JAMB UTME - Mathematics - 1995

In the diagram, the base diameter is 14cm while the height is 12cm. Calculate the total surface area if the cylinder has both a base and a top.[$ \pi \frac{22}{7} $]

836cm2

308cm2

528cm2

154cm2

Volumes

Areas

Lee Azịza

Ajụjụ 6 Ripọtì

Find the distance between two towns p(45$^{o}$N, 30$^{o}$W) and Q(15$^{o}$S, 30$^{o}$W) if the radius of the earth is 7000km. [$\pi = \frac{22}{7}$]

\frac{1100}{3}

km

\frac{2200}{3}

km

\frac{22000}{3}

km

\frac{1100}{3}

km

Coordinate Geometry

Lee Azịza

Ajụjụ 7 Ripọtì

The graph of f(x) = x2 - 5x + 6 crosses the x-axis at the points

(-6, 0), (-1, 0)

(-3, 0), (-2,0)

(-6, 0),(1, 0)

(2, 0), (3, 0)

Graphs Of Linear And Quadratic Functions

Lee Azịza

Ajụjụ 8 Ripọtì

Find the area bounded by the curve $y = 3x^2 - 2x + 1$, the coordinates $x = 1$ and $x = 3$ and the x-axis

24

22

21

20

Areas

Lee Azịza

Ajụjụ 9 Ripọtì

Express $ \frac{5x-12}{(x-2)(x-3)} $ in partial fractions

\frac{2}{x + 2} - \frac{3}{x - 3}

\frac{2}{x - 2} + \frac{3}{x - 3}

\frac{2}{x - 3} - \frac{3}{x - 2}

\frac{5}{x - 3} - \frac{4}{x - 2}

Akọwa Nkọwa

$\frac{5 x - 12}{(x - 2) (x - 3)} = \frac{A}{x - 2} + \frac{B}{x - 3}$

= $\frac{A (x - 3) + B (x - 2)}{(x - 2) (x - 3)}$

⟹5x−12=Ax−3A+Bx−2B

$A + B = 5 . . . (i)$

−(3A+2B)=−12⟹3A+2B=12...(ii)

From (i), $A = 5 - B$

$3 (5 - B) + 2 B = 12$

15−3B+2B=12⟹B=3

A+3=5⟹A=2

$\frac{5 x - 12}{(x - 2) (x - 3)} = \frac{2}{x - 2} + \frac{3}{x - 3}$

Gụọ akwụkwọ nkuzi na Algebraic Expressions (WAEC)

Algebraic Expressions

Lee Azịza

Ajụjụ 10 Ripọtì

The mean and the range of the set of numbers 1.20, 1.00, 0.90, 1.40, 0.80, 1.20, and 1.10 are m and r respectively. Find m + r

1.11

1.69

1.85

2.45

Linear Inequalities

Measures Of Dispersion

Lee Azịza

Ajụjụ 11 Ripọtì

The pie chart above shows the distribution of students in a secondary school class. If 30 students offered French, how many offered CRK?

25

15

10

8

Percentages

Modular Arithmetic

Lee Azịza

Ajụjụ 12 Ripọtì

Factorize completely the expression abx2 + 6y - 3ax - 2byx

(ax - 2y)(bx - 3)

(bx + 3)(2y - ax)

(bx + 3)(ax - 2y)

(ax - 2y)(ax - b)

Solution Of Linear Equations

Algebraic Expressions

Lee Azịza

Ajụjụ 13 Ripọtì

PT is a tangent to the circle TYZX. YT = YX and < PTX = 50o. Calculate < TZY

50o

65o

85o

130o

Akọwa Nkọwa

XTY = 50o = TXY =

\frac{130^{o}}{2}

= 65o

Gụọ akwụkwọ nkuzi na Circles (WAEC)

Circles

Lee Azịza

Ajụjụ 14 Ripọtì

Four members of a social first eleven cricket team are also members of the first fourteen rugby team. How many boys play for at least one of the two teams?

