JAMB UTME - Mathematics - 1986

Question 1 Report

Find the reciprocal of $ \frac{\frac{2}{3}}{\frac{1}{2}+\frac{1}{3}} $

4

5

4

2

5

6

7

Answer Details

$\frac{2}{3}$ = $\frac{2}{3}$
= $\frac{2}{3}$ x $\frac{6}{5}$
= $\frac{4}{5}$
reciprocal of $\frac{4}{5}$ = $\frac{1}{\frac{4}{5}}$
= $\frac{5}{4}$

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Read lesson note on Positive And Negative Integers, Rational Numbers (WAEC)

Fractions, Decimals And Approximations

Positive And Negative Integers, Rational Numbers

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Question 2 Report

Find the median of the numbers 89, 141, 130, 161, 120, 131, 131, 100, 108 and 119

131

125

123

120

Answer Details

Arrange in ascending order
89, 100, 108, 119, |120, 130|, 131, 131, 141, 161
Median = $\frac{120 + 130}{2}$
= 125

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Measures Of Location

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Question 3 Report

make U the subject of the formula $S = \sqrt{\frac{6}{u} - \frac{w}{2}}$

u =

\frac{12}{2 s^{2}}

u =

\frac{12}{2 s + w}

u =	12
	2s²−w

u =

\frac{12}{2 s^{2}}

+ w

Answer Details

S = $\sqrt{\frac{6}{u} - \frac{w}{2}}$
S = $\frac{12 - u w}{2 u}$
2us2 = 12 - uw
u(2s2 + w) = 12
u = $\frac{12}{2 s^{2} + w}$

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Change Of Subject Of A Formula/Relation

Quadratic Equations

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Question 4 Report

Find n if log24 + log27 - log2n = 1

10

14

27

28

Answer Details

log24 + log27 - log2n = 1
= log2(4 x 7) - log2n = 1
= log228 - log2n
= log $\frac{28}{2 n}$
$\frac{28}{2 n}$ = 21
= 2
2n = 28
∴ n = 14

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Logarithms

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Question 5 Report

In 1984, Ike was 24 yrs old and his father was 45 yrs old. In what year was Ike exactly half his father's age?

1981

1979

1982

1978

Answer Details

Let the no. of years be y
24 - y = $\frac{1}{2}$ (45 - y)
45 - y = 2(24 - y)
45 - y = 48 - 2y
2y - y = 48 - 45
∴ y = 3
The exact year = 1984
1984 - 3 = 1981

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Question 6 Report

Factorize completely 8a + 125ax3

(2a + 5x2)(4 + 26ax)

a(2 + 5x)(4 - 10x + 25x2)

(2a + 5x)(4 - 10ax + 25x2)

a(2 + 5x)(4 + 10ax + 25x2)

Algebraic Expressions

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Question 7 Report

If P = 18, Q = 21, R = -6 and S = -4, Calculate $ \frac{(P-Q)^3+S^2}{R^3} + S^2 $

-11

216

11

216

-43

116

43

116

Answer Details

$\frac{(P - Q)^{3} + S^{2}}{R^{3}}$ = $\frac{(18 - 21)^{3} + (- 4)^{2}}{(- 6)^{3}}$
= $\frac{- 27 + 16}{R^{3}}$
= $\frac{- 11}{- 216}$
= $\frac{11}{216}$

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Algebraic Expressions

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Question 8 Report

If Musa scored 75 in biology instead of 57, his average mark in four subjects would have been 60. What was his total mark?

282

240

222

210

Answer Details

Let x represent Musa's total mark when he scores 57 in biology and Let Y represent Musa's total mark when he now scored 75 in biology, if he scored 75 in biology his new total mark will be $\frac{Y}{4}$ = 60, y = 4 x 60 = 240
To get his total mark when he scored 57, subtract 57 from 75 to give 18, then subtract this 18 from the new total mark(ie. 240)
= 240 - 18
= 222

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Question 9 Report

A regular polygon of n sides has 160o as the size of each interior angle. Find n

18

14

16

12

Polynomials

Triangles And Polygons

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Question 10 Report

Simplify $ \frac{0.0324 \times 0.00064}{0.48 \times 0.012} $

3.6 x 102

3.6 x 10-2

3.6 x 10-3

3.6 x 10-4

Answer Details

$\frac{0.0324 \times 0.00064}{0.48 \times 0.012}$ = $\frac{324 \times 10^{- 4} \times 10^{- 5}}{48 x 10^{- 2} x 12 x 10^{- 3}}$
$\frac{324 \times 64 \times 1 0^{- 9}}{48 \times 12 \times 10^{- 5}}$ = 36 x 10-4
= 3.6 x 10-3

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Question 11 Report

If U and V are two distinct fixed points and W is a variable points such that UWV is a right angle, what is the locus of W?

