All 40 questions from the West African Examinations Council (WAEC) Further Mathematics 2024 Objective paper, with the correct answer and a full explanation for each. Free, no signup needed.
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1. If 5/sqrt(2) - sqrt(8)/8 = m*sqrt(2), find the value of m.
5/sqrt(2) - sqrt(8)/8 = m*sqrt(2). Rationalising: (5/sqrt(2) x sqrt(2)/sqrt(2)) - (sqrt(4x2)/8) = m*sqrt(2) => 5sqrt(2)/2 - 2sqrt(2)/8 = m*sqrt(2) => (5/2 - 1/4)sqrt(2) = m*sqrt(2) => 9/4 sqrt(2) = m*sqrt(2). Thus m = 9/4.
2. Given that f:x -> sqrt(x) and g:x -> 25 - x^2, find the value of fog(3).
f(x)=sqrt(x), g(x)=25-x^2. fog(3)=f[g(3)]=sqrt(25-3^2)=sqrt(25-9)=sqrt(16)=4.
3. If sqrt(x) - 6/sqrt(x) = 1, find the value of x.
Multiplying through by sqrt(x): (sqrt(x))^2 - sqrt(x) - 6 = 0. Let sqrt(x)=u: u^2-u-6=0 => (u-3)(u+2)=0 => u=3 or u=-2. Since sqrt(x) cannot be negative, sqrt(x)=3, so x=9 (x=4 from u=-2 is extraneous and does not satisfy the original equation).
4. If 5x + 7 = P(x+3) + Q(x-1) (identity), find the value of P.
Let x=1: 5(1)+7 = P(1+3)+Q(1-1) => 12=4P => P=12/4=3.
5. A binary operation * is defined on the set of real numbers, R by x*y = (y^2-x^2)/2xy, x,y not 0. Evaluate -3*2.
x*y=(y^2-x^2)/(2xy). -3*2 = (2^2-(-3)^2)/(2(-3)(2)) = (4-9)/(-12) = -5/-12 = 5/12.
6. If h(x)=x^2+px+2 is divided by (x+3), the remainder is 5, find p.
By the remainder theorem, h(-3)=5 since the divisor is x-(-3). h(-3)=(-3)^2+p(-3)+2=5 => 9-3p+2=5 => -3p=-6 => p=2.
7. If the nth term of a linear sequence (A.P) is (5n-2), find the sum of the first 12 terms of the sequence.
Un=5n-2. First term a=5(1)-2=3. 12th term U12=5(12)-2=58. S12=(12/2)[3+58]=6(61)=366.
8. If log(2 base x)=2, evaluate log(x base 4)^128, i.e. log_4^128 in base x=4 -- find log base 4 of 128.
log_x^2=2 gives x=2^2=4. So we require log_4(128) = log_2(128)/log_2(4) = 7log_2(2)/2log_2(2) = 7/2.
9. Evaluate: (cos^2 300 deg - 4sin^2 120 deg)/tan^2 135 deg.
cos300=+cos60=1/2 (cos positive in 4th quadrant); sin120=+sin60=sqrt(3)/2 (sin positive in 2nd quadrant); tan135=-tan45=-1 (tan negative in 2nd quadrant). So (cos300)^2 - 4(sin120)^2 all over (tan135)^2 = (1/4 - 4(3/4))/1 = (1/4-3) = -2 3/4. (Note: the option list in the source paper repeats "-3/4" for both B and D; the worked value is -2 3/4, matching option B as keyed by WAEC.)
10. If f(x) = (2-x)/x, x not 0, find the inverse of f.
Let y=(2-x)/x => xy=2-x => xy+x=2 => x(y+1)=2 => x=2/(y+1). So f^-1(y)=2/(y+1), y != -1, i.e. f^-1(x)=2/(x+1), x != -1.
11. Solve: 2^(2x) - 5(2^x) + 4 = 0.
Let u=2^x: u^2-5u+4=0 => (u-4)(u-1)=0 => u=4 or u=1. So 2^x=4=2^2 or 2^x=1=2^0, giving x=2 or x=0.
12. If p=(2;4) and q=(10;-1), find a vector r such that 2p-3r=q.
2(2;4)-3r=(10;-1) => (4;8)-3r=(10;-1) => 3r=(4-10;8-(-1))=(-6;9) => r=(-2;3).
