

By further differentiation of this result, find the Maclaurin series for

3 6e )x−√( + up to and including the term in 2x . [5]

(ii) The second and third terms in the Maclaurin series for 3 6e )xy

−= √( + are

equal to the first and second terms in the series expansion of ( )e ln 1axnx+

respectively. Using appropriate expansions from the List of Formulae

(MF26), find the constants a and n. [4]

4

9 A bank has an account for investors. Interest is added to the account at the end of

each year at a fixed rate of 3% of the amount in the account at the beginning of that

year. Mary and Ben both invest money.

(a) Mary decides to invest $x at the beginning of one year and then a further $x at

the beginning of the second and each subsequent year. She also decides that

she will not draw any money out of the account, but just leave it, and any

interest, to build up.

(i) How much will there be in the account at the end of 1 year, including

the interest? [1]

(ii) Show that, at the end of n years, when the interest for the last year has

been added, she will have a total of $ ( )1031.03 1

3

nx− in her account.

[3]

(iii) If Mary starts investing at the beginning of 2016 with an amount of

$10 000, at the end of which year will she have, for the first time, at

least $190 000 in her account? [4]

(b) Ben decides that, to assist him in his everyday expenses, he will withdraw the

interest as soon as it has been added. He invests $y at the beginning of each

year. Find the total amount of interest he will receive at the end of n years. [4]

10 A curve C has parametric equations

sin 4x a θ= , 22 cosy a θ= ,

where 02

πθ< ≤ and a is a positive constant.

(i) Sketch C, showing clearly the coordinates of any point(s) of intersection with the axes. [2]

(ii) Find the equation of the tangent to C at the point P where 6

πθ = , leaving your

answer in exact form. [4]

(iii) The tangent at P meets the x-axis at A and the normal at P meets the x-axis at

B. Find the exact area of triangle ABP. [6]

5

11 The curve C has equation

2 4 5x xy

x r

λ+ + −=

+,

where λ is a non-zero constant, and a vertical asymptote 1.x =

(i) State the value of r and find the equation of the other asymptote of C. [3]

(ii) Draw a sketch of C for the case when 0λ < . [2]

(iii) By using an algebraic method, find the range of values of λ for which the line

2y x= and C have at least one point in common. [3]

It is now given that 9λ = .

(iv) On a separate diagram from part (ii), sketch the graph of C, indicating clearly

the coordinates of the stationary points. [2]

Another curve D has equation 2 2

2 2

( 4) ( 6)1

4

x y

k

− −+ = .

(v) On the same diagram as part (iv), sketch D for the case when 3k = . [2]

(vi) Deduce the range of values of k for which C and D intersect more than once.

[1]

~ ~ ~ End of Paper ~ ~ ~

2016 H2 Math Promo Exam Suggested Solution

Q Solution

1 Since the coefficients are real, 1 i 3z is another root of the equation.

Method 1

22

2

1 3i 1 3i

1 3i

2 4

z z

z

z z

3 2 12 0z az bz

2 2 4 3 0z z z (By inspection)

Comparing coefficients of 2z , 1a

Comparing coefficients of z, 2b

The other roots are 1 3iz and 3z .

Method 2

Substitute 1 3iz ( or 1 3iz ) into the given eqn,

3 2

1 3i 1 3i 1 3i 12 0a b or equivalent

1 3 3i 9 3 3i 1 2 3i 3 1 3i 12 0a b

4 2 2 3i 0a b a b

Comparing the real parts, 2 4a b

Comparing the imaginary parts, 2 0a b

1a , 2b

See Method 1 for factorization to find the other roots.

2 2

2

3 21

3

x

x

2 2

2

3 2 30

3

x x

x

2

2

4 12 90

3

x x

x

22 3

03 3

x

x x

𝑥 < −√3 or 𝑥 > √3 or 𝑥 = 1.5

-√3 1.5 √3

+ − − +

3 Let the radius of the surface level of the sand be r cm and the depth of the sand be h cm at time

t seconds.

