JAMB UTME - Mathematics - 1990

Swali 1 Ripoti

Each of the interior angles of a regular polygon is 140o. How many sides has the polygon?

9

8

7

5

Triangles And Polygons

Tazama Jibu

Swali 2 Ripoti

If $ \cos \theta = \frac{12}{13} $. Find $ \theta + \cos^2 \theta $

\frac{169}{25}

\frac{25}{169}

\frac{169}{144}

\frac{144}{169}

Maelezo ya Majibu

Cos

θ

=

\frac{12}{13}

x2 + 122 = 132
x2 = 169- 144 = 25
x = 25
= 5
Hence, tan

θ

=

\frac{5}{12}

and cos

θ

=

\frac{12}{13}

If cos2

θ

= 1 +

\frac{1}{t a n^{2} θ}

= 1 +

\frac{1}{\frac{(5)^{2}}{12}}

= 1 +

\frac{1}{\frac{25}{144}}

= 1 +

\frac{144}{25}

=

\frac{25 + 144}{25}

=

\frac{169}{25}

Tazama Jibu

Swali 3 Ripoti

The H.C.F. of a2bx + ab2x and a2b - b2 is

b

a + b

b(a

\div

b)

abx(a2 - b2)

Tazama Jibu

Swali 4 Ripoti

In the figure, YXZ = $30^\circ$, XYZ = $105^\circ$ and XY = 8cm. Calculate YZ

16

\sqrt{2}

cm

8

\sqrt{2}

cm

4

\sqrt{2}

cm

22cm

Maelezo ya Majibu

To find the length of YZ, we can use the sine rule which relates the sides and angles of a triangle. The sine rule states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all sides and angles in the triangle. In other words: a b c --- --- --- = 1 sinA sinB sinC where a, b, and c are the lengths of the sides of the triangle opposite the angles A, B, and C, respectively. Using the sine rule, we can find the length of YZ as follows: YZ XY XY -- = --- = --- sinX sinZ sinY Substituting the given values, we get: YZ 8 8 -- = --- = --- sin30 sin105 sin45 Using a calculator to evaluate the sines, we get: YZ 8 8 -- = --- = --- × (2 + √2) ≈ 8.485 0.5 0.966 0.707 Therefore, YZ is approximately equal to 8.485 cm, which is equivalent to 4√2 cm to two decimal places. Therefore, the correct option is: 4√2 cm.

Soma maelezo ya somo kwenye Introductory Calculus (WAEC)

Soma maelezo ya somo kwenye Angles Of Elevation And Depression (WAEC)

Soma maelezo ya somo kwenye Angles (WAEC)

Introductory Calculus

Angles Of Elevation And Depression

Angles

Tazama Jibu

Swali 5 Ripoti

If f(x - 4) = x2 + 2x + 3, Find, f(2)

6

11

27

51

Solution Of Linear Equations

Algebraic Expressions

Tazama Jibu

Swali 6 Ripoti

The mean of ten positive numbers is 16. When another number is added, the mean becomes 18. Find the eleventh number

3

6

18

38

Tazama Jibu

Swali 7 Ripoti

If the heights of two circular cylinder are in the ratio 2 : 3 and their volumes?

27 : 32

27 : 23

23 : 32

32 : 27

Maelezo ya Majibu

$\frac{h_{1}}{h_{2}}$ = $\frac{2}{3}$
h2 = $\frac{2 h_{1}}{3}$
$\frac{r_{1}}{r_{2}}$ = $\frac{9}{8}$
r2 = $\frac{9 r_{1}}{8}$
v1 = $π$ ( $\frac{9 r_{1}}{8}$ )2( $\frac{2 h_{1}}{3}$ )
= $π$ r1 2h1 x $\frac{27}{32}$
v = $\frac{π r_{1} 2 h_{1} \times \frac{27}{32}}{π r_{1} 2 h_{1}}$ = $\frac{27}{32}$
v2 : v1 = 27 : 32

Soma maelezo ya somo kwenye Matrices And Determinants (JAMB)

Matrices And Determinants

Tazama Jibu

Swali 8 Ripoti

Find the sum of the first twenty terms of the progression log a, log a2, log a3.....

log a20

log a21

log a200

log a210

Sequence And Series

Logarithms

Tazama Jibu

Swali 9 Ripoti

Divide x3 - 2x2 - 5x + 6 by (x - 1)

x2 - x - 6

x2 - 5x + 6

x2 - 7x + 6

x2 - 5x - 6

Algebraic Expressions

Tazama Jibu

Swali 10 Ripoti

What is the equation of the quadratic function represented by the graph?

