WAEC FURTHER MATHEMATICS PAST QUESTIONS AND ANSWER

2010

[latexpage]

1. Find the domain of $f(x) = \frac{x}{3-x}, x \in \mathbb{R}$
Options:
A. $\{x : x \in \mathbb{R}, x \neq 3\}$
B. $\{x : x \in \mathbb{R}, x \neq 1\}$
C. $\{x : x \in \mathbb{R}, x \neq 0\}$
D. $\{x : x \in \mathbb{R}, x \neq -3\}$

2. Find the value of $\cos(60° + 45°)$ leaving your answer in surd form
Options:
A. $\frac{6 + \sqrt{2}}{4}$
B. $\frac{3 + \sqrt{6}}{4}$
C. $\frac{2\sqrt{-6}}{\sqrt{4}}$
D. $\frac{3 – \sqrt{6}}{4}$

3. If $\frac{5}{2\sqrt{-8}\sqrt{8}} = m^2 – \sqrt{m}$, where $m$ is a constant. Find $m$
Options:
A. $\frac{1}{1.2}$
B. $\frac{1}{1.4}$
C. $\frac{2}{1.4}$
D. $\frac{2}{1.2}$

4. If $\frac{16}{3x} = \frac{1}{4}(32^{x-1})$, find the value of $x$
Options:
A. $-1$
B. $-\frac{1}{3}$
C. $-\frac{3}{7}$
D. $-\frac{5}{19}$

5. Simplify $\log_5(8) \log_5(8\sqrt{})$
Options:
A. $-2$
B. $-\frac{1}{2}$
C. $\frac{1}{2}$
D. $2$

6. The coefficient of the 7th term in the binomial expansion of $(2 – \frac{x}{3})^{10}$ in ascending powers of $x$
Options:
A. $\frac{560}{243}$
B. $\frac{841}{243}$
C. $\frac{1120}{243}$
D. $\frac{4481}{243}$

7. The roots of a quadratic equation are $(3 – \sqrt{3})$ and $(3 + \sqrt{3})$. Find its equation
Options:
A. $x^2 – 6x – 9 = 0$
B. $x^2 – 6x + 6 = 0$
C. $x^2 + 6x – 9 = 0$
D. $x^2 + 6x + 6 = 0$

8. If $(x – 3)$ is a factor of $2x^2 – 2x + p$, find the value of constant $p$
Options:
A. $-12$
B. $-6$
C. $3$
D. $6$

9. If $\sin(x) = -\sin(70°)$, $0° < x < 360°$, determine the two possible values of $x$ Options: A. $110°, 250°$ B. $110°, 290°$ C. $200°, 250°$ D. $250°, 290°$ 10. For what values of $x$ is $\frac{x^2 - 9x + 18}{x^2 + 2x - 35}$ undefined Options: A. $6$ or $3$ B. $-18$ or $-9$ C. $-7$ or $5$ D. $-5$ or $7$ 11. Calculate the length of the line joining points $X(3, 5)$ and $Y(5, 1)$, correct to one decimal place Options: A. $4.0$ B. $4.2$ C. $4.5$ D. $5.0$ 12. If $y = 2(2x + \sqrt{x - 2})^2$, find $\frac{dy}{dx}$ Options: A. $\frac{2}{\sqrt{x - 2}}(2x + \sqrt{2x - 2})$ B. $4(2x + \sqrt{x - 2})(2 + \frac{1}{2\sqrt{x}})$ C. $4(2x + \sqrt{x - 2})(2 + \sqrt{x - 2})$ D. $8(2x + \sqrt{x - 2})(2 + \sqrt{x - 2})$ 13. Calculate the acute angle between the lines $3x - 4y + 5 = 0$ and $2x + 3y - 1 = 0$, correct to one decimal place Options: A. $70.6°$ B. $50.2°$ C. $39.8°$ D. $19.4°$ 14. Evaluate $\int_1^2 4x^3 dx$ Options: A. $-\frac{1}{1.2}$ B. $-\frac{15}{16}$ C. $\frac{15}{16}$ D. $\frac{1}{1.2}$ 15. If $\left|\frac{3}{2x} \div \frac{x - 2}{x}\right| = -2$, find the value of $x$ Options: A. $-8$ B. $4$ C. $-4$ D. $8$ 16. Given $P = \{x: x \text{ is a factor of } 6\}$ is the domain of $g(x) = x^2 + 3x - 5$, find the range of $x$ Options: A. $\{-1, 5, 13\}$ B. $\{5, 13, 49\}$ C. $\{1, 2, 3, 6\}$ D. $\{-1, 5, 13, 49\}$ 17. The third term of a geometric progression (G.P) is $10$ and the sixth term is $80$. Find the common ratio Options: A. $2$ B. $3$ C. $4$ D. $8$ 18. Find the axis of symmetry of the curve $y = x^2 - 4x - 12$ Options: A. $x = -2$ B. $y = -2$ C. $x = 2$ D. $y = 2$ 19. Find the equation of the tangent to the curve $y = \frac{4}{x^2 - 12x + 7}$ at point $(2, -1)$ Options: A. $y + 4x - 9 = 0$ B. $y - 4x - 9 = 0$ C. $y - 4x + 9 = 0$ D. $y + 4x + 9 = 0$ 20. The mean age of 15 pupils in a class is $14.2$ years. One new pupil joined the class and the mean changed to $14.1$ years. Calculate the age of the new pupil Options: A. $12.4$ years B. $12.6$ years C. $13.2$ years D. $14.1$ years 21. The distance $s$ metres of a particle from a fixed point at time $t$ seconds is given by $s = 7 + \frac{p}{t^3} + t^2$, where $p$ is a constant. If the acceleration at $t = 3$ secs is $8 \text{ m/s}^{-2}$, find the value of $p$ Options: A. $\frac{1}{3}$ B. $\frac{4}{9}$ C. $\frac{5}{9}$ D. $1$ 22. The probabilities that a husband and wife will be alive in 15 years time are $m$ and $n$ respectively. Find the probability that only one of them will be alive at that time Options: A. $mn$ B. $m + n$ C. $m + n - 2mn$ D. $1 - mn$ 23. In a class of 50 pupils, 35 like Science and 30 like History. What is the probability of selecting a pupil who likes both Science and History? Options: A. $0.10$ B. $0.30$ C. $0.60$ D. $0.70$ 24. $P, Q, R, S$ are points in a plane such that $PQ = 8i - 5j$, $QR = 5i + 7j$, $RS = 7i + 3j$ and $PS = xi + yj$. Find $(x, y)$ Options: A. $(-6, -15)$ B. $(-6, 5)$ C. $(20, 5)$ D. $(20, 15)$ 25. Find the least value of $n$ for which $\binom{3n}{2} > 0$, $n \in \mathbb{R}$
Options:
A. $\frac{1}{3}$
B. $\frac{1}{6}$
C. $\frac{2}{3}$
D. $1$

26. If $\vec{OA} = 3i + 4j$ and $\vec{OB} = 5i – 6j$ where $O$ is the origin and $M$ is the midpoint of $AB$, find $\vec{OM}$
Options:
A. $-2i – 10j$
B. $-2i + 2j$
C. $4i – j$
D. $4i + j$

27. Find the direction cosines of the vector $4i – 3j$
Options:
A. $\frac{9}{10}, \frac{27}{10}$
B. $\frac{17}{27}, -\frac{17}{27}$
C. $\frac{4}{5}, -\frac{3}{5}$
D. $\frac{4}{7}, -\frac{3}{7}$

