JAMB UTME - Mathematics - 1992

Question 1 Report

Find all values of x satisfying the inequality $ -11 \le 4 - 3x \le 28 $

-5

\leq

x v 8

5

\leq

x

\leq

8

-8

\leq

x

\leq

5

-5 < x

\leq

8

Variation

Linear Inequalities

Inequalities

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Question 2 Report

$x$	2	4	6	8
$f$	4	$y$	6	5

If the mean of the above frequency distribution is 5.2, find y

2, 1

1, 2

1, 5

5, 2

Answer Details

Mean

\bar{x}

=

\frac{\sum f x}{\sum f}

=

\frac{5.2}{1}

=

\frac{8 + 4 y + 36 + 40}{4 + y + 6 + 5}

=

\frac{5.2}{1}

=

\frac{84 + 4 y}{15 + y}

= 5.2(15 + y)
= 84 + 4y
= 5.2 x 15 + 5.2y
= 84 + 4y
= 78 + 5.2y
= 84 = 4y
= 5.2y - 4y
= 84 - 78
1.2y = 6
y =

\frac{6}{1.2}

=

\frac{60}{12}

= 5

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Probability

Linear Inequalities

Statistics

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Question 3 Report

Find the sum to infinity to the following series $3 + 2 + \frac{4}{3} + \frac{8}{9} + \frac{16}{17} + .....$

1270

190

18

9

Answer Details

3 + 2 +

\frac{4}{3}

+

\frac{8}{9}

+

\frac{16}{17}

+ .....
a = 3
r =

\frac{2}{3}

s

α

=

\frac{a}{1 - r}

=

\frac{3}{1 - \frac{2}{3}}

=

\frac{3}{\frac{1}{3}}

= 3 x 3
= 9

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Sequence And Series

Progression

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Question 4 Report

A student has 5 courses to take from Mathematics and physics. There are 4 courses in Mathematics and 3 in Physics which he can choose his courses so that he takes exactly two courses in Physics?

48

12

10

7

Sets

Progression

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Question 5 Report

If U = (1, 2, 3, 6, 7, 8, 9, 10) is the universal set. E = (10, 4, 6, 8, 10) and F = is odd. Find (E ∩ F), where means x 1x2 = 26, x the complement of a set

(0)

U

(8)

ϕ

Answer Details

U = (1, 2, 3, 6, 7, 8, 9, 10)
E = (10, 4, 6, 8, 10)
F = (x : x2 = 26, x is odd)
∴ F =

ϕ

Since x2 = 26 = 64
x = + which is even
∴ E ∩ F =

ϕ

Since there are no common elements

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Sets

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Question 6 Report

Five people are to be arranged in a row for a group photograph. How many arrangements are the if a married couple in the group insist on sitting next to each other?

48

24

20

10

Permutation And Combination

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Question 7 Report

Simplify $5\sqrt{18} - 3\sqrt{72} + 4\sqrt{50}$

17

\sqrt{4}

4

\sqrt{17}

17

\sqrt{2}

12

\sqrt{4}

Answer Details

To simplify these expressions, we need to simplify the numbers under the radical sign first by factoring out their perfect square factors. For the first expression 5√18, we can factor out the perfect square factor of 9, which leaves us with 5√2√2√2 or 10√2. For the second expression, we can simplify √72 to √(36*2), and since 36 is a perfect square, we can factor it out, which leaves us with 3√2*6, or 3√2*2√3, which simplifies to 6√6. For the third expression, we can simplify √50 to √(25*2), and since 25 is a perfect square, we can factor it out, which leaves us with 2√2*5, or 2√2*√5, which simplifies to 2√10. Putting it all together, we have: 5√18 = 10√2 - 3√72 = 6√6 - 4√50 = 2√10 Now we can substitute these simplified expressions back into the original expressions: 10√2 - 6√6 + 2√10 To simplify this expression, we can group like terms. The coefficients of √6 are -6 and 0, since there are no other terms with √6. The coefficients of √10 are 2 and 0, since there are no other terms with √10. The coefficient of √2 is 10. So our simplified expression is: 10√2 - 6√6 + 2√10 = 10√2 - 6√6 + 2√10 = 10√2 - 6√(2*3) + 2√(2*5) = 10√2 - 6√2√3 + 2√2√5 = (10-6√3+2√5)√2 Therefore, the answer is 17√2.