y = x2 + x - 2

y = -x2 - x + 2

y = x2 - x - 2

y = -x2 + x + 2

Quadratic Equations

Tazama Jibu

Swali 11 Ripoti

Simplify $ \sqrt{27} + \frac{3}{\sqrt{3}} $

4

\sqrt{3}

\frac{4}{\sqrt{3}}

3

\sqrt{3}

\frac{\sqrt{3}}{4}

Positive And Negative Integers, Rational Numbers

Surds (radicals)

Tazama Jibu

Swali 12 Ripoti

At what value of x is the function x2 + x + 1 minimum?

1

\frac{3}{4}

\frac{5}{3}

9

Quadratic Equations

Tazama Jibu

Swali 13 Ripoti

A cylinder pipe, made of metal is 3cm thick.If the internal radius of the pope is 10cm.Find the volume of metal used in making 3m of the pipe.

153

π

cm3

207

π

cm3

15 300

π

cm3

20 700

π

cm3

Areas

Tazama Jibu

Swali 14 Ripoti

The locus of a point which moves so that it is equidistant from two intersecting straight lines is the

perpendicular bisector of the two lines

angle bisector of the two lines

bisector of the two lines

line parallel to the two lines

Loci

Circles

Coordinate Geometry

Tazama Jibu

Swali 15 Ripoti

The angle of a sector of s circle, radius 10.5cm, is 48 o, Calculate the perimeter of the sector

8.8cm

25.4cm

25.6cm

28.8cm

Coordinate Geometry

Tazama Jibu

Swali 16 Ripoti

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

\frac{3}{11}

\frac{5}{11}

\frac{3}{11}

\frac{4}{11}

Probability

Tazama Jibu

Swali 17 Ripoti

In triangles XYZ and XQP, XP = 4cm, XQ = 5cm and PQ = QY = 3cm. Find ZY

8cm

6cm

4cm

3cm

Triangles And Polygons

Tazama Jibu

Swali 18 Ripoti

Factorize 9(x + y)2 - 4(x - y)2

(x + y)(5x + y)

(x + y)2

(x + 5y)(5x + y)

5(x + y)2

Tazama Jibu

Swali 19 Ripoti

The perimeter of a rectangular lawn is 24m. If the area of the lawn is 35m2; how wide is the lawn?

5cm

7m

12m

14m

Lengths And Perimeters

Tazama Jibu

Swali 20 Ripoti

If $a = 2$, $b = -2$ and $c = -\frac{1}{2}$, evaluate $(ab^2 - bc^2)(a^2c - abc)$

2

-28

-30

-34

Maelezo ya Majibu

(ab2 - bc2)(a2c - abc)
[2(2)2 - (- 2x

\frac{1}{2}

)] [22(-

\frac{1}{2}

) - 2(-2)(-

\frac{1}{2}

)]
[8 =

\frac{1}{2}

][-2 - 2] =

\frac{17}{2}

x 42
= -34

Tazama Jibu

Swali 21 Ripoti

Find the solution of the equation $x - 8\sqrt{x} + 15 = 0$

3, 5

-3, -5

9, 25

-9, 25

Maelezo ya Majibu

x - 8

\sqrt{x}

+ 15 = 0
x + 15 = 8

\sqrt{x}

square both sides = (x + 15)2 = (8

\sqrt{x}

2
x2 + 225 + 30x = 64x
x2 + 225 + 30x - 64x = 0
x2 - 34x + 225 = 0
(x - 9)(x - 25) = 0
x = 9 or 25

Soma maelezo ya somo kwenye Solution Of Linear Equations (WAEC)

Solution Of Linear Equations

Tazama Jibu

Swali 22 Ripoti

Two numbers are removed at random from the numbers 1, 2, 3 and 4. What is the probability that the sum of the numbers removed is even?

