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MID YEAR EXAMINATION 2013________________________________________________________________________________________

ADDITIONAL MATHEMATICS PAPER 1 (2 HOURS)

FORM 5________________________________________________________________________________________

1. Diagram 1 shows the function h : x→ m−xx , x≠0, where m is a constant.

Find the value of m. (2 marks)

Diagram 1

2. Given the functions f ( x )=x−1 and ( x )=kx+2 , find(a) f (5 )(b) the value of k such that gf (5 )=14. (3 marks)

3. Solve the quadratic equation 3 x2+5 x−2=0 . (2 marks)

4. A quadratic equation x2+ px+9=2 x has two equal roots. Find the possible values of p . (3 marks)

5. Find the range of values of x for (2 x−1 ) (x+4 )>4+x . (2 marks)

6. Solve the equation log34 x−log3 (2x−1 )=1. (3 marks)

7. Solve the equation 162 x−3=84 x. (3 marks)

8. Diagram 2 shows a straight line AB.

Diagram 2

Find (a) the midpoint of AB(b) the equation of the perpendicular bisector of AB. (4 marks)

9. A group of 6 students has a total mass of 240 kg. The sum of the squares of their masses is 9654 kg. Find(a) the mean mass of the 6 students(b) the standard deviation. (3 marks)

10. Diagram 3 shows the sectors OAB and ODC with centre 0.

Diagram 3It is given that OA=4cm, the ratio of OA : OD = 2 : 3 and the area of the shaded region is 11.25cm2. Find(a) the length, in cm, of OD(b) θ, in radians. (4 marks)

11. Given the function h ( x )=k x3−4 x2+5 x , find (a) h' ( x )(b) the value of k if h' ' (1)=4 . (4 marks)

12. The first three consecutive terms of an arithmetic progression are −5 ,2x−6∧3 x . Find the common difference of the progression. (3 marks)

13. It is given that 3 ,6 x ,12x2 ,24 x3 , 48x4 , …. is a geometric progression and its sum to infinity is 5. Find(a) the common ratio in terms ofx (b) the positive valuex. (3 marks)

14. The first three terms of an arithmetic progression are 2 , x , 92 (given that all terms

are positive). Find(a) the value of x(b) the sum from the fifth term to the ninth term. (4 marks)

15. The sum of the first n terms of the geometric progression 8, 24, 72, …. is 8744. Find(a) the common ratio of the progression(b) the value of n . (4 marks)

16. The variables x and y are related by the equation ¿5x−1 . A straight line graph is

obtained by plotting x2 y against x , as shown in Diagram 4.

Find the value of p and q . (4 marks)

Diagram 4

17. The variables x and y are related by the equation y=k x 4, where k is a constant. (a) Convert the equation y=k x 4 to linear form.(b) Diagram 5 shows the straight line obtained by plotting log10 y against log10 x .

Diagram 5

Find the value of (i) log10 k(ii) h . (4 marks)

18. Given that ∫ (−5x2+2 x )dx=k x3+x2+c where k and c are constants. Find(a) the value of k(b) the value of c if ∫ (−5x2+2 x )dx=20 when x=20 . (3 marks)

19. Given that ∫−1

3

f (x)dx=11 and ∫−1

3

(3 f ( x )−k x2¿)dx=30¿ , find the value of k. (4 marks)

20. Given that ∫1

5

k (x )dx=4, find

(a) -∫5

1

k (x )dx

(b) ∫1

5

[k ( x)−8]dx (4 marks)

21. Diagram 6 shows the curve y=f (x ) cutting the x-axis at x=a and ¿b .

Diagram 6

Given that the area of the shaded region is 5 unit2, find the value of ∫a

b

2 f ( x )dx. (2 marks)

22. Diagram 7 shows a triangle ABC. Point D lies on BC such that BD : DC = 3 : 2.

Diagram 7

Express the following vectors in terms of x and y.(a) B⃗C(b) A⃗D . (4 marks)

23. Given that m=−10 i+3 j and ¿2 i−k j . Find (a) m−n in the form of x i+ y j (b) the values of k if |m−n|=13 (4 marks)

24. Diagram 8 shows a parallelogram OPQR drawn on a Cartesian plane.

Diagram 8

It is given that O⃗P=6 i+4 j and P⃗Q=−4 i+5 j . Find P⃗R . (3 marks)

~End of the questions paper~

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