28. Yomi was asked to label four seats $S, R, P, Q$. What is the probability he labelled them in alphabetical order?
Options:
A. $\frac{1}{24}$
B. $\frac{1}{6}$
C. $\frac{2}{13}$
D. $\frac{1}{4}$

29. Two forces $(2i – 5j)N$ and $(-3i + 4j)N$ act on a body of mass $5 \text{ kg}$. Find the magnitude of the acceleration of the body in $\text{m/s}^{-2}$
Options:
A. $\frac{2}{\sqrt{5}}$
B. $\frac{5}{2\sqrt{2}}$
C. $\frac{2}{\sqrt{5}}$
D. $\frac{5}{2\sqrt{2}}$

30. Two particles are fired together along a smooth horizontal surface with velocities $4 \text{ m/s}$ and $5 \text{ m/s}$. If they move at $60°$ to each other, find the distance between them in $2$ seconds
Options:
A. $\sqrt{2 \div 61}$
B. $\sqrt{42}$
C. $\sqrt{2 \div 21}$
D. $\sqrt{2 \div 10}$

31. Two forces $\vec{F_1} = (7i + 8j)N$ and $\vec{F_2} = (3i + 4j)N$ act on a particle. Find the magnitude and direction of $\vec{F_1} – \vec{F_2}$
Options:
A. $(\sqrt{42}N, 0°)$
B. $(\sqrt{42}N, 45°)$
C. $(\sqrt{42}N, 90°)$
D. $(\sqrt{42}N, 180°)$

32. A stone is thrown vertically upwards and its height at any time $t$ seconds is $h = 45t – 9t^2$. Find the maximum height reached
Options:
A. $45.25 \text{ m}$
B. $45.50 \text{ m}$
C. $56.00 \text{ m}$
D. $56.25 \text{ m}$

33. Given that $\frac{dy}{dx} = 3x^2 – 4$ and $y = 6$ when $x = 3$, find the equation for $y$
Options:
A. $x^3 – 4x – 9$
B. $x^3 – 4x + 9$
C. $x^3 + 4x – 9$
D. $x^3 + 4x + 9$

34. If $h(x) = x^3 – \frac{1}{x^3}$, evaluate $h(a) – h(\frac{1}{a})$
Options:
A. $-1$
B. $0$
C. $\frac{2}{a^3} – 2a^3$
D. $2a^3 – \frac{2}{a^3}$

35. A company took delivery of 12 vehicles made up of 7 buses and 5 saloon cars for two departments. If the Personnel department is to have at least 3 saloon cars, in how many ways can these vehicles be distributed equally between the departments?
Options:
A. $350$
B. $455$
C. $462$
D. $571$

36. A bicycle wheel of diameter $70 \text{ cm}$ covered a distance of $350 \text{ cm}$ in $2 \text{ seconds}$. How many radians per second did it turn?
Options:
A. $5$
B. $7$
C. $8$
D. $10$

37. The initial velocity of an object is $u = (-\frac{5}{3}) \text{ m/s}^{-1}$. If the acceleration of the object is $a = (3 – 4) \text{ m/s}^{-2}$ and it moved for $3$ seconds, find the final velocity
Options:
A. $(-\frac{14}{15}) \text{ m/s}^{-1}$
B. $(-\frac{2}{1}) \text{ m/s}^{-1}$
C. $(4 – 9) \text{ m/s}^{-1}$
D. $(14 – 9) \text{ m/s}^{-1}$

38. Find the maximum value of $2 + \sin(\theta + 25)$
Options:
A. $1$
B. $2$
C. $3$
D. $4$

39. Simplify $(1 + 2\sqrt{3})^2 – (1 – 2\sqrt{3})^2$
Options:
A. $0$
B. $\frac{8}{\sqrt{3}}$
C. $13$
D. $2 – 4\sqrt{3}$

40. What is the angle between $\vec{a} = (3i – 4j)$ and $\vec{b} = (6i + 4j)$
Options:
A. $13°$
B. $87°$
C. $100°$
D. $110°$

2011

[latexpage]
1. A binary operation * is defined on the set of real numbers $R$, by $a * b = -1$. Find the identity element under the operation *.
A. $-1$
B. $0$
C. $1$
D. $2$

2. Express $75°$ in radians, leaving your answer in terms of $\pi$.
A. $\frac{5\pi}{12}$
B. $\frac{3\pi}{4}$
C. $\frac{5\pi}{6}$
D. $\frac{7\pi}{6}$

3. If $\log_9(3+2x)=1$, find $x$.
A. $-\frac{1}{2}$
B. $-\frac{1}{4}$
C. $\frac{1}{4}$
D. $\frac{1}{2}$

4. Evaluate $\cos(\frac{\pi}{2}+\frac{\pi}{3})$
A. $-\frac{2}{\sqrt{3}}$
B. $-\frac{\sqrt{3}}{2}$
C. $\frac{\sqrt{3}}{4}$
D. $\frac{4\sqrt{3}}{3}$

5. Find the remainder when $5x^3+2x^2-7x-5$ is divided by $(x – 2)$.
A. $-51$
B. $-23$
C. $29$
D. $49$

6. A function is defined by $f(x)=\frac{3x+1}{x^2-1}, x \neq \pm 1$. Find $f(-3)$.
A. $-\frac{1}{4}$
B. $-1$
C. $\frac{4}{5}$
D. $1$

7. Simplify $\frac{8}{27}-\frac{-\sqrt{3}-(4/9)^{-1/2}}{-}$
A. $-\frac{5}{6}$
B. $-\frac{4}{27}$
C. $0$
D. $\frac{2}{9}$

8. Solve $3x^2+4x+1>0$
A. $x<-1, x<-\frac{1}{3}$ B. $x>-1, x>-\frac{1}{3}$
C. $x>\frac{1}{3}, x<-1$ D. $x<\frac{1}{3}, x>-1$

9. The equation of a circle is $3x^2+3y^2+6x-12y+6=0$. Find its radius
A. $1$
B. $3-\sqrt{3}$
C. $11-\sqrt{11}$
D. $6-\sqrt{6}$

10. $f(x)=p+qx$, where $p$ and $q$ are constants. If $f(1) = 7$ and $f(5) = 19$, find $f(3)$.
A. $13$
B. $15$
C. $17$
D. $26$

11. The sum and product of the roots of a quadratic equation are $\frac{4}{7}$ and $\frac{5}{7}$ respectively. Find its equation.
A. $7x^2-4x-5=0$
B. $7x^2-4x+5=0$
C. $7x^2+4x-5=0$
D. $7x^2+4x+5=0$

12. $f(x)=(x^2+3)^2$ is defined on the set of real numbers, $R$. Find the gradient of $f(x)$ at $x = \frac{1}{2}$.
A. $4.0$
B. $6.5$
C. $5.0$
D. $10.6$

13. Find $\lim_{x \to 3} \frac{x+3}{x^2-x-12}$
A. $-1$
B. $-\frac{1}{7}$
C. $\frac{1}{7}$
D. $1$

14. If $y^2+xy-x=0$, find $\frac{dy}{dx}$.
A. $\frac{1-y}{2y}$
B. $\frac{1-2y}{x}$
C. $\frac{1-y}{x+2y}$
D. $\frac{1}{x+2y}$

15. A line is perpendicular to $3x-y+11=0$ and passes through the point $(1, -5)$. Find its equation.
A. $3y – x -14 = 0$
B. $3x + y + 1 = 0$
C. $3y + x + 1 = 0$
D. $3y + x + 14 = 0$