\frac{2}{3}

\frac{2}{15}

\frac{1}{2}

\frac{1}{4}

Maelezo ya Majibu

\begin{array}{cc} 1 & 2 & 3 & 4 \\ 1 (1, 1) & (1, 2) & (1, 3) & (1, 4) \\ 2 (2, 1) & (2, 2) & (2, 3) & (2, 4) \\ 3 (3, 1) & (3, 2) & (3, 3) & (3, 4) \\ 4 (4, 1) & (4, 2) & (4, 3) & (4, 4) \end{array}

sample space = 16
sum of nos. removed are (2), 3, (4), 5
3, (4), 5, (6)
(4), 5, (6), 7
(5), 6, 7, (8)
Even nos. = 8 of them
Pr(even sum) =

\frac{8}{16}

=

\frac{1}{2}

Soma maelezo ya somo kwenye Representation Of Data (JAMB)

Soma maelezo ya somo kwenye Probability (WAEC)

Soma maelezo ya somo kwenye Permutation And Combination (JAMB)

Representation Of Data

Probability

Permutation And Combination

Tazama Jibu

Swali 23 Ripoti

Simplify $ \left(\frac{1}{x^{-1}} + \frac{1}{y^{-1}}\right)^{-1} $

\frac{x}{y}

xy

\frac{x}{y}

(xy)-1

Maelezo ya Majibu

Simplify (

\frac{1}{x^{- 1}} + \frac{1}{y^{- 1}}

)-1 = (

\frac{1}{x^{- 1}} + \frac{1}{y^{- 1}}

)-1
= (x + y)-1 =

\frac{(x)}{y}

=

\frac{x}{y}

Soma maelezo ya somo kwenye Solution Of Linear Equations (WAEC)

Soma maelezo ya somo kwenye Circles (WAEC)

Soma maelezo ya somo kwenye Algebraic Expressions (WAEC)

Solution Of Linear Equations

Circles

Algebraic Expressions

Tazama Jibu

Swali 24 Ripoti

Three children shared a basket of mangoes in such a way that the first child took $ \frac{1}{4} $ of the mangoes and the second $ \frac{3}{4} $ of the remainder. What fraction of the mangoes did the third child take?

\frac{3}{16}

\frac{7}{16}

\frac{9}{16}

\frac{13}{16}

Maelezo ya Majibu

You can use any whole numbers (eg. 1. 2. 3) to represent all the mangoes in the basket.
If the first child takes

\frac{1}{4}

it will remain 1 -

\frac{1}{4}

=

\frac{3}{4}

Next, the second child takes

\frac{3}{4}

of the remainder
which is

\frac{3}{4}

i.e. find

\frac{3}{4}

of

\frac{3}{4}

=

\frac{3}{4}

x

\frac{3}{4}

=

\frac{9}{16}

the fraction remaining now =

\frac{3}{4}

-

\frac{9}{16}

=

\frac{12 - 9}{16}

=

\frac{3}{16}

Soma maelezo ya somo kwenye Number Bases (WAEC)

Number Bases

Tazama Jibu

Swali 25 Ripoti

At what rate would a sum of ₦100.00 deposited for 5 years raise an interest of ₦7.50?

1

\frac{1}{2}

%

2

\frac{1}{2}

%

1.5%

25%

Percentages

Tazama Jibu

Swali 26 Ripoti

Given that $x^2 + y^2 + z^2 = 194$, calculate z if $x = 7$ and $\sqrt{y} = 3$

√10

8

12.2

13.4

Maelezo ya Majibu

Given that x2 + y2 + z2 = 194, calculate z if x = 7 and $\sqrt{y}$ = 3
x = 7
∴ x2 = 49
$\sqrt{y}$ = 3
∴ y2 = 81 = x2 + y2 + z2 = 194
49 + 81 + z2 = 194
130 + z2 = 194
z2 = 194 - 130
= 64
z = $\sqrt{64}$
= 8

Tazama Jibu

Swali 27 Ripoti

4, 16, 30, 20, 10, 14 and 26 are represented on a pie chart. Find the sum of the angles of the bisectors representing all numbers equals to or greater than 16.

48o

84o

92o

276o

Maelezo ya Majibu

Given that 4, 16, 30, 20, 10, 14 and 26
Adding up = 120
nos

\geq

16 are 16 + 30 + 20 + 26 = 92
The requires sum of angles =

\frac{92}{120}

x

\frac{360^{o}}{1}

= 276o

Soma maelezo ya somo kwenye Algebraic Expressions (WAEC)

Algebraic Expressions

Tazama Jibu

Swali 28 Ripoti

If $x + \frac{1}{x} = 4$, find $x^2 + \frac{1}{x^2}$

16

14

12

9

Maelezo ya Majibu

x +

\frac{1}{x}

= 4, find x2 +

\frac{1}{x^{2}}

= (x +

\frac{1}{x}

)2 = x2 +

\frac{1}{x^{2}}

+ 2
x2 +

\frac{1}{x^{2}}

= ( x +

\frac{1}{x^{2}}

)2 - 2
= (4)2 - 2
= 16 - 2
= 14

Soma maelezo ya somo kwenye Solution Of Linear Equations (WAEC)

Solution Of Linear Equations

Tazama Jibu

Swali 29 Ripoti

Find the curved surface area of the frustrum in the figure.

