1AddMaths Revision SPM 2015 [cikgubid/AMF5/W49/Mon/KYRHG]

Nov. 23, 2015PAPER 1

1) Diagram 1 shows a relation of set x to set y.

Diagram 1

(a) Determine the domain of the relation.(b) Is the relation a function? Why?

2) Diagram 2 shows the mapping for the function f−1

and g .

Diagram 2

Given that f ( x )=ax+b and g( x )=b

ax

, calculate the value of a and of b. [a=3

2, b=3

]

3) Given that p and q are the roots of the quadratic equation 2 x2−5 x+10=0 , find the values of

(a)

p+qpq ,

(b) p2+q2. [(a)

12 (b)

−154 ]

4) Given that the equations x2+kx+1=0 and x

2−x+k=0 has common roots, find the value of k and the common root. [k=−2 , x=−1 ]

2

5) Diagram 3 shows the graph of the function y=p−( x+n )2, where p and n are constants.

Diagram 3

Find(a) the value of p,(b) the equation of the axis of symmetry,(c) the value of n and of k. [(a)11 (b) x = –3 (c) n = 3, k = 2]

6) Given that 3 x−2 y+6=0 , find the range of values of x for which −1 .5 < x < 3. [–3< x < 0]

7) Given that ap=2 and a

q=7 , find loga(3 .5 a2 ) in terms of p and q. [q− p+2 ]

8) The sum of the first n terms of the progression

13

,1 ,3 , . .. .. . .is

121 13 .

(a) Find the value of n.

(b) State the value of the (n−1)th term. [(a) 6 (b) ]

9) The straight line y=x+a is perpendicular to the straight line y+( a−1 )x=3 b . Given that the two straight lines intersect at the point (b, c), find the value of a, of b and of c. [a = 2, b = 2, c = 4]

3

10) The variables x and y are related by the equation x2 y=ax2−b , where a and b are constants. A

straight line is obtained by plotting y against

1x2

as shown in Diagram 4.

Diagram 4Find the value of a and of b. [a = 12, b = 3]

11) P(−2,5) , Q(0,3 ) and R(r ,−2) are three points on a straight line. Find(a) the ratio PQ : QR,(b) the value of r.

12) Diagram 5 shows the sector OAB of a circle with centre O. ABCD is a rectangle.

Diagram 5

Given OA = 10 cm and AD = 8 cm, find(a) ∠ AOB in radian,(b) the area of the shaded region. [(a)1.287 rad. (b)16.35 cm2]

13) Find the value of t if ∫1

t 2 x ( x−1)x

dx=4. [3]

4

14) Solve the equation 4 sin xcos x+√3=0 for 0o≤x≤360o

. [120o, 150o, 300o, 330o]

15) It is given that sin α=−4

5 and cos β=24

25 , where α and β are in the same quadrant. Find the value of

(a) sin( α−β ),

(b)cos2( β

2 ). [(a)

−35 (b)

4950 ]

16) Evaluate limn→∞

4 n2

10 n−6n2. [

−23 ]

17) The height of a cylinder is two times its radius. If the radius of the cylinder is increasing at a rate of 3 cm s−1

, calculate the rate of change of its volume at the instant its radius is 6 cm. [648 π cm3 s−1]

18) Diagram 6 shows the curve of y=( x−2)3.

5

Diagram 6

If the area of the shaded region is 20 1

4 units2, find the value of k. [5]

19) ABCD is a parallelogram such that AB→

= i~−2 j

~ and BC→

=2 i~+6 j

~ . Find the unit vector in the

direction of CA→

. [−3

5i~− 4

5j~ ]

20) Box P contains three balls numbered 5, 7 and 9. Box Q contains three cards numbered 4, 6 and 8. A ball is drawn from box P and a card is drawn at random from box Q. Calculate the probability that the sum of the number on the ball and the number on the card is either a prime number of a perfect

square. [79 ]

21) Diagram 7 shows a standard normal distribution graph.

Diagram 7

If P( Z < −m)=0 .2119 , find the area of the shaded region. [0.2881]

6