WAEC FURTHER MATHEMATICS PAST QUESTIONS AND ANSWER

2010

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

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

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

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 12cm, 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

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

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

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°

2017

1. If \log_{y18} = 3, find the value of y.
A. -2
B. -\frac{1}{2}
C. \frac{1}{2}
D. 2

2. A binary operation \Delta is defined on the set of real numbers \mathbb{R} by a \Delta b = \frac{a+b}{\sqrt{ab}}, where a \neq 0, b \neq 0. Evaluate -3 \Delta -1.
A. -\frac{4}{3} - \sqrt{3}
B. -\frac{4}{3}\sqrt{3}
C. -\frac{3}{3\sqrt{4}}
D. -\frac{3}{3\sqrt{4}}

3. Simplify \frac{1}{(1-\sqrt{3})^2}
A. 1 - \frac{1}{2\sqrt{3}}
B. 1 + \frac{1}{2\sqrt{3}}
C. \sqrt{3}
D. 1 + \sqrt{3}

4. If x^2 - kx + 9 = 0 has equal roots, find the values of k.
A. 3, 4
B. \pm 3
C. \pm 5
D. \pm 6

5. Find the coordinates of the centre of the circle 3x^2 + 3y^2 - 4x + 8y - 2 = 0
A. (-2, 4)
B. (-\frac{2}{3}, \frac{4}{3})
C. (\frac{2}{3}, -\frac{4}{3})
D. (2, -4)

6. The function f: x \to \frac{4-2x}{\sqrt{-x}} is defined on the set of real numbers \mathbb{R}. Find the domain of f.
A. x < 2
B. x \leq 2
C. x = 2
D. x > -2

7. Given that f(x) = \frac{x+1}{2}, find f^{-1}(-2).
A. -5
B. -3
C. -\frac{1}{2}
D. 5

8. Given that \frac{6x+m}{2x^2+7x-15} \equiv \frac{4}{x+5} - \frac{2}{2x-3}, find the value of m.
A. 20
B. 12
C. -10
D. -22

9. Find the coefficient of x^4 in the expansion of (1-2x)^6.
A. -320
B. -240
C. 240
D. 320

10. Find the 21st term of the Arithmetic Progression (A.P.): -4, -1.5, 1, 3.5, ...
A. 43.5
B. 46
C. 48.5
D. 51

1. If \log_{y18} = 3, find the value of y.
A. -2
B. -\frac{1}{2}
C. \frac{1}{2}
D. 2

(Previous 10 questions…)

11. How many ways can 6 students be seated around a circular table?
A. 36
B. 48
C. 120
D. 720

12. If \begin{pmatrix}2 & 4 \\ 1 & 3\end{pmatrix}\begin{pmatrix}5 \\ 4\end{pmatrix} = k\begin{pmatrix}17.5 \\ 40.0\end{pmatrix}, find the value of k.
A. 1.2
B. 3.6
C. 0.8
D. 0.5

13. Express \cos 150° in surd form.
A. -\frac{\sqrt{3}}{2}
B. -\frac{\sqrt{3}}{2}
C. -\frac{1}{2}
D. \frac{2}{\sqrt{2}}

14. A straight line 2x+3y=6, passes through the point (-1,2). Find the equation of the line.
A. 2x-3y=2
B. 2x-3y=-2
C. 2x+3y=-4
D. 2x+3y=4

15. \alpha and \beta are the roots of the equation 2x^2 - 3x + 4 = 0. Find \alpha + \beta.
A. -2
B. -\frac{3}{2}
C. \frac{3}{2}
D. 2

16. \alpha and \beta are the roots of the equation 2x^2 - 3x + 4 = 0. Find \frac{\alpha}{\beta} + \frac{\beta}{\alpha}.
A. -\frac{9}{8}
B. -\frac{7}{8}
C. \frac{7}{8}
D. \frac{9}{8}

17. If B = \begin{pmatrix}2 & 1 \\ 5 & 3\end{pmatrix}, find B^{-1}.
A. A = \begin{pmatrix}-3 & 1 \\ -5 & 2\end{pmatrix}
B. A = \begin{pmatrix}3 & 1 \\ -5 & 2\end{pmatrix}
C. A = \begin{pmatrix}3 & -1 \\ -5 & 2\end{pmatrix}
D. A = \begin{pmatrix}-3 & 1 \\ 5 & -2\end{pmatrix}

18. Given that \sin x = \frac{5}{13} and \sin y = \frac{8}{17}, where x and y are acute, find \cos(x+y).
A. \frac{130}{221}
B. \frac{140}{221}
C. \frac{140}{204}
D. \frac{220}{23}

19. A circle with centre (4,5) passes through the y-intercept of the line 5x - 2y + 6 = 0. Find its equation.
A. x^2 + y^2 + 8x - 10y + 21 = 0
B. x^2 + y^2 + 8x - 10y - 21 = 0
C. x^2 + y^2 - 8x - 10y - 21 = 0
D. x^2 + y^2 - 8x - 10y + 21 = 0

20. Given that f(x) = 5x^2 - 4x + 3, find the coordinates of the point where the gradient is 6.
A. (4,1)
B. (4,-2)
C. (1,4)
D. (1,-2)

21. If y = \frac{1+x}{1-x}, find \frac{dy}{dx}.
A. \frac{2}{(1-x)^2}
B. -\frac{2}{(1-x)^2}
C. -\frac{1}{\sqrt{1-x}}
D. \frac{1}{\sqrt{1-x}}

22. Evaluate \int_{-1}^{0} (x+1)(x-2)dx.
A. \frac{7}{6}
B. \frac{5}{6}
C. -\frac{5}{6}
D. -\frac{7}{6}

23. Simplify \frac{128\sqrt{32\sqrt{-2}}}{2\sqrt{}}.
A. \frac{2}{2\sqrt{3}}
B. \frac{3}{2\sqrt{3}}
C. 3
D. 4

24. There are 7 boys in a class of 20. Find the number of ways of selecting 3 girls and 2 boys.
A. 1638
B. 2730
C. 6006
D. 7520

25. The 3rd and 7th term of a Geometric Progression (GP) are 81 and 16. Find the 5th term.
A. \frac{4}{729}
B. \frac{81}{16}
C. 27
D. 36

26. Differentiate \frac{5}{x^3 + x^2}, x \neq 0 with respect to x.
A. 10x + 1
B. 10x + 2
C. x(15x + 1)
D. x(15x + 2)

27. A curve is given by y = 5 - x - \frac{2}{x^2}. Find the equation of its line of symmetry.
A. x = -\frac{41}{8}
B. x = -\frac{1}{4}
C. x = \frac{1}{4}
D. x = \frac{41}{8}

28. In a class of 10 boys and 15 girls, the average score in a Biology test is 90. If the average score for the girls is x, find the average score for the boys in terms of x.
A. \frac{200 - 2x}{3}
B. \frac{225 - 3x}{2}
C. 250 - 2x
D. 250 - 3x

29. A fair die is tossed twice. What is its sample size?
A. 6
B. 12
C. 36
D. 48

30. Given that a = \begin{pmatrix}2 \\ 3\end{pmatrix} and b = \begin{pmatrix}-1 \\ 4\end{pmatrix}, evaluate (2a - \frac{1}{4}b).
A. \begin{pmatrix}17 \\ 4 \\ 7\end{pmatrix}
B. \begin{pmatrix}17 \\ 4 \\ 5\end{pmatrix}
C. \begin{pmatrix}17 \\ 4 \\ 3\end{pmatrix}
D. \begin{pmatrix}17 \\ 4 \\ 2\end{pmatrix}

31. Face | 1 | 2 | 3 | 4 | 5 | 6
Frequency | 12 | 18 | y | 30 | 2y | 45
Given the table above as the results of tossing a fair die 150 times. Find the probability of obtaining a 5.
A. \frac{1}{10}
B. \frac{1}{6}
C. \frac{1}{5}
D. \frac{3}{10}