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Sequence And Series

Surds (radicals)

Areas

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Question 8 Report

Make t the subject of formula $S = ut + \frac{1}{2}at^2$

\frac{1}{a} - u + \sqrt{u^{2} - 2 a s}

\frac{1}{a} u \pm (u^{2} - 2 a s)

\frac{1}{a} u \pm \sqrt{2 a s}

\frac{1}{a} - u + \sqrt{2 a s}

Answer Details

Given S = ut +

\frac{1}{2} a t^{2}

S = ut +

\frac{1}{2} a t^{2}

∴ 2S = 2ut + at2
= at2 + 2ut - 2s = 0
t =

\frac{- 2 u \pm 4 u^{2} + 2 a s}{2 a}

= -2u

π

\frac{\sqrt{u^{2} 4 u^{2} + 2 a s}}{2 a}

= $\frac{1}{a}$ (-u + $\sqrt{U^{2} - 2 a s}$ )

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Change Of Subject Of A Formula/Relation

Statistics

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Question 9 Report

A man invested a sum of ₦280.00 partly at 5% and partly at 4%, if the total interest is ₦12.80 per annum, find the amount invested at 5%.

₦14.00

₦120.00

₦140.00

₦160.00

Modular Arithmetic

Ratio, Proportions And Rates

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Question 10 Report

No. of children	0	1	2	3	4	5	6
No. of families	7	11	6	7	7	5	3

Find the mode and median respectively of the distribution above

2, 1

1, 2

1, 5

5, 2

Probability

Statistics

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Question 11 Report

Solve the equation $y^2 - 11y + 24 = 0$

8,3

4,9

6,4

9,-8

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Question 12 Report

The sketch is the curve of y = ax2 + bx + c. Find a, b and c respectively

1, 0, -4

-2, 2, -4

0, 1, -4

2, -2, -4

-1, 0, 4

Quadratic Equations

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Question 13 Report

If the angles of quadrilateral are (P + 10)o(2P - 30)o(3P + 20)o and 4po, find p

63

40

36

28

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Question 14 Report

$p = \begin{vmatrix} x & 3 & 0 \\ 2 & y & 3 \\ 4 & 2 & 4 \end{vmatrix}$

$q = \begin{vmatrix} x & 2 & z \\ 3 & y & 2 \\ 0 & 3 & z \end{vmatrix}$ Where $p^T$ is the transpose p calculate $/p^T/$ when x = 0, y = 1 and z = 2

48

24

-24

-48

Answer Details

=

| \begin{matrix} 0 & 3 & 0 \\ 2 & 1 & 3 \\ 4 & 2 & 2 \end{matrix} |

PT =

| \begin{matrix} 0 & 2 & 4 \\ 2 & 1 & 3 \\ 0 & 3 & 2 \end{matrix} |

/pT/ =

| \begin{matrix} 0 & 2 & 4 \\ 3 & 1 & 3 \\ 0 & 3 & 2 \end{matrix} |

= 0[2 - 6] - 2[6 - 0] + 4[9 - 0]
= 0 - 12 + 36 = 24

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Matrices And Determinants

Number Bases

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Question 15 Report

x is a point due east of point Y on a coast Z is another point on the coast but 63m due south of y. If the distance ZX is 12Km. Calculate the bearing of Z from X

240o

210o

150o

60o

Bearings

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Question 16 Report

If the scores of 3 students in a test are 5, 6 and 7, find the standard deviation of their scores

\frac{2}{3}

\frac{2}{3} \sqrt{3}

\sqrt{\frac{2}{3}}

\sqrt{\frac{3}{2}}

Measures Of Dispersion

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Question 17 Report

Given that $ \theta $ is an acute angle and $ \sin \theta = \frac{m}{n} $, find $ \cos \theta $

\frac{\sqrt{n^{2} - m^{2}}}{m}

\frac{\sqrt{(n + m) (n - m)}}{n}

\frac{m}{\sqrt{n^{2} - m^{2}}}

\sqrt{\frac{n}{n^{2} - m^{2}}}

Sine, Cosine And Tangent Of An Angle

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Question 18 Report

Evaluate $ \frac{(x-2)(x^2+3x-2)}{x^2-4} $

7

2

3

4

Algebraic Expressions

Quadratic Equations

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Question 19 Report

The probability of an event P is $ \frac{3}{4} $ while that of another event Q is $ \frac{1}{6} $. If the probability of both P and Q is $ \frac{1}{2} $. What is the probability of either P or Q.