Height of the funnel = 2 213 5 12 cm

Using similar triangles,

5

    12

5

12

rr h

h

2 31 25

3 432V r h h

Differentiating both sides wrt t,

2d 25 d

d 144 d

V hh

t t

When 2.5h , d

7d

V

t

225 d

7 2.5144 d

h

t

d 7 1442.05

d 156.25

h

t

(3 s.f.)

the rate at which the depth of the sand falls in the funnel is 2.05 cm s−1 when the depth of the

sand is 2.5 cm.

4(i)

4(ii)

5

13 h

r

4(iii)

5(a)

23

2 1 23 3

1 1

2 22 2

22 2

2

2

2

4 5 1

4 5 1

2 (2 1) 1 ( )4 5 1 2 1

4 4

4(2 1) 1 5 1 ( 1)

2(2 1) 1 2(2 1) 1 5 1 ( 1)

3 3 5 1 5 1 ( 1)

3 1 5 1 ( 1)

n

r n

n n n

r r r n

r n

r r n

n n n nn n n

n n n n n

n n n n n n n

n n n n n

n n n

5(b)(i)

2 2

2 2

2

2 2

1 1LHS

( 1) ( 1)

( 1) ( 1) =

( 1)( 1)

4 = RHS

( 1)

n n

n n

n n

n

n

5(b)(ii) 2 2

2 2 2 2

2 2

2 2

2 2

2 2

2 2

2 2

2 2

1 1 1

4( 1) ( 1) ( 1)

1 1

1 3

1 1

2 4

1 1

3 51

41 1

(2 3) (2 1)

1 1

(2 2) (2 )

1 1

(2 1) (2 1)

n n

r r

r

r r r

n n

n n

n n

2 2

1 5 1 1

4 4 4 (2 1)n n

5(b)(iii) 2 2

2 23

1 1As , 0

4 (2 1)

5 2Thus ,

( 1) 16 9

13 =

144

r

nn n

r

r

6(i) Let r cm be the radius of the cylinder.

2 2r n x

2

2

n xr

2

2

2 2

2

3 2 2

22

2

4 42

4

14 4

2

V r h

n xx

n nx xx

x nx n x

6(ii) 2 2d 112 8

d 2

Vx nx n

x

For stationary value of V , let d

0d

V

x .

2 212 8 0x nx n

6 2 0x n x n

or 6 2

n nx x (rejected 2 0n x )

Stationary value of V

3 22

3

14 4

2 6 6 6

27

n n nn n

n

2

2

d 124 8

2d

Vx n

x

When 6

nx ,

2

2

d 1 224 8 0

2 6d

V n nn

x

By 2nd derivative test, V is maximum when 6

nx .

7(i)

1

Let f ( )

3 5

4 3

4 3 3 5

(4 3) 5 3

5 3

4 3

5 3 3 f ( ) , \

4 3 4

y x

xy

x

xy y x

x y y

yx

y

xx x

x

7(ii) Since

1f f , 2 1 3

f ( ) ff ( ) for , 4

x x x x x

2f

3\

4R

7(iii) The graph of f is symmetrical about the line y x .

7(iv)

g f

g f

3[3, ) and \

4

Since , the function fg exists.

R D

R D

Method 1

3 2fg(2) f 2 2(2 )

= f 16

3(16) 5 =

4(16) 3

53 =

61

You may use your GC to check the accuracy

of your answer.

Method 2

3 2

3 2

3 2

3 2

3 2

fg( ) f 2

3 6 5 =

4 8 3

3(2) 6(2) 5fg(2)

4(2) 8(2) 3

53 =

61

x x x

x x

x x

8(a)

2

21

22

2

1cos 2

1 sin2 1 2

1 1 22

1 1 2 (2 ) ...2

71 2

2

x

x

x x

xx

xx x

x x

8(b)(i) 2

3 6e

3 6e

d2 6e

d

d3e ......

d

x

x

x

x

y

y

yy

x

yy

x

22

2

Differentiat wrt

d d3e

dd

x

x

y yy

xx

When 00, 3 6e 9 3x y

0d d3 3e 1

d d

y y

x x

2 2

2 0

2 2

d d 23 1 3e

d d 3

y y

x x

2

2

2

33 ...

