JAMB UTME - Mathematics - 1980

A cylindrical motor of height 12cm has uniform thickness of 2cm. If the diameter of its outer cross section is 10cm, Find the volume of the constituent material. (take \(\pi\) = \(\frac{22}{7}\)

\(\frac{6600}{7}\)cm³

270cm³

660cm³

\(\frac{4224}{7}\)cm³ Correct answer

\(\frac{1980}{7}\)cm³

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Question 6 Report

The ratio of the areas of similar triangles is necessarily equal to

the ratio of the corresponding sides

the ratio of the squares of corresponding sides Correct answer

the ratio of the corresponding heights of the triangles

half the ratio of the corresponding heights of the triangles

the ratio of the corresponding bases to the heights of the triangles

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Question 7 Report

fG is a given straight line and H is a fixed point. The construction marks shown in the diagram indicate the

perpendicular bisector of the line

perpendicular bisector of th eline JK

perpendicular bisector of the line HI

perpendicular from H to the line FG Correct answer

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Matrices And Determinants

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Question 8 Report

Solve the equation for the positive values of \(\theta\) less than 360^o. 3 tan \(\theta\) + 2 = -1

135^o or 315^o Correct answer

45^o or 135^o

315^o or 180^o

315v + 45^o

360^o or 315^o

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Trigonometry

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Question 9 Report

Rationalize the denominator of the given expression \(\frac{\sqrt{1 + a} - \sqrt{a}}{1 + a + \sqrt{a}}\)

1 + 2a - 2\(\sqrt{a(1 + a)}\) Correct answer

\(\sqrt{1(1 + a)}\)

2a - 2\(\sqrt{a(1 + a)}\)

1 + 2a - 2\(\sqrt{a + b}\)

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Surds (radicals)

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Question 10 Report

In the figure, PS = SR and Rx is a tangent to the circle at R, < QRX is equal to 72^o. Find

20^o

36^o Correct answer

72^o

30^o

15^o

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Circles

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Question 11 Report

In the figure, PS bisects angle QPR. Find the ratio SR:QR.

1:2

1:3

1:4 Correct answer

1:5

1:6

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Question 12 Report

Find the solution of the equation x + 2\(\sqrt{x - 8}\) = 0

(4, 16) Correct answer

(2, 4)

(4, 1)

(1, 16)

(16, 16)

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Surds (radicals)

Quadratic Equations

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Question 13 Report

If PN is perpendicular to QR, find the value of tan x.

\(\frac{5}{9}\)

\(\frac{3}{5}\) Correct answer

\(\frac{3}{4}\)

\(\frac{4}{3}\)

\(\frac{5}{3}\)

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Circles

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Question 14 Report

The median of the set of numbers 4, 9, 4, 13, 7, 14, 10, 17 is

\(\frac{19}{2}\) Correct answer

\(\frac{39}{4}\)

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Number Bases

Sets

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Question 15 Report

Find the product of (2\(\sqrt{y - 3y}\)) and (3y = 2y)

4y = y²

4y + 9y²

4y - 9y² Correct answer

-4y - 9y²

2y + y²

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Probability

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Question 16 Report

In the figure, If PT is parallel to RS, PQ = PT, and angle SQT = 90^o, Find x

35^o

50^o

55^o

70^o Correct answer

80^o

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Triangles And Polygons

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Question 17 Report

Simplify \(\frac{log_{10}8}{log_{10}4}\)

log10²

log10⁴

\(\frac{3}{2}\) Correct answer

None of the above

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Number Bases

Logarithms

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Question 18 Report

What factor is common to all the expressions x² - x, 2x² + x - 1 and x² - 1?

x + 1

x - 1

No common factor Correct answer

(2x - 1)

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Algebraic Expressions

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Question 19 Report

Find, correct to three significant figures, the value of \(\sqrt{41830}\).

205 Correct answer

647

2050

6470

64.7

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Surds (radicals)

Number Bases

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Question 20 Report

Find the missing numerator \(\frac{5}{x + 1}\) - \(\frac{3}{1 - x}\) - \(\frac{7x - 1}{x^2 - 1}\) = \(\frac{?}{x + 1}\).

