WAEC Further Mathematics 2006 past questions and answers

If P = {x : 1 ≤ x ≤ 6} and Q = {x : 2 < x < 9}, where x ∈ R, find P ∩ Q.

1. A.{x : 2 ≤ x ≤ 6}
2. B.{x : 2 ≤ x < 6}
3. C.{x : 2 < x < 6}
4. D.{x : 2 < x ≤ 6}

Simplify √((−1/64)^−2/3)

1. A.−4
2. B.−¼
3. C.½
4. D.4

A binary operation * is defined on the set R, of real numbers by a * b = ab/4. Find the value of √2 * √6.

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

If (x − 3) is a factor of 2x³ + 3x² − 17x − 30, find the remaining factors

1. A.(2x − 5)(x − 2)
2. B.(2x − 5)(x + 2)
3. C.(2x + 5)(x − 2)
4. D.(2x + 5)(x + 2)

Two functions f and g are defined by f : x → 3x − 1 and g : x → 2x³, evaluate fg(−2)

1. A.−49
2. B.−47
3. C.−10
4. D.−9

Given that 1/8^2−3y = 2^y+2, find y.

1. A.⅕
2. B.⅞
3. C.1
4. D.1⅕

Given that (√3 − 5√2)(√3 + √2) = p + q√6, find q.

1. A.4
2. B.−4
3. C.−5
4. D.−7

If f(x) = 1/(2 − x), x ≠ 2, find f⁻¹(−½).

1. A.4
2. B.0
3. C.−2
4. D.−4

Find the coefficient of x⁴ in the binomial expansion of (1 − 2x)⁶.

1. A.320
2. B.240
3. C.−240
4. D.−320

Find the equation of the line passing through (0, −1) and parallel to the y-axis.

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

The roots of the equation 2x² + kx + 5 = 0 are α and β, where k is a constant. If α² + β² = −1, find the values of k.

1. A.±16
2. B.±8
3. C.±4
4. D.±2

Find the sum of the exponential series 96 + 24 + 6 + ..

1. A.144
2. B.128
3. C.72
4. D.64

Evaluate log_0.25 8.

1. A.3/2
2. B.2/4
3. C.2/3
4. D.−3/2

Evaluate lim_x→1 (1 − x)/(x² − 3x + 2)

1. A.1
2. B.½
3. C.0
4. D.−1

The mean age of n men in a club is 50 years. Two men, aged 55 and 63, left the club and the mean age reduced by 1 year. Find the value of n.

1. A.30
2. B.20
3. C.18
4. D.14

A committee of 4 is to be selected from a group of 5 men and 3 women. In how many ways can this be done if the chairman of the committee must be a man?

1. A.15
2. B.40
3. C.70
4. D.175

Simplify ⁿP₄/ⁿC₄

1. A.24
2. B.18
3. C.12
4. D.6

Which of the following matrices is a singular matrix?

1. A.(1 0; 0 1)
2. B.(2 1; 3 2)
3. C.(3 8; 6 16)
4. D.(0 1; 1 0)

Simplify 8ⁿ × 2²ⁿ ÷ 4³ⁿ.

1. A.2⁻ⁿ
2. B.2¹⁻ⁿ
3. C.2n
4. D.2ⁿ⁺¹

The area of a sector of a circle is 3 cm. If the sector subtends an angle of 1.5 radians at the centre, calculate the radius of the circle.

1. A.1cm
2. B.√2cm
3. C.2cm
4. D.4cm

A particle of mass 2.5 kg is moving at a speed of 12 ms⁻¹. If a force of magnitude 10 N acts against it, find how long it takes to come to rest.

1. A.1.5s
2. B.3.0s
3. C.4.0s
4. D.6.0s

Calculate the standard deviation of the distribution below.
Age (in years): 1 − 5, 6 − 10, 11 − 15
Frequency: 3, 5, 2

1. A.1.10
2. B.2.36
3. C.3.50
4. D.7.50

In a firing contest, the probabilities that Kojo and Kwame hit the target are 2/5 and 1/3 respectively. What is the Probability that none of them will hit the target?