The perpendicular bisector of UV

A circle with UV as radius

A line parallel to the line UV

A circle with the line UV as the diameter

Loci

Statistics

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Question 12 Report

If $ \cos \theta = \frac{a}{b} $, find $ 1 + \tan^2 \theta $

\frac{b^{2}}{a^{2}}

\frac{a^{2}}{b^{2}}

\frac{a^{2} + b^{2}}{b^{2} - a^{2}}

\frac{2 a^{2} + b^{2}}{a^{2} + b^{2}}

Answer Details

cos $θ$ = $\frac{a}{b}$ , Sin $θ$ = $\sqrt{\frac{b^{2} - a^{2}}{a}}$
Tan $θ$ = $\sqrt{\frac{b^{2} - a^{2}}{a^{2}}}$ , Tan 2 = $\sqrt{\frac{b^{2} - a^{2}}{a^{2}}}$
1 + tan2 $θ$ = 1 + $\frac{b^{2} - a^{2}}{a^{2}}$
= $\frac{a^{2} + b^{2} - a^{2}}{a^{2}}$
= $\frac{b^{2}}{a^{2}}$

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Sine, Cosine And Tangent Of An Angle

Trigonometry

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Question 13 Report

If x varies directly as y3 and x = 2 when y = 1, find x when y = 5

2

10

125

250

Answer Details

x $α$ y3
x = ky3
k = $\frac{x}{y^{3}}$
when x = 2, y = 1
k = 2
Thus x = 2y3 - equation of variation
= 2(5)3
= 250

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Variation

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Question 14 Report

The people in a city with a population of 0.9 million were grouped according to their ages. Use the diagram to determine the number of people in the 15 - 29 years group

29 x 104

26 x 104

16 x 104

13 x 104

Answer Details

15 - 29 years is represented by 104 $^{\circ}$

Number of people in the group is $\frac{104}{360}$ x 0.9m

= 260000 = 26 x 104

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Number Bases

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Question 15 Report

Of the nine hundred students admitted in a university in 1979, the following was the distribution by state: Anambrs 185, Imo 135, Kaduna 90, Kwara 110, Ondo 155, Oyo 225. In a pie chart drawn to represent this distribution, The angle subtended at the centre by anambra is

50o

65o

74o

88o

Answer Details

Anambra = $\frac{185}{900}$ x $\frac{360}{1}$
= 74o

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Variation

Euclidean Geometry

Sets

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Question 16 Report

If $ y = \frac{x}{x - 3} + \frac{x}{x + 4} $ find y when $ x = -2 $

-

\frac{3}{5}

\frac{3}{5}

-

\frac{7}{5}

\frac{3}{5}

Answer Details

y = $\frac{x}{x - 3}$ + $\frac{x}{x + 4}$ when x = -2
y = $\frac{- 2}{- 5}$ + $\frac{(- 2)}{- 2 + 4}$
= $\frac{- 2}{5}$ + $\frac{- 2}{2}$
= $\frac{4 + - 10}{10}$
= $\frac{- 14}{10}$
= - $\frac{7}{5}$

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Algebraic Expressions

Measures Of Location

Quadratic Equations

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Question 17 Report

A girl walks 45 meters in the direction 050o from a point Q to a point X. She then walks 24 meters in the direction 140o from X to a point Y. How far is she then from Q?

69m

57m

51m

21m

Answer Details

QY = 452 + 242 = 2025 + 576
= 2601
QY = $\sqrt{2601}$
= 51

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Euclidean Geometry

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Question 18 Report

Divide the L.C.M of 48, 64, and 80 by their H.C.F

20

30

48

60

Answer Details

48 = 24 x 3, 64 = 26, 80 = 24 x 5
L.C.M = 26 x 3 x 5
H.C.F = 24
$\frac{2^{6} \times 3 \times 5}{2^{4}}$
= 22 x 3 x 5
= 4 x 3 x 5
= 12 x 5
= 60

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Number Bases

Sets

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Question 19 Report

A man kept 6 black, 5 brown and 7 purple shirts in a drawer. What is the probability of his picking a purple shirt with his eyes closed?