32. From the same table as Question 31, find the mode.
A. 3
B. 4
C. 5
D. 6

33. Given that a = 5i + 4j and b = 3i + 7j, evaluate (3a - 8b).
A. 9i + 44j
B. -9i + 44j
C. -9i - 44j
D. 9i - 44j

34. A force (10i + 4j)N acts on a body of mass 2kg which is at rest. Find the velocity after 3 seconds.
A. (\frac{5i}{3} + \frac{2j}{3})m/s
B. (\frac{10i}{3} + \frac{4j}{3})m/s
C. (5i + 2j)m/s
D. (15i + 6j)m/s

35. Solve \frac{3}{2x} - \frac{3}{x+2} = \frac{3}{x+1} - 27.
A. 1 or 0
B. 1 or 2
C. 1 or -2
D. -1 or 2

36. Find the magnitude and direction of the vector p = (5i - 12j).
A. (13, 113.38°)
B. (13, 067.38°)
C. (13, 025.38°)
D. (13, 157.38°)

37. The velocity, V, of a particle after t seconds, is V = 3t^2 + 2t - 1. Find the acceleration of the particle after 2 seconds.
A. 10 m/s^2
B. 12 m/s^2
C. 14 m/s^2
D. 17 m/s^2

38. Given that f(x) = 2x^2 - 3 and g(x) = x + 1 where x \in \mathbb{R}. Find g \circ f(x).
A. 2(x^2 - 1)
B. 2x^2 + 4x - 1
C. 2x^2 + 6x - 1
D. 3(x^2 - 1)

39. If P = \{n^2 + 1 : n = 0,2,3\} and Q = \{n + 1 : n = 2,3,5\}, find P \cap Q.
A. \{5, 10\}
B. \{4, 6\}
C. \{1, 3\}
D. \{ \}

40. If (2x^2 - x - 3) is a factor of f(x) = 2x^3 - 5x^2 - x + 6, find the other factor.
A. (x - 2)
B. (x - 1)
C. (x + 1)
D. (x + \frac{3}{2})

2018

1. Simplify \frac{3\sqrt{3}}{3\sqrt{3}} + \frac{1}{3\sqrt{3}}
A. \frac{1}{2}
B. 3
C. \frac{2}{3} - \sqrt{}
D. 6

2. Find the domain of g(x) = \frac{\sqrt{4x^2 - 1}}{9x^2 + 1}
A. x : x \in \mathbb{R}, x = \frac{1}{2}
B. x : x \in \mathbb{R}, x \neq \frac{1}{3}
C. x : x \in \mathbb{R}, x = \frac{1}{3}
D. x : x \in \mathbb{R}

3. Given that f(x) = 3x^2 - 12x + 12 and f(x) = 3, find the values of x
A. 1, 3
B. -1, -3
C. 1, -3
D. -1, 3

4. A binary operation * is defined on the set of real numbers by a * b = \frac{a}{b} + \frac{b}{a}. If (\sqrt{x} + 1) * (\sqrt{x} - 1) = 4, find the value of x
A. 6
B. 5
C. 4
D. 3

5. If 4x^2 + 5kx + 10 is a perfect square, find the value of k
A. \frac{5\sqrt{10}}{4}
B. \frac{4 - \sqrt{10}}{4}
C. \frac{5 - \sqrt{10}}{4}
D. \frac{\sqrt{10}}{5}

6. If the polynomial f(x) = 3x^3 - 2x^2 + 7x + 5 is divided by (x - 1), find the remainder
A. -17
B. -7
C. 5
D. 13

7. P = \{1,3,5,7,9\}, Q = \{2,4,6,8,10,12\}, R = \{2,3,5,7,11\} are subsets of U = \{1,2,3,...,12\}. Which statement is true?
A. Q \cap R = \emptyset
B. R \subset P
C. (R \cap P) \subset (R \cap U)
D. n(P' \cap R) = 2

8. If \log_3(a-2) = 3\log_3(b), express a in terms of b
A. a = \frac{b}{3^{-3}}
B. a = \frac{b}{3^{-9}}
C. a = 9b^{3}
D. a = \frac{b}{3^{9}}

9. If \alpha and \beta are roots of 2x^2 - 5x + 6 = 0, find the equation whose roots are (\alpha+1) and (\beta+1)
A. 2x^2 - 9x + 15 = 0
B. 2x^2 - 9x + 13 = 0
C. 2x^2 - 9x - 13 = 0
D. 2x^2 - 9x - 15 = 0

10. Resolve \frac{3x-1}{(x-2)^2}, x \neq 2 into partial fractions
A. \frac{x}{2(x-2)} - \frac{5}{(x-2)^2}
B. \frac{5}{(x-2)} + \frac{x}{2(x-2)^2}
C. \frac{1}{2(x-2)} + \frac{5x}{2(x-2)^2}
D. -\frac{1}{2(x-2)} + \frac{8x}{2(x-2)^2}

(Previous 10 questions…)

11. If \alpha and \beta are roots of 2x^2 + 5x + n = 0, such that \alpha\beta = 2, find the value of n
A. -4
B. -2
C. 2
D. 4

12. Solve \log_2(12x-10) = 1 + \log_2(4x+3)
A. 4.75
B. 4.00
C. 1.75
D. 1.00

13. Find the coefficient of x^3 in the binomial expansion of (x - \frac{3}{x^2})^9
A. 324
B. 252
C. -252
D. -324

14. The general term of an infinite sequence 9, 4, -1, -6,... is u_r = ar + b. Find the values of a and b
A. a = 5, b = 14
B. a = -5, b = 14
C. a = 5, b = -14
D. a = -5, b = -14

15. If \left|\frac{k}{4}\right| + \left|\frac{2-1}{3k}\right| = 6, find the value of the constant k, where k > 0
A. 1
B. 2
C. 3
D. 4

16. How many numbers greater than 150 can be formed from the digits 1, 2, 3, 4, 5 without repetition?
A. 91
B. 191
C. 291
D. 391

17. The first term of a Geometric Progression (GP) is \frac{3}{4}. If the product of the second and third terms is 972, find its common ratio
A. 3
B. 4
C. 6
D. 12

18. If \sin\theta = \frac{3}{5}, 0° < \theta < 90°, evaluate \cos(180° - \theta)
A. \frac{4}{5}
B. \frac{3}{5}
C. -\frac{3}{5}
D. -\frac{4}{5}

19. Find the radius of the circle x^2 + y^2 - 8x - 2y + 1 = 0
A. 9
B. 7
C. 4
D. 3

20. In how many ways can the letters of the word ‘ELECTIVE’ be arranged?
A. 336
B. 1680
C. 6720
D. 20160

21. If the determinant of the matrix \begin{pmatrix} 2 & 3x \\ 5 & x \end{pmatrix} = 13, find the value of x
A. -2
B. -1
C. 1
D. 2

22. Express \frac{13\pi}{4} radians in degrees
A. 495°
B. 225°
C. 585°
D. 135°

23. Find the equation to the circle x^2 + y^2 - 4x - 2y = 0 at the point (1, 3)
A. 2y - x - 5 = 0
B. 2y + x - 5 = 0
C. 2y + x + 5 = 0
D. 2y - x + 5 = 0

24. Given that y = \frac{x(x+1)}{2}, calculate the maximum value of y
A. -2
B. 0
C. 1
D. 2

25. The midpoint of M(4, -1) and N(x, y) is P(3, -4). Find the coordinates of N
A. (2, -3)
B. (2, -7)
C. (-1, -3)
D. (-10, -7)

26. Find the stationary point of the curve y = 3x^2 - 2x^3
A. (1, 0)
B. (-1, 0)
C. (1, 1)
D. (-1, -1)

27. Evaluate \int_{1}^{1} \frac{2x^3 - 4}{x^3} dx
A. -5.5
B. -2.0
C. 2.0
D. 5.5

28. Calculate the standard deviation of \{30, 29, 25, 28, 32, 24\}
A. 2.0
B. 2.8
C. 3.0
D. 3.2

29. Evaluate \int_{-1}^{1} (x+1)^2 dx
A. \frac{8}{3}
B. \frac{7}{3}
C. \frac{5}{3}
D. 2

30. Out of 70 schools, 42 can be attended by boys and 35 by girls. If a pupil is selected at random, find the probability of being from a mixed school
A. \frac{1}{11}
B. \frac{1}{10}
C. \frac{1}{6}
D. \frac{1}{5}

