WAEC Further Mathematics 2008 past questions and answers

Simplify (√3 + √48)/√6

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

Find the range of values of x for which 2x² + 7x − 15 > 0

1. A.x < −3/2 or x > 5
2. B.x < −5 or x > 3/2
3. C.−3/2 < x < 5
4. D.−5 < x < 3/2

A function f is defined on R, the set of real numbers, by f : x → (x + 3)/(x − 1), x ≠ 2. Find f⁻¹

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

The sum of the first n terms of a linear sequence is S_n = n² + 2n. Find the common difference of the sequence.

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

The sum of the first n terms of a linear sequence is S_n = n² + 2n. Determine the general term of the sequence.

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

If f(x) = 2x² − 3x − 1 find the value of x for which f(x) is minimum.

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

The polynomial 2x³ + x² − 3x + p has a remainder 20 when divided by (x − 2). Find the value of constant p.

1. A.8
2. B.6
3. C.−6
4. D.−8

If 2log₄2 = x + 1, find the value of x.

1. A.−2
2. B.−1
3. C.0
4. D.1

Which of the following quadratic curves will not intersect with the x-axis?

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

What is the coordinates of the centre of the circle 5x² + 5y² − 15x + 25y − 3 = 0?

1. A.(15/2, −25/2)
2. B.(3/2, −5/2)
3. C.(−3/2, 5/2)
4. D.(−15/2, 25/2)

Evaluate ∫₁²(2 + 2x − 3x²)dx

1. A.−2
2. B.2
3. C.8
4. D.10

A rectangle has a perimeter of 24m, if its area is to be a maximum, find its dimension.

1. A.12, 12
2. B.6, 6
3. C.4, 8
4. D.9, 3

Express 7π/6 radians in degrees.

1. A.315°
2. B.210°
3. C.105°
4. D.75°

If P = (1 −2; 3 4) and Q = (−2 3; 1 0), find PQ.

1. A.(4 1; −2 9)
2. B.(−4 1; 2 9)
3. C.(−4 3; −2 13)
4. D.(−4 3; −2 9)

Two statements are represented by p and q as follows: p: He is brilliant. q: He is regular in class. Which of the following symbols represents the statement "he is regular in class but dull"?

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

A binary operation *, is defined on the set R, of real numbers by a * b = a² + b + ab. Find the value of x for which 5 * x = 37.

1. A.7
2. B.2
3. C.−2
4. D.−7

Find the derivative of 3x² + 1/x²

1. A.6x + 2x²
2. B.6x + 1/(2x)
3. C.6x − 2/x³
4. D.6x − 1/(2x)

The coefficient of the 5^th term in the binomial expansion of (1 + kx)⁸, in ascending powers of x is 35/8. Find the value of constant k.

1. A.2
2. B.1/2
3. C.−1/2
4. D.−2

Given that f′(x) = 3x² − 6x + 1 and f(3) = 5, find f(x).

1. A.f(x) = x³ − 3x + x + 20
2. B.f(x) = x³ − 3x² + x + 31
3. C.f(x) = x³ − 3x² + x + 2
4. D.f(x) = x³ − 3x² + x − 13

Express 1/(1 − sin 45°) in surd form

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

If |4 x; 5 3| = 32, find the value of x

1. A.4
2. B.2
3. C.−2
4. D.−4

If events A and B are independent and P(A) = 7/12 and P(A∩B) = 1/2, find P(B)

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

Given that AB = 5i + 3j and AC = 2i + 5j, find BC.

1. A.−7i − 8j
2. B.−3i + 2j
3. C.3i − 2j
4. D.3i + 8j

The probability of Jide, Atu and Obu solving a given problem are 1/12, 1/6 and 1/8 respectively. Calculate the probability that only one of them solves the problem.

1. A.1/576
2. B.55/576
3. C.77/576
4. D.167/576

Two forces F₁ = (10N, 020°) and F₂ = (7N, 200°) act on a particle. Find the resultant force.

1. A.(3N, 020°)
2. B.(3N, 200°)
3. C.(17N, 020°)
4. D.(17N, 200°)

The table shows the distribution of marks by some candidates in a test.
Marks: 2, 3, 4, 5, 6, 7, 8
No of students: 5, 7, 9, 6, 3, 6, 4
What is the median score?

