WAEC Further Mathematics 2015 past questions and answers

Simplify: (1 − 2√5)/(2 + 3√2)

1. A.14(2√2 + 6√5 − 4√10)
2. B.(1/14)(2 − 3√2 − 4√5 − 6√10)
3. C.(1/14)(3√2 + 4√5 − 6√10 − 2)
4. D.14(2 + 3√2 − 6√5 + 4√10)

Solve: 2cos x − 1 = 0, for 0 ≤ x ≤ 2π

1. A.(2π/3, 4π/3)
2. B.(π/6, 5π/6)
3. C.(π/5, 2π/5)
4. D.(π/3, 5π/3)

Solve 4(2^x²) = 8^x.

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

If log₃ x = log₉ 3, find the value of x.

1. A.3²
2. B.3^½
3. C.3^1/3
4. D.2^1/3

Find the third term of (x/2 − 1)⁸ in descending order of x.

1. A.x⁷/8
2. B.7x⁶/16
3. C.7x⁵/4
4. D.(35/8)x⁴

Given that f: x → x² and g: x → x + 3 where x ∈ R find fog(2).

1. A.25
2. B.9
3. C.7
4. D.5

Given that P = (−2 1 / 3 4) and Q = (5 −3 / 2 −1), find PQ − QP. (Matrices written row by row; / separates the rows.)

1. A.(0 0 / 0 0)
2. B.(27 12 / 16 −15)
3. C.(−20 −6 / 12 −8)
4. D.(11 12 / 30 −11)

Which of the following is a factor of the polynomial 6x⁴ + 2x³ + 15x + 5?

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

Given that f: x → (2x − 1)/(x + 2), x ≠ −2, find f⁻¹, the inverse of f

1. A.f⁻¹: x → (1 + 2x)/(2 − x), x ≠ 2
2. B.f⁻¹: x → (1 − 2x)/(x + 2), x ≠ −2
3. C.f⁻¹: x → (1 − 2x)/(x − 2), x ≠ −2
4. D.f⁻¹: x → (1 + 2x)/(x − 2), x ≠ −2

If 36, p, 9/4 and q are consecutive terms of an exponential sequence (G. P), find the sum of p and q.

1. A.9/16
2. B.81/16
3. C.9
4. D.9 9/16

Find the minimum value of y = x² + 6x − 12.

1. A.−21
2. B.−12
3. C.−6
4. D.−3

A line passes through the origin and the point (1¼, 2½). What is the gradient of the line?

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

A line passes through the origin and the point (1¼, 2½). Find the y coordinate of the line when x = 4.

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

In how many ways can a committee of five be selected from eight students if two particular students are to be included?

1. A.20
2. B.28
3. C.54
4. D.58

If x = i − 3j and y = 6i + j, calculate the angle between x and y.

1. A.60°
2. B.75°
3. C.81°
4. D.85°

The gradient of a curve at the point (−2, 0) is 3x² − 4x. Find the equation of the curve.

1. A.y = 6x − 4
2. B.y = 6x² − 4 + 12
3. C.y = x³ − 2x²
4. D.y = x³ − 2x² + 16

If α and β are the roots of x² + x − 2 = 0, find the value of (1/α² + 1/β²)

1. A.5/4
2. B.3/4
3. C.1/4
4. D.−3/4

Given that x² + 4x + k ≡ (x + r)² + 1, find the values of k and r

1. A.k = 5, r = −1
2. B.k = 5, r = 2
3. C.k = 2, r = 5
4. D.k = 1, r = 5

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".

1. A.p ∨ q
2. B.~p ∨ q
3. C.p ∧ ~q
4. D.p ∧ q

Forces of magnitude 8 N and 5 N act on a body as shown in the diagram. Calculate, correct to 2 decimal places, the resultant force acting at O.

1. A.14.06 N
2. B.13.00 N
3. C.9.83 N
4. D.8.26 N

Forces of magnitude 8 N and 5 N act on a body as shown in the diagram. Calculate, correct to 2 decimal places, the angle that the resultant makes with the horizontal.

1. A.80.76°
2. B.75.00°
3. C.71.99°
4. D.15.00°

Given that r = 2i − j, s = 3i + 5j and t = 6i − 2j, find the magnitude of (2r + s − t)

1. A.√15
2. B.4
3. C.√24
4. D.√26

The table shows the distribution of marks scored by students in a test.
Marks: 0, 1, 2, 3, 4, 5
No. of candidates: 6, 4, 8, 10, 9, 3
How many candidates scored above the median mark?

1. A.3
2. B.12
3. C.22
4. D.30

The table shows the distribution of marks scored by students in a test.
Marks: 0, 1, 2, 3, 4, 5
No. of candidates: 6, 4, 8, 10, 9, 3
Find the inter-quartile range of the distribution.

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

A mass of 75 kg 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 a constant velocity. [Take g = 9.8 ms⁻²]

1. A.750 N
2. B.745 N
3. C.735 N
4. D.98 N

Each of the 90 students in a class speak at least Igbo and 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.

1. A.28/45
2. B.4/9
3. C.8/45
4. D.1/9

If |1 + 2x, −1 / 6, 3 − x| = −3, find the values of x. (Determinant written row by row; / separates the rows.)

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

Find ∫ (x³ + 5x + 1)/x³ dx.

1. A.x² + 10x + c
2. B.x + (5/3)x³ + x⁴ + c
3. C.x − 5x² − 2x² + c
4. D.x − 5/x − 1/(2x²) + c

A particle starts from rest and moves in a straight line such that its acceleration after t seconds is given by a = (3t − 2) ms⁻². Find the other time when the velocity would be zero.

