WAEC Further Mathematics 2013 past questions and answers

Solve Sin θ = tan θ.

1. A.200°
2. B.90°
3. C.60°
4. D.0°

Given that a^5/6 × a^−1/n = 1, solve for n.

1. A.−6.00
2. B.−1.20
3. C.0.83
4. D.1.20

Express log ¼ + log ½ in terms of log 2.

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

If f(x) = x² and g(x) = Sin x, find g o f.

1. A.Sin²x
2. B.Sin x²
3. C.(Sin x)x²
4. D.x Sin x

Find the third term in the expansion of (a − b)⁶ in ascending powers of b.

1. A.−15a⁴b²
2. B.15a⁴b²
3. C.−15a³b³
4. D.15a³b³

If √x + √(x + 1) = √(2x + 1), find the possible value of x.

1. A.1 and −1
2. B.1 and 2
3. C.1 and 2
4. D.0 and −1

If α and β are the roots of the equation 2x² − 6x + 5 = 0, evaluate β/α + α/β.

1. A.24/5
2. B.8/5
3. C.5/8
4. D.5/24

Given that f(x) = 2x³ − 3x² − 11x + 6 and f(3) = 0, factorize f(x).

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

Find the equation of the line that is perpendicular to 2y + 5x − 6 = 0 and bisects the line joining the points P(4, 3) and Q(−6, 1).

1. A.y + 5x + 3 = 0
2. B.2y − 5x − 9 = 0
3. C.5y + 2x − 8 = 0
4. D.5y − 2x − 12 = 0

Differentiate x² + xy − 5 = 0 with respect to x.

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

The fourth term of an exponential sequence is 192 and its ninth term is 6. Find the common ratio of the sequence.

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

Find the range of values of x for which x² + 4x + 5 is less than 3x² − x + 2.

1. A.x > −½, x > 3
2. B.x < −½, x > 3
3. C.−½ ≤ x ≤ 3
4. D.−½ < x < 3

Given that dy/dx = √x, find y.

1. A.2x^3/2 + c
2. B.(2/3)x^3/2 + c
3. C.−½ ≤ x ≤ 3
4. D.(2/3)x² + c

Given that P = (y − 2 y − 1 / y − 4 y + 2) and |P| = −23, find the value of y. (Matrix written row by row; / separates the rows.)

1. A.−4
2. B.−3
3. C.−1
4. D.2

An object is thrown vertically upwards from the top of a cliff with a velocity of 25 ms⁻¹. Find the time in seconds, when it is 20 meters above the cliff. [Take g = 10 ms⁻²]

1. A.0 and 1
2. B.0 and 4
3. C.0 and 5
4. D.1 and 4

Evaluate ∫₀² (8x − 4x²) dx.

1. A.−16
2. B.−16/3
3. C.16/3
4. D.16

Find the coordinates of the point which divides the line joining P(−2, 3) and Q(4, 9) internally in the ratio 2 : 3.

1. A.(5⅔, ⅖)
2. B.(⅖, 5⅖)
3. C.(⅖, 2⅖)
4. D.(−⅖, 5⅖)

The angle subtended by an arc of a circle at the centre is π/3 radians. If the radius of the circle is 12 cm, calculate the perimeter of the major sector.

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

The function f : R → R is defined by f(x) = 3x + 2 : x > 4; 3x − 2 : x = 4; 5x − 3 : x < 4. Find f(4) − f(−3).

1. A.28
2. B.26
3. C.−26
4. D.−28

A committee consists of 5 boys namely Kofi, John, Ojo, Ozo and James and 3 girls namely: Rose, Ugo and Ama. In how many ways can a subcommittee consisting of 3 boys and 2 girls be chosen, if Ozo must be on the sub-committee?

1. A.35
2. B.30
3. C.18
4. D.12

Forces 50N and 80N act on a body as shown in the diagram. Find, correct to the nearest whole number, the horizontal component of the resultant force.

1. A.13N
2. B.43N
3. C.57N
4. D.95N

The sales of five sales girls on a certain day are as follows: GH¢26.00, GH¢39.00, GH¢33.00, GH¢25.00, GH¢37.00. Calculate the standard deviation if the mean sale is GH¢32.00.

1. A.GH¢5.65
2. B.GH¢5.66
3. C.GH¢6.55
4. D.GH¢6.56

A circular ink blot on a piece of paper increases its area at the rate of 4 mm²/s. Find the rate of increase of the radius of the blot when the radius is 8 mm. [Take π = 22/7].