2!

13 ...

3

y x x

x x

8(b)(ii)

2 2

2 2 2 2

2 22

22

e ln 1 1 ... ...2! 2

1 ... ..2 2

...2

...2

axax nx

nx ax nx

a x n xax nx

n xnx anx

nnx an x

Comparing the terms,

1nx x n

2 22 21 1

2 3 2 3

n nan x x an

Sub 1n , 1 1 1 1 5

2 3 3 2 6a a

Alternative:

1

2

3 6e

d 13 6e 6e

d 2

x

x x

y

y

x

d3 6e 3e

d

d3e

d

x x

x

y

x

yy

x

9(a)

(i)

$1.03x

9(a)

(ii)

Mary’s account:

Beginning of year End of year

1st yr x 1.03x

2nd yr 1.03𝑥 + 𝑥 1.03(1.03𝑥 + 𝑥)

= 𝑥(1.03 + 1.032)

3rd yr 1.032𝑥 + 1.03𝑥 + 𝑥

𝑥(1.03 + 1.032

+ 1.033)

⋮ ⋮ ⋮ nth yr 𝑥(1.03 + 1.032 + ⋯

+ 1.03𝑛)

(i) Amount at the end of nth year is

1.03 1.03 1 1031.03 1

1.03 1 3

n

nxx

9(a)

(iii)

When 𝑥 = 10000,

103 100001.03 1 190000

3

571.03 1

103

57ln(1.03) ln 1

103

14.9006

n

n

n

n

Miss Lee will have at least $190 000 in her account at the end of 2030.

9(b) Ben’s account:

Amt at the beginning

of each year

Interest received at the

end of each year

1st yr y 0.03𝑦

2nd yr 2𝑦 0.03(2𝑦)

3rd yr 3𝑦 0.03(3𝑦)

⋮ ⋮ ⋮ nth yr 𝑛𝑦 0.03(𝑛𝑦)

Total interest received

0.03 1 2 3 ...y n

0.03 12

0.015 (1 )

ny n

y n n

10(i) sin 4x a , 22 cosy a , where 0 .

2

10(ii)

sin 4

d4 cos 4

d

x a

xa

22 cos

d4 cos sin

d

2 sin 2

y a

ya

a

d 2 sin 2 4 cos sin

d 4 cos 4 4 cos 4

d sin 2 cos sin

d 2cos 4 cos 4

y a aor

x a a

yor

x

When 6

,

sind 33

2d 22cos

3

y

x

3sin

3 2x a a

2 32 cos

6 2y a a

Equation of tangent at the point P

3 3 3

2 2 2

3 3 3

2 4 2

3 3

2 4

y a x a

y x a a

y x a

10(iii) Equation of normal at point P

3 2 3

2 23

2 3

23

2 5

23

y a x a

y x a a

y x a

At point A,

When y = 0,

3 30

2 4

3

2

x a

x a

At point B,

When y = 0,

2 50

23

5 3

4

x a

x a

Area of Triangle ABP 1 5 3 3 3

2 4 2 2a a a

2

221 3units

16

a

11(i) 2 4 5x xy

x r

Since 1x is an asymptote,

1 0 1r r

1 55

1 1

x xy x

x x

The eqn of the oblique asymptote is 5y x .

11(ii)

11(iii) Let

2 4 52

1

x xx

x

2

2

4 5 2 ( 1)

6 5 0

x x x x

x x

There are at least one common point 𝑏2 − 4𝑎𝑐 ≥ 0

36 4(5 ) 0

5 9

4

Since ≠ 0, the range of values of is

4 0 0or (Accept 4 and 0 ).

11(iv)

& (v)

11(vi) 𝑘 > 6