16. Solve $\frac{9}{2x+1} = \frac{81}{3x+2}$
A. $-\frac{3}{4}$
B. $-\frac{2}{3}$
C. $\frac{4}{5}$
D. $\frac{3}{2}$

17. The inverse of a function is given by $f^{-1}: x \to \frac{x+1}{4}$.
A. $f: x \to 4x-1$
B. $f: x \to 4x+1$
C. $f: x \to \frac{4x-1}{4}$
D. $f: x \to \frac{x-1}{2}$

18. If $(\frac{3}{7})^{2x}(\frac{2}{3}) = (\frac{12}{29})$, find $x$.
A. $5$
B. $6$
C. $7$
D. $8$

19. The fourth term of a geometric sequence is $2$ and the sixth term is $8$. Find the common ratio.
A. $\pm 1$
B. $\pm 2$
C. $\pm 3$
D. $\pm 4$

20. What percentage increase in the radius of a sphere will cause its volume to increase by $45\%$?
A. $13\%$
B. $15\%$
C. $23\%$
D. $25\%$

21. Evaluate $\frac{1}{1-\sin 60°}$, leaving your answer in surd form.
A. $1-\sqrt{3}$
B. $2-\sqrt{3}$
C. $4-2\sqrt{3}$
D. $4+2\sqrt{3}$

22. Find the equation of a circle with centre $(-3, -8)$ and radius $\frac{4}{\sqrt{6}}$.
A. $x^2-y^2-6x+16y+23=0$
B. $x^2+y^2+6x+16y-23=0$
C. $x^2+y^2+6x-16y+23=0$
D. $x^2+y^2-6x+16y+23=0$

23. Determine the coefficient of $x^2$ in the expansion of $(a+3x)^6$.
A. $\frac{18}{a^2}$
B. $\frac{45}{a^4}$
C. $\frac{135}{a^4}$
D. $\frac{1215}{a^2}$

24. The mean of $2, 5, (x + 2), 7$ and $9$ is $6$. Find the median.
A. $5.5$
B. $6.0$
C. $6.5$
D. $7.0$

25. The probability that Kofi and Ama hit a target in a shooting competition are $\frac{1}{6}$ and $\frac{1}{9}$ respectively. What is the probability that only one of them hit the target?
A. $\frac{1}{54}$
B. $\frac{13}{54}$
C. $\frac{20}{27}$
D. $\frac{41}{54}$

26. In how many ways can $3$ prefects be chosen out of $8$ prefects?
A. $6$
B. $24$
C. $56$
D. $336$

27. Find the standard deviation of the numbers $3,6,2,1,7$ and $5$.
A. $2.00$
B. $2.16$
C. $2.50$
D. $2.56$

28. The table shows the distribution of marks of students in a class. Find the upper class boundary of the modal class.
Marks | No of students
5-7 | 4
8-10 | 7
11-13 | 26
14-16 | 41
17-19 | 14
20-22 | 8
A. $13.5$
B. $16$
C. $16.5$
D. $22.5$

29. If $^3C_2 = 15$, find the value of $x$?
A. $2$
B. $4$
C. $5$
D. $6$

30. Four doctors and two nurses are to sit round a circular table. In how many ways can this be done if the nurses are to sit together?
A. $48$
B. $60$
C. $240$
D. $720$

31. A basket contains $3$ red and $1$ white identical balls. A ball is drawn from the basket at random. Calculate the probability that it is either white or red.
A. $\frac{1}{3}$
B. $\frac{1}{2}$
C. $\frac{3}{4}$
D. $1$

32. A force of $200N$ acting on a body of mass $20kg$ initially at rest causes it to move a distance of $320m$ along a straight line for $t$ secs. Find the value of $t$.
A. $4s$
B. $6s$
C. $8s$
D. $10s$

33. Two forces $10N$ and $15N$ act on an object at an angle of $120°$ to each other. Find the magnitude of the resultant.
A. $\sqrt{5.5}N$
B. $\sqrt{5.7}N$
C. $\sqrt{7.5}N$
D. $\sqrt{7.7}N$

34. A body of mass $25kg$ changes its speed from $15m/s$ to $35m/s$ in $5$ seconds by the action of an applied force $F$. Find the value of $F$.
A. $100N$
B. $375N$
C. $500N$
D. $600N$

35. A particle starts from rest and moves in a straight line such that its velocity, $v$, at time $t$ seconds is given by $v=(3t^2-2t)m/s$. Calculate the distance covered in the first $2$ seconds.
A. $2m$
B. $4m$
C. $6m$
D. $8m$

36. A particle starts from rest and moves in a straight line such that its velocity, $v$, at time $t$ seconds is given by $v=(3t^2-2t)m/s$. Determine the acceleration when $t = 2$ secs.
A. $4m/s^2$
B. $6m/s^2$
C. $8m/s^2$
D. $10m/s^2$

37. Given that $\vec{q}=9i+6j$ and $\vec{r}=4i-6j$, which of the following statements is true?
A. $\vec{r}$ and $\vec{q}$ are collinear
B. $\vec{r}$ and $\vec{q}$ are perpendicular
C. The magnitude of $\vec{r}$ is $\sqrt{52}$ units
D. The projection of $\vec{r}$ on $\vec{q}$ is $\sqrt{117}$ units.

38. The functions $f$ and $g$ are defined on the set, $R$, of real numbers by $f:x \to \frac{x^2-x-6}{1}$ and $g:x \to x-1$. Find $f \circ g(3)$.
A. $-8$
B. $-6$
C. $-4$
D. $-3$

39. Find the unit vector in the direction of $(-5i + 12j)$.
A. $\frac{1}{13}(-5i-12j)$
B. $\frac{1}{13}(5i-12j)$
C. $\frac{1}{13}(-5i+12j)$
D. $\frac{1}{13}(5i+12j)$

40. Find, correct to two decimal places, the acute angle between $\vec{p}=(\frac{13}{14})$ and $\vec{q}=(\frac{12}{5})$.
A. $23.52°$
B. $24.50°$
C. $29.52°$
D. $29.82°$

2012

[latexpage]
WAEC 2012 Mathematical Questions

1. Which of the following sets is equivalent to $(P \cup Q) \cap (P \cup Q’)$?
A. $P$
B. $P \cap Q$
C. $P \cup Q$
D. $\emptyset$

2. Simplify: $\frac{\cos2\theta – 1}{\sin2\theta}$
A. $-\tan\theta$
B. $-\cos\theta$
C. $\tan\theta$
D. $\cos\theta$

3. Solve the inequality $x^2 – 2x \geq 3$
A. $-1 \leq x \leq 3$
B. $x \geq 3$ and $x \leq -1$
C. $x \geq 3$ or $x < -1$ D. $-1 \leq x < 3$ 4. Given $6^{-\sqrt{2}}, 3^{-\sqrt{2}}, 6^{-\sqrt{3}}, 9^{-\sqrt{2}}, ...$ as the first four terms of an exponential sequence (G.P), find the 8th term in its simplest form. A. $\frac{27}{2^{-\sqrt{2}}}$ B. $\frac{27}{6^{-\sqrt{2}}}$ C. $\frac{81}{2^{-\sqrt{2}}}$ D. $\frac{81}{6^{-\sqrt{2}}}$ 5. Given $\sin x = -\frac{\sqrt{3}}{2}$ and $\cos x > 0$, find $x$.
A. $300°$
B. $240°$
C. $120°$
D. $60°$

6. Evaluate $\log_{10}(\frac{1}{3} + \frac{1}{4}) + 2\log_{10}2 + \log_{10}(\frac{3}{7})$
A. $-3$
B. $0$
C. $\frac{5}{6}$
D. $1$