16

π \sqrt{10}

cm2

20

π \sqrt{10}

cm2

24

π \sqrt{10}

cm2

36

π \sqrt{10}

cm2

Maelezo ya Majibu

\frac{x}{4} = \frac{6 + x}{6}

6x = 4(6 + x) = 24 + 4x

x = 12 = c = $π R L - π L$

= $π (6) \sqrt{18^{2}} + 6^{2} - π \times 4 \times \sqrt{12^{2}} + 4^{2}$

= 6 $π \sqrt{360} - 4 π \sqrt{160}$

= 36 $π \sqrt{10} - 16 π \sqrt{10}$

= 20 $π \sqrt{10}$ cm2

Soma maelezo ya somo kwenye Matrices And Determinants (JAMB)

Matrices And Determinants

Tazama Jibu

Swali 30 Ripoti

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

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

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

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

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

Maelezo ya Majibu

\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}

Soma maelezo ya somo kwenye Sine, Cosine And Tangent Of An Angle (WAEC)

Soma maelezo ya somo kwenye Matrices And Determinants (JAMB)

Soma maelezo ya somo kwenye Algebraic Expressions (WAEC)

Sine, Cosine And Tangent Of An Angle

Matrices And Determinants

Algebraic Expressions

Tazama Jibu

Swali 31 Ripoti

In the figure, PQR is a semicircle. Calculate the area of the shaded region.

125

\frac{2}{7}

cm2

149

\frac{2}{7}

cm2

234

\frac{1}{7}

cm2

267

\frac{1}{2}

cm2

Areas

Tazama Jibu

Swali 32 Ripoti

A car painter charges ₦40.00 per day for himself and ₦10.00 per day for his assistant. if a fleet of cars were painted for ₦2000.00 and the painter worked 10days more than his assistant, how much did the assistant receive?

₦32.00

₦320.00

₦420.00

₦1680.00

Percentages

Positive And Negative Integers, Rational Numbers

Modular Arithmetic

Areas

Tazama Jibu

Swali 33 Ripoti

If a2 + b2 = 16 and 2ab = 7.Find all the possible values of (a - b)

3, -3

2, -2

1, -1

3, -1

Maelezo ya Majibu

a2 + b2 = 16 and 2ab = 7
To find all possible values = (a - b)2 + b2 - 2ab
Substituting the given values = (a - b)2
= 16 - 7
= 9
(a - b) =

\pm

9
=

\pm

3
OR a - b = 3, -3

Soma maelezo ya somo kwenye Number Bases (WAEC)

Soma maelezo ya somo kwenye Quadratic Equations (WAEC)

Number Bases

Quadratic Equations

Tazama Jibu

Swali 34 Ripoti

Correct 241.34(3 x 10-3)2 to 4 significant figures

0.0014

0.001448

0.0022

0.002172

Fractions, Decimals And Approximations

Tazama Jibu

Swali 35 Ripoti

If the shadow of a pole 7m high is 1/2 its length what is the angle of elevation of the sun, correct to the nearest degree?

90o

63o

60o

26o

0o

Introductory Calculus

Bearings

Tazama Jibu

Swali 36 Ripoti

Below are the scores of a group of students in a test:

Scores	1	2	3	4	5	6
No. of students	1	4	5	6	$x$	2

If the average scores is 3.5, find the value of x.

1

2

3

4

Maelezo ya Majibu

\begin{array}{cc} S c o r e s & 1 & 2 & 3 & 4 & 5 & 6 \\ No. of students & 1 & 4 & 5 & 6 & x & 2 \end{array}

Average = 3.5
3.5 =

\frac{(1 \times 1) + (2 \times 4) + (3 \times 5) + (4 \times 6) + 5 x + (6 \times 2)}{1 + 4 + 5 + 6 + x + 2}

