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PAPER 1

1. Diagram shows part of the relation f(x)

State

(a) the type ofthe relation,

(b) the range.

Answer:

(a) Many-to-one

(b) f(x) -3

x

f(x)f(x) = (x2)23

(2, -3).

1

1

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2. Given the function f(x) = 3x, g(x) = px2q andgf(x) = 3x218x + 5. Find

(a) the value ofpand q.(b) gf(- 2).

Answer:

(a) gf(x) = p(3x)2q= px26px + 9pq

Compare to 3x218x + 5

p = 3, 9pq = 5 9(3)q = 5

q = 22

(b) f(-2) = 3(2) = 5 gf(- 2) = g(5) = 3(5)222

= 53

OR gf (-2) = 3(-2) 218(-2) + 5 = 53

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2

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2 Both correct

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3. A function g is defined as g(x) = 7x4. Find

(a) g -1,

(b) the value of hif g -1(h) = 2

Answer:

(a) g -1 =

(b) g -1(h) = =

h = 10

x + 4

7

Using the principle:

ax + b

c

cxb

a=

1

h + 4

72

2

M1

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4. A quadratic equation with rootspand qis

x2+ mx + m = 0wherep, qand mare constants.

Expresspin terms of q.

Answer:

(xp) (xq) = 0

x2(p + q) x + pq = 0

Compare tox2+ mx + m = 0

(p + q ) = m and pq = m

pq = (p + q) pq + p = q

p =q

1 + q 3

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M2

OR:

Use SOR/PORmethod:

SOR =p + q

POR =pq

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5. Find the range of values of x for 4x2 3 4x

Answer:

4x2 + 4x3 0

Let 4x2+ 4x3 = 0

(2x1) (2x + 3) = 0

x = ; x = 2 ?

x 2 ; x

- 2 x

a = 4 > 0minimum graph

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M2

3

M2

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6. Diagram shows the graph of a quadratic function

y = f(x). The straight line y = 15 is a tangent of thecurve y = f(x).

(a) State the equation of the axis of symmetry of the

curve.

(b) Express f(x) in the form of (x + p)2+ q, where p

and q are constants.

Answer:

(a) x =

x = 2

(b) (x2) 15

q = 15

y = 15

x2 6

y = f(x)

y

2 + 62

(x + p) + q

x + p = 0x = p (axis of symmetry)q = optimum value (either minimum or

maximum based on the graph)

the mid-point of the two

points on the curve that

intersect at x-axis

2

2

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0

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7. Solve the equation: 16x =

Answer:

24(x) =

24x = 2 [2 3(2 x)]

24x = 216 + 3x

4x =5 + 3x

x = 5

2

8 2 x

2

23(2 x) M1

M2

3

M1

Use the concept of the rules of

equation of indices

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8. Solve the equation: 2 logx4 + logx8 = 7

Answer:

logx 42+ logx 8 = 7

logx 16(8) = 7 OR logx 16(8) = logx x 7

128 = x7

27

= x7

OR 27

=

(1

/x )7

( )7 = x7

x =

Use the concept of the rules of

equation of logarithms.

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3

Converting log to indices

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9. The nth term of an arithmetic progression is given by

Tn = 123n. Find the common difference of theprogression.

Answer:

a = T1= 123(1)

= 9T2= 123(2)

= 6

d = T2 a= 69= 3

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3

M2

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10. Given that 12, 6, 3 is a geometric progression, findthe sum of the first 7 terms after the 3th.term of the

progression

Answer:

a = 12

r =

S7* = S10S3

12 [1( )10] 12 [1( )3]

= 2.977

1 1=

OR:

T4 = 12 ( )3 = 12/8

S7* = (12/8) [1( )7

]= 2.977

S7* = Sum from T4 to T10 or

first 7 terms after the 3rd.term

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M2

3

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11. Given 0.471 + 0.000471 + 0.000000471 + = -------Find the value of k.