31. Calculate Spearmann’s rank correlation coefficient for the given rankings
A. 0.2
B. 0.5
C. 0.6
D. 0.7

32. Given \vec{a} = i - 3j, \vec{b} = -2i + 5j, and \vec{c} = 3i - j, calculate |\vec{a} - \vec{b} + \vec{c}|
A. \sqrt{13}
B. \sqrt{13}/3
C. \sqrt{13}/6
D. \sqrt{13}/9

33. Probability of obtaining a head and a six when a fair coin and a die are tossed together
A. \frac{1}{12}
B. \frac{1}{3}
C. \frac{1}{2}
D. \frac{2}{3}

34. If \vec{OX} = \begin{pmatrix} -7 \\ 6 \end{pmatrix} and \vec{OY} = \begin{pmatrix} 16 \\ -11 \end{pmatrix}, find \vec{YX}
A. \begin{pmatrix} 9 \\ -5 \end{pmatrix}
B. \begin{pmatrix} -23 \\ -5 \end{pmatrix}
C. \begin{pmatrix} 9 \\ 17 \end{pmatrix}
D. \begin{pmatrix} -23 \\ 17 \end{pmatrix}

35. A body of mass 28g, initially at rest, is acted upon by a force F Newtons. If it attains a velocity of 5.4 \text{m/s} in 18 seconds, find the value of F
A. 0.0082 \text{N}
B. 0.0084 \text{N}
C. 0.082 \text{N}
D. 0.084 \text{N}

36. Find the angle between forces of magnitude 7 \text{N} and 4 \text{N} if their resultant has a magnitude of 9 \text{N}
A. 39.45°
B. 73.40°
C. 75.34°
D. 106.60°

37. Find the constant term in the binomial expansion (2x^2 + \frac{1}{x})^9
A. 84
B. 168
C. 336
D. 672

38. A particle starts from rest and moves through a distance S = 12t^2 - 2t^3 metres in time t seconds. Find its acceleration in 1 second
A. 24 \text{m/s}^2
B. 18 \text{m/s}^2
C. 12 \text{m/s}^2
D. 10 \text{m/s}^2

39. A car is moving at 120 \text{km/h}. Find its speed in \text{m/s}
A. 33.3 \text{m/s}
B. 66.6 \text{m/s}
C. 99.9 \text{m/s}
D. 120.0 \text{m/s}

40. Two functions f and g are defined on the set of real numbers by f: x \to x^2 + 1 and g: x \to x - 2. Find f \circ g
A. x^2 + 4x - 5
B. x^2 - 4x + 5
C. x^2 - 1
D. x - 1

2019

1. Solve: \frac{8}{x-2} = 4 \frac{3x}{1}
A. -2
B. -1
C. 1
D. 2

2. Solve: \frac{p^2}{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

3. Evaluate \tan 75°, leaving the answer in surd form
A. \frac{3+\sqrt{2}}{1}
B. \frac{3+\sqrt{1}}{1}
C. \frac{3-\sqrt{1}}{1}
D. \frac{3-\sqrt{2}}{1}

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

5. If \binom{n}{2} = 15, find the value of n
A. 8
B. 7
C. 6
D. 5

6. For operation * defined by x * y = x + y - xy on set T = \{-1, 0, ..., 5\}, which operation(s) give an image in T?
I. 2 * 5 II. 3 * 5 III. 3 * 4
A. I only
B. II only
C. I and III only
D. II and III only

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

8. Linear transformation T: (x, y) \to (-x + y, -4y). Find image of Q(-3, 2)
A. Q'(5, -8)
B. Q'(-8, 5)
C. Q'(5, -3)
D. Q'(-5, -8)

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

10. Symbolic representation of “If the grass is green and the sky is not blue, then the birds do not fly”
A. (r \wedge \neg p) \to q
B. (r \wedge q) \to p
C. (r \wedge \neg p) \to q
D. (r \wedge \neg p) \to q

11. Given \frac{1}{x^2-4} = \frac{P}{x+2} + \frac{Q}{x-2}, find P + Q
A. \frac{3}{2}
B. 1
C. \frac{1}{2}
D. 0

12. Sum of first 20 terms of sequence -7, -3, 1, ...
A. 620
B. 660
C. 690
D. 1240

13. Solve \sqrt{6^{4x^2+1}} = 13x for x > 0
A. \frac{6}{5}
B. \frac{25}{24}
C. \frac{24}{25}
D. \frac{5}{6}

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

15. If P = \begin{pmatrix} 2 & -4 \\ 3 & 1 \end{pmatrix}, Q = \begin{pmatrix} 6 \\ 8 \end{pmatrix}, and PQ = k \begin{pmatrix} 4 & -5 \\ -20 \end{pmatrix}, find k
A. -\frac{5}{4}
B. -\frac{4}{5}
C. \frac{4}{5}
D. \frac{5}{4}

16. Second and fourth terms of geometric progression are \frac{2}{9} and \frac{8}{81} respectively. Find the sixth term
A. \frac{81}{32}
B. \frac{9}{8}
C. \frac{1}{4}
D. \frac{32}{729}

17. Point X and Y on horizontal base with X 96m east of building, Y west. Angle of elevation from X is 30°, from Y is 50°. Calculate Y’s distance from building base
A. 30m
B. 32m
C. 42m
D. 50m

18. Find coordinates on curve y = 3x^2 - 2x - 5 where tangent is parallel to y = -5 = 8x
A. (-\frac{5}{3}, 0)
B. (0, -\frac{5}{3})
C. (0, \frac{5}{3})
D. (\frac{5}{3}, 0)

19. Mean of 2, 5, (x+1), (x+2), 7, 9 is 6. Find the median
A. 6.5
B. 6.0
C. 5.5
D. 5.0

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

21. Particle velocity V at time t given by V = 3t^2 - 6t. Acceleration in 3rd second
A. 12m
B. 16m
C. 64m
D. 96m

22. Particle velocity V at time t given by V = 3t^2 - 6t. Acceleration in 3rd second
A. 0 \text{ ms}^{-2}
B. 3 \text{ ms}^{-2}
C. 6 \text{ ms}^{-2}
D. 9 \text{ ms}^{-2}

23. Constant term in (2x^2 + \frac{1}{x^2})^4
A. 10
B. 12
C. 24
D. 42

24. Inequality represented by graph’s shaded portion
A. 2y + x - 3 < 0
B. 2y - x - 3 < 0
C. 2y - x + 3 < 0
D. 2y + x + 3 < 0

25. 35 \text{N} force acts on 5 \text{ kg} body for 2 \text{ seconds}. Change in momentum
A. 70 \text{ kg ms}^{-1}
B. 55 \text{ kg ms}^{-1}
C. 50 \text{ kg ms}^{-1}
D. 35 \text{ kg ms}^{-1}

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

27. Given P and Q are non-empty subsets of universal set U, find P \cap (Q \cup Q')
A. P
B. P'
C. Q
D. Q'

28. Coefficient of term in [2x + \frac{3y}{4}]^3 in descending powers of x
A. \frac{27}{64}y^2
B. \frac{27}{8}y^2
C. 8y^2
D. 9y^2

29. Coordinates of circle center 3x^2 + 3y^2 - 6x + 9y - 5 = 0
A. (-3, \frac{9}{2})
B. (-1, \frac{3}{2})
C. (1, -\frac{3}{2})
D. (3, -\frac{9}{2})

30. Evaluate \int_0^1 \sqrt{x} dx
A. 3
B. 9
C. 18
D. 27

31. Function f: x \to x^2 + px + q, turning point at x = -3, remainder -6 when divided by (x+2). Find q
A. 6
B. 2
C. -2
D. -8