\frac{1}{96}

\frac{1}{8}

\frac{5}{6}

\frac{11}{12}

Probability

Surds (radicals)

Probability

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Question 20 Report

In the figure, if XZ is 10cm, calculate RY in cm.

1010(1 -

\frac{1}{\sqrt{2}}

)

10(1 -

\frac{1}{\sqrt{3}}

)

10(1 -

\frac{1}{\sqrt{2}}

)

10[1 -

\sqrt{3}

]

Circles

Triangles And Polygons

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Question 21 Report

For an arithmetical sequence, the first term is 2 and the common difference is 3. Find the sum of the first 11 terms

157

187

197

200

Sequence And Series

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Question 22 Report

If the binary operation $ \ast $ is defined by $ m \ast n = mn + m + n $ for any real number m and n, find the identity of the elements under this operation

e = 1

e = -1

e = -2

e = 0

Answer Details

m

*

n = mn + m + n
m

*

e = me + m + e, e

*

m = e
∴ me + m + e, m(e + 1)e - e = 0
e + 1 = 0
∴ e = -1

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Binary Operations

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Question 23 Report

What is the value of x satisfying the equation $ \frac{4^{2x}}{4^{3x}} = 2 $?

-2

-

\frac{1}{2}

\frac{1}{2}

2

Answer Details

\frac{4^{2 x}}{4^{3 x}}

= 2
42x - 3x = 2
4-x = 2
(22)-x
= 21
Equating coefficients: -2x = 1
x = -

\frac{1}{2}

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Question 24 Report

Find n if 34n = 100112

5

6

7

8

Number Bases

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Question 25 Report

Determine the distance on the earth's surface between two town P (latoN, Long 20oN) and Q(Lat 60oN, Long 25oW) (Radius of the earth = 6400km)

\frac{800 π}{9}

km

\frac{800 \sqrt{3 π}}{9}

km

800

π

km

800

\sqrt{3 π}

km

Coordinate Geometry

Angles And Intercepts On Parallel Lines

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Question 26 Report

What is the n-th term of the sequence 2, 6, 12, 20...?

4n - 2

2(3n - 1)

n2 + n

n2 + 3n + 2

Sequence And Series

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Question 27 Report

If two dice are thrown together, what is the probability of obtaining at least a score of 10?

\frac{1}{6}

\frac{1}{12}

\frac{5}{6}

\frac{11}{12}

Answer Details

The total sample space when two dice are thrown together is 6 x 6 = 36

\begin{matrix} 1 & 2 & 3 & 4 & 5 & 6 \\ 1. & 1.1 & 1.2 & 1.3 & 1.4 & 1.5 & 1.6 \\ 2 & 2.1 & 2.2 & 2.3 & 2.4 & 2.5 & 2.6 \\ 3 & 3.1 & 3.2 & 3.3 & 3.4 & 3.5 & 3.6 \\ 4 & 4.1 & 4.2 & 4.3 & 4.4 & 4.5 & 4.6 \\ 5 & 5.1 & 5.2 & 5.3 & 5.4 & 5.5 & 5.6 \\ 6 & 6.1 & 6.2 & 6.3 & 6.4 & 6.5 & 6.6 \end{matrix}

At least 10 means 10 and above
P(at least 10) =

\frac{6}{36}

=

\frac{1}{6}

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Probability

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Question 28 Report

The radius of a circle is given as 5cm subject to an error of 0.1cm. What is the percentage error in the area of the circle?

\frac{1}{25}

\frac{1}{4}

4

25

Answer Details

Area =

π r^{2} = π \times 5 \times 5 = 25 π

Area error =

π \times {0.1}^{2} = 0.01 π

∴

% error =

\frac{0.01 π}{25 π} \times 10

= \frac{0.01}{25} \times 100 = \frac{1}{25}

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Representation Of Data

Percentages

Probability

Fractions, Decimals, Approximations And Percentage

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Question 29 Report

A cliff on the bank of a river is 300 meter high. if the angle of depression of a point on the opposite side of the river is $60^\circ$, find the width of the river.