7. For triangle QRS, $\vec{QR} = (3i+2j)$ and $\vec{SR} = (-5i+3j)$, find $\vec{SQ}$
A. $8i + j$
B. $2i – j$
C. $-2i – 3j$
D. $-8i – j$

8. If $(x + 1)$ is a factor of the polynomial $x^3 + px^2 + x + 6$, find the value of $p$
A. $-8$
B. $-4$
C. $4$
D. $8$

9. A polynomial is defined by $f(x+1) = x^3 + px^2 – 4x + 2$, find $f(2)$
A. $-8$
B. $-2$
C. $2$
D. $8$

10. The equation of a circle is $3x^2 + 3y^2 + 24x – 12y = 15$. Find its radius
A. $2$
B. $3$
C. $4$
D. $5$

11. If the midpoint of the line joining $(1 – k, -4)$ and $(2, k + 1)$ is $(-k, k)$, find the value of $k$
A. $-4$
B. $-3$
C. $-2$
D. $-1$

12. Evaluate $\int_{-2}^{3} (3x^2 – 2x – 12) dx$
A. $-30$
B. $-18$
C. $-6$
D. $6$

13. If $y = x^3 – x^2 – x + 6$, find the values of $x$ at the turning point
A. $\frac{1}{2}, 3$
B. $\frac{1}{3}, -\frac{1}{2}$
C. $1, -\frac{1}{3}$
D. $1, \frac{1}{3}$

14. Given $P = \begin{pmatrix} 2 & 5 \\ 1 & -3 \end{pmatrix}$ and $Q = \begin{pmatrix} 4 & 1 \\ -8 & -2 \end{pmatrix}$, find $(2P – Q)$
A. $\begin{pmatrix} -6 & 3 \\ 17 & 1 \end{pmatrix}$
B. $\begin{pmatrix} -2 & 4 \\ 9 & 1 \end{pmatrix}$
C. $\begin{pmatrix} 0 & 9 \\ -6 & -8 \end{pmatrix}$
D. $\begin{pmatrix} 0 & 9 \\ 10 & -4 \end{pmatrix}$

15. A binary operation, $\Delta$, is defined on the set of real numbers by $a \Delta b = a + b + 4$. Find the identity element
A. $4$
B. $2$
C. $\frac{1}{4}$
D. $-4$
16. The marks obtained by 10 students in a test are: 3, 7, 6, 2, 8, 5, 9, 1, 4 and 10. Find the mean mark
A. $4.50$
B. $5.50$
C. $6.50$
D. $6.75$

17. For the same test scores, find the variance
A. $8.25$
B. $8.50$
C. $9.00$
D. $9.17$

18. If $r$ denotes the correlation coefficient between two variables, which of the following is always true?
A. $0 < r \leq 1$ B. $-1 \leq r < 1$ C. $-1 < r \leq 0$ D. $-1 \leq r \leq 1$ 19. A stone is dropped from a height of 45m. Find the time it takes to hit the ground [g = $10 \text{m/s}^2$] A. $3.0$ seconds B. $4.5$ seconds C. $5.0$ seconds D. $9.0$ seconds 20. Differentiate $\frac{x}{x+1}$ with respect to $x$ A. $\frac{1}{x+1}$ B. $\frac{1}{(x+1)^2}$ C. $\frac{1-x}{x+1}$ D. $\frac{1-x}{(x+1)^2}$ 21. Two forces 10N and 6N act in the directions $060°$ and $330°$ respectively. Find the x-component of their resultant A. $\frac{5\sqrt{3}-3}{3}$ B. $\frac{3-5\sqrt{3}}{3}$ C. $\frac{5-3\sqrt{3}}{3}$ D. $\frac{3\sqrt{3}-5}{3}$ 22. Find the unit vector in the direction of the vector $-12i+5j$ A. $\frac{-12i}{13} - \frac{5j}{13}$ B. $\frac{-1i}{13} + \frac{5j}{13}$ C. $\frac{-12i}{13} + \frac{5j}{13}$ D. $\frac{-5i}{13} + \frac{12j}{13}$ 23. In computing the mean of 8 numbers, a boy mistakenly used 17 instead of 25 as one of the numbers and obtained 20 as the mean. Find the correct mean A. $19$ B. $21$ C. $23$ D. $24$ 24. Given $^nP_r = 90$ and $^nC_r = 15$, find the value of $r$ A. $2$ B. $3$ C. $5$ D. $6$ 25. Which of the following is NOT a measure of central tendency? A. Mean B. Variance C. Median D. Mode 26. A fair die is tossed twice. Find the probability of obtaining a 3 and a 5 A. $\frac{5}{12}$ B. $\frac{2}{3}$ C. $\frac{1}{18}$ D. $\frac{1}{36}$ 27. If $P(x - 3) + Q(x + 1) = 2x + 3$, find the value of $(P + Q)$ A. $0$ B. $1$ C. $2$ D. $3$ 28. Find the values of $x$ at the point of intersection of the curve $y = x^2 + 2x - 3$ and the line $y + x = 1$ A. $(1, -2)$ B. $(0, 4)$ C. $(2, -3)$ D. $(1, -4)$ 29. Find the constant term in the binomial expansion of $(2x - \frac{3}{x})^8$ A. $90720$ B. $1296$ C. $1120$ D. $672$ 30. A straight line makes intercepts of $-3$ and $2$ on the x- and y-axes respectively. Find the equation of the line A. $2x + 3y + 6 = 0$ B. $3x - 2y - 6 = 0$ C. $-3x + 2y - 6 = 0$ D. $-2x + 3y - 6 = 0$ 31. Find the number of different arrangements of the word IKOTITINA A. $30240$ B. $60840$ C. $120960$ D. $362880$ 32. Find the acute angle between the lines $2x + y = 4$ and $-3x + y + 7 = 0$ A. $40°$ B. $44°$ C. $45°$ D. $54°$ 33. A box contains 4 red and 3 blue identical balls. If two are picked at random, one after the other without replacement, find the probability that one is red and the other is blue A. $\frac{4}{7}$ B. $\frac{2}{7}$ C. $\frac{1}{7}$ D. $\frac{1}{12}$ 34. The distance $s$ in metres covered by a particle in $t$ seconds is $s = \frac{3}{2}t^2 - 3t$. Find its acceleration A. $1 \text{ m/s}^2$ B. $2 \text{ m/s}^2$ C. $3 \text{ m/s}^2$ D. $4 \text{ m/s}^2$ 35. The angle of a sector of a circle is $0.9$ radians. If the radius of the circle is $4$ cm, find the length of the arc of the sector A. $3.6$ cm B. $7.6$ cm C. $8.0$ cm D. $11.6$ cm 36. From the diagram, which of the following represents the vector $V$ in component form? A. $-\frac{5\sqrt{3}}{3}i + 5j$ B. $3i + \frac{3}{\sqrt{3}}j$ C. $i + \frac{2}{\sqrt{3}}j$ D. $-\frac{2}{\sqrt{3}}i - j$ 37. From the diagram, $h[g(3)]$ is A. $s$ B. $\beta$ C. $s, \beta$ D. $s, w, \beta$ 38. $g \circ h$ is A. one-to-one B. onto C. a relation D. a series 39. The diagram is a velocity-time graph of a moving object. Calculate the distance travelled when the acceleration is zero A. $300$ m B. $90$ m C. $150$ m D. $210$ m 40. Simplify $\frac{x^{3n+1}}{x^{2n+\frac{5}{2}}} \left(x^{2n-3}\right)^{\frac{1}{2}}$ A. $0$ B. $-\frac{1}{2}$ C. $1$ D. $10$