-1

x - 1

\(\frac{3(1 - 5x)}{x + 1}\)

1 Correct answer

3(1 - 5x)

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Algebraic Fractions

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Question 21 Report

A solid cylinder of radius 3cm has a total surface area of 36\(\pi\)cm2. Find its height

2cm

3cm Correct answer

4cm

5cm

6cm

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Volumes

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Question 22 Report

If it is given that 5^{x + 1} + 5^x = 150, then the value of x is equal to

2 Correct answer

\(\frac{1}{2}\)

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Indices

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Question 23 Report

(\(\frac{x^a}{x^b}\))^{a + b} by (\(\frac{x^{a + b}}{x^{a - b}}\))^{\(\frac{a^2}{b}\)}

x^-a2

x^b2

x^{a2 - b2}

\(\frac{1}{x^{a2 + b2}}\) Correct answer

x^{b2 - a2}

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Number Bases

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Question 24 Report

A set of data contains a total of 130 items which are divided into six groups for display on a pie chart. If one group contains 26 items, then the sector representing this group on the pie chart contains an angle x^o at the centre of the circle where x is

36^o

60^o

70^o

72^o Correct answer

75^o

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Sets

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Question 25 Report

In triangle FGB, < G = 90^o, < H = 60^o, while in triangle XYZ, < X = 60^o and < Y = 30^o. Form XYZ write down the ratio equal to \(\frac{FG}{FH}\)

\(\frac{YX}{ZX}\)

\(\frac{YX}{YZ}\)

\(\frac{ZX}{YZ}\)

\(\frac{YZ}{XY}\) Correct answer

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Question 26 Report

The angles marked in the figure are measured in degrees. Find x.

45^o

90^o

60^o

15^o

30^o Correct answer

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Angles

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Question 27 Report

Find the area of the curved surface of a cone whose base radius is 6cm and whose height is 8cm. (take \(\pi\) = \(\frac{22}{7}\))

188.57cm² Correct answer

1320cm²

188cm²

188.08cm²

10cm²

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Areas

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Question 28 Report

If x varies inversely as y, and y varies directly as the square root of z, and z varies directly \(\frac{1}{w^2}\) write down in words how x varies with w

x varies inversely as w²

x varies directly as w² Correct answer

x varies directly as w

x varies inversely as w

x varies directly as square root of w

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Question 29 Report

Evaluate \((2^{0} + 4^{-\frac{1}{2}})^{2}\)

\(\frac{1}{4}\)

\(\frac{5}{4}\)

\(\frac{9}{4}\) Correct answer

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Number Bases

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Question 30 Report

In the figure, FGHJ is a circle of radius 3cm centre O. FOH, GOJ are perpendicular diameters. With G as centre of an arc of a circle is drawn to pass through F and H. Find the length of the perimeter of the lunar portion shaded.

3\(\pi\)\(\sqrt{2 - 1}\)cm

\(\frac{9}{2}\)\(\pi\)cm

3 \(\pi\)(1 + \(\frac{2}{2}\)) Correct answer

3(1 + \(\frac{2}{2}\))

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Circles

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Question 31 Report

Simplify the given expression \(\sqrt{\frac{1 - cos x}{1 + cos x}}\)

\(\frac{1 - cos x}{sin x}\) Correct answer

1 - cos x

sin x

1 + cos x

\(\frac{1 + cos x}{sin x}\)

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Question 32 Report

When a dealer sells a bicycle for N81 he makes a profit of 80%. What did he pay for the bicycle?

N73

N74.52

N75 Correct answer

N87.48

N75.52

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Positive And Negative Integers, Rational Numbers

Areas

Financial Arithmetic

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Question 33 Report

If \(\theta\) = \(\frac{3}{2}\) and \(\theta\) is less than 90^o, calculate tan \(\frac{(90 - \theta)}{cos^2\theta}\)

\(\frac{1}{2}\)

\(\frac{4}{\sqrt{3}}\) Correct answer

\(\sqrt{3}\)

4\(\sqrt{3}\)

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Question 34 Report

Find the area of the shades segments in the figure

\(\sqrt{3}

4 \(\pi - \sqrt{3}\)

-\(\frac{2}{3} \pi\)

\(\frac{2\pi}{3}\) -3 Correct answer

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Introductory Calculus

Modular Arithmetic

Surds (radicals)

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Question 35 Report

Find the roots of the equation 10x² - 13x - 3 = 0

x = \(\frac{3}{5}\) or -\(\frac{1}{2}\)

x = \(\frac{3}{10}\) or -1

x = \(\frac{3}{10}\) or 1

x = \(\frac{1}{5}\) or \(\frac{-3}{2}\)

x = -\(\frac{1}{5}\) or \(\frac{3}{2}\) Correct answer

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Quadratic Equations

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Question 36 Report

In the figure, PQ\\SR, ST\\, ST\\RQ, PS = 7cm, PT = 7cm, SR = 4cm. Find the ratio of the area QRST to the area of PQRS.

