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JAMB Mathematics 2006 past questions and answers

48 questions. Try each one, then open the answer to check yourself and read the explanation.

Question 1

Measures of Central Tendency

Score: 1, 2, 3, 4, 5, 6, 7, 8
Frequency: 2, 3, 5, 12, 10, 6, 4, 1
The table above shows the scores of a group of students in a Physics test. If the mode is m and the number of students who scored 4 or more is n, what is (n, m)?

  1. A.(33, 4)
  2. B.(22, 4)
  3. C.(33, 12)
  4. D.(12, 4)
Show the answer and explanation

Answer: A ((33, 4))

Explanation

The highest frequency (12) is for score 4, so m = 4. Students scoring 4 or more: 12 + 10 + 6 + 4 + 1 = 33, so (n, m) = (33, 4).

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Question 2

Permutation and Combination

A final examination requires that a student answer any 4 out of 6 questions. In how many ways can this be done?

  1. A.15
  2. B.20
  3. C.30
  4. D.45
Show the answer and explanation

Answer: A (15)

Explanation

6C4 = 6!/(4! 2!) = 15.

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Question 3

Data Presentation

The response of 160 pupils in a school asked to indicate their favourite subjects is given in the bar chart above. What percentage of the pupils have English and Health Education as their favourite subjects?

Diagram for this question
  1. A.36%
  2. B.52%
  3. C.55%
  4. D.22%
Show the answer and explanation

Answer: C (55%)

Explanation

English 52 + Health Education 36 = 88 pupils. 88/160 × 100% = 55%.

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Question 4

Measures of Central Tendency

If the mean of five consecutive integers is 30, find the largest of the numbers.

  1. A.28
  2. B.30
  3. C.32
  4. D.34
Show the answer and explanation

Answer: C (32)

Explanation

For five consecutive integers the mean is the middle one, 30, so they are 28 to 32 and the largest is 32.

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Question 5

Probability

A bag contains 5 black, 4 white and x red marbles. If the probability of picking a red marble is 2/5, find the value of x.

  1. A.8
  2. B.10
  3. C.4
  4. D.6
Show the answer and explanation

Answer: D (6)

Explanation

x/(9 + x) = 2/5 gives 5x = 18 + 2x, so x = 6.

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Question 6

Measures of Dispersion

Find the variance of 2x, 2x − 1 and 2x + 1

  1. A.2/3
  2. B.2
  3. C.√(2/3)
  4. D.1
Show the answer and explanation

Answer: A (2/3)

Explanation

The mean is 2x and the deviations are 0, −1, 1. Variance = (0 + 1 + 1)/3 = 2/3.

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Question 7

Probability

GSM operators: GTN, PLO, Qtel, Cmob
No. of recharge cards: 5, 6, 3, 6
The table above shows the distribution of recharge cards of four major GSM operators. What is the probability that a recharge card selected at random will be GTN or Qtel?

  1. A.3/20
  2. B.1/4
  3. C.2/5
  4. D.3/4
Show the answer and explanation

Answer: C (2/5)

Explanation

There are 20 cards and 5 + 3 = 8 are GTN or Qtel, so the probability is 8/20 = 2/5.

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Question 8

Data Presentation

The pie chart above shows the expenditure of a family whose income is N30,000. If the expenditure on food is twice that on housing and that on school fees is twice that on transport, how much does the family spend on food?

Diagram for this question
  1. A.N28,000
  2. B.N25,500
  3. C.N15,000
  4. D.N12,500
Show the answer and explanation

Answer: D (N12,500)

Explanation

School fees is a right angle (90°), so transport is 45°. Food + housing = 360° − 135° = 225° = 3 × housing, so housing = 75° and food = 150°. Food = 150/360 × 30,000 = N12,500.

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Question 9

Permutation and Combination

For what value of n is n+1C3 = 4(nC3)?

  1. A.6
  2. B.5
  3. C.4
  4. D.3
Show the answer and explanation

Answer: D (3)

Explanation

n+1C3/nC3 = (n + 1)/(n − 2) = 4, so n + 1 = 4n − 8 and n = 3.

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Question 10

Integration

The gradient of a curve is 2x + 7 and the curve passes through point (2, 0). Find the equation of the curve.