\frac{1}{7}

\frac{11}{18}

\frac{7}{18}

\frac{7}{11}

Probability

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Question 20 Report

The figure is a solid with the trapezium PQRS as its uniform cross-section. Find its volume

120m2

576m3

816m3

1056m3

Answer Details

Volume of solid = cross section x H

Since the cross section is a trapezium

= $\frac{1}{2} (6 + 11) \times 12 \times 8$

= 6 x 17 x 8 = 816m3

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Volumes

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Question 21 Report

Factorize (4a + 3)2 - (3a - 2)2

(a + 1)(a + 5)

(a - 5)(7a - 1)

(a + 5)(7a + 1)

a(7a + 1)

Algebraic Expressions

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Question 22 Report

Simplify $ \frac{9^{\frac{1}{3}} \times 27^{-\frac{1}{3}}}{3^{-\frac{1}{6}} \times 3^{\frac{2}{3}}} $

1

3

1

3

9

Answer Details

\frac{9^{\frac{1}{3}} \times 27^{- \frac{1}{3}}}{3^{- \frac{1}{6}} \times 3^{\frac{2}{3}}}

=

\frac{(3^{2})^{\frac{1}{3}} \times (3^{3})^{- \frac{1}{3}}}{3^{- \frac{1}{6}} \times 3^{\frac{2}{3}}}

=

\frac{3^{\frac{2}{3}} \times 3^{\frac{2}{3}}}{3^{- \frac{1}{6}} \times 3^{\frac{2}{3}}}

=

\frac{3^{\frac{5}{6}}}{3^{- \frac{5}{6}}}

= 1

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Representation Of Data

Probability

Number Bases

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Question 23 Report

Find all real numbers $x$ which satisfy the inequality $\frac{1}{3}(x + 1) - 1 > \frac{1}{5}(x + 4)$

x < 11

x < -1

x > 6

x > 11

Answer Details

$\frac{1}{3}$ (x + 1) - 1 > $\frac{1}{5}$ (x + 4)
= $\frac{x + 1}{3}$ - 1 > $\frac{x + 4}{5}$
$\frac{x + 1}{3}$ - $\frac{x + 4}{5}$ - 1 > 0
= $\frac{5 x + 5 - 3 x - 12}{15}$
= 2x - 7 > 15
= 2x > 12
= x > 11

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Inequalities

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Question 24 Report

Simplify $ \frac{1}{x-2} + \frac{1}{x+2} + \frac{2x}{x^2-4} $

2x

(x−2)(x+2)(x²−4)

2x

x²−4

x

x²−4

4x

x²−4

Answer Details

$\frac{1}{x - 2}$ + $\frac{1}{x + 2}$ + $\frac{2 x}{x^{2} - 4}$
= $\frac{(x + 2) + (x - 2) + 2 x}{(x + 2) (x - 2)}$
= $\frac{4 x}{x^{2} - 4}$

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Algebraic Expressions

Algebraic Fractions

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Question 25 Report

If 5(x + 2y) = 5 and 4(x + 3y) = 16, find 3(x + y)

7

1

3

27

Algebraic Expressions

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Question 26 Report

A number of pencils were shared out among Bisi, Sola and Tunde in the ratio of 2 : 3 : 5 respectively. If Bisi got 5, how many were share out?

15

25

30

50

Answer Details

Let x r3epresent total number of pencils shared

B : S : T = 2 + 3 + 5 = 10

2 : 3 : 5

= $\frac{2}{10}$ x y

= 5

2y =5

2y = 50

∴ y = $\frac{50}{2}$

= 25

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Sine, Cosine And Tangent Of An Angle

Financial Arithmetic

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Question 27 Report

The ages of Tosan and Isa differ by 6 and the product of their ages is 187. Write their ages in the form (x, y), where x > y.