Answer:

a = 0.471

r = 0.001

S = =

=

k = 157

k

333

0.471

10.001

k

333

157

333

3

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M1

Use the conceptofthe sum of infinity.

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13. Given that y = x3 (3x + 1)4, find the value of

when x = 1. dx

Answer:

= (3x + 1)4 3x2+ x3 (4)(3x + 1)3(3)

x = 1; = [3(1) + 1]4 3(1)2 + (1)3 (4)[3(1) + 1]3(3)

= 144

M2 3

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dy

dx

dy

dy

dx

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14. The variablesxand yare related by the equation

y = .The diagram below shows the straight line graph

obtained by plotting log10yagainstx.

(a) Express the equation in its linear form used to obtain

the straight line graph as shown in the diagram.

(b) Find the value of p.

Answer:

(a) log10 y = log10 p x log103

(b) log10 p = 2

p = 100

p

3 x

3 x

p

log10y

x

(0,2)

0

2

1

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15. Given that y =

(a) find the value of dy/dx, when x = 4 ,

(b) the approximate change in y when x increases

from 4 to 4.01

Answer:

(a)

2x + 1

(x3)2

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2

, x = 3

(2x + 1) (2) (2)(x3) (1) (x3)2

(x3)4dy/dx=

[2(4) + 1] (2) (43) (1)(43)2(2)

(43)4=

= 16

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15 (b) Continue...

(b) x = 4

x = 0.01

dy/dx= 16

y = dy/dxx x

y = 16 (0.01)

y = 0.16

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2

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16. Given y = 5h i + (h2) j is a non-zero vector and is

parallel to x-axis. Find the value of h.

Answer:

5h i + (h2) j = x i + 0 j

( parallel to x-axis, vertical component of the vector = 0 )

h2 = 0h = 2

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2

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17. Diagram shows the position of the point A, B and C relative

to origin, O.

Given B(1, 8), OA = 7i + 2j and OC = AB.

Find in terms of i and j,(a) OC,

(b) CB.

Answer:

(a) AB = OA + OB= (7 i + 2 j) + (i + 8 j)= 6 i + 6 j

OC = 9/2i + 9/2j(b) CB = OC + OB

= 9/2i + 9/2j + (i + 8 j)= 11/2i + 7/2j

2

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2

O

y

x

. A

C .. B

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19. Given sin A = 3/5, cos B = 12/13. If both the angle A

and B are at the same quarter, find the value of

(a) sin (A + B),

(b) tan (A B)

Answer:

(a) sin A cos B + cos A sin B

= (3/5) (12/13) + (4/5) (5/13)

= 56/65

tan A tan B

(3/4)(5/12)

A

53

4

5

12

13

B

1 + tan A tan B(b)

1+ (3/4) (5/12)= = 7/16

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2

2 M1

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20. Point A lies on a curve y = 2x4x, find thecoordinates of point A where the gradient of the

normal at point A is -1/7.

Answer:

dy/dx= mt= 8x31

8x31= 7

8x3= 8

x = 1

y = 2(1)4

1= 1

A = ( 1, 1)

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3

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21. The standard deviation of a set of six numbers is 15.Given that the sum of square for the set of numbers is

144. Find the new mean when a member 10 is addedto this set.

Answer:

N = 62 = 15

x2= 144 x = x (N)

= 3 (6) = 18

xnew= 18 + 1015 =144

6 x2

x 2 = 9x = 3

x new= 28/7= 4

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3

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22. A set of positive integers consists of 6, 7, k, 1, 8, 3, 3.

(a) Find the value of k if the mean of the data is 5.

(b)State the range of the values ofk if the median of

the data isk.

Answer:

(a) 28 + k

7= 5

k = 7

(b) 1, 3, 3, k, 6, 7, 8

3 k 6

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2

1

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23. A Proton Wira can accommodate 1 driver and 3 adult.

Find the number of different ways the selection can

be made from 3 men and 4 women if,

(a) there are no restriction,(b) the driver must be a man.

Answer:

(a) 7C4

= 35

(b) 3C1 x6C3

= 60

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