WAEC Further Mathematics 2014 past questions and answers

If 1/5^−y = 25(5^{4 − 2y}), find the value of y.

1. A.4
2. B.2
3. C.−4
4. D.−5

Simplify (1 − sin θ)(1 + sin θ).

1. A.sin²θ
2. B.sec²θ
3. C.tan²θ
4. D.cos²θ

Given that the lines 3x + 4y + 6 = 0 and 4x − by + 3 = 0 are perpendicular, find the value of b.

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

Given that x * y = (x + y)/2, x o y = x²/y and (3 * b) o 48 = 1/3, find b, where b > 0.

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

If f(x) = 3x³ + 8x² + 6x + k, and f(2) = 1, find the value of k.

1. A.−67
2. B.−61
3. C.61
4. D.67

If 8^x ÷ (1/4)^y = 1 and log₂ (x − 2y) = 1, find the value of (x − y).

1. A.5/4
2. B.3/5
3. C.1
4. D.2/3

Simplify (1 + √8)/(3 − √2).

1. A.7 + √2
2. B.7 + 7√2
3. C.1 − 7√2
4. D.1 + √2

Using the binomial expansion (1 + x)⁶ = 1 + 6x + 15x² + 20x³ + 15x⁴ + 6x⁵ + x⁶, find, correct to three decimal places, the value of (1.98)⁶.

1. A.64.245
2. B.61.255
3. C.60.255
4. D.60.245

If (x + 2) and (3x − 1) are factors of 6x³ + x² − 19x + 6, find the third factor.

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

If 2, (k + 1), 8, … form an exponential sequence (GP), find the values of k.

1. A.−3 and 5
2. B.5 and −5
3. C.3 and −3
4. D.−5 and 3

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 2/3, find the value of k.

1. A.5
2. B.6
3. C.8
4. D.10

If T = (−2 −5 / 3 8), find T⁻¹, the inverse of T. (Matrices written row by row; / separates the rows.)

1. A.(−8 −5 / 3 2)
2. B.(−8 −5 / 3 −2)
3. C.(−8 −5 / −3 2)
4. D.(−8 −5 / −3 −2)

A function is defined by h: x → 2 − 1/(2x − 3), x ≠ 3/2. Find h⁻¹(x), the inverse of h.

1. A.h⁻¹(x) = (3x − 4)/(2x − 7), x ≠ 7/2
2. B.h⁻¹(x) = (3x − 7)/(2x − 4), x ≠ 2
3. C.h⁻¹(x) = (2x − 7)/(4x − 3), x ≠ 3/4
4. D.h⁻¹(x) = (4x − 7)/(2x − 4), x ≠ 2

A function is defined by h: x → 2 − 1/(2x − 3), x ≠ 3/2. Find h⁻¹(½).

1. A.6
2. B.11/6
3. C.11/4
4. D.5/3

The radius of a sphere is increasing at a rate of 3 cm s⁻¹, find the rate of increase in the surface area, when the radius is 2cm.

1. A.8π cm²s⁻¹
2. B.16π cm²s⁻¹
3. C.24π cm²s⁻¹
4. D.48π cm²s⁻¹

The table shows the distribution of ages of members of a cultural Troupe.
Age (years): 10 − 14, 15 − 19, 20 − 24, 25 − 29, 30 − 34
Frequency: 6, 8, 14, 10, 12
What is the class-mark of the median class?

1. A.19
2. B.22
3. C.24
4. D.32

The table shows the distribution of ages of members of a cultural Troupe.
Age (years): 10 − 14, 15 − 19, 20 − 24, 25 − 29, 30 − 34
Frequency: 6, 8, 14, 10, 12
In which group is the upper quartile?

1. A.(15 − 19)
2. B.(20 − 24)
3. C.(25 − 29)
4. D.(30 − 34)

The table shows the distribution of ages of members of a cultural Troupe.
Age (years): 10 − 14, 15 − 19, 20 − 24, 25 − 29, 30 − 34
Frequency: 6, 8, 14, 10, 12
Find the mean of the distribution.

1. A.23.4
2. B.23.6
3. C.24.3
4. D.24.6

If Px² + (P + 1)x + P = 0 has equal roots, find the values of P.

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

Integrate (x − 1/x)² with respect to x.

1. A.(1/3)(x − 1/x)³ + C
2. B.x³/3 − x − 1/x³ + C
3. C.x³/3 − 2x + 1/x³ + C
4. D.x³/3 − 2x − 1/x + C

Given that AB = (4, 3) and AC = (2, −3) (column vectors), find |BC|.

1. A.4√2
2. B.6√2
3. C.2√10
4. D.4√10

Find the angle between (5i + 3j) and (3i − 5j).

1. A.180°
2. B.90°
3. C.45°
4. D.0°

Find the coefficient of x³ in the binomial expansion of (3x + 4)⁴ in ascending powers of x.

1. A.432
2. B.194
3. C.144
4. D.108

If a fair coin is tossed four times, what is the probability of obtaining at least one head?

1. A.½
2. B.¼
3. C.13/16
4. D.15/16

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.