(12, 6)

(23, 17)

(17, 11)

(18, 12)

Solution Of Linear Equations

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Question 28 Report

An arc of a circle of radius 6cm is 8cm long. Find the area of the sector

5

\frac{1}{3}

cm2

24cm2

36cm2

84cm2

Answer Details

Radius of the circle r = 6cm, Length of the arc = 8cm
Area of sector = $\frac{θ}{360}$ x 2 $π$ r2........(i)
Length of arc = $\frac{θ}{360}$ x 2 $π$ r........(ii)
from eqn. (ii) $θ$ = $\frac{240}{π}$ , subt. for $θ$ in eqn (i)
Area x $\frac{240}{1}$ x $\frac{1}{360}$ x $\frac{π 6}{1}$
= 24cm2

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Coordinate Geometry

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Question 29 Report

An open rectangular box externally measures 4m x 3m x 4m. Find the total cost of painting the box externally if it costs ₦2.00 to paint one square meter

₦96.00

₦112.00

₦136.00

₦160.00

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Question 30 Report

In the diagram, PQ and RS are chords of a circle centre O which meet at T outside the circle. If TP = 24cm. TQ = 8cm and TS = 12cm, find TR.

16cm

14cm

12cm

8cm

Answer Details

PT x QT = TR x TS

24 x 8 = TR x 12

TR = $\frac{24 \times 8}{12}$

= = 16cm

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Introductory Calculus

Loci

Inequalities

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Question 31 Report

Factorize x2 + 2a + ax + 2x

(x + 2a)(x + 1)

(x + 2a)(x - 1)

(x2 - 1)(x - a)

(x + 2)(x + a)

Algebraic Expressions

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Question 32 Report

If $ \frac{a}{c} = \frac{c}{d} = k $, find the value of $ \frac{3a^2-ac+c^2}{3b^2-bd+d^2} $

3k2

2k - k2

\frac{3 b^{2} k^{2} - b k^{2} d + d k^{2}}{3 b^{2} - b d + d^{2}}

k2

Answer Details

$\frac{a}{c}$ = $\frac{c}{d}$ = k
∴ $\frac{a}{b}$ = bk
$\frac{c}{d}$ = k
∴ c = dk
= $\frac{3 a^{2} - a c + c^{2}}{3 b^{2} - b d + d^{2}}$
= $\frac{3 (b k)^{2} - (b k) (d k) + d k^{2}}{3 b^{2} - b d + a^{2}}$
= $\frac{3 b^{2} k^{2} - b k^{2} d + d k^{2}}{3 b^{2} - b d + d^{2}}$
k = $\frac{3 b^{2} k^{2} - b k^{2} d + d k^{2}}{3 b^{2} - b d + d^{2}}$

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Circles

Algebraic Expressions

Quadratic Equations

Graphs Of Linear And Quadratic Functions

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Question 33 Report

Find the total surface area of solid cone of radius $2\sqrt{3}$ cm and slanting side $4\sqrt{3}$

8

\sqrt{3} π

cm2

24

π

cm2

15

\sqrt{3} π

cm2

36

π

cm2

Answer Details

Total surface area of a solid cone
r = 2 $\sqrt{3}$
= $π r^{2}$ + $π$ rH
H = 4 $\sqrt{3}$ , $π$ r(r + H)
∴ Area = $π$ 2 $\sqrt{3}$ [2 $\sqrt{3}$ + 4 $\sqrt{3}$ ]
= $π$ 2 $\sqrt{3}$ (6 $\sqrt{3}$ )
= 12 $π$ x 3
= 36 $π$ cm2

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Circles

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Question 34 Report

Solve the equation 3x2 + 6x - 2 = 0

x = -1

\pm

\frac{\sqrt{3}}{3}

x = -1

\pm

\frac{\sqrt{15}}{3}

x = -2

\pm

2

x = 3

\pm

\frac{\sqrt{3}}{15}

Answer Details

3x2 + 6x - 2 = 0
Using almighty formula i.e. x = $\frac{b \pm \sqrt{b^{2} - 4 a c}}{2 a}$
a = 3, b = 6, c = -2
x = $\frac{- 6 \pm \sqrt{6^{2} - 4 (3) (- 2)}}{2 (3)}$
x = $\frac{6 \pm \sqrt{36 - 24}}{6}$
x = $\frac{6 \pm \sqrt{60}}{6}$
x = $\frac{- 6 \pm \sqrt{4 \times 15}}{6}$
x = $\frac{- 1 \pm \sqrt{15}}{3}$

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Quadratic Equations

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Question 35 Report

In the figure, $ \triangle $PQT is isosceles. PQ = QT, SRQ = $35^\circ$, TPQ = $20^\circ$ and PQR is a straight line.Calculate TSR

