1 2019 May WASSCE Elective Maths Paper 1

1. Solve \(8^{x - 2} = 4^{3x}\).
A. -1
B. -2
C. 1
D. 2

2. Evaluate \(\tan 75^\circ\), leaving the answer in surd form (radicals).
A. \(\sqrt{3} + 1\)
B. \(\sqrt{3} - 1\)
C. \(\sqrt{3} + 2\)
D. \(\sqrt{3} - 2\)

3. Solve \(\displaystyle \frac{p}{2} + \frac{k}{3} = 5\) and \(2p - k = 6\) simultaneously.
A. p = 6, k = 6
B. p = -6, k = -6
C. p = 6, k = -6
D. p = -6, k = 6

4. Rationalize: \(\displaystyle \frac{1}{\sqrt{2} + 1}\).
A. \(1 - \sqrt{2}\)
B. \(\frac{1 - \sqrt{2}}{2}\)
C. \(\sqrt{2} - 1\)
D. \(\frac{\sqrt{2} - 1}{2}\)

5. If \(^nC_2 = 15\), find the value of \(n\).
A. 6
B. 5
C. 8
D. 7

6. An operation \((*)\) is defined on the set \(\mathbf{T} = \{-1, 0, \ldots, 5\}\) by \(x (*) y = x + y - xy\). Which of the following operation(s) will given an image which is an element of \(\mathbf{T}\)?
I. \(2 (*) 5\)
II. \(3 (*) 2\)
III. \(3 (*) 4\)
A. I and III only
B. II and III only
C. I only
D. II only

7. Given that \(g: x \to 3x\) and \(f: x \to \cos x\), find the value of \(g \circ f(20^\circ)\).
A. 0.94
B. 0.50
C. 2.82
D. 2.60

8. A linear transformation is defined by \(\mathbf{T}:(x, y) \to (-x + y, -4y)\). Find the image, Q’ of Q(-3, 2) under \(\mathbf{T}\).
A. Q’(5, 3)
B. Q’(-5, -8)
C. Q’(5, -8)
D. Q’(-8, 5)

9. If \(g:r \to 5 - 2r\), \(r\) is a real number, find the image of -3.
A. 11
B. -9
C. 13
D. -1

10. Consider the following statements:
p: Birds fly
q: The sky is blue
r: The grass is green
What is the symbolic representation of “If the grass is green and the sky is not blue, then the birds do not fly”?
A. \((r \land \sim q) \implies \sim p\)
B. \((r \land p) \implies q\)
C. \((r \land \sim p) \implies \sim q\)
D. \((r \land q) \implies \sim p\)

11. Given that \(\displaystyle \frac{1}{x^2 - 4} \equiv \frac{P}{x + 2} + \frac{Q}{x - 2}, x \neq \pm 2\), find the value of \((P + Q)\).
A. \(\frac12\)
B. 0
C. \(\frac32\)
D. 1

12. Find the sum of teh first 20 terms of teh sequence: -7, -3, 1, …
A. 690
B. 620
C. 1240
D. 660

13. Find the value of \(x\) for which \(6(\sqrt{4x^2 + 1}) = 13x\), where \(x > 0\).
A. \(\frac{24}{25}\)
B. \(\frac{5}{6}\)
C. \(\frac65\)
D. \(\frac{25}{24}\)

14. Calculate the distance between points (-2, -5) and (-1, 3).
A. \(\sqrt{17}\) units
B. \(\sqrt 5\) units
C. \(\sqrt{73}\) units
D. \(\sqrt{65}\) units

15. If \(\displaystyle \pmatrix{ 2 & 3 \\ -4 & 1}, Q = \pmatrix{ 6 \\ 8}\) and \(\displaystyle PQ = k \pmatrix{45 \\ -20}\), find the value of \(k\).
A. \(\frac45\)
B. \(-\frac54\)
C. \(\frac54\)
D. \(-\frac45\)

16. The second and fourth terms of an exponential sequence (G.P.) are \(\frac29\) and \(\frac{8}{81}\) respectively. Find the sixth term of the sequence.
A. \(\frac14\)
B. \(\frac{32}{729}\)
C. \(\frac{81}{32}\)
D. \(\frac98\)

17. Points X and Y are on the same horizontal base as teh foot of a building such that X is 96 m due east of the building and Y is due west. If the angle of elevation of the top of the building from X is \(30^\circ\) and that of Y is \(60^\circ\), calculate the distance of Y from the building.
A. 32 m
B. 42 m
C. 50 m
D. 30 m

18. Find the coordinates of the point on the curve \(y = 3x^2 - 2x - 5\), where the tangent is parallel to the line \(y - 5 - 8x\).
A. \((0, \frac53)\)
B. \((\frac53, 0)\)
C. \((-\frac53, 0)\)
D. \((0, -\frac53)\)

19. If the mean of \(2, 5, (x + 1), (x + 2), 7\) and \(9\) is \(6\), find the median.
A. 5.5
B. 6.5
C. 5.0
D. 6.0

20. Calculate the mean deviation of 5, 8, 2, 9 and 6.
A. 4
B. 2
C. 5
D. 3

A particle starts from rest and moves in a straight line such that its velocity \(v\) ms^-1 at time \(t\) seconds is given by \(v = 3t^2 - 6t\).