2013

[latexpage]
1. A binary operation * is defined on the set of real numbers, $\mathbb{R}$, by
$x * y = x + y – xy$. If the identity element under the operation * is 0, find the inverse of
$x \in \mathbb{R}$.
A. $\frac{-x}{1-x}, x \neq 1$
B. $\frac{1}{1-x}, x \neq 1$
C. $\frac{-1}{1-x}, x \neq 1$
D. $\frac{x}{1-x}, x \neq 1$

2. Solve: $\sin\theta = \tan\theta$
A. $200°$
B. $90°$
C. $60°$
D. $0°$

3. Given that $a^{5/6} \times a^{-1/n} = 1$, solve for $n$.
A. $-6.00$
B. $-1.20$
C. $0.83$
D. $1.20$

4. Express $\log_{1/8} + \log_{1/2}$ in terms of $\log_2$.
A. $3 \log 2$
B. $4 \log 2$
C. $-3 \log 2$
D. $-4 \log 2$

5. If $f(x) = \frac{x^2}{2}$ and $g(x) = \sin x$, find $g \circ f$.
A. $\sin^2 x$
B. $\sin(x^2/2)$
C. $(\sin x)^{x^2/2}$
D. $x \sin x$

6. Find the third term in the expansion of $(a-b)^6$ in ascending powers of $b$.
A. $-\frac{15}{a^4b^2}$
B. $\frac{15}{a^4b^2}$
C. $-\frac{15}{a^3b^3}$
D. $\frac{15}{a^3b^3}$

7. If $\sqrt{x – \sqrt{x+1}} = \sqrt{2x+1}$, find the possible values of $x$.
A. $1$ and $-1$
B. $-1$ and $2$
C. $1$ and $2$
D. $0$ and $-1$

8. If $\alpha$ and $\beta$ are the roots of the equation $2x^2 – 6x + 5 = 0$, evaluate $\frac{\beta}{\alpha} + \frac{\alpha}{\beta}$.
A. $\frac{24}{5}$
B. $\frac{8}{5}$
C. $\frac{5}{8}$
D. $\frac{5}{24}$

9. Given that $f(x) = 2x^3 – 3x^2 – 11x + 6$ and $f(3) = 0$, factorize $f(x)$.
A. $(x – 3)(x – 2)(2x + 2)$
B. $(x + 3)(x – 2)(x – 1)$
C. $(x – 3)(x + 2)(2x – 1)$
D. $(x + 3)(x – 2)(2x – 1)$

10. Find the equation of the line that is perpendicular to $2y + 5x – 6 = 0$ and bisects the line joining the points $P(4, 3)$ and $Q(-6, 1)$.
A. $y + 5x + 3 = 0$
B. $2y – 5x – 9 = 0$
C. $5y + 2x – 8 = 0$
D. $5y – 2x – 12 = 0$

11. Differentiate $x^2 + xy – 5 = 0$

A. $\frac{-2x+y}{x}$

B. $\frac{2x-y}{x}$

C. $\frac{-x}{2x+y}$

D. $\frac{2x+y}{x}$

12. The fourth term of an exponential sequence is 192 and its ninth term is 6. Find the common ratio of the sequence.

A. $\frac{1}{3}$

B. $\frac{1}{2}$

C. $2$

D. $3$

13. Find the range of values of $x$ for which $\frac{x^2+4x+5}{3x^2-x+2}$ is less than 1

A. $x > -\frac{1}{2}, x > 3$

B. $x < -\frac{1}{2}, x > 3$

C. $-\frac{1}{2} \leq x \leq 3$

D. $-\frac{1}{2} < x < 3$ 14. Given that $\frac{dy}{dx} = x^{-\frac{1}{2}}$, find $y$ A. $\frac{2}{3}x^{\frac{3}{2}} + c$ B. $2x^{\frac{3}{2}} + c$ C. $\frac{3}{2}x^{\frac{3}{2}} + c$ D. $\frac{2}{3}x^2 + c$ 15. Given that $P = \begin{pmatrix} \frac{y-2}{y-4} \\ \frac{y-1}{y+2} \end{pmatrix}$ and $|P| = -23$, find the value of $y$ A. -4 B. -3 C. -1 D. 2 16. An object is thrown vertically upwards from the top of a cliff with a velocity of $25\frac{m}{s}$. Find the time, in seconds, when it is 20 metres above the cliff. $[g=10\frac{m}{s^2}]$ A. 0 and 1 B. 0 and 4 C. 0 and 5 D. 1 and 4 17. Evaluate $\int_{0}^{2} (8x-4x^2)dx$ A. $-16$ B. $-\frac{16}{3}$ C. $\frac{16}{3}$ D. $16$ 18. Find the coordinates of the point which divides the line joining $P(-2, 3)$ and $Q(4, 9)$ internally in the ratio $2 : 3$ A. $(\frac{5}{2/3}, \frac{2}{5})$ B. $(\frac{2}{5}, \frac{5}{2/5})$ C. $(\frac{2}{5}, \frac{22}{5})$ D. $(-\frac{2}{5}, \frac{5}{2/5})$ 19. The angle subtended by an arc of a circle at the centre is $\frac{\pi}{3}$ radians. If the radius of the circle is $12$cm, calculate the perimeter of the major arc A. $4(6+5\pi)$ B. $4(6+2\pi)$ C. $4(3+3\pi)$ D. $4(3+5\pi)$ 20. The function $f:F \to R = f(x) = \begin{cases} 3x+2 & : x > 4 \\ 3x-2 & : x = 4 \\ 5x-3 & : x < 4 \end{cases}$. Find $f(4) - f(-3)$ A. 28 B. 26 C. -26 D. -28 21. A committee consists of 5 boys (Kofi, John, Ojo, Ozo and James) and 3 girls (Rose, Ugo and Ama). In how many ways can a sub-committee consisting of 3 boys and 2 girls be chosen, if Ozo must be on the sub-committee? A. 35 B. 30 C. 18 D. 12 22. Forces $50N$ and $80N$ act on a body as shown in the diagram. Find, correct to the nearest whole number, the horizontal component of the resultant force A. $13N$ B. $43N$ C. $57N$ D. $95N$ 23. The sales of five salesgirls on a certain day are as follows: $GH¢ 26.00, GH¢ 39.00, GH¢ 33.00, GH¢ 25.00$ and $GH¢ 37.00$. Calculate the standard deviation if the mean sale is $GH¢ 32.00$ A. $GH¢ 5.65$ B. $GH¢ 5.66$ C. $GH¢ 6.5$ D. $GH¢ 6.56$ 24. A circular ink blot on a piece of paper increases its area at the rate $4\frac{mm^2}{s}$. Find the rate of the radius of the blot when the radius is $8mm$. $[\pi=\frac{22}{7}]$ A. $0.20 \frac{mm}{s}$ B. $0.08 \frac{mm}{s}$ C. $0.25 \frac{mm}{s}$ D. $0.05 \frac{mm}{s}$ 25. Express $\frac{x^2+x+4}{(1-x)(x^2+1)}$ in partial fractions A. $\frac{x}{x^2+1} + \frac{x+4}{1-x}$ B. $\frac{3}{1-x} + \frac{2x+1}{x^2+1}$ C. $\frac{x}{1-x} + \frac{x+4}{x^2+1}$ D. $\frac{3}{1-x} + \frac{2x+2}{x^2+1}$ 26. Two bodies of masses $3kg$ and $5kg$ moving with velocities $2 \frac{m}{s}$ and $V \frac{m}{s}$ respectively in opposite directions collide. If they move together after collision with velocity $3.5 \frac{m}{s}$ in the direction of the $5kg$ mass, find the value of $V$ A. $7.8 \frac{m}{s}$ B. $6.8 \frac{m}{s}$ C. $5.6 \frac{m}{s}$ D. $4.6 \frac{m}{s}$ 27. The equation of a circle is $x^2 + y^2 - 8x + 9y + 15 = 0$. Find its radius A. 5 B. $\frac{1}{2}\sqrt{15}$ C. $\frac{1}{2}\sqrt{85}$ D. $\sqrt{85}$ 28. A particle is acted upon by two forces $6N$ and $3N$ inclined at an angle of $120°$ to each other. Find the magnitude of the resultant force A. $\sqrt{18-3} N$ B. $27 N$ C. $24 N$ D. $\sqrt{3-3} N$ 29. If $s = 3i - j$ and $t = 2i + 3j$, find $(t-3s).(t+3s)$ A. -77 B. -71 C. -53 D. -41 30. If $2\sin^2\theta = 1 + \cos\theta$, $0° \leq \theta \leq 90°$, find $\theta$ A. $30°$ B. $45°$ C. $60°$ D. $90°$ 31. Find the upper quartile of the following scores: $41, 29, 17, 2, 12, 33, 45, 18, 43$ and $5$ A. 45 B. 41 C. 33 D. 21 32. Given that $P = \begin{pmatrix} 3/2 \\ 4/x \end{pmatrix}$, $Q = \begin{pmatrix} 1 \\ -2 \end{pmatrix}$, $R = \begin{pmatrix} -5 \\ -8 \end{pmatrix}$ and $PQ = R$, find the value of $x$ A. -5 B. -2 C. 2 D. 5 33. Two out of ten tickets on sale for a raffle draw are winning tickets. If a guest bought two tickets, what is the probability that both tickets are winning tickets? A. $\frac{1}{80}$ B. $\frac{1}{45}$ C. $\frac{1}{20}$ D. $\frac{1}{10}$ 34. $P$ and $Q$ are the points $(3, 1)$ and $(7, 4)$ respectively. Find the unit vector along $PQ$ A. $\begin{pmatrix} 4/3 \end{pmatrix}$ B. $\begin{pmatrix} 0.6 \\ 0.8 \end{pmatrix}$ C. $\begin{pmatrix} 0.8 \\ 0.6 \end{pmatrix}$ D. $\begin{pmatrix} -0.8 \\ 0.6 \end{pmatrix}$ 35. If $g(x) = \frac{x+1}{x-2}, x \neq -2$, find $g^{-1}(2)$ A. 3 B. 2 C. $\frac{3}{4}$ D. -3 36. Calculate the mean deviation of $1, 2, 3, 4, 5, 5, 6, 7, 8, 9$ A. 2 B. 3 C. 4 D. 5 37. If $V = \begin{pmatrix} -2 \\ 4 \end{pmatrix}$ and $U = \begin{pmatrix} -1 \\ 5 \end{pmatrix}$, find $|U+V|$ A. $\sqrt{3/10}$ B. $\sqrt{82}$ C. 15 D. $\sqrt{2/5}$ 38. Find the equation of the straight line that passes through $(2, -3)$ and perpendicular to the line $3x - 2y + 4 = 0$ A. $2y - 3x = 0$ B. $3y - 2x + 5 = 0$ C. $3y + 2x + 5 = 0$ D. $2y - 3x - 5 = 0$ 39. If $\frac{n!}{(n-3)!} \div \frac{n!}{(n-2)!} = 1$, find the value of $n$ A. 8 B. 7 C. 6 D. 5 40. A body is kept at rest by three forces $F_1 = (10N, 030°)$, $F_2 = (10N, 150°)$ and $F_3$. Find $F_3$ A. $(12N, 090°)$ B. $(10N, 270°)$ C. $(10N, 180°)$ D. $(10N, 120°)$