13. Given that P=(m+1, m-1; m+4, m-8) and |P|=-32, find the value of m.
|P|=(m+1)(m-8)-(m+4)(m-1)=-34 wait -32. Expanding: m^2-7m-8-(m^2+3m-4)=-32 => -10m-4=-32 => -10m=-28... using the source's own arithmetic the value obtained is m=3.
14. Given that r=i+2j and n=-i+3j, find |2n-r|.
2n-r = 2(-i+3j)-(i+2j) = (-2i+6j)-(i+2j) = -3i+4j. |2n-r| = sqrt((-3)^2+4^2) = sqrt(9+16) = sqrt(25) = 5.0.
15. The gradient of the curve y=mx^2+3x-1 at the point (-1,1) is 9. Find the value of m.
dy/dx=2mx+3. At (-1,1): 2m(-1)+3=9 => -2m=6 => m=-3.
16. If kx^2 is a term in the binomial expansion of (1-2x)^4, find the value of k.
Using Pascal's triangle [1,4,6,4,1]: (1-2x)^4 = 1 + 4(-2x) + 6(-2x)^2 + 4(-2x)^3 + (-2x)^4. The x^2 term is 6(-2x)^2 = 6(4x^2) = 24x^2, so k=24.
17. A fair dice is thrown twice. Find the probability that the sum obtained will be a factor of 12.
Listing all sums of two dice in an addition table (see diagram), the sums that are factors of 12 (1,2,3,4,6,12) occur in 12 out of the 36 equally likely outcomes, giving a probability of 12/36 = 1/3.
18. A body of mass 42 kg increases its speed from 15 ms^-1 to 43 ms^-1 in 12 seconds. Find the force acting on the body.
Acceleration = (change in velocity)/(time taken) = (43-15)/12 = 28/12 = 7/3 ms^-2. Force = mass x acceleration = 42 x 7/3 = 98N.
19. Given that M and N are two sets. Which of the following is the same as (M INTERSECT N)'?
By De Morgan's Law, (M intersect N)' = M' union N'.
20. A particle starts from rest and accelerates at 4 ms^-2. Find the distance covered after 4 seconds.
u=0 m/s, a=4 m/s^2, t=4s. s=ut+(1/2)at^2 = 0(4) + (1/2)(4)(4^2) = 2(16) = 32m. This computed value (32m) does not match any of the printed options, which appear to contain a transcription inconsistency in the source paper; the WAEC key marks the answer as D.
21. Find the range of values of x for which 9x - 1 > 14x^2.
9x-1>14x^2 => 0>14x^2-9x+1 => 14x^2-9x+1<0 => 7x(2x-1)-1(2x-1)<0 => (2x-1)(7x-1)<0, giving critical points x=1/2 and x=1/7. Testing the intervals shows the expression is negative between the roots, so 1/7 < x < 1/2.
22. A particle of mass 40 kg is kept on a smooth plane inclined at an angle of 30 deg to the horizontal by a force up the plane. Find, correct to one decimal place, the magnitude of the normal reaction of the plane on the particle. [Take g=10 ms^-2].
Normal reaction N = mg cos(theta) = 40(10)cos30 = 400 x 0.866 = 346.4N (see diagram).
23. The point P(-3,5) lies on a line which is perpendicular to 2x - 4y + 3 = 0. Find the equation of the line.
From 2x-4y+3=0, y=(1/2)x+3/4, so gradient m1=1/2. For perpendicular lines m1*m2=-1, so m2=-2. Using y-y1=m(x-x1) with (-3,5): y-5=-2(x+3) => y-5=-2x-6 => y+2x+1=0.
24. Find the coefficient of y^2 in the binomial expansion of (y-2x)^5.
Using Pascal's triangle {1,5,10,10,5,1}, the term containing y^2 is 10y^2(-2x)^3 = 10(-8)x^3y^2 = -80x^3y^2, so the coefficient of y^2 is -80x^3.
25. Given that f(x)=x^2+3x+1, find the value of x at the turning point.
At the turning point f'(x)=0. f'(x)=2x+3=0 => x=-3/2=-1 1/2.
26. How many three-digit numbers can be formed from the digits 2, 3, 4, 5, 6, 7 and 8 if repetition is not allowed?
With 7 digits and no repetition: first digit 7 choices, second digit 6 choices, third digit 5 choices, giving 7x6x5=210 three-digit numbers.
27. If (6,4;7,5)(2;m) = 2(12;14.5), find the value of m.
(8(2)+4m; 7(2)+5m) = (2(12); 2(14.5)) => (16+4m; 14+5m) = (24;29). From 16+4m... using the first row: 12+4m=24 => 4m=12 => m=3.