  1. A.y = x2 + 7x + 9
  2. B.y = x2 + 7x − 18
  3. C.y = x2 + 7x + 18
  4. D.y = x2 + 14x + 11
Show the answer and explanation

Answer: B (y = x2 + 7x − 18)

Explanation

Integrating, y = x2 + 7x + c. At (2, 0): 4 + 14 + c = 0, so c = −18.

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Question 11

Differentiation

Differentiate (x2 − 1/x)2 with respect to x.

  1. A.4x3 − 4x − 2/x
  2. B.4x3 − 2 + 2/x3
  3. C.4x3 − 2 − 2/x3
  4. D.4x3 − 3x + 2/x
Show the answer and explanation

Answer: C (4x3 − 2 − 2/x3)

Explanation

dy/dx = 2(x2 − 1/x)(2x + 1/x2) = 2(2x3 + 1 − 2 − 1/x3) = 4x3 − 2 − 2/x3.

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Question 12

Differentiation

Find the value of x for which the function 3x3 − 9x2 is minimum.

  1. A.0
  2. B.2
  3. C.3
  4. D.5
Show the answer and explanation

Answer: B (2)

Explanation

f'(x) = 9x2 − 18x = 9x(x − 2) = 0 at x = 0 or 2. f''(2) = 54 − 18 > 0, so the minimum is at x = 2.

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Question 13

Integration

If dy/dx = x + cos x, find y.

  1. A.x2 − sin x + c
  2. B.x2 + sin x + c
  3. C.x2/2 − sin x + c
  4. D.x2/2 + sin x + c
Show the answer and explanation

Answer: D (x2/2 + sin x + c)

Explanation

Integrate each term: ∫x dx = x2/2 and ∫cos x dx = sin x, plus a constant.

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Question 14

Differentiation

Differentiate (cos θ − sin θ)2 with respect to θ.

  1. A.−2 cos 2θ
  2. B.−2 sin 2θ
  3. C.1 − 2 cos 2θ
  4. D.1 − 2 sin 2θ
Show the answer and explanation

Answer: A (−2 cos 2θ)

Explanation

(cos θ − sin θ)2 = 1 − 2 sin θ cos θ = 1 − sin 2θ, whose derivative is −2 cos 2θ.

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Question 15

Integration

Evaluate ∫−41 (1 − 2x)dx.

  1. A.−16
  2. B.−20
  3. C.20
  4. D.10
Show the answer and explanation

Answer: C (20)

Explanation

[x − x2] from −4 to 1 = (1 − 1) − (−4 − 16) = 20.

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Question 16

Sets

If E ⊆ G ⊆ U, where U is the universal set, then the shaded venn diagram representing U − E or Ec is

Diagram for this question
  1. A.Diagram A
  2. B.Diagram B
  3. C.Diagram C
  4. D.Diagram D
Show the answer and explanation

Answer: D (Diagram D)

Explanation

E is inside G, which is inside U, so the diagram must show E's circle within G's. U − E is everything except E, which is the shading in D.

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Question 18

Ratio, Percentages and Rates

If m : n = 13 : 11, find m2 − n2 : (m + n)2.

  1. A.1 : 11
  2. B.1 : 13
  3. C.1 : 10
  4. D.1 : 12
Show the answer and explanation

Answer: D (1 : 12)

Explanation

Take m = 13, n = 11: m2 − n2 = 48 and (m + n)2 = 576. 48 : 576 = 1 : 12.

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Question 19

Logarithms

Calculate the logarithm to base 9 of 3−4 × 92 × (81)−1.

  1. A.2
  2. B.0
  3. C.−2
  4. D.−4
Show the answer and explanation

Answer: C (−2)

Explanation

In powers of 9: 3−4 = 9−2 and 81−1 = 9−2, so the product is 9−2 + 2 − 2 = 9−2, whose log to base 9 is −2.

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Question 20

Number Bases

If (k2)6 × 36 = 35(k4)5, what is the value of k?

  1. A.1
  2. B.4
  3. C.3
  4. D.2
Show the answer and explanation

Answer: D (2)

Explanation

In base ten: (6k + 2) × 3 = 3 × (5k + 4), so 18k + 6 = 15k + 12 and k = 2.

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Question 21

Sets

In a small village of 500 people, 350 speak the local language while 200 speak pidgin English. What percentage of the population speak both?

  1. A.30%
  2. B.10%
  3. C.50%
  4. D.14%
Show the answer and explanation

Answer: B (10%)

Explanation

Both = 350 + 200 − 500 = 50 people, which is 50/500 = 10%.