Use the information to answer questions 21 and 22.

21. Calculate the distance in 4 seconds.
A. 16 m
B. 96 m
C. 12 m
D. 64 m

22. Calculate the acceleration in the 3rd second.
A. 3 ms^-2
B. 9 ms^-2
C. 0 ms^-2
D. 6 ms^-2

23. Find the constant term in the binomial expansion of \(\displaystyle \left( 2x^2 + \frac{1}{x^2} \right)^4\).
A. 12
B. 42
C. 10
D. 24

24. Which of these inequalities is represented by the shaded prtion of the graph?
A. \(2y - x + 3 < 0\)
B. \(2y + x + 3 < 0\)
C. \(2y + x - 3 < 0\)
D. \(2y - x - 3 < 0\)

25. A 35 N force acts on a body of mass 5 kg for 2 seconds. Calculate the change in momentum of the body.
A. 50 kgms^-1
B. 70 kgms^-1
C. 35 kgms^-1
D. 55 kgms^-1

26. Solve, correct to three significant figures, \((0.3)^x = (0.5)^8\).
A. 0.461
B. 4.61
C. 0.0130
D. 4.606

27. Given that P and Q are two non-empty subsets of the universal set \(\mathbf{\mu}\), find \(P \cap (Q \cup Q')\).
A. P’
B. Q’
C. P
D. Q

28. Find the coefficient of the third term in the binomial expansion of \(\displaystyle \left( 2x + \frac{3y}{4} \right)^3\) in descending powers of \(x\).
A. \(\frac{27}{8}y^2\)
B. \(9y^2\)
C. \(\frac{27}{64} y^2\)
D. \(8y^2\)

29. Find the coordinates of the centre of the circle \(3x^2 + 3y^2 - 6x + 9y - 5 = 0\).
A. \((1, -\frac32 )\)
B. \((3, -\frac92)\)
C. \((-3, \frac92)\)
D. \((-1, \frac32)\)

30. Evaluate \(\displaystyle \int_0^9 \sqrt{x}\ dx\).
A. 9
B. 3
C. 27
D. 18

31. The function \(f: x \to x^2 + px + q\) has a turning point when \(x = -3\) and a remainder of \(-6\) when divided by \((x + 2)\). Find the value of \(q\).
A. -2
B. 6
C. -8
D. 2

32. If \(y = (5 - x)^{-3}\), find \(\displaystyle \frac{dy}{dx}\).
A. \(\displaystyle \frac{3}{(5 - x)^4}\)
B. \(\displaystyle \frac{-15}{(5 - x)^4}\)
C. \(\displaystyle \frac{15}{(5 - x)^4}\)
D. \(\displaystyle \frac{-3}{(5 - x)^4}\)

33. Which of the following vectors is perpendicular to \(\displaystyle \pmatrix{-1 \\ 3}\)?
A. \(\displaystyle \pmatrix{ 1 & 3 }\)
B. \(\displaystyle \pmatrix{ 3 & 1 }\)
C. \(\displaystyle \pmatrix{ -3 & 1 }\)
D. \(\displaystyle \pmatrix{ 1 & -3 }\)

34. Find, correct to the nearest degree, the angle between \(\mathbf{p} = 12 \mathbf{i} - 5 \mathbf{j}\) and \(\mathbf{q} = 4 \mathbf{i} + 3\mathbf{j}\).
A. 75°
B. 59°
C. 76°
D. 60°

35. Find the area between the line \(y = x + 1\) and the \(x\)-axis from \(x = -2\) to \(x = 0\).
A. 2 square units
B. 1 square unit
C. 5 square units
D. 4 square units

36. How many numbers greater than 200 can be formed from the digits 1, 2, 3, 4, 5 if no digit is to be repeated in any particular number?
A. 288
B. 50
C. 300
D. 60

37. The probabilities that John and Jane will pass an examination are 0.9 and 0.7 respectively. Find the probability that at least one of them will pass the examination.
A. 0.72
B. 0.97
C. 0.28
D. 0.67

38. Given that X and Y are independed events such that \(p(X) = 0.5, p(Y) = m\) and \(p(X \cup Y) = 0.75\), find the value of \(m\).
A. 0.4
B. 0.3
C. 0.6
D. 0.5

39. A uniform beam, \(PQ\), is 100 cm long and weighs 35 N. It is placed on a support at a point 40 cm from \(P\). If weights of 54 N and F N are attached at \(P\) and \(Q\) respectively in order to keep it in a horizontal position, calculate, correct to the nearest whole number, the value of F.
A. 35
B. 30
C. 69
D. 60

40. Evaluate: \(\displaystyle \lim_{x \to 1} \frac{1 - x}{x^2 - 3x + 2}\).
A. \(\frac12\)
B. -1
C. 1
D. \(-\frac12\)