# Additional Mathematics - Paper 1 by nklye

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```									ADDITIONAL MATHEMATICS
SPM 2012
PAPER 1
( PART TWO )

LAST KOPEK REVISION
14.      The variables x and y are related by the equation
p
y = x . The diagram below shows the straight line graph
3
obtained by plotting log10 y against x.
p
(a) Express the equation x in its linear form used to obtain
3
the straight line graph as shown in the diagram.

(b) Find the value of p.

(a) log10 y = log10 p – x log10 3   √1                         (0,2)

(b) log10 p = 2     √ M1                                   0
x

p = 100    √2
15.   Given that y = 2x + 1 , x = 3
(x – 3)2
(a) find the value of dy/dx, when x = 4 ,
(b) the approximate change in y when x increases
from 4 to 4.01

(a) dy/dx = (2x + 1) (2) (x – 3) (1) – (x – 3) (2)
2
√ M1
(x – 3)4
[2(4) + 1] (2) (4 – 3) (1) – (4 – 3)2 (2)
=
(4 – 3)4
= 16 √ 2
15 (b) Sambungan

(b) x = 4
∂x = 0.01    √ M1

dy/dx = 16

∂y = dy/dx x ∂x
∂y = 16 (0.01)      √ M1

∂y = 0.16    √2
16.   Given y = 5h i + (h – 2) j is a non-zero vector and is
parallel to x-axis. Find the value of h.

5h i + (h – 2) j = x i + 0 j
( parallel to x-axis, vertical component of the vector = 0 )

h–2=0         √ M1

h=2         √2
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 = ¾ AB.
Find in terms of i and j,
(a) OC,
(b) CB.

C.
.B
(a) AB = − OA + OB
.A
= − (7 i + 2 j) + (i + 8 j) √ M1
x
=−6i+6j                                   O

OC = −9/2 i + 9/2 j √ 2
(b) CB = − OC + OB
= − 9/2 i + 9/2 j + (i + 8 j) √ M1
= − 11/2 i + 7/2 j √ 2
18.   Diagram shows a unit circle. T (−h, k) is a point on
the rotating ray of an angle θ. Find in terms of h, the
value of
(a) cos θ.
(b) sin 2θ.
y

(-h, k)
k
θ
(a) cos θ = −h            θ
x
√1           −p               O

(b) sin 2θ = 2 sinθ cosθ
= 2 (k) (−p) √ M1
= −2pk √ 2
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)

5
A
(a) sin A cos B + cos A sin B                4
= (3/5) (12/13) + (4/5) (5/13) √ M1         13
5
= 56/65 √ 2                              B
tan A − tan B                           12
(b)
1 + tan A tan B
= (3/4) – (5/12) √ M1 = 7/16 √ 2
1 + (3 /4) (5 /12)
20.     Point A lies on a curve y = 2x4 – x, find the
coordinates of point A where the gradient of the
normal at point A is -1/7.

dy/dx = mt = 8x3 − 1    √ M1

8x3 – 1 = 7 √ M2
8x3 = 8
x=1
y = 2(1)4 – 1
=1
A = ( 1, 1) √ 3
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 added
to this set.

N=6                       x2 =9
σ2 = 15                   x =3
∑ x2 = 144                  ∑x = x (N)
144                     = 3 (6) = 18
15 =     −x2    √ M1
6                   ∑xnew = 18 + 10     √ M2

x new = 28/7 = 4   3
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 of k if the median of
the data is k.

(a)   28 + k       √ M1
=5
7
k=7    √2

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

3≤k≤6      √1
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.

(a) 7C4
= 35

(b) 3C1 x 6C3
= 60
THE END
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