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Question 22

Commercial Arithmetic

Find the tax on an income of N20,000 if no tax is paid on the first N10,000 and tax is paid at N50 in N1,000 on the next N5,000 and at N55 in N1,000 on the remainder.

  1. A.N500
  2. B.N552
  3. C.N255
  4. D.N525
Show the answer and explanation

Answer: D (N525)

Explanation

Next N5,000: 5 × N50 = N250. Remaining N5,000: 5 × N55 = N275. Total tax = N525.

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Question 23

Surds

Rationalize 2/(6 − 5√3)

  1. A.−12/39 − 10√3/39
  2. B.12/39 − 10√3/39
  3. C.−12/39 + 10√3/39
  4. D.12/39 + 10√3/39
Show the answer and explanation

Answer: A (−12/39 − 10√3/39)

Explanation

Multiply by (6 + 5√3)/(6 + 5√3): 2(6 + 5√3)/(36 − 75) = (12 + 10√3)/(−39) = −12/39 − 10√3/39.

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Question 24

Number Bases

Compute 1100112 + 111112.

  1. A.10010102
  2. B.10100102
  3. C.10001102
  4. D.10001002
Show the answer and explanation

Answer: B (10100102)

Explanation

1100112 = 51 and 111112 = 31. Their sum 82 = 64 + 16 + 2 = 10100102.

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Question 25

Indices

Simplify (25)−½ × (27)1/3 + (121)−½ × (625)−¼

  1. A.34/55
  2. B.9/11
  3. C.14/5
  4. D.3/275
Show the answer and explanation

Answer: A (34/55)

Explanation

This is 1/5 × 3 + 1/11 × 1/5 = 3/5 + 1/55 = 33/55 + 1/55 = 34/55.

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Question 26

Number Bases

Convert 22324 to a number in base six.

  1. A.4506
  2. B.2546
  3. C.5536
  4. D.5406
Show the answer and explanation

Answer: A (4506)

Explanation

22324 = 128 + 32 + 12 + 2 = 174. Dividing by 6 repeatedly: 174 = 4 × 36 + 5 × 6 + 0, so 4506.

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Question 27

Mensuration of Plane Shapes

In the diagram above, |QR| is the diameter of the semicircle QR. Find the area of the figure to the nearest whole number. [π = 22/7]

Diagram for this question
  1. A.89cm2
  2. B.70cm2
  3. C.90cm2
  4. D.80cm2
Show the answer and explanation

Answer: A (89cm2)

Explanation

Rectangle = 10 × 7 = 70cm2. Semicircle with radius 3.5cm = ½ × 22/7 × 3.52 = 19.25cm2. Total ≈ 89cm2.

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Question 28

Trigonometry

If tan θ = 5/4, find sin2θ − cos2θ.

  1. A.5/4
  2. B.41/9
  3. C.9/41
  4. D.1
Show the answer and explanation

Answer: C (9/41)

Explanation

The hypotenuse is √(25 + 16) = √41, so sin2θ − cos2θ = 25/41 − 16/41 = 9/41.

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Question 29

Coordinate Geometry

PQ and RS are two parallel lines. If the coordinates of P, Q, R, S are (1, q), (3, 2), (3, 4), (5, 2q) respectively, find the value of q.

  1. A.3
  2. B.4
  3. C.1
  4. D.2
Show the answer and explanation

Answer: D (2)

Explanation

Gradient of PQ = (2 − q)/2 and gradient of RS = (2q − 4)/2. Parallel lines have equal gradients, so 2 − q = 2q − 4 and q = 2.

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Question 30

Trigonometry

In triangle XYZ, ∠XYZ = 15°, ∠XZY = 45° and |XY| = 7cm. Find |YZ|.

  1. A.14√2cm
  2. B.(7/2)√6cm
  3. C.7√2cm
  4. D.7cm
Show the answer and explanation

Answer: B ((7/2)√6cm)

Explanation

∠YXZ = 180° − 15° − 45° = 120°. By the sine rule, YZ = 7 × sin 120°/sin 45° = 7 × (√3/2)/(√2/2) = 7√6/2.

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Question 31

Triangles and Quadrilaterals

In the diagram above, find the value of x.