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

33. Vector perpendicular to \begin{pmatrix} -1 \\ 3 \end{pmatrix}
A. \begin{pmatrix} -3 \\ 1 \end{pmatrix}
B. \begin{pmatrix} 1 \\ 3 \end{pmatrix}
C. \begin{pmatrix} 1 \\ -3 \end{pmatrix}
D. \begin{pmatrix} 1 \\ 3 \end{pmatrix}

34. Angle between p = 12i - 5j and q = 4i + 3j, to nearest degree
A. 59°
B. 60°
C. 75°
D. 76°

35. Area between y = x + 1 and x-axis from x = -2 to x = 0
A. 5 square units
B. 4 square units
C. 2 square units
D. 1 square unit

36. Numbers > 200 formed from digits 1,2,3,4,5 without repetition
A. 50
B. 60
C. 288
D. 300

37. Probability John (0.9) and Jane (0.7) pass exam, probability at least one passes
A. 0.28
B. 0.67
C. 0.72
D. 0.97

38. Independent events X and Y, P(X) = 0.5, P(Y) = m, P(X \cup Y) = 0.75, find m
A. 0.6
B. 0.5
C. 0.4
D. 0.3

39. Uniform beam PQ, 100m long, 35N, supported 40cm from P, weights 54N and FN at P and Q to keep horizontal. Find F
A. 69
B. 60
C. 35
D. 30

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

2020

1. A binary operation * is defined on the set of real number, \mathbb{R}, by x * y = x^2 - y^2 + xy, where x, y \in \mathbb{R}. Evaluate (3 - \sqrt{2}) * (2 - \sqrt{2})

A. 1 - \sqrt{6}
B. \sqrt{6} - 1
C. \sqrt{6}
D. 1 + \sqrt{6}

2. Find the inverse of \begin{pmatrix} 3 & 1 \\ 5 & 2 \end{pmatrix}

A. \begin{pmatrix} 5 & -3 \\ 1 & 2 \end{pmatrix}
B. \begin{pmatrix} 2 & -1 \\ -5 & 3 \end{pmatrix}
C. \begin{pmatrix} -5 & -1 \\ 2 & 3 \end{pmatrix}
D. \begin{pmatrix} 5 & 2 \\ 1 & 3 \end{pmatrix}

3. If \cos x = -0.7133, find the values of x between 0^\circ and 360^\circ

A. 44.5^\circ, 224.5^\circ
B. 123.5^\circ, 190.5^\circ
C. 135.5^\circ, 213.5^\circ
D. 135.5^\circ, 224.5^\circ

4. If \int_{0}^{3} (px^2 + 16)dx = 129. Find the value of p

A. 9
B. 8
C. 7
D. 6

5. If \begin{pmatrix} p+q & 0 \\ 1 & p-q \end{pmatrix} = \begin{pmatrix} 2 & 0 \\ 1 & 8 \end{pmatrix} Find the values of p and q

A. p = 5, q = 3
B. p = 5, q = -3
C. p = -5, q = -3
D. p = -5, q = 3

6. Given that X : \mathbb{R} \to \mathbb{R} is defined by x = \frac{y+1}{5-y}, y \in \mathbb{R}, find the domain of x

A. \{y : y \in \mathbb{R}, y \neq 0\}
B. \{y : y \in \mathbb{R}, y \neq 1\}
C. \{y : y \in \mathbb{R}, y \neq 5\}
D. \{y : y \in \mathbb{R}, y \neq 7\}

7. Simplify \frac{5\sqrt{2}+3}{4-10\sqrt{2}}

A. \frac{2}{3\sqrt{5}} + \frac{5}{6\sqrt{2}} + 2
B. \frac{2}{3\sqrt{5}} + \frac{5}{6\sqrt{2}} + \frac{1}{2\sqrt{10}} + 2
C. \frac{2}{3\sqrt{5}} + \frac{5}{6\sqrt{2}} + \frac{1}{2\sqrt{10}}
D. \frac{2}{3\sqrt{5}} - \frac{5}{6\sqrt{2}} + \frac{1}{2\sqrt{10}} + 2

8. If \frac{6x+k}{2x^2+7x-15} = \frac{4}{x+5} - \frac{2}{2x-3}. Find the value of k

A. -21
B. -22
C. -24
D. -25

9. Differentiate \frac{x}{x+1} with respect to x

A. \frac{x}{x+1}
B. -\frac{1}{x+1}
C. \frac{1-x}{(x+1)^2}
D. \frac{1}{(x+1)^2}

10. Given that 2x + 3y - 10 and 3x = 2y - 11, calculate the value of (x - y)

A. 5
B. 3
C. -3
D. -5

11. If V = p \log_x(M + N), express N in terms of X, P, M and V

A. N = \frac{X}{V/p} - M
B. N = \frac{X}{p/V} - M
C. N = \frac{X}{V/p} + M
D. N = \frac{X}{p/V} + M

12. Determine the coefficient of x^3 in the binomial expansion of (1 + \frac{1}{2}x)^6

A. \frac{5}{8}
B. \frac{5}{6}
C. \frac{5}{4}
D. \frac{5}{2}

13. Given that P = \{x : 1 \leq x \leq 6\} and Q = \{x : 2 < x < 10\}. Where x are integers, find n(P \cap Q)

A. 4
B. 6
C. 8
D. 10

14. If \cos X = \frac{3}{5} and \cos Y = \frac{24}{25}, where X and Y are acute, find the value of \cos(X + Y)

A. \frac{117}{125}
B. \frac{24}{25}
C. \frac{3}{5}
D. \frac{7}{25}

15. Find the median of the numbers 9, 7, 5, 2, 12, 9, 9, 2, 10, 10, and 18

A. 7
B. 9
C. 10
D. 11

16. Calculate the probability that the product of two numbers selected at random with replacement from the set \{-5, -2, 4, 8\} is positive

A. \frac{2}{3}
B. \frac{1}{2}
C. \frac{1}{3}
D. \frac{1}{6}

17. Find the angle between i + 5j and 5i - j

A. 0^\circ
B. 45^\circ
C. 60^\circ
D. 90^\circ

18. Given that F = 3i - 12j, R = 7i + 5j and N = pi + qj are forces acting on a body, if the body is in equilibrium, find the values of p and q

A. p = -10, q = 7
B. p = -10, q = -7
C. p = 10, q = -7
D. p = -10, q = 7

19. A stone was dropped from the top of a building 40m high. Find, correct to one decimal place, the time it took the stone to reach the ground. [Take g = 9.8\text{ms}^{-2}]

A. 2.9 seconds
B. 2.8 seconds
C. 2.6 seconds
D. 1.4 seconds

20. In which of the following series can the formula S = \frac{a}{1-r} where a is the first term and r is the common ratio, be used to find the sum of all the terms?

A. 4 + 8 + 16 + 32 + ...
B. \frac{1}{2} + 2\frac{1}{2} + 12\frac{1}{2} + 62\frac{1}{2} + ...
C. \frac{4}{81} + \frac{2}{27} + \frac{1}{9} + \frac{1}{6} + ...
D. 128 + 64 + 32 + 16 + ...