25

21

16

3

Sets

Lee Azịza

Ajụjụ 15 Ripọtì

Class Interval	Frequency	Class boundaries	Class Mid-point
$1.5 - 1.9$	$2$	$1.45 - 1.95$	$1.7$
$2.0 - 2.4$	$21$	$1.95 - 2.45$	$2.2$
$2.5 - 2.9$	$4$	$2.45 - 2.95$	$2.7$
$3.0 - 2.9$	$15$	$2.95 - 3.45$	$3.2$
$3.5 - 3.9$	$10$	$3.45 - 3.95$	$3.7$
$4.0 - 4.4$	$5$	$3.95 - 4.45$	$4.2$
$4.5 - 4.9$	$3$	$4.45 - 4.95$	$4.7$

The median of the distribution above is

4.0

3.4

3.2

3.0

Akọwa Nkọwa

Median =

\frac{a + b}{f m}

(

\frac{1}{2} \sum f - C F_{b}

)
= 2.95 +

\frac{0.5}{15}

(2.-7)
= 2.95 +

\frac{0.5}{15}

x 13
= 2.95 + 0. 43
= 3.38
= 3.4

Lee Azịza

Ajụjụ 16 Ripọtì

Simplify $ \frac{x^2 - 1}{x^3 + 2x^2 - x - 2} $

\frac{1}{x + 2}

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

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

x−1x+2

\frac{1}{x - 2}

Akọwa Nkọwa

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

=

\frac{(x - 1) (x + 1)}{(x - 1) (x + 2) (x + 1)}

=

\frac{1}{x + 2}

Lee Azịza

Ajụjụ 17 Ripọtì

In the diagram, find PQ if the area of triangle PQR is 35cm2

7cm

10cm

14cm

17cm

Lee Azịza

Ajụjụ 18 Ripọtì

A worker's present salary is ₦24,000 per annum. His annual increment is 10% of his basic salary at the beginning of the third year?

₦28,800

₦29,040

₦31,200

₦31,944

Lee Azịza

Ajụjụ 19 Ripọtì

Let $p$ be a probability function on set $S$, where $S = (a_1, a_2, a_3, a_4)$. Find $P(a_1)$ if $P(a_2) = \frac{1}{3}$, $p(a_3) = \frac{1}{6}$ and $p(a_4) = \frac{1}{5}$

\frac{7}{3}

\frac{2}{3}

\frac{1}{3}

\frac{3}{10}

Probability

Lee Azịza

Ajụjụ 20 Ripọtì

Which of the following binary operations is cumulative in the set of integers?

a

*

b = a + 2b

a

*

b = a + b - ab

a

*

b = a2 + b

a

*

b =

\frac{a (b + 1)}{2}

Binary Operations

Lee Azịza

Ajụjụ 21 Ripọtì

Two perpendicular lines PQ and QR intersect at (1, -1). If the equation of PQ is x - 2y + 4 = 0, find the equation of QR

x + 2y - 1= -0

2x + y - 3 = 0

x - 2y - 3 = 0

2x + y - 1

Algebraic Expressions

Lee Azịza

Ajụjụ 22 Ripọtì

Evaluate $ \frac{(18)^{\frac{3}{4}}-(27)^3}{3\times 2^3} $

27

1

\frac{1}{3}

\frac{1}{8}

Number Bases

Modular Arithmetic

Lee Azịza

Ajụjụ 23 Ripọtì

Find the sum to infinity of the following sequence 1. $ \frac{9}{10} $, 2. ($ \frac{9}{10} $), 3. ($ \frac{9}{10} $)

\frac{1}{10}

\frac{9}{10}

\frac{10}{9}

10

Sequence And Series

Lee Azịza

Ajụjụ 24 Ripọtì

Express the product of 0.0014 and 0.011 in standard form

1.54 x 10-2

1.54 x 10-3

1.54 x 10-4

1.54 x 10-5

Fractions, Decimals And Approximations

Lee Azịza

Ajụjụ 25 Ripọtì

A school boy lying on the ground 30m away from the foot of a water tank tower observes that the angle of elevation of the top of the tank 60o. Calculate the height of the water tank.

60m

30

\sqrt{3}

m

20

\sqrt{3}

m

10

\sqrt{3}

m

Lee Azịza

Ajụjụ 26 Ripọtì

In a triangle XYZ, < YXZ = 44° and < XYZ = 112°. Calculate the acute angle between the internal bisectors of < XYZ and < XZY.

