WAEC Further Mathematics 2023 past questions and answers

Express 3/(3 − √6) in the form x + m√y.

1. A.3 + 3√6
2. B.3 + √6
3. C.3 − √6
4. D.3 − 3√6

If (1/9)^2x−1 = (1/81)^2−3x, find the value of x.

1. A.3/4
2. B.5/8
3. C.−5/8
4. D.−3/4

If (x − 5) is a factor of x³ − 4x² − 11x + 30, find the remaining factors.

1. A.(x − 3) and (x + 2)
2. B.(x − 3) and (x − 2)
3. C.(x + 3) and (x − 2)
4. D.(x + 3) and (x + 2)

Consider the statements:
x: The school bus arrived late
y: The students walked down to school.
Which of the following can be represented by y ⇒ x?

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

Evaluate: ∫₀¹ x(x² − 2)² dx.

1. A.1/7
2. B.6/7
3. C.1⅙
4. D.3⅙

In how many ways can a committee of 3 women and 2 men be chosen from a group of 7 men and 5 women?

1. A.720
2. B.500
3. C.350
4. D.210

If α and β are the roots of 7x² + 12x − 4 = 0, find the value of αβ/(α + β)².

1. A.36/7
2. B.7/36
3. C.−7/36
4. D.−36/7

If f: x → 2 tan x and g: x → √(x² + 8), find (g o f)(45°)

1. A.6
2. B.3√2
3. C.4
4. D.2√3

An exponential sequence (G.P.) is given by 9/2, 3/4, 1/8, …. Find its sum to infinity

1. A.4⅕
2. B.5⅖
3. C.6¾
4. D.13½

Given that M is the midpoint of T(2, 4) and Q(−8, 6), find the length MQ.

1. A.√24 units
2. B.√26 units
3. C.√28 units
4. D.√30 units

Given that y² + xy = 5, find dy/dx.

1. A.−y/(2y − x)
2. B.−y/(2y + x)
3. C.y/(2y − x)
4. D.y/(2y + x)

Given that P = {x : 2 ≤ x ≤ 8} and Q = {x : 4 < x ≤ 12} are subsets of the universal set μ = {x : x ∈ R}, find (P ∩ Q′).

1. A.{x : 4 ≤ x ≤ 8}
2. B.{x : 2 ≤ x ≤ 4}
3. C.{x : 4 < x < 8}
4. D.{x : 2 < x ≤ 4}

The probabilities that Atta and Tunde will hit a target in a shooting contest are 1/6 and 1/9 respectively. Find the probability that only one of them will hit the target.

1. A.1/54
2. B.13/54
3. C.20/27
4. D.41/54

The table shows the marks obtained by students in a test.
Marks: 1, 2, 3, 4, 5
Frequency: 2, k, 1, 1, 2
If the mean mark is 3, find the value of k.

1. A.1
2. B.2
3. C.3
4. D.4

If 3x² + px + 12 = 0 has equal roots, find the values of p.

1. A.±3
2. B.±4
3. C.±6
4. D.±12

Simplify: (log √27 − log √8)/(log 3 − log 2).

1. A.−1/4
2. B.−3/2
3. C.1/4
4. D.3/2

A particle began to move at 27 ms⁻¹ along a straight line with constant retardation of 9 ms⁻². Calculate the time it took the particle to come to a stop.

1. A.4 sec.
2. B.3 sec.
3. C.2 sec.
4. D.1 sec.

In how many ways can four Mathematicians be selected from six?

1. A.15
2. B.60
3. C.90
4. D.360

A linear transformation on the oxy plane is defined by P: (x, y) → (2x + y, −2y). Find P². (Matrices written row by row; / separates the rows.)

1. A.(4 1 / 0 4)
2. B.(4 0 / 1 4)
3. C.(4 0 / 0 4)
4. D.(4 4 / 0 0)

The velocity of a body of mass 4.56 kg increases from (10 ms⁻¹, 060°) to (50 ms⁻¹, 060°) in 16 seconds. Calculate the magnitude of the force acting on it.

1. A.36.5 N
2. B.17.1 N
3. C.11.4 N
4. D.5.7 N

Find the fifth term in the binomial expansion of (q + x)⁷.