21. If the binomial expansion of (1 + 3x)^6 is used to evaluate (0.97)^6, find the value of x

A. 0.03
B. 0.01
C. -0.01
D. -0.03

22. Find the nth term of the linear sequence (A.P) (5y + 1), (2y + 1), (1 - y), ...

A. (8 + 3n)y + 1
B. 8y + 3n + 1
C. (8 - 3n)y + 1
D. 8y - 3n + 1

23. A circle with centre (5, -4) passes through the point (5, 0). Find its equation

A. x^2 + y^2 + 10x + 8y + 25 = 0
B. x^2 + y^2 + 10x - 8y - 25 = 0
C. x^2 + y^2 - 10x + 8y + 25 = 0
D. x^2 + y^2 - 10x - 8y - 25 = 0

24. Calculate, correct to two decimal places, the area enclosed by the line 3x - 5y + 4 = 0 and the axes

A. 0.50 square units
B. 0.51 square units
C. 0.53 square units
D. 0.54 square units

25. In how many ways can the letters of the word MEMBER be arranged?

A. 720
B. 360
C. 180
D. 90

26. Which of the following is not an equation of a circle?

A. 3x^2 + 3y^2 + 5x + 7y = 5
B. x^2 + y^2 + 5x + 4y = 0
C. 5x^2 + 5y^2 - 16 = 0
D. x^2 - y^2 + 3x - 5y = 2

27. A function f defined by f : x \to x^2 + px + q is such that f(3) = 6 and f(3) = 0. Find the value of q

A. -9
B. -6
C. 15
D. 21

28. In what interval is the function f : x \to 2x - x^2 increasing?

A. x = 1
B. x < 1
C. x > 1
D. 0 < x < 1

29. A force of 230N acts in its direction 065^\circ. Find its horizontal component

A. 208.5N
B. 197.2N
C. 118.5N
D. 97.2N

30. Calculate the variance of \sqrt{2}, (1 + \sqrt{2}) and (2 + \sqrt{2})

A. 0
B. \sqrt{2/3}
C. \frac{2}{3}
D. 2

31. A three-digit odd number less than 500 is to be formed from 1,2,3,4 and 5. If repetition of digits is allowed, in how many ways can this be done?

A. 125
B. 75
C. 60
D. 36

32. The variables x and y are such that y = 2x^3 - 2x^2 - 5x + 5. Calculate the corresponding change in y and x changes from 2.00 to 2.05

A. 0.58
B. 0.95
C. 1.48
D. 1.95

33. A bag contains 5 red and 5 blue identical balls. Three balls are selected at random without replacement. Determine the probability of selecting balls alternating in color

A. \frac{7}{18}
B. \frac{5}{18}
C. \frac{5}{36}
D. \frac{1}{36}

34. The distance(s) in metres covered by a particle in motion at any time, t seconds, is given by S = 120t - 16t^2. Find in metres, the distance covered by the body before coming to rest

A. 220
B. 222
C. 223
D. 225

35. P(3,4) and Q(-3, -4) are two points in a plane. Find the gradient of the line that is normal to the line PQ

A. \frac{4}{3}
B. \frac{3}{4}
C. -\frac{3}{4}
D. -\frac{4}{3}

36. Find the unit vector in the direction opposite to the resultant of forces F_1 = (-2i - 3j) and F_2 = (5i - j)

A. \frac{1}{5}(-3i - 4j)
B. \frac{1}{5}(-3i + 4j)
C. \frac{1}{5}(3i - 4j)
D. \frac{1}{5}(3i + 4j)

37. If the sum of the roots of 2x^2 + 5mx + n = 0 is 5, find the value of m

A. -2.5
B. -2.0
C. 2.0
D. 2.5

38. If \log_5(\frac{125}{x^3y\sqrt{3}}) is expressed in the values of p, q and k respectively

A. 3, -\frac{1}{3}, 5
B. -\frac{1}{3}, 3, 5
C. 3, -\frac{1}{3}, 3
D. 3, -\frac{1}{3}, 3

39. Consider the statements:
x: Birds fly
y: The sky is blue
Which of the following statements can be represented as x \to y?

A. When birds fly, the sky is blue
B. Birds fly if and only if the sky is blue
C. Either the bird is flying or the sky is blue
D. When the sky is blue, the bird flies

2021

1. Simplify \left( \frac{1}{2-\sqrt{3}} + \frac{2}{2+\sqrt{3}} \right)^{-1}
A. -\frac{1}{33}(6 + \sqrt{3})
B. -\frac{1}{33}(6 - \sqrt{3})
C. \frac{1}{33}(6 + \sqrt{3})
D. \frac{1}{33}(6 - \sqrt{3})

2. For what range of values of x is x^2 - 2x - 3 \leq 0
A. \{x: -1 \leq x \leq 3\}
B. \{ x: -3 \leq x \leq 1\}
C. \{ x: -3 \leq x \leq -1\}
D. \{ x: 1 \leq x \leq 3\}

3. Given that M = \begin{pmatrix} 3 \\ -1 \\ 2 \\ 4 \end{pmatrix} and N = \begin{pmatrix} 5 \\ -2 \\ 6 \\ -3 \end{pmatrix}, calculate (3M - 2N)
A. \begin{pmatrix} 1 \\ 1 \\ 6 \\ 18 \end{pmatrix}
B. \begin{pmatrix} -1 \\ 1 \\ -6 \\ 18 \end{pmatrix}
C. \begin{pmatrix} 1 \\ -1 \\ 6 \\ -18 \end{pmatrix}
D. \begin{pmatrix} -1 \\ -1 \\ -6 \\ -18 \end{pmatrix}

4. Simplify \frac{1}{3} \log 8 + \frac{1}{3} \log 64 - 2 \log 6
A. \log \frac{2}{7}
B. \log 2
C. \log \frac{2}{9}
D. \log 9

5. Solve \left(\frac{1}{9}\right)^{x+2} = 243^{x-2}
A. \frac{7}{5}
B. \frac{6}{7}
C. -\frac{7}{6}
D. -\frac{6}{7}

6. g(x) = 2x + 3 and f(x) = 3x^2 - 2x + 4, find f\{g(-3)\}
A. 37
B. 1
C. -3
D. -179

7. Using binomial expansion of (1 + x)^6 = 1 + 6x + 15x^2 + 20x^3 + 6x^5 + x^6, find, correct to three decimal places, the value of (1.998)^6
A. 63.616
B. 63.167
C. 62.628
D. 62.629

8. In how many ways can 8 persons be seated on a bench if only three seats are available?
A. 100
B. 125
C. 336
D. 427

9. If \alpha and \beta are the roots of 3x^2 - 7x + 6 = 0, find \frac{1}{\alpha} + \frac{1}{\beta}
A. \frac{7}{6}
B. \frac{7}{3}
C. \frac{14}{5}
D. \frac{14}{3}

10. If f(x) = 4x^3 + px^2 + 7x - 23 is divided by (2x -5), the remainder is 7. Find the value of p
A. -7.0
B. -8.0
C. -9.6
D. 9

11. For what value of k is 4x^2 - 12x + k a perfect square?
A. -9
B. -\frac{9}{4}
C. \frac{9}{4}
D. 9

12. A binary operation * is defined on the set of real numbers, R, by p * q = \frac{q^2 - p^2}{2pq}. Find 3 * 2
A. \frac{13}{12}
B. \frac{5}{12}
C. -\frac{5}{12}
D. -\frac{1}{2}

13. Find the inverse of \begin{pmatrix} 4 \\ -3 \\ 2 \\ -2 \end{pmatrix}
A. \begin{pmatrix} 1 \\ -1.5 \\ 1 \\ -2 \end{pmatrix}
B. \begin{pmatrix} 1 \\ 1.5 \\ -1 \\ -2 \end{pmatrix}
C. \begin{pmatrix} -2 \\ 1.5 \\ 1 \\ 1 \end{pmatrix}
D. \begin{pmatrix} -2 \\ 1.5 \\ -1 \\ 1 \end{pmatrix}

14. Given that P = \{ x: 0 \leq x \leq 36, x \text{ is a factor of } 36 \text{ divisible by } 3\} and Q = \{ x: 0 \leq x \leq 36, x \text{ is an even number and a perfect square}\}, find P \cap Q
A. \{1,4,9,36\}
B. \{3,9.36\}
C. \{9,36\}
D. \{36\}

15. A body of mass 15\text{kg} is placed on a smooth plane which is inclined at 60° to the horizontal. If the box is at rest, calculate the normal reaction to the plane. [Take g = 10\text{m/s}^2]
A. 129.9\text{N}
B. 75\text{N}
C. 60.0\text{N}
D. 7.5\text{N}

16. A fair die is tossed 60 times and the results are recorded in the table
Number of die: 1, 2, 3, 4, 5, 6
Frequency: 15, 10, 14, 2, 8, 11
Find the probability of obtaining a prime number.
A. \frac{7}{30}
B. \frac{1}{6}
C. \frac{7}{15}
D. \frac{8}{15}