100

75√3 m

100√3m

200√3m

Trigonometry

Areas

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Question 30 Report

Simplify $\frac{(1.25 \times 10^{-4}) \times (2.0 \times 10^{-1})}{(6.25 \times 10^{5})}$

4.0 x 10-3

5.0 x 10-2

2.0 x 10-1

5.0 x 10-3

Answer Details

\frac{(1.25 \times 10^{- 4}) \times (2.0 \times 10^{- 1})}{(6.25 \times 10^{5})}

=

\frac{1.25 \times 2}{6.25}

x 104 - 1 - 5

\frac{2.50}{6.25}

x 10-2 =

\frac{250}{625}

x 10-2
0.4 x 10-2 = 4.0 x 10-3

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Number Bases

Sets

Logarithms

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Question 31 Report

What is the perpendicular distance of a point (2, 3) from the line 2x - 4y + 3 = 0?

\frac{\sqrt{5}}{2}

\frac{\sqrt{5}}{20}

\frac{5}{\sqrt{13}}

6

Answer Details

2x - 4y + 3 = 0
Required distance =

\frac{(2 \times 2) + 3 (- 4) + 3}{\sqrt{2^{2}} + (- 4)^{2}}

=

\frac{4 - 12 + 3}{\sqrt{20}}

=

\frac{- 5}{- 2 \sqrt{5}}

= \frac{\sqrt{5}}{2}

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Euclidean Geometry

Coordinate Geometry

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Question 32 Report

In the figure, PQr is a semicircle while PQ and Qr are chords. QS is the perpendicular from Q to the diameter PR. What is the expression for QS?

QS = PS

\sqrt{S R}

QS =

\sqrt{(P S S R)}

QS = 2

\sqrt{(P S S R)}

QS =

\frac{1}{\sqrt{2}}

\sqrt{(P S S R)}

Polynomials

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Question 33 Report

If x + 1 is a factor of x3 + 3x2 + kx + 4, find the value of k

6

-6

8

-8

Algebraic Expressions

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Question 34 Report

a student blows a balloon and its volume increases at a rate of $ \pi(20 - t^2)\text{ cm}^3\text{ s}^{-1} $ after t seconds. If the initial volume is 0 cm³, find the volume of the balloon after 2 seconds

37.00 $π$

37.33 $π$

40.00 $π$

42.67 $π$

Integration

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Question 35 Report

The diagram is a circle with centre O. Find the area of the shaded portion.

9(

π - 2) c m^{2}

18

π c m^{2}

9

π c m^{2}

36

π c m^{2}

Answer Details

To find the area of the shaded portion, we need to subtract the area of the triangle from the area of the sector. First, we need to find the radius of the circle. Since the diameter is given as 6cm, the radius is half of it which is 3cm. Next, we need to find the angle of the sector. We can do this by using the properties of the isosceles triangle. The angle at the centre of the circle is twice the angle at the circumference of the circle. So, the angle at the centre of the circle is: 2 x (180 - 90) = 180 degrees This means that the angle of the sector is 180 degrees. Now we can calculate the area of the sector: Area of sector = (angle/360) x πr^2 = (180/360) x π(3)^2 = 4.5π cm^2 Next, we need to find the area of the triangle. Since the triangle is isosceles, the base angles are equal, and we can use trigonometry to find the height. The base of the triangle is equal to the diameter of the circle which is 6cm. Using trigonometry, we know that: tan(45) = height/base height = base x tan(45) height = 6 x 1 height = 6cm Now we can calculate the area of the triangle: Area of triangle = (1/2) x base x height = (1/2) x 6 x 6 = 18 cm^2 Finally, we can calculate the area of the shaded portion by subtracting the area of the triangle from the area of the sector: Area of shaded portion = Area of sector - Area of triangle = 4.5π - 18 = (9/2)(π - 2) cm^2 Therefore, the answer is 9(π−2)cm2.