12o

56o

68o

78o

Angles

Lee Azịza

Ajụjụ 27 Ripọtì

Find T in terms of K, Q and S if S = 2r$ \pi $QT + K)

\frac{S^{2}}{2 π r^{2} Q} - \frac{k}{Q}

\frac{S^{2}}{2 π r^{2} Q}

- k

\frac{S^{2}}{4 π r^{2} Q} - \frac{k}{Q}

\frac{s^{2}}{4 π r^{2} Q}

Akọwa Nkọwa

\frac{s^{2}}{4 r^{2}}

= QT

π

+ KT

\frac{s^{2}}{4 r^{2}}

- k

π

= QT

π

T = $\frac{s^{2}}{4 Q π r^{2}}$ - k

Gụọ akwụkwọ nkuzi na Quadratic Equations (WAEC)

Quadratic Equations

Lee Azịza

Ajụjụ 28 Ripọtì

The 4th term of an A.P is 13 while the 10th term is 31. Find the 21st term

175

85

64

45

Sequence And Series

Lee Azịza

Ajụjụ 29 Ripọtì

Class Interval	Frequency	Class boundaries	Class Mid−point
$1.5-1.9$	$2$	$1.45-1.95$	$1.7$
$2.0-2.4$	$21$	$1.95-2.45$	$2.2$
$2.5-2.9$	$4$	$2.45-2.95$	$2.7$
$3.0-2.9$	$15$	$2.95-3.45$	$3.2$
$3.5-3.9$	$10$	$3.45-3.95$	$3.7$
$4.0-4.4$	$5$	$3.95-4.45$	$4.2$
$4.5-4.9$	$3$	$4.45-4.95$	$4.7$

Find the mode of the distribution above to find the mode of the distribution.

3.2

3.3

3.7

4.2

Akọwa Nkọwa

Mode = a + (b - a)(fm - Fb)

2Fm - Fa - Fb

= 3.0 + $\frac{(3.4 - 3) (15 - 4)}{2 (15) - 4 - 10}$

= 3 + $\frac{(6.4) (11)}{30 - 14}$

= 3 + $\frac{4.4}{16}$

= 3 + 0.275

= 3.275

= 3.3cm

Gụọ akwụkwọ nkuzi na Representation Of Data (JAMB)

Gụọ akwụkwọ nkuzi na Measures Of Dispersion (JAMB)

Representation Of Data

Measures Of Dispersion

Lee Azịza

Ajụjụ 30 Ripọtì

Simplify $ \frac{\sqrt{12}-\sqrt{3}}{\sqrt{12}+\sqrt{3}} $

\frac{1}{3}

9

16cm

3

Number Bases

Sets

Fractions, Decimals, Approximations And Percentage

Lee Azịza

Ajụjụ 31 Ripọtì

The variance of the scores 1, 2, 3, 4, 5 is

1.25

1.40

2.05

3.00

Measures Of Dispersion

Lee Azịza

Ajụjụ 32 Ripọtì

class	1−3	4−6	7−9
Frequency	5	8	5

Find the standard deviation of the data using the table above

5

\(\sqrt{6}\0

\frac{5}{3}

\sqrt{5}

Akọwa Nkọwa

\begin{matrix} class intervals & F r e (F) & class-marks(x) & F x & (x - x) & (x - x)^{2} & F (x - x)^{2} \\ 1 - 3 & 5 & 10 & 10 & - 3 & 9 & 90 \\ 4 - 6 & 8 & 40 & 40 & 0 & 0 & 0 \\ 7 - 9 & 5 & 40 & 40 & 3 & 9 & 360 \\ 18 & 90 & 450 \end{matrix}

x =

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

=

\frac{90}{18}

= 5
S.D =

\frac{\sum f (x - x)^{2}}{\sum f}

=

\frac{450}{18}

=

\sqrt{25}

= 5

Lee Azịza

Ajụjụ 33 Ripọtì

\[ \begin{vmatrix} -2 & 1 & 1 \\ 2 & 1 & k \\ 1 & 3 & -1 \end{vmatrix} = 23 \]

1

2

3

4

Number Bases

Lee Azịza

Ajụjụ 34 Ripọtì

Age in years	13	14	15	16	17
No. of students	3	10	30	42	15

The frequency distribution above shows the ages of students in a secondary school. In a pie chart constructed to represent the data, the angles corresponding to the 15 years old is

27o

30o

54o

108o

Lee Azịza

Ajụjụ 35 Ripọtì

Solve the inequality $ (x - 3)(x - 4) \le 0 $

3

\leq

x

\leq

4

3 < x < 4

3

\leq

x < 4

3 < x

\leq

4

Linear Inequalities

Inequalities

Lee Azịza

Ajụjụ 36 Ripọtì

Use the graph of the curve $y = f(x)$to solve the inequality $f(x) \le 0$

-1

\geq

x

\geq

1, x

\geq

2

x

\leq

-1, 1

\geq

x

\geq

2

x

\geq

-1, 1

\geq

x

\geq

2

x

\geq

2, -1

\geq

x

\geq

1

Akọwa Nkọwa

-1

\geq

x

\geq

1, 1 < x

\geq

2

Combining solutions

= x $\leq$ 1; 1 $\geq$ x $\geq$ 2

Gụọ akwụkwọ nkuzi na Linear Inequalities (WAEC)

Linear Inequalities

Lee Azịza

Ajụjụ 37 Ripọtì

For what value of x is the tangent to the curve $y = x^2 - 4x + 3$ parallel to the x-axis?