17. If 2y^2 + 7 = 3y - xy, find \frac{dy}{dx}
A. -\frac{y}{4x+y-3}
B. \frac{y}{4x+y-3}
C. -\frac{y}{4x+x-3}
D. \frac{y}{4x+x-3}

18. Three forces, F_1 (8\text{N}, 030°), F_2 (10\text{N}, 150°) and F_3 (K\text{N}, 240°) are in equilibrium. Find the value of K
A. 5\sqrt{3}
B. 6\sqrt{2}
C. 6\sqrt{3}
D. 9\sqrt{3}

19. In \triangle PQR, \overline{PQ} = 5i - 2j and \overline{QR} = 4i + 3j. Find \overline{RP}
A. -i - 5j
B. -9 - j
C. i + 5j
D. -9i + j

20. A stone is thrown vertically upward and distance, S metres after t seconds is given by S = 12t + \frac{5}{2}t^2 - t^3. Calculate the maximum height reached.
A. 418.5\text{m}
B. 56.0\text{m}
C. 31.5\text{m}
D. 30.0\text{m}

21. A stone is thrown vertically upward and distance, S metres after t seconds is given by S = 12t + \frac{5}{2}t^2 - t^3. Calculate the distance travelled in the third second.
A. 5.5\text{m}
B. 14.5\text{m}
C. 26.0\text{m}
D. 30.0\text{m}

22. Given that F_1(x) = x^3 \sqrt{x}, find f(x)
A. \frac{2}{9} x^{9/2} + c

23. If (1 - 2x)^4 = 1 + px + qx^2 - 32x^3 + 16^4, find the value of (q - p)
A. -32
B. -16
C. 16
D. 32

24. If \sin x = \frac{12}{13} and \sin y = \frac{4}{5}, where x and y are acute angles, find \cos(x + y)
A. \frac{48}{65}
B. \frac{13}{15}
C. -\frac{33}{65}
D. -\frac{48}{65}

25. The first term of an AP is 4 and the sum of the first three terms is 18. Find the product of the first three terms.
A. 292
B. 272
C. 192
D. 172

26. A committee consists of 6 boys and 4 girls. In how many ways can a sub-committee consisting of 3 boys and 2 girls be formed if one particular boy and one particular girl must be on the sub-committee?
A. 120
B. 80
C. 56
D. 30

27. If \sqrt{5}\cos x + \sqrt{15}\sin x = 0, for 0° < x < 360°, find the values of x.
A. 30° and 150°
B. 150° and 210°
C. 150° and 330°
D. 210° and 330°

28. If 2i + pj and 4i - 2j are perpendicular, find the value of p.
A. 2
B. 3
C. 4
D. 5

29. Consider the following statements:
X: Benita is polite
Y: Benita is neat
Z: Benita is intelligent
Which of the following symbolizes the statement: “Benita is neat if and only if she is neither polite nor intelligent”?
A. y \iff \neg x \lor z
B. y \iff \neg x \lor \neg z
C. y \iff \neg x \land \neg z
D. y \iff \neg x \land z

30. A bag contains 8 red, 4 blue and 2 green identical balls. Two balls are drawn randomly from the bag without replacement. Find the probability that the balls drawn are red and blue.
A. \frac{12}{91}
B. \frac{16}{91}
C. \frac{30}{91}
D. \frac{32}{91}

31. The gradient of y = 3x^2 + 11x + 7 at P(x,y) is -1. Find the coordinates of P.
A. (-3, -2)
B. (-2,-3)
C. (-2,3)

32. Find the equation of the normal to the curve y = 2x^2 - 5x + 10 at P(1, 7).
A. y+x-3 = 0
B. y-x+6 = 0
C. y - x - 6 = 0
D. y -x+ 3 = 0

33. Find the value of the derivative of y = 3x^2(2x +1) with respect to x at the point x = 2.
A. 72
B. 84
C. 96
D. 120

34. Find the radius of the circle 2x^2 - 4x + 2y^2 - 6y -2 = 0.
A. \frac{17}{4}
B. \frac{17}{2}
C. \frac{17}{\sqrt{2}}
D. \frac{\sqrt{17}}{2}

35. Given that f: x \to x^2 - x + 1 is defined on the Set Q = \{ x : 0 \leq x < 20, x \text{ is a multiple of } 5\}. Find the set of range of F.
A. \{21, 91, 221\}
B. \{21, 91, 221, 381\}
C. \{1,21, 91, 221\}
D. \{1,21, 91, 221,381\}

36. If \frac{15-2x}{(x+4)(x-3)} = \frac{R}{(x+4)} + \frac{9}{7(x-3)}, find the value of R.
A. -\frac{32}{7}
B. -\frac{23}{7}
C. \frac{23}{7}
D. \frac{32}{7}

37. The table shows the distribution of marks obtained by some students in a test
Marks | Frequency
0-9 | 4
10-19 | 12
20-29 | 16
30-39 | 6
40-49 | 2
What is the upper class boundary of the upper quartile class?
A. 49.5
B. 39.5
C. 29.5
D. 19.5

38. The table shows the distribution of marks obtained by some students in a test
Marks | Frequency
0-9 | 4
10-19 | 12
20-29 | 16
30-39 | 6
40-49 | 2
Find the modal class mark.
A. 4.5
B. 14.5
C. 24.5
D. 34.5

2022

1. A binary operation \Delta is defined on the set of real numbers R, by x\Delta y = \frac{x+y-\sqrt{xy}}{4}, where x, y\in R. Find the value of 4\Delta3
A. 16
B. 8
C. 4
D. 2

2. \left(\frac{3\sqrt{6}+54\sqrt{5}}{\sqrt{3^5}}\right)^{-1}
A. \frac{5\sqrt{3}}{6}
B. \frac{3\sqrt{15}}{6}
C. \frac{5\sqrt{6}}{12}
D. \frac{5\sqrt{3}}{12}

3. If \log_{10}(3x-1)+\log_{10}4=\log_{10}(9x+2), find the value of x
A. \frac{1}{3}
B. 1
C. 2
D. 3

4. Simplify \frac{9^{3^{n+1}}-3^{n+2}}{3^{n+1}-3^n}
A. 3
B. 9
C. 27
D. 81

5. Consider the following statement:
x: All wrestlers are strong
y: Some wresters are not weightlifters.
Which of the following is a valid conclusion?
A. All strong wrestlers are weightlifters
B. Some strong wrestlers are not weightlifters
C. Some weak wrestlers are weightlifters
D. All weightlifters are wrestlers

6. The functions f:x \to 2x^2 + 3x -7 and g:x \to 5x^2 + 7x - 6 are defined on the set of real numbers, R. Find the values of x for which 3f(x) = g(x).
A. x = -3 or -5
B. x = -3 or 5
C. x = 3 or -5
D. x = 3 or 5

7. Express \frac{4\pi^2}{\pi} radians in degrees.
A. 288º
B. 200º
C. 144º
D. 120º

8. 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

9. Which of the following is the semi-interquartile range of a distribution?
A. Mode – Median
B. Highest score – Lowest score
C. \frac{1}{2} (Upper Quartile – Median)
D. \frac{1}{2} (Upper Quartile – Lower Quartile)

10. Evaluate \int_{-1}^{0} (x + 1)(x - 2) dx
A. 7/6
B. 5/6
C. -5/6
D. -7/6

11. If 36, p, \frac{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\frac{9}{16}

12. Differentiate \frac{5}{x^3}+\frac{x^2}{x}, x \neq 0 with respect to x.
A. 10x + 1
B. 10x + 2
C. x(15x + 1)
D. x(15x + 2)

13. Given that \frac{8x+m}{x^2-3x-4} \equiv \frac{5}{x+1}+\frac{3}{x-4}
A. 23
B. 17
C. -17
D. 17

14. If x^2+y^2-2x-6y+5=0, evaluate \frac{dy}{dx} when x=3 and y=2.
A. 2
B. -2
C. -4
D. 4

15. Evaluate \int_{0}^{1} \frac{x^2}{(x^3+2)^3}
A. \frac{56}{12}
B. \frac{65}{12}
C. 12
D. 65

16. Given \left|\begin{matrix}2&1&-3\\4&-6&k\\3&-26&-26\end{matrix}\right| = 15. Solve for k.
A. -8
B. -5
C. -4
D. -3

17. A linear transformation T is defined by T: (x,y) \to (3x - y, x + 4y). Find the image of (2, -1) under T.
A. (7, -2)
B. (5, -2)
C. (-2, 7)
D. (-7, 2)