1. A.3.5
2. B.4.0
3. C.4.5
4. D.5.0

The table shows the distribution of marks by some candidates in a test.
Marks: 2, 3, 4, 5, 6, 7, 8
No of students: 5, 7, 9, 6, 3, 6, 4
Find, correct to one decimal place, the mean of the distribution.

1. A.5.5
2. B.5.3
3. C.5.2
4. D.4.7

The table shows the distribution of marks by some candidates in a test.
Marks: 2, 3, 4, 5, 6, 7, 8
No of students: 5, 7, 9, 6, 3, 6, 4
If a student is selected at random, what is the probability that she scored at least 6 marks?

1. A.3/40
2. B.1/4
3. C.13/40
4. D.27/40

Express r = (12, 210°) in the form ai + bj

1. A.6(−i + √3j)
2. B.6(−i − √3j)
3. C.6(i − √3j)
4. D.6(i + √3j)

A test consists of 12 questions out of which candidates are to answer 10. If the first 6 questions are compulsory, in how many ways can each candidate select her questions?

1. A.40
2. B.25
3. C.15
4. D.10

A body starts from rest and moves in a straight line with a uniform acceleration of 5ms⁻². How far, in m, does it go in 10 seconds?

1. A.50
2. B.250
3. C.350
4. D.500

If n items are arranged two at a time, the number obtained is 20. Find the value of n.

1. A.5
2. B.10
3. C.15
4. D.40

If p = (2, −2) and q = (3, 4), find |q − ½p|

1. A.2√2
2. B.√13
3. C.5
4. D.√29

Find the value of constant k for which a = 4i − kj and b = 3i + 8j are perpendicular.

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

A pack of cards has 6 of them with a ⊗ written on them, 5 with ⊕ and 9 with ∅. What is probability that single card drawn at random is not a ⊕?

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

The initial and final velocities of an object of mass 5kg are u = (1, 3) and v = (4, 7) respectively. Find the magnitude of its change in momentum.

1. A.25
2. B.15
3. C.3√7
4. D.√10

If y = x² − 6x + 11 is written in the form y = a(x − h)² + k, find the value of (a + h + k)

1. A.−4
2. B.−3
3. C.0
4. D.6

The distance between P(x, 7) and Q(6, 19) is 13 units. Find the values of x

1. A.1 or −7
2. B.1 or 7
3. C.1 or 11
4. D.5 or −5

In the diagram below, forces P, Q and 50N are acting on a body at M. If the system is in equilibrium, calculate, in terms of θ, the magnitude of P.

1. A.50cosθ/sin(θ + 45)°
2. B.50cosθ/cos(θ + 45)°
3. C.50sinθ/cos(θ + 45)°
4. D.50sinθ/sin(θ + 45)°

A function f is defined on the set R, of real numbers by f : x → px² + qx + 2, where p and q are constants. If f(−2) = 0 and f(1) = 3, find f(−4)

If sin A = 3/5 and cos B = 15/17, where A is obtuse and B is acute, find the value of cos(A + B).

Differentiate, with respect to x, x³ − 2x from the first principle

A straight line passes through the point P(−1, 3). Another line which passes through Q(−4, 4) intersects the first line at a point R(k, 5), where k is constant. If ∠PRQ = 90° find the values of K.

In a hotel the breakfast menu is a choice between yam (Y) or plantain (P) or both. The Venn diagram shows the choices made by 25 guests of the Hotel. (a) Find the value of x. (b) What is the probability that a guest chosen at random chose only one of the two?

The table shows the distribution of marks obtained by some candidates in a test
Marks: 10 − 14, 15 − 24, 25 − 29, 30 − 39, 40 − 44, 45 − 49
No. of candidates: 14, 30, 22, 18, 12, 4
Draw a histogram for the distribution.

A car travelling at a velocity of 50 km h⁻¹, covers a distance of 20km. If it was accelerating at 6km h⁻², calculate, correct to one decimal place, the time the car took to cover the distance.

