WAEC Further Mathematics 2022 past questions and answers

A binary operation Δ is defined on the set of real numbers, R, by x Δ y = √(x + y − xy/4), where x, y ∈ R. Find the value of 4 Δ 3.

1. A.16
2. B.8
3. C.4
4. D.2

Simplify: ((3√6 + √54)/(√5(3√5)))⁻¹

1. A.5√3/6
2. B.3√15/6
3. C.5√6/12
4. D.5√3/12

If log₁₀(3x − 1) + log₁₀4 = log₁₀(9x + 2), find the value of x.

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

Simplify: (9 × 3ⁿ⁺¹ − 3ⁿ⁺²)/(3ⁿ⁺¹ − 3ⁿ)

1. A.3
2. B.9
3. C.27
4. D.81

Consider the following statements:
x: All wrestlers are strong
y: Some wrestlers are not weightlifters
Which of the following is a valid conclusion?

1. A.All strong wrestlers are weightlifters
2. B.Some strong wrestlers are not weightlifters
3. C.Some weak wrestlers are weightlifters
4. D.All weightlifters are wrestlers

The functions f: x → 2x² + 3x − 7 and g: x → 5x² + 7x − 6 are defined on the set of real numbers, R. Find the values of x for which 3f(x) = g(x).

1. A.x = −3 or −5
2. B.x = −3 or 5
3. C.x = 3 or −5
4. D.x = 3 or 5

Express 4π/5 radians in degrees.

1. A.288°
2. B.200°
3. C.144°
4. D.120°

Given that (8x + m)/(x² − 3x − 4) ≡ 5/(x + 1) + 3/(x − 4), find the value of m.

1. A.23
2. B.17
3. C.−17
4. D.−23

If x² + y² − 2x − 6y + 5 = 0, evaluate dy/dx when x = 3 and y = 2.

1. A.−2
2. B.2
3. C.4
4. D.−4

Evaluate ∫₀¹ x²(x³ + 2)³ dx.

1. A.56/12
2. B.65/12
3. C.12
4. D.65

Evaluate: ⁴P₂ + ⁴C₂ − ⁴P₃.

1. A.18
2. B.6
3. C.−6
4. D.−18

Find the coefficient of x² in the binomial expansion of (x + 2/x²)⁵.

1. A.10
2. B.40
3. C.32
4. D.80

Given that P = {x : x is a multiple of 5}, Q = {x : x is a multiple of 3} and R = {x : x is an odd number} are subsets of μ = {x : 20 ≤ x ≤ 35}, find (P ∪ Q) ∩ R.

1. A.{20, 21, 25, 30, 33}
2. B.{21, 25, 27, 33, 35}
3. C.{20, 21, 25, 27, 33, 35}
4. D.{21, 25, 27, 30, 33, 35}

A particle moving with a velocity of 5 ms⁻¹ accelerates at 2 ms⁻². Find the distance it covers in 4 seconds.

1. A.16 m
2. B.26 m
3. C.36 m
4. D.46 m

If U_n = kn² + pn, U₁ = −1, U₅ = 15, find the values of k and p.

1. A.k = −1, p = 2
2. B.k = −1, p = −2
3. C.k = 1, p = −2
4. D.k = 1, p = 2

In how many ways can six persons be paired?

1. A.5
2. B.10
3. C.15
4. D.25

Solve: 3^2x−2 − 28(3^x−2) + 3 = 0.

1. A.x = −2 or x = 1
2. B.x = 0 or x = −3
3. C.x = 2 or x = 1
4. D.x = 0 or x = 3

Given that P = (−4, −5) and Q = (2, 3), express PQ in the form (k, θ), where k is the magnitude and θ the bearing.

1. A.(10 units, 063°)
2. B.(9 units, 049°)
3. C.(10 units, 037°)
4. D.(9 units, 027°)

If PQ = −2i + 5j and RQ = −i − 7j, find PR.

