JAMB UTME - Mathematics - 1990

Tambaya 1 Rahoto

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

Duba Amsa

Tambaya 2 Rahoto

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

Duba Amsa

Tambaya 3 Rahoto

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

Duba Amsa

Tambaya 4 Rahoto

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

\frac{169}{25}

\frac{25}{169}

\frac{169}{144}

\frac{144}{169}

Bayanin Amsa

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}

Duba Amsa

Tambaya 5 Rahoto

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

√10

8

12.2

13.4

Bayanin Amsa

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

Duba Amsa

Tambaya 6 Rahoto

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

Duba Amsa

Tambaya 7 Rahoto

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}

Bayanin Amsa

\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}

Karanta bayanin darasi akan Representation Of Data (JAMB)

Karanta bayanin darasi akan Probability (WAEC)

Karanta bayanin darasi akan Permutation And Combination (JAMB)

Representation Of Data

Probability

Permutation And Combination

Duba Amsa

Tambaya 8 Rahoto

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

16

14

12

9

Bayanin Amsa

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

Karanta bayanin darasi akan Solution Of Linear Equations (WAEC)

Solution Of Linear Equations

Duba Amsa

Tambaya 9 Rahoto

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

Duba Amsa

Tambaya 10 Rahoto

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

0.0014

0.001448

0.0022

0.002172

Fractions, Decimals And Approximations

Duba Amsa

Tambaya 11 Rahoto

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

3, -3

2, -2

1, -1

3, -1

Bayanin Amsa

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

Karanta bayanin darasi akan Number Bases (WAEC)

Karanta bayanin darasi akan Quadratic Equations (WAEC)

Number Bases

Quadratic Equations

Duba Amsa

Tambaya 12 Rahoto

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

Bayanin Amsa

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.

Karanta bayanin darasi akan Introductory Calculus (WAEC)

Karanta bayanin darasi akan Angles Of Elevation And Depression (WAEC)

Karanta bayanin darasi akan Angles (WAEC)

Introductory Calculus

Angles Of Elevation And Depression

Angles

Duba Amsa

Tambaya 13 Rahoto

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

Duba Amsa

Tambaya 14 Rahoto

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

Bayanin Amsa

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

Karanta bayanin darasi akan Algebraic Expressions (WAEC)

Algebraic Expressions

Duba Amsa

Tambaya 15 Rahoto

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)

Duba Amsa

Tambaya 16 Rahoto

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

6

11

27

51

Solution Of Linear Equations

Algebraic Expressions

Duba Amsa

Tambaya 17 Rahoto

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

9

8

7

5

Triangles And Polygons

Duba Amsa

Tambaya 18 Rahoto

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

3, 5

-3, -5

9, 25

-9, 25

Bayanin Amsa

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

Karanta bayanin darasi akan Solution Of Linear Equations (WAEC)

Solution Of Linear Equations

Duba Amsa

Tambaya 19 Rahoto

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)

Bayanin Amsa

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)

Karanta bayanin darasi akan Sequence And Series (WAEC)

Sequence And Series

Duba Amsa

Tambaya 20 Rahoto

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

(x + y)(5x + y)

(x + y)2

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

5(x + y)2

Duba Amsa

Tambaya 21 Rahoto

Simplify 3 log69 + log612 + log664 - log672

5

7776

log631

(7776)6

Logarithms

Duba Amsa

Tambaya 22 Rahoto

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

Duba Amsa

Tambaya 23 Rahoto

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

2

-28

-30

-34

Bayanin Amsa

(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

Duba Amsa

Tambaya 24 Rahoto

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

Duba Amsa

Tambaya 25 Rahoto

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}

Bayanin Amsa

\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}

Karanta bayanin darasi akan Sine, Cosine And Tangent Of An Angle (WAEC)

Karanta bayanin darasi akan Matrices And Determinants (JAMB)

Karanta bayanin darasi akan Algebraic Expressions (WAEC)

Sine, Cosine And Tangent Of An Angle

Matrices And Determinants

Algebraic Expressions

Duba Amsa

Tambaya 26 Rahoto

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}

Bayanin Amsa

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}

Karanta bayanin darasi akan Number Bases (WAEC)

Number Bases

Duba Amsa

Tambaya 27 Rahoto

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

Bayanin Amsa

\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

Karanta bayanin darasi akan Matrices And Determinants (JAMB)

Matrices And Determinants

Duba Amsa

Tambaya 28 Rahoto

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

9r2

12

\sqrt{3 r}

13r

\sqrt{13 r}

Bayanin Amsa

\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

Karanta bayanin darasi akan Surds (radicals) (WAEC)

Surds (radicals)

Duba Amsa

Tambaya 29 Rahoto

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

8cm

6cm

4cm

3cm

Triangles And Polygons

Duba Amsa

Tambaya 30 Rahoto

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

\frac{x}{y}

xy

\frac{x}{y}

(xy)-1

Bayanin Amsa

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}

Karanta bayanin darasi akan Solution Of Linear Equations (WAEC)

Karanta bayanin darasi akan Circles (WAEC)

Karanta bayanin darasi akan Algebraic Expressions (WAEC)

Solution Of Linear Equations

Circles

Algebraic Expressions

Duba Amsa

Tambaya 31 Rahoto

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

Duba Amsa

Tambaya 32 Rahoto

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

Bayanin Amsa

\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

Duba Amsa

Tambaya 33 Rahoto

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

Duba Amsa

Tambaya 34 Rahoto

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

27 : 32

27 : 23

23 : 32

32 : 27

Bayanin Amsa

$\frac{h_{1}}{h_{2}}$ = $\frac{2}{3}$
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