56.77

56.105 Correct answer

28:105

28:49

56:49

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Mensuration

Triangles And Polygons

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Question 37 Report

A man and his wife went to buy an article costing N400. The woman had 10% of the cost and the man 40% of the remainder. How much did they have altogether?

N216

N200

N184 Correct answer

N144

N100

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Question 38 Report

Which of the following is NOT a factor of 12\(^{4}\) - 5\(^{4}\)?

49 Correct answer

169

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Logarithms

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Question 39 Report

Make c the subject of formula v = 1 - \(\frac{a}{5}\)(b + \(\frac{3c}{7}\))

[\(\frac{7}{3}\) + \(\frac{5}{a}\)(v - 1)] + b Correct answer

[\(\frac{7b}{3}\) + \(\frac{5}{a}\)(v - 1)]

\(\frac{7}{3}\)[b + \(\frac{5}{a}\)(v - 1)]

\(\frac{7}{3}\)[b + \(\frac{5v - 1}{a}\)]

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Question 40 Report

Three numbers, x, y and z are connected by the relationships y = \(\frac{4x}{9}\) + 1. If x = 99, find z

6\(\frac{1}{3}\)

21 Correct answer

176\(\frac{4}{9}\)

None of the above

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Solution Of Linear Equations

Simple Operations On Algebraic Expressions

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Question 41 Report

P and Q are fixed points and X is a variable point which moves so that angle PXQ = 45o. What is the locus of x?

A pair of straight lines parallel to PQ

The perpendicular bisector of PQ Correct answer

An arc of a circle passing through P and Q

A circle with diameter PQ

The bisector of angle PXQ

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Loci

Euclidean Geometry

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Question 42 Report

List all integer values of x satisfying the inequality -1 < 2x - 5 \(\leq\) 5

<2, 3, 4, 5

3, 4, 5 Correct answer

2, 3, 4

3, 4

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Inequalities

Areas

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Question 43 Report

Write the decimal number 39 to base 2

10011 base 2 Correct answer

110111 base 2

111001 base 2

100101 base 2

19.5 base 2

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Number Bases

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Question 44 Report

Which of the formula below represents the general terms of the following set of numbers(-1, \(\frac{2}{3}\), -\(\frac{1}{2}\), \(\frac{2}{5}\)......) for n = 1, 2, 3, 4.......?

\(\frac{2}{n + 1}\)

(-1)^{n + 1} \(\frac{2}{n + 1}\)

(-1)ⁿ \(\frac{2}{n + 1}\) Correct answer

\(\frac{n}{2n - 1}\)

(-1)ⁿ\(\frac{n}{2n - 1}\)

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Algebraic Fractions

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Question 45 Report

What will be the value of k so that the quadratic equation kx² - 4x + 1 = 0 has two equal roots?

4 Correct answer

\(\frac{1}{4}\)

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Quadratic Equations

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Question 46 Report

Simplify 10² + log₁₀5

500

2 log₁₀ 5

log₁₀5 x 10¹⁰⁰ Correct answer

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Lengths And Perimeters

Logarithms

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Question 47 Report

If a function is defined by f(x + 1) = 3x2 - x + 4, Find f(0).

8 Correct answer

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Solution Of Linear Equations

Functions And Relations

Algebraic Expressions

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Question 48 Report

The expression x3 - 4x2 + cx + d is such that x + 1 is its factor, and its value is 1 when x is -2. Find c and d.

c = 4 and d = -9

c = -4 and d = 9

c = -20 and d = -15 Correct answer

c = 20 and d = -15

c = -20 and d = 15

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Algebraic Expressions

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Question 49 Report

In a school, there are 35 students in class 2A and 40 in class 2B. The mean score for class 2 in an English Literature examination is 60.0 and that for 2B in the same paper is 52.5. Find, to one place of decimal, the mean for the combined classes

2.5

56.0 Correct answer

56.2

56.3

6.5

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Integration

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