\frac{3.5}{1}

=

\frac{1 + 8 + 15 + 24 + 5 x + 12}{18 + x}

\frac{3.5}{1}

=

\frac{60 + 5 x}{18 + x}

60 + 5x = 3.5(18

\div

x)
60 + 5x = = 63 + 1.5x
5x - 1.5x = 63 - 60
1.5x = 3
x =

\frac{3}{1.5}

\frac{30}{15}

= 2

Tazama Jibu

Swali 37 Ripoti

Simplify $4\frac{3}{4} - 6\frac{1}{4}$

-7

\frac{7}{8}

\frac{- 2}{7}

\frac{- 10}{21}

\frac{10}{21}

Maelezo ya Majibu

4

\frac{3}{4}

- 6

\frac{1}{4}

\frac{19}{4}

-

\frac{25}{4}

............(A)

\frac{21}{5}

-

\frac{5}{4}

.............(B)
Now work out the value of A and the value of B and then find the value

\frac{A}{B}

A =

\frac{19}{4}

-

\frac{25}{4}

=

\frac{- 6}{4}

B =

\frac{21}{5}

x

\frac{5}{20}

=

\frac{105}{20}

=

\frac{21}{4}

But then

\frac{A}{B}

=

\frac{- 6}{4}

\frac{21}{4}

=

\frac{- 6}{4}

\div

\frac{21}{4}

=

\frac{- 6}{4}

x

\frac{4}{21}

=

\frac{- 24}{84}

=

\frac{- 2}{7}

Soma maelezo ya somo kwenye Solution Of Linear Equations (WAEC)

Soma maelezo ya somo kwenye Areas (WAEC)

Solution Of Linear Equations

Areas

Tazama Jibu

Swali 38 Ripoti

An arc of length 22cm subtends an angle of θ at the center of the circle. What is the value of θ if the radius of the circle is 15cm?[Take π = 22/7]

70o

84o

96o

156o

168o

Circles

Tazama Jibu

Swali 39 Ripoti

Simplify 3 log69 + log612 + log664 - log672

5

7776

log631

(7776)6

Logarithms

Tazama Jibu

Swali 40 Ripoti

Find the sum of the first 18 terms of the progression 3, 6, 12......

3(217 - 1)

3(218 - 1)

3(218 + 1)

3(217 - 1)

Maelezo ya Majibu

3 + 6 + 12 + .....18thy term
1st term = 3, common ratio

\frac{6}{3}

= 2
n = 18, sum og GP is given by Sn = a

\frac{(r^{n} - 1)}{r - 1}

s18 = 3

\frac{(2^{18} - 1)}{2 - 1}

= 3(217 - 1)

Soma maelezo ya somo kwenye Sequence And Series (WAEC)

Sequence And Series

Tazama Jibu

Swali 41 Ripoti

Simplify $ \sqrt{160r^2} + \sqrt{71r^4} + \sqrt{100r^2} $

9r2

12

\sqrt{3 r}

13r

\sqrt{13 r}

Maelezo ya Majibu

\sqrt{160 r^{2} + 71 r^{4} + 100 r^{8}}

Simplifying from the innermost radical and progressing outwards we have the given expression

\sqrt{160 r^{2} + 71 r^{4} + 100 r^{8}}

=

\sqrt{160 r^{2} + 81 r^{4}}

\sqrt{160 r^{2} + 9 r^{2}}

=

\sqrt{169 r^{2}}

= 13r

Soma maelezo ya somo kwenye Surds (radicals) (WAEC)

Surds (radicals)

Tazama Jibu

Swali 42 Ripoti

A sector of a circle of radius 7cm has an area of 44cm2. Calculate the angle of the sector correct to the nearest degree [Take π = 22/7]

6o

26o

52o

103o

206o

Circles

Indices, Logarithms And Surds

Tazama Jibu

Swali 43 Ripoti

Simplify and express in standard form $ \frac{0.00275 \times 0.0064}{0.025 \times 0.08} $

8.8 x 10-1

8.8 x 10-2

8.8 x 10-3

8.8 x 103

Modular Arithmetic

Tazama Jibu

Swali 44 Ripoti

Find the length of a side of a rhombus whose diagonals are 6cm and 8cm

8cm

5cm

4cm

3cm

Lengths And Perimeters

Tazama Jibu

Swali 45 Ripoti

What must be added to 4x2 - 4 to make it a perfect square?

)

\frac{- 1}{x^{2}}

\frac{1}{x^{2}}

1

-1

Simple Operations On Algebraic Expressions

Quadratic Equations

Tazama Jibu

Swali 46 Ripoti

If sin x = cos 50o, then x equals

40o

45o

50o

90o

130o

Trigonometry

Tazama Jibu






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