2014

[latexpage]
WAEC 2014 Mathematics Questions

1. If $\frac{1}{5-y} = 25\left(\frac{5}{4-2y}\right)$, find the value of y.
A. 4
B. 2
C. -4
D. -5

2. Simplify: $\frac{(1-\sin\theta)(1+\sin\theta)}{\text{something}}$
A. $\sin^2\theta$
B. $\sec^2\theta$
C. $\tan^2\theta$
D. $\cos^2\theta$

3. Given that $3x+4y+6=0$ and $4x-by+3=0$ are perpendicular, find the value of b.
A. 4
B. 3
C. $\frac{1}{3}$
D. $\frac{1}{4}$

4. Given that $x * y = \frac{x+y}{2}$, $x \circ y = \frac{x^2}{y}$ and $(3*b) \circ 48 = \frac{1}{3}$, find b, where b > 0.
A. 8
B. 6
C. 5
D. 4

5. If $f(x) = 3x^3 + 8x^2 + 6x + k$ and $f(2) = 1$, find the value of k.
A. -67
B. -61
C. 61
D. 67

6. If $8^x \div (1/4)^y = 1$ and $\log_2(x-2y) = 1$, find the value of (x – y).
A. $\frac{5}{4}$
B. $\frac{3}{5}$
C. 1
D. $\frac{2}{3}$

7. Simplify $1 + \frac{8}{\sqrt{3} – 2\sqrt{}}$
A. $7 + \frac{2\sqrt{}}{}$
B. $7 + \frac{7}{2\sqrt{}}$
C. $\frac{1-7}{2\sqrt{}}$
D. $1 + \frac{2\sqrt{}}{}$

8. Using the binomial expansion $(1+x)^6 = 1+6x+15x^2+20x^3+15x^4+6x^5+x^6$, find, correct to 3 dp, the value of $(1.98)^6$.
A. 64.245
B. 61.255
C. 60.255
D. 60.245

9. If $(x+2)$ and $(3x-1)$ are factors of $6x^3 + x^2 – 19x + 6$, find the third factor.
A. $2x-3$
B. $3x+1$
C. $x-2$
D. $3x+2$

10. If 2, (k+1), 8, … form an exponential sequence (GP), find the values of k.
A. -3 and 5
B. 5 and -5
C. 3 and -3
D. -5 and 3

11. A box contains 5 red and k blue balls. A ball is selected at random from the box. If the probability of selecting a blue ball is $\frac{2}{3}$, find the value of k.
A. 5
B. 6
C. 8
D. 10

12. If $\frac{x+P}{(x-1)(x-3)} = \frac{Q}{x-1} + \frac{2}{x-3}$, find the value of (P + Q).
A. -2
B. -1
C. 0
D. 1

13. Find the derivative of $\frac{3x^3+1}{\sqrt{3}}$ with respect to x.
A. $\frac{3x^3}{(3x^3+1)}$
B. $\frac{3x^2}{(3x^3+1)^{1/2}\sqrt{3}}$
C. $\frac{3x^3x^2+1}{\sqrt{3}}$
D. $\frac{3x^2}{(3x^2+1)^2}$

14. If $T = \left(-\frac{2}{3} -\frac{5}{8}\right)$, find $T^{-1}$, the inverse of T.
A. $\left(-\frac{8}{3} -\frac{5}{2}\right)$
B. $\left(-\frac{8}{3} -\frac{5}{-2}\right)$
C. $\left(-\frac{8}{-3} -\frac{5}{2}\right)$
D. $\left(-\frac{8}{-3} -\frac{5}{-2}\right)$