1. A.−3i + 12j
2. B.−3i − 12j
3. C.−i + 12j
4. D.i − 12j

The table shows the distribution of the distance (in km) covered by 40 hunters while hunting.
Distance (km): 3, 4, 5, 6, 7, 8
Frequency: 5, 4, x, 9, 2x, 1
If a hunter is selected at random, find the probability that the hunter covered at least 6 km.

1. A.3/5
2. B.2/5
3. C.3/8
4. D.9/40

The table shows the distribution of the distance (in km) covered by 40 hunters while hunting.
Distance (km): 3, 4, 5, 6, 7, 8
Frequency: 5, 4, x, 9, 2x, 1
Find the mode of the distribution

1. A.5
2. B.6
3. C.7
4. D.8

If g(x) = √(1 − x²), find the domain of g(x).

1. A.x < −1 or x > 1
2. B.x ≤ −1 or x ≥ 1
3. C.−1 ≤ x ≤ 1
4. D.−1 < x < 1

Find the coefficient of x³y² in the binomial expansion of (x − 2y)⁵.

1. A.−80
2. B.−10
3. C.40
4. D.80

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.

1. A.−1/2, 8
2. B.1/2, −8
3. C.−1/2, −8
4. D.1/2, 8

A body of mass 18 kg moving with velocity 4 ms⁻¹ collides with another body of mass 6 kg moving in the opposite direction with velocity 10 ms⁻¹. If they stick together after collision, find their common velocity.

1. A.−½ ms⁻¹
2. B.⅓ ms⁻¹
3. C.2 ms⁻¹
4. D.3 ms⁻¹

The mean heights of three groups of students consisting of 20, 16 and 14 students each are 1.67 m, 1.50 m and 1.40 m respectively. Find the mean height of all the students.

1. A.1.63 m
2. B.1.54 m
3. C.1.52 m
4. D.1.42 m

Find, correct to the nearest degree, the acute angle formed by the lines y = 2x + 5 and 2y = x − 6.

1. A.76°
2. B.53°
3. C.37°
4. D.14°

Solve: 4sin²θ + 1 = 2, where 0° < θ < 180°.

1. A.60°, 120°
2. B.30°, 150°
3. C.30°, 120°
4. D.60°, 150°

The probability that a student will graduate from a college is 0.4. If 3 students are selected from the college, what is the probability that at least one student will graduate?

1. A.0.06
2. B.0.22
3. C.0.78
4. D.0.80

The equation of a circle is given as 2x² + 2y² − x − 3y − 41 = 0. Find the coordinates of its centre.

1. A.(−¼, ¾)
2. B.(¼, ¾)
3. C.(−½, 3/2)
4. D.(−½, −3/2)

The gradient of a function at any point (x, y) is 2x − 6. If the function passes through (1, 2), find the function.

1. A.y = x² − 6x − 5
2. B.y = x² − 6x + 5
3. C.y = x² − 6x − 3
4. D.y = x² − 6x + 7

A particle of mass 3 kg moving along a straight line under the action of a F N, covers a line distance, d, at time, t, such that d = t² + 3t. Find the magnitude of F at time t.

1. A.0 N
2. B.2 N
3. C.3(2t + 3) N
4. D.6 N

If α and β are the roots of x² + mx − n = 0, where m and n are constants, form the equation whose roots are 1/α and 1/β.

1. A.mnx² − n²x − m = 0
2. B.mx² − nx + 1 = 0
3. C.nx² − mx + 1 = 0
4. D.nx² − mx − 1 = 0

A particle is acted upon by forces F = (10 N, 060°), P = (15 N, 120°) and Q = (12 N, 200°). Express the force that will keep the particle in equilibrium in the form xi + yj, where x and y are scalars.