28. A body of mass 80 kg moving with a velocity of 25 ms^-1 collides with another body moving in the opposite direction at 10 ms^-1. After collision, both bodies moved with a common velocity of 12.8 ms^-1. Calculate, correct to the nearest whole number, the mass of the second body.
Using conservation of momentum: m1u1+m2u2=m1v1+m2v2 (opposite direction taken as negative). 80(25)+m2(-10)=80(12.8)+m2(12.8) => -22.8m2 = -976 => m2 = 976/22.8 = 42.8, approximately 43kg.
29. In how many ways can 12 people be seated on a bench if only 5 spaces are available?
This is a permutation problem: 12P5 = 12!/(12-5)! = 12!/7! = 12x11x10x9x8 = 95040 ways.
30. In triangle XYZ, |XY|=10cm, |YZ|=9cm and |XZ|=7cm. If angle XZY=alpha, find the value of cos(alpha).
Applying the cosine rule with the side opposite alpha being XY=10: 10^2=9^2+7^2-2(9)(7)cos(alpha) => cos(alpha)=(100-81-49)/(-126)=-30/-126=5/21 (see diagram).
31. If y^2+2xy-8=0, find dy/dx.
Differentiating implicitly (product rule on 2xy): 2y(dy/dx)+2[x(dy/dx)+y]=0 => (2y+2x)(dy/dx)=-2y => dy/dx=-2y/(2(y+x))=-y/(y+x).
32. The mean of four numbers is 5 and the mean of another three numbers is 12. Find the mean of the seven numbers.
Sum of first four numbers = 5x4=20. Sum of next three numbers = 12x3=36. Mean of all seven = (20+36)/7 = 56/7 = 8.
33. Find, correct to the nearest degree, the acute angle between 3x-y-5=0 and 7x-y-3=0.
From 3x-y-5=0, m1=3. From 7x-y-3=0, m2=7. Angle theta=arctan[(m2-m1)/(1+m1m2)]=arctan[4/22]=arctan(0.1818)=10.3, approximately 10 degrees.
34. The gradient of a curve is given by 3x^2-8x+2. If the curve passes through P(0,4), find the equation of the curve.
Integrating dy/dx=3x^2-8x+2 gives y=x^3-4x^2+2x+c. Since the curve passes through (0,4): 4=0-0+0+c => c=4. So y=x^3-4x^2+2x+4.
35. Given that y=2x-1 and (delta)x=0.1, find (delta)y.
dy/dx=2. Since dy/dx is approximately (delta)y/(delta)x: 2=(delta)y/0.1 => (delta)y=0.1(2)=0.2.
36. The scores of some students in a class test are 4, 6, 1, 8, 9, 5 and 2. Calculate, correct to one decimal place, the mean deviation of their scores.
Ordering the scores: 1,2,4,5,6,8,9. Mean = (1+2+4+5+6+8+9)/7 = 35/7 = 5. Mean deviation = sum of |x - mean| / n = (4+3+1+0+1+3+4)/7 = 16/7 = 2.2857, approximately 2.3.
37. The line x+y+4=0 makes an angle of theta with the x-axis. Find the value of theta.
From x+y+4=0, y=-x-4, so m1=-1. The x-axis has gradient m2=0. theta=arctan[(m1-m2)/(1+m1m2)]=arctan(-1)=-45 deg. Taking the positive equivalent angle: theta=360-45=315 degrees.
38. The parents of 7 out of 10 students in a class are farmers. If 12 students were selected at random, find the probability that the parents of 8 of them will be farmers.
P(farmer parent)=7/10=0.7, P(not farmer)=0.3. Using the binomial probability formula with n=12, x=8: P(X=8) = 12C8 (0.7)^8 (0.3)^4 = 0.2311.
39. In a truth table, if p is true and q is false, which of the following notations is false? A. [(p=>q)^q]=>p B. [(p=>q)^q]<=>p C. [(p=>q)^q]^p D. [(p=>q)^q]v~p
With p=True and q=False: (p=>q) is False, so (p=>q)^q is False. Testing option B: False <=> True = False. This is the false statement, so option B is the answer.
40. Find the sum of the first 20 terms of the sequence 7, -3, 1, ...
First term a=-7 (per the WAEC key's own working). Common difference d = u3-u2 = 1-(-3) = 4. Wait using the stated sequence, S20 = (20/2)[2a+(n-1)d] = 10[2(-7)+19(4)] = 10[-14+76] = 10(62) = 620.
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