1. A.0.25 mm/s
2. B.0.20 mm/s
3. C.0.08 mm/s
4. D.0.05 mm/s

Express (x² + x + 4)/((1 − x)(x² + 1)) in partial fractions.

1. A.x²/(x² + 1) + (x + 4)/(1 − x)
2. B.3/(1 − x) + (2x + 1)/(x² + 1)
3. C.x²/(1 − x) + (x + 4)/(x² + 1)
4. D.3/(1 − x) + (2x + 2)/(x² + 1)

Two bodies of masses 3kg and 5kg moving with velocities 2 ms⁻¹ and V ms⁻¹ respectively in opposite directions collide. If they move together after collision with velocity 3.5 ms⁻¹ in the direction of the 5kg mass, find the value of V.

1. A.7.8 ms⁻¹
2. B.6.8 ms⁻¹
3. C.5.6 ms⁻¹
4. D.4.6 ms⁻¹

The equation of a circle is x² + y² − 8x + 9y + 15 = 0. Find its radius.

1. A.√5
2. B.½√15
3. C.½√85
4. D.√85

A particle is acted upon by two forces 6N and 3N inclined at an angle of 120° to each other. Find the magnitude of the resultant force.

1. A.18√3 N
2. B.27N
3. C.24N
4. D.3√3 N

If s = 3i − j and t = 2i + 3j, find (t + 3s) · (t − 3s).

1. A.−77
2. B.−71
3. C.−53
4. D.−41

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

1. A.30°
2. B.45°
3. C.60°
4. D.90°

Find the upper quartile of the following scores: 41, 29, 17, 2, 12, 33, 45, 18, 21, 43 and 5.

1. A.45
2. B.41
3. C.33
4. D.21

Given that P = (3 4 / 2 x), Q = (1 3 / −2 4), R = (−5 25 / −8 26) and PQ = R, find the value of x. (Matrices written row by row; / separates the rows.)

1. A.−5
2. B.−2
3. C.2
4. D.5

Two out of 10 tickets on sale for a raffle draw are winning tickets. If a guest bought two tickets, what is the probability that both tickets are winning ticket?

1. A.1/90
2. B.1/45
3. C.1/20
4. D.1/10

P and Q are the points (3, 1) and (7, 4) respectively. Find the unit vector along PQ. (Column vectors written as (x, y).)

1. A.(4, 3)
2. B.(0.6, 0.8)
3. C.(0.8, 0.6)
4. D.(−0.8, −0.6)

If g(x) = (x + 1)/(x + 2), (x ≠ −2), find g⁻¹(2).

1. A.3
2. B.2
3. C.−2
4. D.−3

Calculate the mean deviation of 1, 2, 3, 4, 5, 5, 6, 7, 8, 9.

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

If V = (−2, 4) and U = (−1, 5) (column vectors), find |U + V|.

1. A.3√10
2. B.√82
3. C.√46
4. D.2√5

Find the equation of the straight line that passes through (2, −3) and perpendicular to the line 3x − 2y + 4 = 0.

1. A.2y − 3x = 0
2. B.3y − 2x + 5 = 0
3. C.3y + 2x + 5 = 0
4. D.2y − 3x − 5 = 0

If ⁿC₃/ⁿP₂ = 1, find the value of n.

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

A body is kept at rest by three forces F₁ = (10N, 030°), F₂ = (10N, 150°) and F₃. Find F₃.

1. A.(12N, 090°)
2. B.(10 N, 270°)
3. C.(10 N, 180°)
4. D.(10N, 120°)

(a) If the coefficient of x² and x³ in the expansion of (p + qx)⁷ are equal, express q in terms of p.
(b) A man makes a weekly contribution into a fund. In the first week, he paid ₦180.00, second week ₦260.00, third week ₦340.00 and so on. How much would he have contributed in 16 weeks?

Calculate the gradient of the curve x³ + y³ − 2xy = 11 at the point (2, −1).

A side of a rectangle is three times the other. If the perimeter increases by 2%, find the percentage increase in the area of the rectangle.

The line 2y = x + 3 meets the circle x² + y² − 2x + 6y − 15 = 0 at point M and N, where N is in the first quadrant. Find the coordinates of M and N.

Three school prefects are to be chosen from four girls and five boys. What is the probability that: (a) Only boys will be chosen; (b) more girls than boys will be chosen?