1. A.21q⁴x³
2. B.35q⁵x²
3. C.35q³x⁴
4. D.21q²x²

Given that P = (x 4 / 3 7), Q = (x 3 / 1 2x) and the determinant of Q is three more than that of P, find the values of x. (Matrices written row by row; / separates the rows.)

1. A.−2, −3/2
2. B.−2, 3/2
3. C.2, −3/2
4. D.2, 3/2

Solve 6 sin 2θ tan θ = 4, where 0° < θ < 90°.

1. A.35.26°
2. B.30.00°
3. C.19.47°
4. D.18.43°

Calculate, correct to one decimal place, the angle between 5i + 12j and −2i + 3j.

1. A.54.8°
2. B.56.3°
3. C.66.4°
4. D.76.3°

The distance S metres moved by a body in t seconds is given by S = 5t³ − (19/2)t² + 6t − 4. Calculate the acceleration of the body after 2 seconds.

1. A.19 ms⁻²
2. B.21 ms⁻²
3. C.31 ms⁻²
4. D.41 ms⁻²

Find the equation of the normal to the curve y = 3x² + 2 at point (1, 5).

1. A.6y − x − 29 = 0
2. B.6y + x − 31 = 0
3. C.y − 6x + 1 = 0
4. D.y − 6x − 1 = 0

Adu's scores in five subjects in an examination are 85, 84, 83, 86 and 87. Calculate the standard deviation.

1. A.1.4
2. B.1.6
3. C.1.8
4. D.2.0

Differentiate f(x) = 1/(1 − x²)⁵ with respect to x.

1. A.10x/(1 − x²)⁶
2. B.5x/(1 − x²)⁶
3. C.−5x/(1 − x²)⁶
4. D.−10x/(1 − x²)⁶

If X and Y are two independent events such that P(X) = 1/8 and P(X ∪ Y) = 5/8, find P(Y).

1. A.4/7
2. B.3/7
3. C.4/21
4. D.1/6

The table shows the operation * on the set {x, y, z, w}.
Row x: x * x = y, x * y = z, x * z = x, x * w = w
Row y: y * x = z, y * y = w, y * z = y, y * w = x
Row z: z * x = x, z * y = y, z * z = z, z * w = w
Row w: w * x = w, w * y = x, w * z = w, w * w = z
Find the identity element.

1. A.x
2. B.y
3. C.z
4. D.w

Find the radius of the circle 2x² + 2y² − 4x + 5y + 1 = 0

1. A.√5/6
2. B.5/6
3. C.33/4
4. D.√33/4

Evaluate: ∫(2x + 1)³ dx.

1. A.8(2x + 1)² + k
2. B.6(2x + 1)² + k
3. C.⅙(2x + 1)⁴ + k
4. D.⅛(2x + 1)⁴ + k

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 ms⁻²]

1. A.2.0
2. B.2.5
3. C.3.0
4. D.4.0

If m and (m + 4) are the roots of 4x² − 4x − 15 = 0, find the equation whose roots are 2m and (2m + 8).

1. A.x² + 8x + 15 = 0
2. B.x² − 8x − 15 = 0
3. C.x² + 2x + 15 = 0
4. D.x² − 2x − 15 = 0

Find the coefficient of the 6^th term in the binomial expansion of (1 − 2x/3)¹⁰ in ascending powers of x.

1. A.−896x⁶/9
2. B.−896x⁶/27
3. C.−896x⁵/9
4. D.−896x⁵/27

An exponential sequence (G.P.) is given by 8√2, 16√2, 32√2, …. Find the n^th term of the sequence

1. A.8n√2
2. B.8√2ⁿ
3. C.√(2^{(n + 3)})
4. D.2^{(n + 2)}√2

Given that r = (10 N, 200°) and n = (16 N, 020°), find (3r − 2n).

1. A.(62 N, 200°)
2. B.(62 N, 020°)
3. C.(62 N, 240°)
4. D.(62 N, 280°)

Given that sin x = 4/5 and cos y = 12/13, where x is an obtuse angle and y is an acute angle, find the value of sin (x − y)

1. A.16/65
2. B.48/65
3. C.56/65
4. D.63/65

If ⁹C_x = 4[⁷C_x−1], find the values of x.