15. A function is defined by $h:x \to 2 – \frac{1}{2x-3}, x \neq \frac{3}{2}$. Find $h^{-1}$, the inverse of h.
A. $\frac{3x-4}{2x-7}, x \neq \frac{7}{2}$
B. $\frac{3x-7}{2x-4}, x \neq 2$
C. $\frac{2x-7}{4x-3}, x \neq \frac{3}{4}$
D. $\frac{4x-7}{2x-4}, x \neq 2$

16. A function is defined by $h:x \to 2 – \frac{1}{2x-3}, x \neq \frac{3}{2}$. Find $h^{-1}\left(\frac{1}{2}\right)$.
A. 6
B. $\frac{11}{6}$
C. $\frac{11}{4}$
D. $\frac{5}{3}$

17. The radius of a sphere is increasing at a rate $3\text{cm}\cdot\text{s}^{-1}$. Find the rate of increase in the surface area, when the radius is 2cm.
A. $8\pi\text{cm}^2\cdot\text{s}^{-1}$
B. $16\pi\text{cm}^2\cdot\text{s}^{-1}$
C. $24\pi\text{cm}^2\cdot\text{s}^{-1}$
D. $48\pi\text{cm}^2\cdot\text{s}^{-1}$

18. Age in years
10 – 14: 6
15 – 19: 8
20 – 24: 14
25 – 29: 10
30 – 34: 12
What is the class mark of the median class?
A. 17
B. 22
C. 27
D. 32

19. For the same age distribution as previous question, in which group is the upper quartile?
A. 15 – 19
B. 20 – 24
C. 25 – 29
D. 30 – 34

20. Find the mean of the distribution for the same age data.
A. 23.4
B. 23.6
C. 24.3
D. 24.6

21. If $P x^2 + (P+1)x + P = 0$ has equal roots, find the values of P.
A. -1 and $-\frac{1}{3}$
B. 1 and $-\frac{1}{3}$
C. -1 and $\frac{1}{3}$
D. 1 and $\frac{1}{3}$

22. Integrate $\left(x – \frac{1}{x}\right)^2$ with respect to x.
A. $\frac{1}{3}\left(x – \frac{1}{x}\right)^3 + c$
B. $\frac{x^3}{3} – \frac{x}{1/x^3} + c$
C. $\frac{x^3}{3} – 2x + \frac{1}{x^3} + c$
D. $\frac{x^3}{3} – 2x – \frac{1}{x} + c$

23. Given that $AB = \begin{pmatrix} 4 \\ 3 \end{pmatrix}$ and $AC = \begin{pmatrix} 2 \\ -3 \end{pmatrix}$, find $|BC|$.
A. $\frac{4}{\sqrt{2}}$
B. $\frac{6}{\sqrt{2}}$
C. $\frac{2}{\sqrt{10}}$
D. $\frac{4}{\sqrt{10}}$

24. Find the angle between $5i + 3j$ and $3i – 5j$.
A. 180°
B. 90°
C. 45°
D. 0°

25. Find the coefficient of $x^3$ in the binomial expansion of $(3x+4)^4$ in ascending powers of x.
A. 432
B. 194
C. 144
D. 108

26. If a fair coin is tossed four times, what is the probability of obtaining at least one head?
A. $\frac{1}{2}$
B. $\frac{1}{4}$
C. $\frac{13}{16}$
D. $\frac{15}{16}$

27. Forces 90N and 120N act in the directions 120° and 240° respectively. Find the resultant of these forces.
A. $-45(2i + \sqrt{2}j)$
B. $60(\sqrt{3}i + 7j)$
C. $30(7i + \sqrt{3}j)$
D. $-15(7i + \sqrt{3}j)$

28. The deviations from the mean of a set of numbers are $(k+3)^2, (k+7), -2, k$ and $(k+2)^2$, where k is a constant. Find the value of k.
A. 3
B. 2
C. -2
D. -3

29. Find the equation of a circle with centre (2, -3) and radius 2 units.
A. $x^2 + y^2 – 4x + 6y + 9 = 0$
B. $x^2 + y^2 + 4x – 6y – 9 = 0$
C. $x^2 + y^2 + 4x + 6y – 9 = 0$
D. $x^2 + y^2 + 4x – 6y + 9 = 0$

30. The first term of a linear sequence is 9 and the common difference is 7. If the nth term is 380, find the value of n.
A. 45
B. 54
C. 56
D. 65

31. For what values of m is $9y^2 + my + 4$ a perfect square?
A. $\pm 2$
B. $\pm 3$
C. $\pm 6$
D. $+12$

32. A particle accelerates at $12\text{m}\cdot\text{s}^{-2}$ and travels a distance of 250m in 6 seconds. Find the initial velocity of the particle.
A. 5.7 $\text{m}\cdot\text{s}^{-1}$
B. 6.0 $\text{m}\cdot\text{s}^{-1}$
C. 60.0 $\text{m}\cdot\text{s}^{-1}$
D. 77.5 $\text{m}\cdot\text{s}^{-1}$

33. In how many ways can 9 people be seated on a bench if only 3 places are available?
A. 1200
B. 504
C. 320
D. 204

34. Find the variance of 1, 2, 0, -3, 5, -2, 4.
A. $\frac{52}{7}$
B. $\frac{40}{7}$
C. $\frac{32}{7}$
D. $\frac{27}{7}$

35. If the points (-1, t-1), (t, t-3) and (t-6, 3) lie on the same straight line, find the values of t.
A. t = -2 and 3
B. t = 2 and -3
C. t = 2 and 3
D. t = -2 and -3

36. A ball is thrown vertically upwards with a velocity of $15\text{m}\cdot\text{s}^{-1}$. Calculate the maximum height reached. [g = $10\text{m}\cdot\text{s}^{-2}$]
A. 15.25m
B. 13.25m
C. 11.25m
D. 10.25m

37. Find the distance between the points (2, 5) and (5, 9).
A. 4 units
B. 5 units
C. 12 units
D. 14 units

38. Find $\lim_{x \to 3} \frac{x^3 + x^2 – 12x}{x^2 – 9}$.
A. $\frac{7}{2}$
B. 0
C. $-\frac{7}{2}$
D. -7

39. Evaluate $\frac{\tan 120° + \tan 30°}{\tan 120° – \tan 60°}$.
A. $\sqrt{3} + \sqrt{2}$
B. $\frac{2}{3}$
C. $\frac{1}{3}$
D. $-\frac{2}{\sqrt{3}}$

40. Express (14N, 240°) as a column vector.
A. $\begin{pmatrix} -7 \\ -7\sqrt{3} \end{pmatrix}$
B. $\begin{pmatrix} 7\sqrt{3} \\ 7\sqrt{3} \end{pmatrix}$
C. $\begin{pmatrix} -7\sqrt{3} \\ -7 \end{pmatrix}$
D. $\begin{pmatrix} 7 \\ -7\sqrt{3} \end{pmatrix}$

2015

[latexpage]
1. Simplify $\frac{1-2\sqrt{5}}{2+3\sqrt{2}}$
A. $\frac{1}{4}(2^2 – \sqrt{+6} – \sqrt{5} – \sqrt{-4} – \sqrt{10})$
B. $\frac{1}{14}(2-3^2 – \sqrt{-4} – \sqrt{5} – \sqrt{-6} – \sqrt{10})$
C. $\frac{1}{14}(3^2 – \sqrt{+4} – \sqrt{5} – \sqrt{-6} – \sqrt{10} – 2)$
D. $\frac{1}{4}(2+3^2 – \sqrt{-6} – \sqrt{5} + \sqrt{4} – \sqrt{10})$