3

2

1

0

Statistics

Lee Azịza

Ajụjụ 38 Ripọtì

If $a \ast b = +\sqrt{ab}$, evaluate $2 \ast (12 \ast 27)$

12

9

6

2

Lee Azịza

Ajụjụ 39 Ripọtì

The length of a notebook 15cm, was measured as 16.8cm. Calculate the percentage error to 2 significant figures.

12.00%

11.00%

10.71%

0.12%

Percentages

Lee Azịza

Ajụjụ 40 Ripọtì

P is on the locus of points equidiatant from two given points X and Y. UV is a straight line throuh Y parallel to the locus. If < PYU is 40°, find < XPY.

100o

80 o

50 o

40 o

Akọwa Nkọwa

< PYU = < YPQ = 40°(Alternative) = PQ

∥

UV

< XPQ = < YPQ = 40°, Since QX = QY

therefore < XPY = 80°

Gụọ akwụkwọ nkuzi na Loci (WAEC)

Loci

Lee Azịza

Ajụjụ 41 Ripọtì

The derivative of cosec x is

tan x cosec x

-cot x cosec x

tan x sec x

-cot x sec x

Matrices And Determinants

Introductory Calculus

Lee Azịza

Ajụjụ 42 Ripọtì

Determine the value of x in the figure.

134

^{\circ}

81

^{\circ}

53

^{\circ}

46

^{\circ}

Angles

Lee Azịza

Ajụjụ 43 Ripọtì

Two variables x and y are such that $ \frac{dy}{dx} = 4x - 3 $ and y = 5 when x = 2. Find y in terms of x

2x2 - 3x + 5

2x2 - 3x + 3

2x2 - 3x

4

Solution Of Linear Equations

Lee Azịza

Ajụjụ 44 Ripọtì

Calculate 33105 - 14425

13135

21315

43025

11035

Number Bases

Lee Azịza

Ajụjụ 45 Ripọtì

If S = (x : x2 = 9, x > 4), then S is equal to

4

{0}

\oplus

{

\oplus

}

Algebraic Fractions

Lee Azịza

Ajụjụ 46 Ripọtì

Convert 3.1415926 to 5 decimal places

3.14160

3.14159

0.31415

3.14200

Fractions, Decimals And Approximations

Modular Arithmetic

Lee Azịza

Ajụjụ 47 Ripọtì

A die has four of it's faces coloured white and the remaining two coloured black. What is the probability that when the die is thrown two consecutive time, the top face will be white in both cases?

\frac{2}{3}

\frac{1}{9}

\frac{4}{9}

\frac{1}{36}

Probability

Lee Azịza

Class Interval	Frequency	Class boundaries	Class Mid-point
\(1.5 - 1.9\)	\(2\)	\(1.45 - 1.95\)	\(1.7\)
\(2.0 - 2.4\)	\(21\)	\(1.95 - 2.45\)	\(2.2\)
\(2.5 - 2.9\)	\(4\)	\(2.45 - 2.95\)	\(2.7\)
\(3.0 - 2.9\)	\(15\)	\(2.95 - 3.45\)	\(3.2\)
\(3.5 - 3.9\)	\(10\)	\(3.45 - 3.95\)	\(3.7\)
\(4.0 - 4.4\)	\(5\)	\(3.95 - 4.45\)	\(4.2\)
\(4.5 - 4.9\)	\(3\)	\(4.45 - 4.95\)	\(4.7\)

Class Interval	Frequency	Class boundaries	Class Mid−point
\(1.5-1.9\)	\(2\)	\(1.45-1.95\)	\(1.7\)
\(2.0-2.4\)	\(21\)	\(1.95-2.45\)	\(2.2\)
\(2.5-2.9\)	\(4\)	\(2.45-2.95\)	\(2.7\)
\(3.0-2.9\)	\(15\)	\(2.95-3.45\)	\(3.2\)
\(3.5-3.9\)	\(10\)	\(3.45-3.95\)	\(3.7\)
\(4.0-4.4\)	\(5\)	\(3.95-4.45\)	\(4.2\)
\(4.5-4.9\)	\(3\)	\(4.45-4.95\)	\(4.7\)






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