1. A.3
2. B.2
3. C.−2
4. D.−3

Find the equation of a circle with centre (2, −3) and radius 2 units.

1. A.x² + y² − 4x + 6y + 9 = 0
2. B.x² + y² + 4x − 6y − 9 = 0
3. C.x² + y² + 4x + 6y − 9 = 0
4. D.x² + y² + 4x − 6y + 9 = 0

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.

1. A.45
2. B.54
3. C.56
4. D.65

For what values of m is 9y² + my + 4 a perfect square?

1. A.±2√3
2. B.±6√2
3. C.+6
4. D.+12

A particle accelerates at 12 ms⁻² and travels a distance of 250m in 6 seconds. Find the initial velocity of the particles.

1. A.5.7 ms⁻¹
2. B.6.0 ms⁻¹
3. C.6.0 ms⁻¹
4. D.77.5 ms⁻¹

In how many ways can 9 people be seated on a bench if only 3 places are available?

1. A.1200
2. B.504
3. C.320
4. D.204

Find the variance of 1, 2, 0, −3, 5, −2, 4.

1. A.52/7
2. B.40/7
3. C.32/7
4. D.27/7

If the points (−1, t − 1), (t, t − 3) and (t − 6, 3) lie on the same straight line, find the values of t.

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

A ball is thrown vertically upwards with a velocity of 15 ms⁻¹. Calculate the maximum height reached. [Take g = 10 ms⁻²]

1. A.15.25m
2. B.13.25m
3. C.11.25m
4. D.10.25m

Find the distance between the points (2, 5) and (5, 9).

1. A.4 units
2. B.5 units
3. C.12 units
4. D.14 units

Find lim_x→3 ((x³ + x² − 12x)/(x² − 9)).

1. A.7/2
2. B.0
3. C.7/−2
4. D.−7

Evaluate (tan 120° + tan 30°)/(tan 120° − tan 60°).

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

Express (14N, 240°) as a column vector. (Column vectors written as (x, y).)

1. A.(−7, −7√3)
2. B.(7√3, 7√3)
3. C.(−7√3, −7)
4. D.(7, −7√3)

Solve (log₂m)² − log₂m³ = 10.

A binary operation * is defined on the set of rational numbers by m * n = (m² − n²)/(2mn), n ≠ 0 and m ≠ 0. (a) Find −3 * 2. (b) Show whether or not * is associative.

If α and β are the roots of 3x² + 5x + 1 = 0, evaluate 27(α³ + β³).

Find the gradient of xy² + x²y = 4xy at the point (1, 3).

The position vectors of points P, Q and R are 11i + j, 5i + (13/3)j and 2i + 6j respectively. (a) Show that P, Q and R lie on a straight line. (b) Find the ratio |PQ| : |QR|.

A committee of 3 is formed from a panel of 5 men and 3 women. Find the: (a) number of ways of forming the committee; (b) probability that at least one woman is on the committee.

The table shows the distribution of the lengths of 20 iron rods measured in metres.
Length (m): 1.0 − 1.1, 1.2 − 1.3, 1.4 − 1.5, 1.6 − 1.7, 1.8 − 1.9
Frequency: 2, 3, 8, 5, 2
Using an assumed mean of 1.45, calculate the mean of the distribution.

Find the direction of the resultant of the forces in the diagram.

(a) Differentiate (x − 3)(x² + 5) with respect to x.
(b) If (x + 1)² is a factor of f(x) = x³ + ax² + bx + 3, where a and b are constants, find the: (i) values of a and b; (ii) zeros of f(x).

(a) Express (5 + √2)/(3 − √2) − (5 − √2)/(3 + √2) in the form a + b√2.
(b) Solve the following equations simultaneously using the determinant method: 3x − y − z = −2; x + 5y + 2z = 5; 2x + 3y + z = 0.

(a) If f(x) = (x − 3)/(2x − 1), x ≠ ½ and g(x) = (x − 1)/(x + 1), x ≠ −1, find g o f.
(b)(i) Sketch the curve y = 9x − x³. (ii) Calculate the total area bounded by the x-axis and the curve y = 9x − x³.

The probabilities that Kofi, Kwasi and Ama will pass a certain examination are 9/10, 4/5 and x respectively. If the probability that only one of them will pass examination is 9/50, find the: (a) value of x; (b) probability that at least one of them will pass examination.

(a) Given that x = 3i − j, y = 2i + kj and the cosine of the angle between x and y is √5/5, find the values of the constant k.
(b) In the quadrilateral ABCD, AB = (5, −1), AC = (−6, −9) and BD = (4, −7) (column vectors). Show whether or not ABCD is a parallelogram.

(a) A(−1, 2), B(3, 5) and C(4, 8) are the vertices of triangle ABC. Forces whose magnitudes are 5N and 3√10 N act along AB and CB respectively. Find the direction of the resultant of the forces.
(b) A particle starts from rest and moves in a straight line. It attains a velocity of 20 ms⁻¹ after covering a distance of 8 metres. Calculate: (i) its acceleration; (ii) the time it will take to cover a distance of 40 metres.