18. Evaluate \frac{4P^2+4C^2-4P^3}{P}
A. 18
B. 6
C. -6
D. -18

19. Find the coefficient of x^2 in the binomial expansion of (x+\frac{2}{x^2})^5
A. 10
B. 40
C. 32
D. 80

20. Given that P = \{x: x \text{ is a multiple of 5}\}, Q = \{x: x \text{ is a multiple of 3}\} and R = \{x: x \text{ is an odd number}\} are subsets of \mu = \{x: 20 \leq x \leq 35\}, (P\cup Q)\cap R.
A. \{20, 21, 25, 30, 33\}
B. \{21, 25, 27, 33, 35\}
C. \{20, 21, 25, 27, 33, 35\}
D. \{21, 25, 27, 30, 33, 35\}

21. A particle moving with a velocity of 5m/s accelerates at 2m/s^2. Find the distance it covers in 4 seconds.
A. 16m
B. 26m
C. 36m
D. 46m

22. If U_n = kn^2 + pn, U_1 = -1, U_5 = 15, find the values of k and p.
A. k = -1, p = 2
B. k = -1, p = -2
C. k = 1, p = -2
D. k = 1, p = 2

23. In how many ways can six persons be paired?
A. 5
B. 10
C. 15
D. 20

24. Solve: \frac{3}{2x-2} - 28(\frac{3}{x-2}) + 3 = 0
A. x = -2 or x = 1
B. x = 0 or x = -3
C. x = 2 or x = 1
D. x = 0 or x = 3

25. Given that P = (-4, -5) and Q = (2,3), express \vec{PQ} in the form (k,\theta), where k is the magnitude and \theta the bearing.
A. (10 \text{ units}, 053º)
B. (9 \text{ units}, 049º)
C. (10 \text{ units}, 037º)
D. (9 \text{ units}, 027º)

26. If \vec{PQ} = -2i + 5j and \vec{RQ} = -i - 7j, find \vec{PR}
A. -3i + 12j
B. -3i - 12j
C. -i + 12j
D. i - 12j

27. The table shows the distribution of the distance (in km) covered by 40 hunters while hunting. If a hunter is selected at random, find the probability that the hunter covered at least 6km.
A. \frac{3}{5}
B. \frac{2}{5}
C. \frac{3}{8}
D. \frac{9}{40}

28. The table shows the distribution of the distance (in km) covered by 40 hunters while hunting. What is the mode of the distribution?
A. 5
B. 6
C. 7
D. 8

29. If g(x) = \sqrt{1-x^2}, find the domain of g(x)
A. x < -1 or x > 1
B. x \leq -1 or x \geq 1
C. -1 \leq x \leq 1
D. -1 < x < 1

30. Find the coefficient of x^3y^2 in the binomial expansion of (x-2y)^5
A. -80
B. 10
C. 40
D. 80

31. The first, second and third terms of an exponential sequence (G.P) are (x - 4), (x + 2), and (3x + 1) respectively. Find the values of x.
A. -\frac{1}{2}, 8
B. \frac{1}{2}, -8
C. -\frac{1}{2}, -8
D. \frac{1}{2}, 8

32. A body of mass 18kg moving with velocity 4ms^{-1} collides with another body of mass 6kg moving in the opposite direction with velocity 10ms^{-1}. If they stick together after the collision, find their common velocity.
A. \frac{1}{2} m/s
B. \frac{1}{3} m/s
C. 2m/s
D. 3m/s

33. The mean heights of three groups of students consisting of 20, 16 and 14 students each are 1.67m, 1.50m and 1.40m respectively. Find the mean height of all the students.
A. 1.63m
B. 1.54m
C. 1.52m
D. 1.42m

34. Find correct to the nearest degree, the acute angle formed by the lines y = 2x + 5 and 2y = x - 6
A. 76°
B. 53°
C. 37°
D. 14°

35. Solve: 4\sin^2\theta + 1 = 2, where 0° < \theta < 180°
A. 60° or 120°
B. 30° or 150°
C. 30° or 120°
D. 60° or 150°

36. Find the range of values of x for which 2x^2 + 7x - 15 \geq 0.
A. x \leq -5 or x \geq \frac{3}{2}
B. x \geq -5 or x \leq \frac{3}{2}
C. -5 \leq x \leq \frac{3}{5}
D. \frac{3}{5} \leq x \leq -5

37. The probability that a student will graduate from college is 0.4. If 3 students are selected from the college, what is the probability that at least one student will graduate?
A. 0.06
B. 0.22
C. 0.78
D. 0.80

38. The equation of a circle is given as 2x^2 + 2y^2 - x - 3y - 41 = 0. Find the coordinates of its centre.
A. (-\frac{1}{4}, \frac{3}{4})
B. (\frac{1}{4}, \frac{3}{4})
C. (-\frac{1}{2}, \frac{3}{2})
D. (-\frac{1}{2}, -\frac{3}{2})

39. The gradient of a function at any point (x,y) 2x - 6. If the function passes through (1,2), find the function.
A. x^2 - 6x - 5
B. x^2 - 6x + 5
C. x^2 - 6x - 3
D. x^2 - 6x + 7

40. A particle of mass 3kg moving along a straight line under the action of a F N, covers a line distance, d, at time, t, such that d = t^2 + 3t. Find the magnitude of F at time t.
A. 0N
B. 2N
C. 3(2t + 3)N
D. 6N

2023

1. Calculate, correct to one decimal place, the angle between 5\vec{i} + 12\vec{j} and -2\vec{i} + 3\vec{j}
A. 56.3°
B. 76.3°
C. 66.4°
D. 54.8°

2. Find the equation of the normal to the curve y = \frac{3x^2}{2} + 2 at point (1, 5).
A. 6y - x - 29 = 0
B. 6y + x - 31 = 0
C. y - 6x - 1 = 0
D. y - 6x + 1 = 0

3. The distance S metres moved by a body in t seconds is given by S = 5t^3 - \frac{19}{2}t^2 + 6t - 4. Calculate the acceleration of the body after 2 seconds
A. 19 m/s^2
B. 21 m/s^2
C. 41 m/s^2
D. 31 m/s^2

4. Evaluate \int_0^1 x(x^2-2)^2 dx
A. \frac{6}{7}
B. \frac{11}{6}
C. \frac{1}{7}
D. \frac{31}{6}

5. Given that \sin x = \frac{4}{5} and \cos y = \frac{12}{13}, where x is an obtuse angle and y is an acute angle, find the value of \sin(x - y).
A. \frac{63}{65}
B. \frac{48}{65}
C. \frac{56}{65}
D. \frac{16}{65}

6. If (\frac{1}{9})^{2x-1} = (\frac{1}{81})^{2-3x} find the value of x
A. -\frac{5}{8}
B. -\frac{3}{4}
C. \frac{3}{4}
D. -\frac{5}{8}

7. The table shows the operation * on the set {x, y, z, w}. [Table omitted]
Find the identity of the element.
A. W
B. Y
C. Z
D. X

8. Find the radius of the circle 2x^2 + 2y^2 - 4x + 5y + 1 = 0
A. \frac{3\sqrt{3}}{4}
B. \frac{5}{\sqrt{6}}
C. \frac{5}{6}
D. \frac{33}{4}

9. Given that M is the midpoint of T(2, 4) and Q(-8, 6), find the length of MQ.
A. \sqrt{26} units
B. \sqrt{28} units
C. \sqrt{24} units
D. \sqrt{30} units

10. A particle began to move at 27m/s along a straight line with constant retardation of 9m/s^2. Calculate the time it took the particle to come to a stop.
A. 3 sec
B. 2 sec
C. 4 sec
D. 1 sec

11. Find the fifth term in the binomial expansion of (q + x)^7.
A. 21q^2x^5
B. 21q^4x^3
C. 35q^3x^4
D. 35q^5x^2

12. Given that P = \{x \mid 2 \leq x \leq 8\} and Q = \{x \mid 4 < x \leq 12\} are subsets of the universal set \mu = \{x \mid x \in \mathbb{R}\}, find P \cap Q^1.

