S4011 WASSCE

May/June 2015

Further Mathematics/Mathematics (Elective) 1

Objective Test

1 1/2 hours

40 marks

1. Simplify $latex \displaystyle \frac{1 – 2\sqrt{5}}{2 + 3\sqrt{2}}$.

A. $latex 14(2\sqrt{2} + 6\sqrt{5} – 4\sqrt{10})$

B. $latex \frac{1}{14}(2 – 3\sqrt{2} – 4\sqrt{5} – 6\sqrt{8})$

C. $latex \frac{1}{14}(3\sqrt{2} + 4\sqrt{5} – 6\sqrt{10} – 2)$

D. $latex 14(2 + 3\sqrt{2} – 6\sqrt{5} + 4\sqrt{10})$

2. Solve $latex 2 \cos x – 1 = 0$ for $latex 0 \leq x \leq 2\pi$.

A. $latex (2\pi/3, 4\pi/3)$

B. $latex (\pi/6, 5\pi/6)$

C. $latex (\pi/5, 2\pi/5)$

D. $latex (\pi/3, 5\pi/3)$

3. Solve $latex 4(2^{x^2}) = 8^x$.

A. 1 and 2

B. 1 and -2

C. -1 and 2

D. -1 and 2

4. If $latex \log_3 x = \log_x 3$, find the value of x.

A. 3^{2}

B. 3^{1/2}

C. 3^{1/3}

D. 2^{1/3}

5. Find the third term of $latex \left( \frac{x}{2} – 1 \right)^8$ in descending order of x.

A. x^{3}/8

B. 7x^{6}/16

C. 7x^{5}/4

D. 35x^{4}/8

6. Given that $latex f:x \to x^2$ and $latex g:x \to x + 3$ where $latex x \in \mathbb{R}$, find $latex f \circ g(2)$.

A. 25

B. 9

C. 7

D. 5

7. Given that $latex \displaystyle \frac{2x}{(x + 6)(x + 3)} \equiv \frac{P}{x + 6} + \frac{Q}{x + 3}$, find P and Q.

A. P = 4 and Q = 2

B. P = 2 and Q = 4

C. P = 4 and Q = -2

D. P = -2 and Q = 4

8. Given that $latex \mathbf{P} = \left( \begin{array}{cc} -2 & 1 \\ 3 & 4 \end{array} \right)$ and $latex \mathbf{Q} = \left( \begin{array}{cc} 5 & -3 \\ 2 & -1 \end{array} \right)$, find $latex \mathbf{PQ – QP}$.

A. $latex \left( \begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array} \right)$

B. $latex \left( \begin{array}{cc} 27 & 12 \\ 16 & -15 \end{array} \right)$

C. $latex \left( \begin{array}{cc} -20 & -6 \\ 12 & -8 \end{array} \right)$

D. $latex \left( \begin{array}{cc} 11 & 12 \\ 30 & -11 \end{array} \right)$

9. Which of the following is a factor of the polynomial 6x^{4} + 2x^{3} + 15x + 5?

A. 3x + 1

B. x + 1

C. 2x + 1

D. x + 2

10. Given that $latex \displaystyle f:x \to \frac{2x – 1}{x + 2}, x \neq -2$, find $latex f^{-1}$, the inverse of f.

A. $latex \displaystyle \frac{1 + 2x}{2 – x}, x \neq 2$

B. $latex \displaystyle \frac{1 – 2x}{x + 2}, x \neq -2$

C. $latex \displaystyle \frac{1 – 2x}{x – 2}, x \neq 2$

D. $latex \displaystyle \frac{1 + 2x}{x + 2}, x \neq -2$

11. If 36, p, 9/4 and q are consecutive terms of an exponential sequence (G.P.), find the sum of p and q.

A. 9/16

B. 81/16

C. 9

D. 9 9/16

12. Find the minimum value of y = x^{2} + 6x -12.

A. -21

B. -12

C. -6

D. -3

A line passes through the origin and the point (1 1/4, 2 1/2).

Use this information to answer questions 13 and 14.

13. What is the gradient of the line?

A. 1

B. 2

C. 3

D. 4

14. Find the y coordinate of the line when x = 4.

A. 2

B. 4

C. 6

D. 8

15. In how many ways can a committee of five be selected from eight students if two particular students are to be included?

A. 20

B. 28

C. 54

D. 58

16. If x = i – 3j and y = 6i + j, calculate the angle between x and y.

A. 60°

B. 75°

C. 81°

D. 85°

17. The gradient of a curve at the point (-2, 0) is 3x^{2} – 4×. Find the equation of the curve.

A. y = 6x – 4

B. y = 6x^{2} – 4x + 12

C. y = x^{3} – 2x^{2}

D. y = x^{3} – 2x^{2} + 16

18. If $latex \alpha$ and $latex \beta$ are the roots of x^{2} + x – 2 = 0, find the value of $latex \left(\frac{1}{\alpha^2} + \frac{1}{\beta^2} \right)$.