The table shows the distribution of ages of 22 students in a school.
Age (years): 12 − 14, 15 − 17, 18 − 20, 21 − 23, 24 − 26
Freq.: 6, 10, 3, 2, 1
Using an assumed mean of 19, calculate, correct to three significant figures, the: (a) mean age; (b) standard deviation of the distribution.

The initial velocity of a particle of mass 0.1 kg is 40 ms⁻¹ in the direction of the unit vector j. The velocity of the particle changed to 30 ms⁻¹ in the direction of the unit vector i. Find the change in momentum.

A stone is dropped vertically downwards from the top of a tower of height 45m with a speed of 20 ms⁻¹. Find the: (a) time it takes to reach the ground; (b) speed with which it hits the ground. [Take g = 10 ms⁻²]

(a) Differentiate (x² + 1)/(x + 1)² with respect to x.
(b)(i) Evaluate the determinant (1 2 −1 / 2 3 −1 / −1 1 3) (rows separated by /).
(ii) Using the answer in 9(b)(i), solve the system of equations: x + 2y − z = 4; 2x + 3y − z = 2; −x + y + 3z = −1.

(a) Using a scale of 2 cm to 30° on the x-axis and 2 cm to 0.2 units on the y-axis, on the same graph sheet, draw the graphs of y = sin 2x and y = cos x for 0° ≤ x ≤ 210° at intervals of 30°.
(b) Using the graphs in 10(a), find the truth set of: (i) sin 2x = 0; (ii) sin 2x − cos x = 0.

(a) The sum of the first three terms of a decreasing exponential sequence (GP) is equal to 7 and the product of these three terms is equal to 8. Find the: (i) common ratio; (ii) first three terms of the sequence.
(b) Using the trapezium rule with ordinates at x = 1, 2, 3, 4 and 5, calculate, correct to two decimal places, the value of ∫₁⁵ (x + 2/x²) dx.

(a) Find the maximum and minimum points of the curve y = 2x³ − 3x² − 12x + 4.
(b) Sketch the curve in 12(a) above.

The table gives the relationship between the height, in meters, of a plant and the number of days it is left to grow.
Number of days (x): 10, 20, 30, 40, 50, 60, 70, 80
Height (y): 1.0, 1.1, 1.2, 1.4, 1.6, 1.8, 2.0, 2.3
(a) Using a scale of 2cm to represent 0.5 units on the y-axis and 2 cm to represent 10 units on the x-axis, draw a scatter diagram for the information. (b) Find x̄ the mean of x, and ȳ the mean of y, and plot (x̄, ȳ) on the diagram. (c) Draw the line of best fit to pass through (x̄, ȳ) and (10, 1). (d) From the graph, find the: (i) equation of the line of best fit; (ii) height of plant in 75 days.

(a) A bag contains 5 blue, 4 green and 3 yellow balls. All the balls are identical except for colour. Three balls are drawn at random without replacement. Find the probability that: (i) all three balls have the same colour; (ii) two balls have the same colour.
(b) The table shows the ranks of the marks scored by 7 candidates in Physics and Chemistry tests.
Physics: 6, 5, 4, 3, 2, 7, 1
Chemistry: 7, 6, 2, 4, 1, 5, 3
Calculate the Spearman's rank correlation coefficient.

(a) The probability that a man wins a race is 0.8. In four different races, what is the probability that he wins: (i) all races; (ii) no race; (iii) at most 3 races?
(b) A class consists of 5 girls and 10 boys. If a committee of 5 is chosen at random from the class find the probability that: (i) 3 boys are selected; (ii) at least one girl is selected.

(a) Three vectors a, b and c are (8, 3), (6, −5) and (2, 3) (column vectors) respectively. Find the vector d such that |d| = √41 and d is in the direction of a + b − 2c.
(b) The coordinates of A and B are (3, 4) and (3, n) respectively. If O is the origin and ∠AOB = 30°, find, correct to 2 decimal places, the values of n.

(a) A particle is under the action of forces P = (4N, 030°) and R = (10N, 300°). Find the force that will keep the particle in equilibrium.
(b) A train travelling at 45 ms⁻¹ is brought to rest after covering a distance of 1500m. Find the: (i) time taken to come to rest; (ii) uniform retardation.

The displacement S metres of the particle from a fixed point O at time t seconds is given by S = t² − 6t + 5.
(a) On a graph sheet, draw a displacement-time graph for the interval 0 ≤ t ≤ 6.
(b) From the graph, find the: (i) time at which the velocity is zero; (ii) average velocity over the interval 0 ≤ t ≤ 4; (iii) total distance covered in the interval 0 ≤ t ≤ 5.