2. Solve: $2\cos x – 1 = 0$
A. $\left(\frac{2\pi}{3}, \frac{4\pi}{3}\right)$
B. $\left(\frac{\pi}{6}, \frac{5\pi}{6}\right)$
C. $\left(\frac{\pi}{5}, \frac{2\pi}{5}\right)$
D. $\left(\frac{\pi}{3}, \frac{5\pi}{3}\right)$

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

4. If $\log_3 x = \log_9 3$, find the value of $x$
A. $\frac{3}{2}$
B. $\frac{3}{1/2}$
C. $\frac{3}{1/3}$
D. $\frac{2}{1/3}$

5. Find the 3rd term of $(x^2 – 1)^8$ in descending order of $x$
A. $x^{7/8}$
B. $\frac{7}{16}x^6$
C. $\frac{7}{4}x^5$
D. $\frac{35}{8}x^4$

6. Given $f: x \to x^2$ and $g: x \to x+3$, where $x \in \mathbb{R}$, find $f \circ g(2)$
A. 25
B. 9
C. 7
D. 5

7. Given $\frac{2x}{(x+6)(x+3)} = \frac{P}{x+6} + \frac{Q}{x+3}$, 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 $P = \begin{pmatrix} -2 & 3 \\ 1 & 4 \end{pmatrix}$ and $Q = \begin{pmatrix} 5 & 2 \\ -3 & -1 \end{pmatrix}$, find $PQ – QP$
A. $\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$
B. $\begin{pmatrix} 27 & 16 \\ 12 & -15 \end{pmatrix}$
C. $\begin{pmatrix} -20 & 12 \\ -6 & -8 \end{pmatrix}$
D. $\begin{pmatrix} 11 & 30 \\ 12 & -11 \end{pmatrix}$

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 $f: x \to \frac{2x-1}{x+2}, x \neq -2$, find $f^{-1}$, the inverse of $f$
A. $f^{-1}: x \to \frac{1+2x}{2-x}, x \neq 2$
B. $f^{-1}: x \to \frac{1-2x}{x+2}, x \neq -2$
C. $f^{-1}: x \to \frac{1-2x}{x-2}, x \neq 2$
D. $f^{-1}: x \to \frac{1+2x}{x+2}, x \neq -2$

11. If $36, p, \frac{9}{4}, q$ are consecutive terms of an exponential sequence (G.P.). Find the sum of $p$ and $q$
A. $\frac{9}{16}$
B. $\frac{81}{16}$
C. $9$
D. $\frac{9}{9/16}$

12. Find the minimum value of $y = x^2 + 6x – 12$
A. -21
B. -12
C. -6
D. -3

13. A line passes through the origin and the point $(1\frac{1}{4}, 2\frac{1}{2})$, what is the gradient of the line?
A. 1
B. 2
C. 3
D. 4

14. A line passes through the origin and the point $(1\frac{1}{4}, 2\frac{1}{2})$. 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 5 be selected from 8 students if 2 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 $\frac{3}{x^2 – 4x}$. 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 $\alpha$ and $\beta$ are the roots of $x^2 + x – 2 = 0$, find the value of $\frac{1}{\alpha^2} + \frac{1}{\beta^2}$
A. $\frac{5}{4}$
B. $\frac{3}{4}$
C. $\frac{1}{4}$
D. $-\frac{3}{4}$

19. Given that $x^2 + 4x + k = (x+r)^2 + 1$, find the value 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. $p \lor q$
B. $\lnot p \lor q$
C. $p \land (\lnot q)$
D. $p \land q$

21. Given $r = 2i – j$, $s = 3i + 5j$, and $t = 6i – 2j$, find the magnitude of $2r + s – t$
A. $\sqrt{15}$
B. 4
C. $\sqrt{24}$
D. $\sqrt{26}$

22. [Marks distribution table]
How many candidates scored above the median score?
A. 3
B. 12
C. 22
D. 30

23. [Marks distribution table]
Find the interquartile range of the distribution
A. 4
B. 3
C. 2
D. 1

24. A mass of 75kg 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 constant velocity. [g = 9.8m/s²]
A. 750N
B. 745N
C. 735N
D. 98N

25. 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 Igbo only
A. $\frac{28}{45}$
B. $\frac{4}{9}$
C. $\frac{8}{45}$
D. $\frac{1}{9}$

26. If $\left|\frac{1+2x}{6} – \frac{1}{3-x}\right| = -3$, find the values of $x$
A. $x = 3, -2$
B. $x = 4, -\frac{2}{3}$
C. $x = -4, \frac{3}{2}$
D. $x = 4, -\frac{3}{2}$

27. Find $\int \frac{x^3 + 5x + 1}{x^3} dx$
A. $x^2 + 10x + c$
B. $x + \frac{5}{3}x^3 + \frac{1}{4}x^4 + c$
C. $\frac{x-5}{x^2} – \frac{2}{3}x^3 + c$
D. $x – \frac{5}{x} – \frac{1}{2}x^2 + c$

28. 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. $\left(\frac{5}{2}, \frac{2}{5}\right)$
B. $\left(\frac{2}{5}, \frac{5}{2}\right)$
C. $\left(-\frac{2}{5}, \frac{5}{2}\right)$
D. $\left(\frac{2}{5}, \frac{4}{1/5}\right)$

29. A particle starts from rest and moves in a straight line such that its acceleration after $t$ seconds is given by $a = (3t-2)$ m/s². Find the other time when the velocity would be zero
A. $\frac{1}{3}$ seconds
B. $\frac{3}{4}$ seconds
C. $\frac{4}{3}$ seconds
D. 2 seconds

30. A particle starts from rest and moves in a straight line such that its acceleration after $t$ secs is given by $a = (3t-2)$ m/s². Find the distance covered after 3 secs
A. 10m
B. 9m
C. $\frac{13}{3}$m
D. $\frac{9}{2}$m

31. Given that $y = 4 – 9x$ and $\Delta x = 0.1$, calculate $\Delta y$
A. 9.0
B. 0.9
C. -0.3
D. -0.9

32. Four fair coins are tossed once. Calculate the probability of having equal heads and tails
A. $\frac{1}{4}$
B. $\frac{3}{8}$
C. $\frac{1}{2}$
D. $\frac{15}{16}$

33. 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

34. Simplify: $\frac{n C_r}{n C_{r-1}}$
A. $\frac{n(n-r)}{r}$
B. $\frac{n}{r(n-r)}$
C. $\frac{1}{r(n-r)}$
D. $\frac{n+1-r}{r}$

35. If $2\sin^2\theta = 1 + \cos\theta$, $0° \leq \theta \leq 90°$, find the value of $\theta$
A. $90°$
B. $60°$
C. $45°$
D. $30°$

36. A 24N force acts on a body such that it changes its velocity from 5m/s to 9m/s in 2 secs. If the body is travelling in a straight line, calculate the distance covered in the period
A. 22m
B. 18m
C. 14m
D. 10m

37. The sum, $S_n$, of a sequence is given by $S_n = 2n^2 – 5$. Find the 6th term
A. 112
B. 67
C. 45
D. 22

38. Forces $F_1 = (8N, 030°)$ and $F_2 = (10N, 150°)$ act on a particle. Find the horizontal component of the resultant force
A. 1.7N
B. 4.5N
C. 9.0N
D. 13.0N

39. Forces of magnitude 8N and 5N act on a body as shown. Calculate, correct to 2 d.p., the resultant force acting at O
A. 14.06N
B. 13.00N
C. 9.83N
D. 8.26N

40. Forces of magnitude 8N and 5N act on a body as shown. Calculate, correct to 2 dp, the angle that the resultant makes with the horizontal
A. 80.76°
B. 75.00°
C. 71.99°
D. 15.00°

2016