1. A.17.55i + 13.78j
2. B.17.55i − 13.78j
3. C.−17.55i + 13.78j
4. D.−17.55i − 13.78j

Evaluate: lim_x→−2 (x³ + 8)/(x + 2)

1. A.8
2. B.2
3. C.4
4. D.12

If f(x − 1) = x³ + 3x² + 4x − 5, find f(2)

1. A.61
2. B.25
3. C.20
4. D.13

The length of the line joining points (x, 4) and (−x, 3) is 7 units. Find the value of x.

1. A.4√3
2. B.2√6
3. C.3√2
4. D.2√3

A binary operation * is defined on the set T = {−2, −1, 1, 2} by p * q = p² + 2pq − q², where p, q ∈ T.
(a) Copy and complete the table (row p, column q; blanks to be filled):
Row −2: q = −2: __, q = −1: 7, q = 1: __, q = 2: −8
Row −1: q = −2: __, q = −1: 2, q = 1: −2, q = 2: __
Row 1: q = −2: −7, q = −1: __, q = 1: __, q = 2: 1
Row 2: q = −2: __, q = −1: −1, q = 1: __, q = 2: __
(b) Using the table in 1(a), find the value of p such that (−2 * p) * 2 = −7.

Solve 2^(2y+1) − 5(2^y) + 2 = 0

Solve 3cos 2x − sin x = 0 for 0° ≤ x ≤ 360°.

The probability that Abiola will be late to office on a given day is 2/5. In a given working week of six days, find, correct to four significant figures, the probability that he will: (a) be late for only 3 days; (b) not be late in the week; (c) be late throughout the six days.

The table shows the scores obtained by a group of artistes in Vocal (X) and Instrumental (Y) music competion.
Vocal (X): 63, 69, 72, 59, 82, 91, 95, 68
Instrumental (Y): 58, 61, 67, 51, 53, 79, 92, 57
Calculate the Spearman's rank correlation coefficient between the scores.

A body of mass 18 kg is suspended by an inextensible string from a rigid support and is pulled by a horizontal force F until the angle of inclination of the string to the vertical is 35°. If the system is in equilibrium, calculate the: (a) value of F; (b) tension in the string. [Take g = 10ms⁻²]

Given that p = (8 N, 030°) and q = (9 N, 150°), find, in component form, the unit vector along (p − q).

Given that ⁿC₄, ⁿC₅ and ⁿC₆ are the first 3 terms of a linear sequence (A. P.), find the: (a) values of n; (b) common differences of the sequence.

A solid rectangular block has a base which measures 3x cm by 2x cm. The height of the block is y cm and its volume is 72 cm³. (a) Express y in terms of x. (b) Find: (i) an expression for the total surface area of the block in terms of x only; (ii) the value of x for which the total surface area has a stationary value.

(a) Find the binomial expansion of (1 + 2x)⁷ and (1 − 2x)⁷.
(b) Using the result in 11(a), find, correct to three decimal places, the value of (1.2)⁷ − (0.8)⁷.

A basket contains 12 fruits: orange, apple and avocado pear, all of the same size. The number of oranges, apples and avocado pear forms three consecutive integers. Two fruits are drawn one after the other without replacement. Calculate the probability that: (a) the first is an orange and the second is an avocado pear; (b) both are of the same fruit; (c) at least one is an apple.

(a) A particle initially at rest moves in a straight line with an acceleration of (10t − 4t²) ms⁻². Find the: (i) velocity of the particle after t seconds; (ii) average acceleration of the particle during the 4^th second.
(b) A load of mass 120 kg is placed on a lift. Calculate the reaction between the floor of the lift and the load when the lift moves upwards: (i) at a constant velocity; (ii) with an acceleration of 3 ms⁻². [Take g = 10ms⁻²]

The vectors 6i + 8j and 8i − 6j are parallel to OP and OQ respectively. If the magnitude of OP and OQ are 80 units and 120 units respectively, express: (a) OP and OQ in terms of i and j; (b) |PQ|, in the form c√k, where c and k are constants.