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

The equation of the line of best fit for variables x and y is y = 19.33 + 0.42x, where x is the independent variable. Estimate the value of y when x = 15

1. A.18.91
2. B.19.74
3. C.25.63
4. D.38.23

Find the coordinates of the point on the curve y = x² + 4x − 2, where the gradient is zero.

1. A.(−2, 10)
2. B.(−2, 2)
3. C.(−2, −2)
4. D.(−2, −6)

Find the least value of the function f(x) = 3x² + 18x + 32.

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

A force of 32 Newtons is applied to an object of mass m kg which is at rest on a smooth horizontal surface. If the acceleration produced is 8ms⁻², find the value of m

1. A.16
2. B.12
3. C.6
4. D.4

Given that (2 −3; 1 4)(−6; p) = (3; −26). Find the value of p.

1. A.−8
2. B.−5
3. C.−4
4. D.−3

Find the coordinates of the centre of the circle 4x² + 4y² − 5x + 3y − 2 = 0.

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

The parallelogram PQRS has vertices P(−2,3), Q (1,4), R (2,6) and S (−1,5). Find the coordinates of the point of intersection of the diagonals.

1. A.(−1,5)
2. B.(−½, 3½)
3. C.(0, 4½)
4. D.(1½, 5)

The mean and median of integers x, y, z and t are 5 and z respectively. If x < y < z < t and y = 4, find (x + t)

1. A.12
2. B.11
3. C.10
4. D.8

If a = (3; 2) and b = (−3; 5), find a vector c such that 4a + 3c = b.

1. A.(3; −1)
2. B.(−5; −1)
3. C.(−5; 1)
4. D.(−5; −9)

A lift moving upwards with a uniform acceleration of 5ms⁻², carries a body of mass p kg. If the reaction on the floor is 480 N, find the value of p. (take g = 10ms⁻²)

1. A.32
2. B.36
3. C.48
4. D.64

A particle is projected vertically upwards from a height 45 metres above the ground with a velocity of 40ms⁻¹. How long does it take to hit the ground? [Take g = 10ms⁻²]

1. A.1s
2. B.3s
3. C.7s
4. D.9s

Two forces, each of magnitude 16 N, are inclined to each other at an angle of 60°. Calculate the magnitude of their resultant.

1. A.16
2. B.16√3
3. C.18
4. D.18√3

ABCD is a square. Forces of magnitude 14N, 4N, 2N and 2√2N act along the sides AB, BC, CD and DA respectively. Find in Newtons the magnitude of resultant of the forces.

1. A.14.11
2. B.13.81
3. C.12.06
4. D.11.05

The sum of the 2^nd and 5th terms of an arithmetic progression (A.P) is 42. If the difference between the 6^th and the 3^rd terms is 12, find the (a) common difference; (b) first term; (c) 20^th term.

If 2^2x−3y = 32 and log_yx = 2, find the values of x and y.

If (x + 2) and (x − 1) are factors of f(x) = 6x⁴ + mx³ − 13x² + nx + 14, find the; (a) values of m and n; (b) remainder when f(x) is divided by (x + 1).

Find the equation of the tangent to the curve y = (x − 1)/(2x + 1), x ≠ −½, at the point (1, 0).

(a) There are six points in a plane. How many triangles can be formed with the points?
(b) A family of six is to be seated in a row. In how many ways can this be done, if the father and mother are not to sit together?

The table shows the distribution of the ages of a group of people in a village.
Ages (in years): 15 − 18, 19 − 22, 23 − 26, 27 − 30, 31 − 34, 35 − 38
Freq.: 40, 33, 25, 10, 8, 4
Using an assumed mean of 24.5, calculate the mean of the distribution.

A body of mass 5kg resting on a smooth horizontal plane, is acted upon by force 6i + 2j, 5i + 4j and 4 − j. Calculate the;
(a) velocity of the body;
(b) magnitude of its velocity, after 4 seconds.