The magnitude of a force xi + 15j is 17N. Find the: (a) possible value of x; (b) directions of the forces, correct to the nearest degree.

(a) The sum of the first n terms of a sequence is given by S_n = 5n²/2 + 5n/2. Write down the first four terms of the sequence and find an expression for the nth term.
(b) The equation of a circle is given by x² + y² − 10x − 8y + 25 = 0. (i) Show that the circle touches the X-axis. (ii) Find the coordinates of the point of contact.

(a) If α and β are the roots of the equation 2x² + 5x − 6 = 0, find the equation whose roots are (α − 2) and (β − 2).
(b) Given that ∫₀^k(x² − 2x)dx = 4/3, find the values of k.

(a) Express (2x² − 5x + 1)/(x³ − 4x² + 3x) in partial fractions.
(b) If |x − 3 −4 3|
|5 2 2|
|2 −4 6 − x| = −24, find the value of x.

(a) The point P(3, −5) is rotated through an angle 60° anticlockwise about the origin. (i) Obtain the matrix for the rotation. (ii) Find the image P′ of point P under the rotation.
(b) A linear transformation is given by N: (x, y) → (2x + 3y, 3x − y). (i) Write down the matrix N of the transformation. (ii) If N² + aN + bI = O, where a, b ∈ R, I is the 2 × 2 unit matrix and O is the 2 × 2 null matrix, find the values of a and b.

A survey indicated that 65% of the families in an area have cars. Find, correct to three decimal places, the probability that among 7 families selected at random in the area (a) exactly 5; (b) 3 or 4; (c) at most 2 of them have cars.

The following table shows the distribution of marks obtained by some students in an examination.
Marks: 0 − 9, 10 − 19, 20 − 29, 30 − 39, 40 − 49, 50 − 59, 60 − 69, 70 − 79, 80 − 89, 90 − 99
No. of students: 50, 50, 40, 60, 100, 100, 50, 25, 15, 10
(a) Construct a cumulative frequency table for the distribution. (b) Draw an ogive for the distribution. (c) Use your graph in (b) to determine the (i) semi-interquartile range; (ii) number of students who failed, if the pass mark for the examination is 37; (iii) probability that a student selected at random scored between 20% and 60%.

(a) Five (5) female and seven (7) male teachers applied for 4 vacancies in a Junior High School. The teachers are equally qualified. Find the number of ways of employing the 4 teachers. If (i) there is no restriction; (ii) at least 2 of them are females.
(b) The table shows the positions awarded to 7 contestants by Judges x and y in a competition.
Contestant: P, Q, R, S, T, U, V
Judges X: 2, 7, 1, 3, 6, 5, 4
Judges Y: 4, 6, 2, 3, 7, 1, 5
(i) Calculate, correct to one decimal place, the Spearman's rank correlation coefficient. (ii) interprete your answer in (b)(i) above.

The position vectors of points P, Q and R with respect to the origin are (4i − 5j), (i + 3j) and (−5i + 2j) respectively. If PQRM is a parallelogram, find: (a) the position vector of M; (b) |PM| and |PQ|; (c) the acute angle between PM and PQ, correct to one decimal place; (d) the area of PQRM.

(a) Forces F₁ = (−5, 4)N, F₂ = (2, 5)N, F₃ = (2, −1)N, F₄ = (3, −5)N, act on a body, Find the: (i) resultant of these forces; (ii) fifth force that will keep the system in equilibrium.
(b) A body, moving at 20ms⁻¹ accelerates uniformly at 2.5ms⁻² for 4 seconds. It continues the journey at this speed for 8 seconds, before coming to rest t seconds with a uniform retardation. If the ratio of acceleration to retardation is 3:4 (i) sketch the velocity - time graph of the journey; (ii) find t, (iii) find the total distance of the journey.

(a) An object is thrown up a smooth plane inclined at an angle of 30° to the horizontal. If the plane is 15m long and the object comes to rest at the top, find the (i) initial speed of the object (ii) time taken to reach the top.
(b) Force of magnitudes 5N, 5√3 N, 10N, 5√3 and 5N act on a body P, of mass 5kg as shown in the diagram. Find the (i) magnitude of the resultant force; (ii) acceleration of the body.