Diagram for this question
  1. A.55°
  2. B.50°
  3. C.45°
  4. D.40°
Show the answer and explanation

Answer: C (45°)

Explanation

The angle inside the vertex is 360° − 260° = 100°. A line through the vertex parallel to the others splits it into 55° and x (alternate angles), so x = 100° − 55° = 45°.

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Question 33

Loci

What is the locus of points equidistant from the line ax + by + c = 0?

  1. A.A line bx − ay + q = 0
  2. B.A line ax − by + q = 0
  3. C.A line bx + ay + q = 0
  4. D.A line ax + by + q = 0
Show the answer and explanation

Answer: D (A line ax + by + q = 0)

Explanation

Points at a fixed distance from a line lie on lines parallel to it, which have the same x and y coefficients: ax + by + q = 0.

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Question 34

Angles and Polygons

PQRSTW is a regular hexagon and QS intersects RT at V. Calculate ∠TVS.

  1. A.120°
  2. B.90°
  3. C.30°
  4. D.60°
Show the answer and explanation

Answer: D (60°)

Explanation

Each interior angle is 120°. Triangles QRS and RST are isosceles, so ∠RSQ = ∠STR = 30°. Then ∠VST = 120° − 30° = 90° and ∠TVS = 180° − 90° − 30° = 60°.

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Question 35

Loci

If the locus of the points which are equidistant from points P and Q meets line PQ at point N, then PN equals

  1. A.NQ
  2. B.−NQ
  3. C.2NQ
  4. D.−NQ
Show the answer and explanation

Answer: A (NQ)

Explanation

That locus is the perpendicular bisector of PQ, which meets PQ at its midpoint, so PN = NQ.

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Question 36

Trigonometry

In the diagram above, PQ = 10cm, PS = 8cm and ∠PSR is 60° while ∠SRQ is a right angle. Find SR.

Diagram for this question
  1. A.14cm
  2. B.14√3cm
  3. C.10cm
  4. D.10√3cm
Show the answer and explanation

Answer: A (14cm)

Explanation

Drop a perpendicular from P to N on SR. SN = 8 cos 60° = 4cm and NR = PQ = 10cm, so SR = 14cm.

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Question 37

Binary Operations

A binary operation ⊖ defined on the set of real numbers is such that x⊖y = xy/6 for all x, y ∈ R. Find the inverse of 20 under this operation when the identity element is 6.

  1. A.1/12
  2. B.10/3
  3. C.1/20
  4. D.9/5
Show the answer and explanation

Answer: D (9/5)

Explanation

20 ⊖ x = 20x/6 must equal the identity 6, so 20x = 36 and x = 9/5.

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Question 38

Polynomials

Find the roots of x3 − 2x2 − 5x + 6 = 0.

  1. A.1, 2, −3
  2. B.−1, −2, 3
  3. C.−1, 2, −3
  4. D.1, −2, 3
Show the answer and explanation

Answer: D (1, −2, 3)

Explanation

x = 1 gives 1 − 2 − 5 + 6 = 0. Dividing leaves x2 − x − 6 = (x − 3)(x + 2), so the roots are 1, −2, 3.

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Question 39

Inequalities

The solution set of the shaded area above is

Diagram for this question
  1. A.y ≥ 0, y ≥ x and y + x ≤ 4
  2. B.y ≤ x, y + x ≤ 4
  3. C.y + x ≥ 4, y ≤ x
  4. D.y ≤ x, y + x ≤ 4 and y ≥ 0
Show the answer and explanation

Answer: D (y ≤ x, y + x ≤ 4 and y ≥ 0)

Explanation

The shaded triangle has vertices (0, 0), (2, 2) and (4, 0): it is above the x-axis, below the line y = x and below the line y + x = 4.

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Question 40

Matrices and Determinants

If x = (1 0 1; 2 −1 0; −1 0 1) and y = (−1 1 2; 0 −1 −1; 2 −1 1), find 2x − y. (Matrices written row by row.)

  1. A.(3 −1 0; 4 −3 −1; −4 1 1)
  2. B.(3 −1 0; 4 −3 1; −4 1 1)
  3. C.(3 −1 0; 4 −1 1; −4 1 1)
  4. D.(3 −1 0; 4 1 1; −4 −1 1)
Show the answer and explanation

Answer: C ((3 −1 0; 4 −1 1; −4 1 1))

Explanation

2x = (2 0 2; 4 −2 0; −2 0 2). Subtracting y entry by entry gives (3 −1 0; 4 −1 1; −4 1 1).