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

A particle starts from rest and moves in a straight line such that its acceleration after t seconds is given by a = (3t − 2) ms⁻². Find the distance covered after 3 seconds

1. A.10 m
2. B.9 m
3. C.13/3 m
4. D.9/2 m

Given that y = 4 − 9x and Δx = 0.1, calculate Δy.

1. A.9.0
2. B.0.9
3. C.−0.3
4. D.−0.9

Four fair coins are tossed once. Calculate the probability of obtaining equal number of heads and tails.

1. A.1/4
2. B.3/8
3. C.1/2
4. D.15/16

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.

1. A.24
2. B.23
3. C.21
4. D.19

Simplify: ⁿC_r ÷ ⁿC_r−1

1. A.n(n − r)/r
2. B.n/(r(n − r))
3. C.1/(r(n − r))
4. D.(n − r + 1)/r

If 2 sin² θ = 1 + Cos θ, 0° ≤ θ ≤ 90°, find the value of θ.

1. A.90°
2. B.60°
3. C.45°
4. D.30°

A 24 N force acts on a body such that it changes its velocity from 5 ms⁻¹ to 9 ms⁻¹ in 2 seconds. If the body is travelling in a straight line, calculate the distance covered during the period.

1. A.22 m
2. B.18 m
3. C.14 m
4. D.10 m

The sum, S_n, of a sequence is given by S_n = 2n² − 5. Find the 6th term.

1. A.112
2. B.67
3. C.45
4. D.22

Forces F₁ (8 N, 030°) and F₂ (10 N, 150°) act on a particle. Find the horizontal component of the resultant force.

1. A.1.7 N
2. B.4.5 N
3. C.9.0 N
4. D.13.0 N

A binary operation Δ is defined on the set of real numbers, R, by a Δ b = a³ − b³. Without using calculator, find the value of (√3 + √2) Δ (√3 − √2), leaving the answer in surd form.

Points (2, 1) and (6, 7) are opposite verticals of a square which is inscribed in a circle. Find the: (a) centre of the circle; (b) equation of the circle.

If f″(x) = 2, f′(1) = 0 and f(0) = −8, find f(x).

Solve: tan(2x − 15)° − 1 = 0, for values of x such that 0° ≤ x ≤ 360°.

A car moving with an initial velocity, u, travels in a straight line with a constant acceleration of 3 ms⁻² until it attains a velocity of 33 ms⁻¹ after 6 seconds. Calculate the distance travelled by the car.

Five finalists in a beauty pageant were ranked by two judges X and Y as shown in the table.
Finalist: Anne, Linda, Susan, Rose, Erica
X: 1, 4, 3, 5, 2
Y: 3, 2, 4, 5, 1
Calculate the Spearman's rank correlation co-efficient.

There are 8 boys and 6 girls in a class. If two students are selected at random from the class, find the probability that they are of: (a) The same sex; (b) Different sexes.

Forces of magnitude 3N, 4 N and 2 N act along the vectors j, −i + j and i + j respectively. Calculate, correct to one decimal place, the magnitude of the resultant of the forces.

(a) The functions f: x → x² + 1 and g: x → 5 − 3x are defined on the set of real numbers, R. (i) State the domain of f⁻¹, the inverse of f. (ii) Find g⁻¹(2).
(b) Evaluate ∫₀² (x + 3)/√(x² + 6x + 9) dx

(a) If (√5 + 4)/(3 − 2√5) − (2 + √5)/(4 − 2√5) = a + b√5, find the values of a and b.
(b)(i) Evaluate: |2 −1 2 / 1 3 4 / 1 2 1| (rows separated by /). (ii) Using the result in 10(b)(i), find, correct to two decimal places the value of x in the system of equations: 2x − y + 2z + 5 = 0, x + 3y + 4z − 1 = 0, x + 2y + z + 2 = 0.

(a)(i) Write down the Binomial expansion of (1 + x)⁴. (ii) Use the result in 11(a)(i) to evaluate, correct to three decimal places (5/4)⁴.
(b) The first, second and fifth terms of a linear sequence (A. P.) are three consecutive terms of an exponential sequence (G. P.). If the first term of the linear sequence is 7, find its common difference.

The table shows the distribution of heights of a group of people.
Height (m): 0.4 − 0.5, 0.6 − 0.9, 1.0 − 1.2, 1.3 − 1.4, 1.5 − 1.7
No. of People: 2, 8, 12, 6, 6
(a) Draw a histogram to illustrate the distribution. (b) Using an assumed mean of 1.1 m, find, correct to one decimal place, the mean height of the group.

(a) Edem and his wife were invited to a dinner by a family of 5. They all sat in such a way that Edem sat next to his wife. Find the number of ways of seating them in a row.
(b) A bag contains 4 red and 5 black identical balls. If 5 balls are selected at random from the bag, one after the other with replacement, find the probability that: (i) a red ball was picked 3 times; (ii) a black ball was picked at most 2 times.

(a) Given that AB = (4, 5) and BC = (−3, 5) (column vectors), find the: (i) angle between the vectors AB and BC;
(b) P, Q, R and M are points in the OXY plane. If PQ = 2i + 8j, PR = 11i − 12j and M divides QR internally in the ratio 3 : 7, find PM.

(a) A bucket full of water with a mass of 8 kg is pulled out of a well with a light inextensible rope. Find its acceleration when the tension in the rope is 150 N. [Take g = 10 ms⁻²]
(b) A mass of 12 kg is acted upon by a force F, changing its speed from 15 ms⁻¹ to 25 ms⁻¹ after covering a distance of 50 m. Find the: (i) value of F; (ii) distance covered when its speed is 35 ms⁻¹