An object is projected vertically upwards with a velocity of 80 ms⁻¹. Find the;
(a) maximum height reached; (b) time taken to return to the point of projection. [Take g = 10 ms⁻²]

(a) simplify (√75 − 3)/(√3 + 1), leaving your answers in the form a + b√c, where a, b and c are rational numbers.
(b) The points (7,3), (2,8) and (−3, 3) lie on a circle. Find the: (i) equation; (ii) radius, of the circle.

(a) The gradient of the tangent to the curve y = 4x³ at points P and Q is 108. Find the coordinates of P and Q.
(b) Given that  = 45°, B̂ = 30°, sin(A + B) = sin A cos B + sin B cos A and cos(A + B) = cos A cos B − sin A sin B
(i) show that: sin 15° = (√6 − √2)/4 and cos 15° = (√6 + √2)/4
(ii) hence find tan 15°

(a) Using the substitution U = 5 − x², evaluate ∫₁² x/√(5 − x²) dx
(b) If y = px² + qx, dy/dx = 7 and d²y/dx² = 6, find the values of p and q.

Question 12 is for candidates in Sierra Leone and Gambia
(a) Given that A = (3 4; 2 5) and X = (3 4), evaluate XA
(b) A linear transformation T maps A = (3 4) → A¹(−6, 8) and B(5, 2) → B(−10, 4). Find the: (i) Matrix of T, (ii) Image of the point k(1, −1) under T.

(a) A man P has 5 red, 3 blue and 2 white buses. Another man Q has 3 red, 2 blue and 4 white buses. A bus owned by P is involved in an accident with a bus belonging to Q. Calculate the probability that the two buses are not of the same colour.
(b) A man travels from Nigeria to Ghana by air and from Ghana to Liberia by ship. He returns by the same means. He has 6 airlines and 4 shipping lines to choose from. In how many ways can he make his journey without using the same airline or shipping line twice?

The table below shows the distribution of hours spent at work by the employees of a factory in a week.
Time (hours): 20 − 29, 30 − 39, 40 − 49, 50 − 59, 60 − 69, 70 − 79
No. of Persons: 8, 11, 23, 25, 8, 5
(a) Draw an ogive for the distribution.
(b) Using your graph, estimate the (i) median; (ii) lower quartile; (iii) 40th percentile; (iv) number of employees that spent at least 50 hours 30 minutes.

(a) Two pupils are chosen at random from a group of 4 boys and 5 girls. Find the probability that the two pupils chosen would be boys.
(b) Twenty percent of the total production of transistors produced by a machine are below standard. If a random sample of 6 transistors produced by the machine is taken, what is the probability of getting; (i) exactly 2; (ii) exactly 1; (iii) at least 2; (iv) at most 2, standard transistor(s)?

(a) A body of mass 15 kg is suspended at a point P by two light inextensible strings XP and YP. The strings are inclined at 60° and 40° respectively to the downward vertical. Find, correct to two decimal places, the tensions in the strings. [Take g = 10 ms⁻²]
(b) The height h metres, of a ball thrown into the air is 2 + 20t + kt², after t seconds. If it takes 2 seconds for the ball to reach its highest point, find: (i) the value of k; (ii) its highest point from the point of throw.

(a) A ball P moving with velocity 2u ms⁻¹, collides with a similar ball Q, of different mass, which is at rest. After collision, Q moves with velocity u ms⁻¹ and P with velocity ½u ms⁻¹, in the opposite direction. Find the ratio of the masses of P and Q.
(b) Two forces of magnitudes 3N and 7N have a resultant of magnitude 5N. Calculate, correct to one decimal place, the angle between the two forces.
(c) AB = (−4; 6) and CB = (2; −3) are two vectors in the XY plane. If V is the mid-point of AB, find CV.

(a) A body of mass 5kg is placed on a smooth plane inclined at an angle 30° to the horizontal. Find the magnitude of the force; (i) acting parallel to the plane; (ii) required to keep the body in equilibrium. [Take g = 10 ms⁻²]
(b) A uniform plank PQ of length 10m and mass m kg rests on two supports A and B, where |PA| = |BQ| = 1m. A load of mass 8kg is placed on the plank at point C such that |AC| = 3.5 m. If the reaction at B is 100 N, calculate the (i) value of m; (ii) reaction at A. [Take g = 10ms⁻²]