The volume of a cube is increasing at the rate of 3½ cm³s⁻¹. Find the rate of change of the side of the base when its length is 6 cm.

The inverse of a function f is given by f⁻¹(x) = (5x − 6)/(4 − x), x ≠ 4. Find the: (a) function, f(x) (b) value of x for which f(x) = 5.

The first term of an Arithmetic Progression is −8, the last term is 52 and the sum of terms is 286. Find the: (a) number of terms in the series; (b) common difference.

The table shows the distribution of the heights (cm) of 60 seedlings in a vegetable garden.
Heights (cm): 0.1 − 0.3, 0.4 − 0.6, 0.7 − 0.9, 1.0 − 1.4, 1.5 − 1.9, 2.0 − 2.2, 2.3 − 2.5
Frequency: 6, 9, 12, 15, 3, 6, 9
(a) Draw a histogram for the distribution. (b) Use the histogram to estimate the modal height of the seedlings.

There are 6 boys and 8 girls in a class. If five students are selected from the class, find the probability that more girls than boys are selected.

A bus travels with a velocity of 6 ms⁻¹. It then accelerates uniformly and travels a distance of 70 m. If the final velocity is 20 ms⁻¹, find, correct to one decimal place, the: (a) acceleration; (b) time to travel this distance.

P is the mid-point of NO and equidistant from MN and MO. If MN = 8i + 3j and MO = 14i − 5j; find MP. (MN, MO and MP are vectors.)

(a) Find the derivative of 4x − 7/x² with respect to x, from first principle.
(b) Given that tan P = 3/(x − 1) and tan Q = 2/(x − 1), find tan (P − Q).

(a) A quadratic polynomial, g(x) has (2x + 1) as a factor. If g(x) is divided by (x + 1) and (x − 2), the remainders are −6 and −5 respectively. Find: (i) g(x); (ii) the zeros of g(x).
(b) Find the third term when (x/2 − 1)⁸ is expanded in descending powers of x.

(a) Express (8x² + 8x + 9)/((x − 1)(2x + 3)²) in partial fractions.
(b) The coordinates of the centre and circumference of a circle are (−2, 5) and 6π units respectively. Find the equation of the circle.

The table shows the distribution of marks scored by some candidates in an examination
Marks: 11 − 20, 21 − 30, 31 − 40, 41 − 50, 51 − 60, 61 − 70, 71 − 80, 81 − 90, 91 − 100
Number of candidates: 5, 39, 14, 40, 57, 25, 11, 8, 1
(a) Construct a cumulative frequency table for the distribution. (b) Draw a cumulative frequency curve for the distribution. (c) Use the curve to estimate the: (i) number of candidates who scored marks between 24 and 58; (ii) lowest mark for distinction, if 12% of the candidates passed with distinction.

(a) A bag contains 16 identical balls of which 4 are green. A boy picks a ball at random from the bag and replaces it. If this is repeated 5 times, what is the probability that he: (i) did not pick a green ball; (ii) picked a green ball at least three times?
(b) The deviations from a mean of values from a set of data are −2, (m − 1), (m² + 1), −1, 2, (2m − 1) and −2. Find the possible values of m.

A particle of mass 2 kg moves under the action of a constant force, F N, with an initial velocity (3i + 2j) ms⁻¹ and a velocity of (15i − 4j) ms⁻¹ after 4 seconds. Find the: (a) acceleration of the particle; (b) magnitude of the force F; (c) magnitude of the velocity of the particle after 8 seconds, correct to three decimal places.

(a) A see-saw pivoted at the middle is kept in balance by weights of Richard, John and Philip such that only Richard whose mass is 60 kg sits on one side. If they sit at distances 2m, 3m and 4m respectively from the pivot and Philip is 15 kg, find the mass of John.
(b) A body of mass 12 kg rests on a rough plane inclined at an angle of 30° to the horizontal. The coeffient of friction between the body and the plane is 2/3. A force of magnitude P Newtons acts on the body along the inclined plane. Find the value of P, if the body is at the point of moving: (i) down the plane; (ii) up the plane.