20

^{\circ}

55

^{\circ}

75

^{\circ}

140

^{\circ}

Answer Details

Given $△$ isosceles PQ = QT, SRQ = 35 $^{\circ}$

TPQ = 20 $^{\circ}$

PQR = is a straight line

Since PQ = QT, angle P = angle T = 20 $^{\circ}$

Angle PQR = 180 $^{\circ}$ - (20 + 20) = 140 $^{\circ}$

TQR = 180 $^{\circ}$ - 140 $^{\circ}$ = 40 $^{\circ}$ < on a straight line

QSR = 180 $^{\circ}$ - (40 + 35) $^{\circ}$ = 105 $^{\circ}$

TSR = 180 $^{\circ}$ - 105 $^{\circ}$

= 75 $^{\circ}$

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Question 36 Report

At what points does the straight line y = 2x + 1 intersect the curve y = 2x2 + 5x - 1?

(-2, -3) and(

\frac{1}{2}

, 0)

(1, 0), (1, 3)

(4, 0) and (0,1)

(2, 0) and (0,1)

Graphs Of Linear And Quadratic Functions

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Question 37 Report

Two chords QR and NP of a circle intersect inside the circle at x. If RQP = 37o, RQN = 49o and QPN = 35o, find PRQ

35o

37o

49o

59o

Polynomials

Euclidean Geometry

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Question 38 Report

Simplify $\left(\frac{1}{\sqrt{5}+\sqrt{3}}-\frac{1}{\sqrt{5}-\sqrt{3}}\right)x\frac{1}{\sqrt{3}}$

√3

√5

2√3

7

-2

-1

Answer Details

( $\frac{1}{\sqrt{5} + \sqrt{3}} - \frac{1}{\sqrt{5} - \sqrt{3}}$ x $\frac{1}{\sqrt{3}}$ )
$\frac{1}{\sqrt{5} + \sqrt{3}} - \frac{1}{\sqrt{5} - \sqrt{3}}$ = $\frac{\sqrt{5} - \sqrt{3} - (\sqrt{5} + \sqrt{3})}{(\sqrt{5} + \sqrt{3}) (5 \sqrt{3})}$
= $\frac{- 2 \sqrt{3}}{5 - 3}$
= $\frac{- 2 \sqrt{3}}{7}$

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Number Bases

Surds (radicals)

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Question 39 Report

Evaluate (212)3 - (121)3 + (222)3

(313)3

(1020)3

(1222)3

Number Bases

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Question 40 Report

Three boys shared some oranges. The first received 1/3 of the oranges and the second received 2/3 of the remaining. If the third boy received the remaining 12 oranges, how many oranges did they share

60

54

48

42

Number Bases

Modular Arithmetic

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Question 41 Report

The table below gives the scores of a group of students in a Mathematical test.

Scores	1	2	3	4	5	6	7	8
Frequency	2	4	7	14	12	6	4	1

If the mode in m and the number of students who scored 4 or less is s. What is (s, m)?

(27, 4)

(14,4)

(13,4)

(4,4)

Representation Of Data

Number Bases

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Question 42 Report

In $ \triangle $XYZ, determine the cosine of angle Z.

3

39

29

36

14

5

7

5

Answer Details

cos z = $\frac{y^{2} + x^{2} - z^{2}}{2 y x}$

= $\frac{9 + 36 - 16}{2 (3) (6)}$

= $\frac{29}{36}$

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Circles

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Question 43 Report

Find the smallest number by which 252 can be multiplied to obtain a perfect square

2

3

5

7

Introductory Calculus

Number Bases

Surds (radicals)

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Question 44 Report

PQ and PR are tangents from P to a circle centre O as shown in the figure. If QRP = $34^\circ$, find the angle marked x

34

^{\circ}

56

^{\circ}

68

^{\circ}

112

^{\circ}

Answer Details

From the circle centre 0, if PQ & PR are tangents from P and QRP = 34 $^{\circ}$

Then the angle marked x i.e. QOP

34 $^{\circ}$ x 2 = 68 $^{\circ}$

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Circles

Triangles And Polygons

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Question 45 Report

Find the values of x which satisfy the equation 16x - 5 x 4x + 4 = 0

1 and 4

-2 and 2

0 and 1

-1 and 0

Solution Of Linear Equations

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Question 46 Report

Find the probability that a number selected at random from 40 to 50 is a prime

2

11

5

11

3

11

4

11

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Question 47 Report

Udoh deposited ₦150.00 in the bank. At the end of 5 years the simple interest on the principal was ₦55.00. At what rate per annum was the interest paid?

11%

7