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Circles

Areas

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Question 36 Report

Evaluate logban if b = an

nn

n

\frac{1}{n}

\frac{1}{n^{2}}

Surds (radicals)

Logarithms

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Question 37 Report

Evaluate the integral $ \int_{\frac{\pi}{12}}^{\frac{\pi}{4}} 2 \cos 2x\, dx $

-

\frac{1}{2}

-1

\frac{1}{2}

1

Answer Details

let I =

\int_{\frac{π}{12}}^{\frac{π}{4}}

2 cos 2x dx
= 2(sin 2x)

\frac{π}{4}

(sin 2x)

\frac{π}{4}

\frac{π}{12}

(2)

\frac{π}{12}

= -1 -

\frac{1}{2}

=

\frac{1}{2}

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Integration

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Question 38 Report

ind then equation line through (5, 7) parallel to the line 7x + 5y = 12

5x + 7y = 120

7x + 5y = 70

x + y = 7

15x + 17y = 90

Answer Details

Equation (5, 7) parallel to the line 7x + 5y = 12
5Y = -7x + 12
y =

\frac{- 7 x}{5}

+

\frac{12}{5}

Gradient =

\frac{- 7}{5}

∴ Required equation =

\frac{y - 7}{x - 5}

=

\frac{- 7}{5}

i.e. 5y - 35 = -7x + 35
5y + 7x = 70

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Question 39 Report

If $x = 3 - \sqrt{3}$, find $x^2 + \frac{36}{x^2}$

9

18

24

27

Answer Details

x = 3 -

\sqrt{3}

x2 = (3 -

\sqrt{3}

)2
= 9 + 3 - 6

\sqrt{34}

= 12 - 6

\sqrt{3}

= 6(2 -

\sqrt{3}

)
∴ x2 +

\frac{36}{x^{2}}

= 6(2 -

\sqrt{3}

) +

\frac{36}{6 (2 - \sqrt{3})}

6(2 -

\sqrt{3}

) +

\frac{6}{2 - \sqrt{3}}

= 6(-

\sqrt{3}

) +

\frac{6 (2 + \sqrt{3})}{(2 - \sqrt{3}) (2 + \sqrt{3})}

= 6(2 -

\sqrt{3}

) +

\frac{6 (2 + \sqrt{3})}{4 - 3}

6(2 -

\sqrt{3}

) + 6(2 +

\sqrt{3}

) = 12 + 12
= 24

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Statistics

Quadratic Equations

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Question 40 Report

The angle between latitudes 30oS and 13oN is

17o

33o

43o

53o

Angles

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Question 41 Report

If in the diagram above, FG is parallel to KM, find the value of x

75o

95o

105o

125o

Angles And Intercepts On Parallel Lines

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Question 42 Report

If x = (all prime factors of 44) and y = (all prime factors of 60), the elements of X ∪ Y and X ∩ Y respectively are

(2, 4, 3, 5, 11) and (4)

(4, 3, 5, 11) and (3, 4)

(2, 5, 11) and (2)

(2, 3, 5, 11) and (2)

Sets

Algebraic Fractions

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Question 43 Report

$p = \begin{vmatrix} x & 3 & 0 \\ 2 & y & 3 \\ 4 & 2 & 4 \end{vmatrix}$

$q = \begin{vmatrix} x & 2 & z \\ 3 & y & 2 \\ 0 & 3 & z \end{vmatrix}$

pq is equivalent to

ppT

pp-1

qp

pp

Answer Details

=

| \begin{matrix} 0 & 3 & 0 \\ 2 & 1 & 3 \\ 4 & 2 & 2 \end{matrix} |

Q =

| \begin{matrix} 0 & 2 & 4 \\ 3 & 1 & 2 \\ 0 & 3 & 2 \end{matrix} |

= pT
pq = ppT

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Matrices And Determinants

Quadratic Equations

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Question 44 Report

Factorize 9p2 - q2 + 6qr - 9r2

(3p - 3q + r)(3p - q - 3r)

(6p - 3q - 3r)(3p - q - 4r)

(3p - q + 3r)(3p - q - 3r)

(3q - p + 3r)(3q - p + 3r)

Algebraic Expressions

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Question 45 Report

Obtain a maximum value of the function $f(x)=x^3-12x+11$.

-5

-2

2

27

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Question 46 Report

Ice forms on a refrigerator ice box at rate of (4 - 0.61)g per minute after 1 minute. If initially, there were 2g of ice, find the mass of ice formed in 5 minutes.

19.5

17.0

14.5

12.5