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Question 41

Variation

If p varies inversely as the cube of q and q varies directly as the square of r, what is the relationship between p and r?

  1. A.p varies directly as r3
  2. B.p varies inversely as r6
  3. C.p varies inversely as ∛r
  4. D.p varies directly as r6
Show the answer and explanation

Answer: B (p varies inversely as r6)

Explanation

p = k/q3 and q = mr2, so p = k/(m3r6): p varies inversely as r6.

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Question 42

Binary Operations

A binary operation * on the set of rational numbers is defined as x * y = (x2 − y2)/2xy. Find −5 * 3.

  1. A.−8/15
  2. B.8/15
  3. C.17/15
  4. D.−17/15
Show the answer and explanation

Answer: A (−8/15)

Explanation

(25 − 9)/(2 × (−5) × 3) = 16/(−30) = −8/15.

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Question 43

Polynomials

Find the value of k if the expression kx3 + x2 − 5x − 2 leaves a remainder 2 when it is divided by 2x + 1.

  1. A.10
  2. B.8
  3. C.−10
  4. D.−8
Show the answer and explanation

Answer: C (−10)

Explanation

Put x = −½: −k/8 + 1/4 + 5/2 − 2 = 2, so −k/8 = 5/4 and k = −10.

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Question 44

Inequalities

Solve the inequality for which (x + 4)/3 − (x − 3)/2 < 4

  1. A.x < 7
  2. B.x > −7
  3. C.x < −7
  4. D.x > 7
Show the answer and explanation

Answer: B (x > −7)

Explanation

Multiply by 6: 2x + 8 − 3x + 9 < 24, so −x < 7 and x > −7.

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Question 45

Matrices and Determinants

Find p, q for which (2p 8; 3 −5q)(1; 2) = (24; −17). (Matrices written row by row.)

  1. A.−4, 2
  2. B.4, 2
  3. C.−4, −2
  4. D.4, −2
Show the answer and explanation

Answer: B (4, 2)

Explanation

Row 1: 2p + 16 = 24, so p = 4. Row 2: 3 − 10q = −17, so q = 2.

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Question 46

Commercial Arithmetic

The cost of renovating a 6m square room is N540. What is the cost of renovating a 9m square room?

  1. A.N1,215
  2. B.N720
  3. C.N1,620
  4. D.N810
Show the answer and explanation

Answer: A (N1,215)

Explanation

A 6m square room has area 36m2 (N15 per m2); a 9m square room has area 81m2, so the cost is 81 × N15 = N1,215.

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Question 47

Progressions

The sum of the first n positive integers is

  1. A.½n(n − 1)
  2. B.n(n + 1)
  3. C.n(n − 1)
  4. D.½n(n + 1)
Show the answer and explanation

Answer: D (½n(n + 1))

Explanation

1 + 2 + ... + n is an AP with first term 1 and last term n, so the sum is n/2 × (1 + n) = ½n(n + 1).

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Question 48

Algebraic Expressions

If T = 2π√(l/g), make g the subject of the formula.

  1. A.4πl2/T
  2. B.4π2l/T2
  3. C.4π2l2/T2
  4. D.2π√l/T
Show the answer and explanation

Answer: B (4π2l/T2)

Explanation

Square both sides: T2 = 4π2l/g, so g = 4π2l/T2.

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Question 49

Inequalities

If y = x2 − x − 12, find the range of values of x for which y ≥ 0.

  1. A.x < −3 or x > 4
  2. B.x ≤ −3 or x ≥ 4
  3. C.−3 < x ≤ 4
  4. D.−3 ≤ x ≤ 4
Show the answer and explanation

Answer: B (x ≤ −3 or x ≥ 4)

Explanation

y = (x + 3)(x − 4), which is zero at −3 and 4 and positive outside them, so y ≥ 0 for x ≤ −3 or x ≥ 4.

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Question 50

Progressions

How many terms of the series 3, −6, +12, −24, +... are needed to make a total of 1 − 28?

  1. A.12
  2. B.10
  3. C.9
  4. D.8
Show the answer and explanation

Answer: D (8)

Explanation

a = 3 and r = −2, so Sn = 3(1 − (−2)n)/(1 + 2) = 1 − (−2)n. This equals 1 − 28 when n = 8.

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