A. 5/4

B. 3/4

C. 1/4

D. -3/4

19. Given that $latex x^2 + 4x + k \equiv (x + r)^2 + 1$, find the values of k and r.

A. k = 5, r = -1

B. k = 5, r = 2

C. k = 2, r = 5

D. k = -1, r = 5

20. Given the statements:

p: the subject is difficult;

q: I will do my best.

Which of the following is equivalent to *Although the subject is difficult, I will do my best.*

A. $latex p \lor q$

B. $latex \sim p \lor q$

C. $latex p \land (\sim q)$

D. $latex p \land q$

Forces of magnitude 8N and 5N act on a body as shown in the diagram.

Use this information to answer questions 21 and 22.

21. Calculate, correct to 2 decimal places, the resultant force acting at O.

A. 14.06 N

B. 13.00 N

C. 9.83 N

D. 8.26 N

22. Calculate, correct to 2 decimal places, the angle that the resultant makes with the horizontal.

A. 80.76°

B. 75.00°

C. 71.99°

D. 15.00°

23. Given that r = 2i – j, s = 3i + 5j and t = 6i – 2j, find the magnitude of (2r + s – t).

A. $latex \sqrt{15}$

B. 4

C. $latex \sqrt{24}$

D. $latex \sqrt{26}$

The table above shows the distribution of marks scored by students in a test.

Use the information to answer Questions 24 and 25.

24. How many candidates scored above the median mark?

A. 3

B. 12

C. 22

D. 30

25. Find the inter-quartile range of the distribution.

A. 4

B. 3

C. 2

D. 1

26. A mass of 75 kg is placed on a lift. Find the force exerted by the floor of the lift on the mass when the lift is moving up with a constant velocity. [Take g = 9.8ms^{-2}]

A. 750 N

B. 745 N

C. 735 N

D. 98 N

27. Each of the 90 students in a class speak at least Igbo or Hausa. If 56 students speak Igbo and 50 speak Hausa, find the probability that a student selected at random from the class speaks only Igbo.

A. 28/45

B. 4/9

C. 8/45

D. 1/9

28. If $latex left| \begin{array}{cc} 1 + 2x & -1 \\ 6 & 3 – x \end{array} \right| = -3$, find the values of x.

A. x = 3, -2

B. x = 4, -2/3

C. x = -4, 3/2

D. x = 4, -3/2

29. Find $latex \int \frac{x^3 + 5x + 1}{x^3} dx$.

A. x^{2} + 10x + c

B. x + 5/3 x^{3} + x^{4} + c

C. x – 5x^{2} – 2x^{3} + c

D. x – 5/x – 1/2x^{2} + c

30. Find the coordinates of the point which divides the line joining P(-2, 3) and Q(4, 6) internally in the ratio 2:3.

A. (5 2/5, 2/5)

B. (2/5, 5 2/5)

C. (-2/5, 5 2/5)

D. (2/5, 2 2/5)

A particle starts from rest and moves in a straight line such that its acceleration after t seconds is given by a = (3t – 2) ms^{-2}.

Use this information to answer Questions 31 and 32.

31. Find the other time when the velocity would be zero.

A. 1/3 s

B. 3/4 s

C. 4/3 s

D. 2 s

32. Find the distance covered after 3 seconds.

A. 10 m

B. 9 m

C. 13/3 m

4. 9/2 m

33. Given that y = 4 – 9x and $latex \Delta x$ = 0.1, calculate $latex \Delta y$.

A. 9.0

B. 0.9

C. -0.3

D. -0.9

34. Four fair coins are tossed once. Calculate the probability of obtaining equal numbers of heads and tails.

A. 1/4

B. 3/8

C. 1/2

D. 15/16

35. In calculating the mean of 8 numbers, a boy mistakenly used 17 instead of 25 as one of the numbers. If he obtained 20 as the mean, find the correct mean.

A. 24

B. 23

C. 21

D. 19

36. Simplify: ^{n}C_{r} + ^{n}C_{r – 1}.

A. $latex \frac{n(n – r)}{r} $

B. $latex \frac{n}{r(n – r)} $

C. $latex \frac{1}{r(n – r)} $

D. $latex \frac{n – r + 1}{r} $

37. If $latex 2 \sin^2 \theta = 1 + \cos \theta, 0^\circ \leq \theta \leq 90^\circ$, find the value of $latex \theta$.

A. 90°

B. 60°

C. 45°

D. 30°

38. A 24 N force acts on a body such that it changes its velocity from 5 ms^{-1} to 9 ms^{-1} in 2 seconds. If the body is travelling in a straight line, calculate the distance covered during this period.

A. 22 m

B. 18 m

C. 14 m

D. 10 m

39. The sum, S_{n} of a sequence is given by S_{n} = 2n^{2} – 5. Find the nth term.

A. 112

B. 67

C. 45

D. 22

40. Forces F_{1}(8 N, 030°) and F_{2}(10 N, 150°) act on a particle. Find the horizontal component of the resultant force.

A. 1.7 N

B. 4.5 N

C. 9.0 N

D. 13.0 N