A. \{x \mid 4 < x < 8\}
B. \{x \mid 2 < x \leq 4\}
C. \{x \mid 2 \leq x \leq 4\}
D. \{x \mid 4 \leq x \leq 8\}

13. Consider the statements:
x: The school bus arrived late
y: The student walked down to school

Which of the following can be represented by y \Rightarrow x?

A. The school bus arrived early and Kate ran to school
B. Mary walked to school because the school bus arrived late
C. Either the school bus arrived late or Maryam walked to school
D. Emmanuella did not go to school because the school bus arrived late

14. Differentiate f(x) = \frac{1}{(1-x^2)^5} with respect to x.
A. \frac{-5x}{(1-x^2)^6}
B. \frac{-10x}{(1-x^2)^6}
C. \frac{5x}{(1-x^2)^6}
D. \frac{10x}{(1-x^2)^6}

15. Express \frac{3}{3-\sqrt{6}} in the form x + m\sqrt{y}
A. 3 - 3\sqrt{6}
B. 3 + 3\sqrt{6}
C. 3 + \sqrt{6}
D. 3 - \sqrt{6}

16. The table shows the mark obtained by students in a test.
[Table with marks 1,2,3,4,5 and frequencies 2,k,1,1,2]
If the mean mark is 3, find the value of k.
A. 4
B. 1
C. 2
D. 3

17. Simplify: \frac{\log\sqrt{27} - \log\sqrt{8}}{\log 3 - \log 2}
A. \frac{3}{2}
B. -\frac{1}{4}
C. -\frac{3}{2}
D. \frac{1}{4}

18. Given that r = (10\text{ N}, 200°) and n = (16\text{ N}, 020°), find (3r - 2n).
A. (62\text{ N}, 240°)
B. (62\text{ N}, 200°)
C. (62\text{ N}, 280°)
D. (62\text{ N}, 020°)

19. Solve 6\sin 2\theta \tan \theta = 4, where 0^\circ < \theta < 90^\circ

A. 18.43^\circ
B. 30.00^\circ
C. 35.26^\circ
D. 19.47^\circ

20. An exponential sequence (G.P.) is given by 8\sqrt{2}, 16\sqrt{2}, 32\sqrt{2}, .... Find the nth term of the sequence
A. 8\cdot 2^{n-\frac{1}{2}}
B. 2^{n+2}\sqrt{2}
C. 2^{n-\frac{3}{2}}
D. 8n\sqrt{2}

21. If f:x \mapsto 2\tan x and g:x \mapsto \sqrt{x^2+8}, find (g \circ f)(45^\circ)

A. 4
B. 2\sqrt{3}
C. 6
D. 3\sqrt{2}

22. A uniform beam PQ of length 80 cm and weight 60 N rests on a support at X where |PX| = 30 cm. If the body is kept in equilibrium by a mass m kg which is placed at P, calculate the value of m [Take g = 10 m/s^2]
A. 2.0
B. 3.0
C. 2.5
D. 4.0

23. An exponential sequence (G.P.) is given by \frac{9}{2}, \frac{3}{4}, \frac{1}{8}, .... Find its sum to infinity.
A. \frac{25}{2}
B. \frac{41}{5}
C. \frac{13}{12}
D. \frac{63}{4}

24. Adu’s scores in five subjects in an examination are 85, 84, 83, 86 and 87. Calculate the standard deviation.
A. 2.0
B. 1.4
C. 1.8
D. 1.6

25. In how many ways can a committee of 3 women and 2 men be chosen from a group of 7 men and 5 women?
A. 500
B. 350
C. 720
D. 210

26. Evaluate: \int(2x+1)^3 dx
A. 8(2x+1)^2 + k
B. 6(2x+1)^2 + k
C. \frac{1}{8}(2x+1)^4 + k
D. \frac{1}{6}(2x+1)^4 + k

27. If \alpha and \beta are the roots of 7x^2 + 12x - 4 = 0, find the value of \frac{\alpha \beta}{(\alpha + \beta)^2}
A. \frac{7}{36}
B. -\frac{36}{7}
C. \frac{36}{7}
D. -\frac{7}{36}

28. If 3x^2+px+12=0 has equal roots, find the values of p.
A. ±12
B. ±3
C. ±4
D. ±6

29. Given that \frac{3x+4}{(x-2)(x+3)} \equiv \frac{P}{x+3} + \frac{Q}{x-2}, find the value of Q.
A. 2
B. -2
C. 1
D. -1

30. The velocity of a body of mass 4.56 kg increases from (10m/s, 060°) to (50m/s, 060°) in 16 seconds. Calculate the magnitude of force acting on it.
A. 17.1 N
B. 11.4 N
C. 36.5 N
D. 5.7 N

31. A linear transformation on the oxy plane is defined by P:(x,y)→(2x+y,-2y). Find P^2
A. \begin{bmatrix} 4 & 1 \\ 0 & 4 \end{bmatrix}
B. \begin{bmatrix} 4 & 0 \\ 4 & 0 \end{bmatrix}
C. \begin{bmatrix} 4 & 0 \\ 0 & 4 \end{bmatrix}
D. \begin{bmatrix} 4 & 0 \\ 1 & 4 \end{bmatrix}

32. Given that y^2+xy=5, find \frac{dy}{dx}.
A. \frac{y}{2y+x}
B. \frac{-y}{2y+x}
C. \frac{-y}{2y-x}
D. \frac{y}{2y+x}

33. If X and Y are two independent events such that P(X) = \frac{1}{8} and P(X \cup Y) = \frac{5}{8}, find P(Y)
A. \frac{1}{6}
B. \frac{4}{7}
C. \frac{4}{21}
D. \frac{3}{7}

34. A function f is defined by f:x \mapsto \frac{x+2}{x-3}, x \neq 3. Find the inverse of f.
A. \frac{x+3}{x-2}, x \neq 2
B. \frac{x-3}{x+2}, x \neq -2
C. \frac{3x-2}{x+1}, x \neq -1
D. \frac{3x+2}{x-1}, x \neq 1

35. The probabilities that Atta and Tunde will hit a target in a shooting contest are \frac{1}{6} and \frac{1}{9} respectively. Find the probability that only one of them will hit the target.
A. \frac{1}{54}
B. \frac{41}{54}
C. \frac{20}{27}
D. \frac{13}{54}

36. Given that P=\begin{bmatrix} x & 3 \\ 4 & 7 \end{bmatrix} Q=\begin{bmatrix} x & 1 \\ 3 & 2x \end{bmatrix} and the determinant of Q is three more than that of P, find the values of x.
A. -2, \frac{3}{2}
B. 2, \frac{3}{2}
C. -2, -\frac{3}{2}
D. 2, -\frac{3}{2}

37. If m and (m + 4) are the roots of 4x^2-4x-15=0, find the equation whose roots are 2m and (2m + 8).
A. x^2+8x-15=0
B. x^2-2x-15=0
C. x^2-8x-15=0
D. x^2+2x+15=0

38. Find the coefficient of the 6th term in the binomial expansion of (1-\frac{2x}{3})^{10} in ascending powers of x.
A. -\frac{896x^6}{9}
B. -\frac{896x^5}{9}
C. -\frac{896x^5}{27}
D. -\frac{896x^6}{27}

39. In how many ways can four Mathematicians be selected from six?
A. 90
B. 60
C. 15
D. 360

40. If (x-5) is a factor of x^3-4x^2-11x+30, find the remaining factors.
A. (x+3) and (x-2)
B. (x-3) and (x+2)
C. (x-3) and (x-2)
D. (x+3) and (x+2)

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