JAMB UTME - Mathematics - 1995

Swali 1 Ripoti

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

1.25

1.40

2.05

3.00

Measures Of Dispersion

Tazama Jibu

Swali 2 Ripoti

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

Maelezo ya Majibu

< PYU = < YPQ = 40°(Alternative) = PQ

∥

UV

< XPQ = < YPQ = 40°, Since QX = QY

therefore < XPY = 80°

Soma maelezo ya somo kwenye Loci (WAEC)

Loci

Tazama Jibu

Swali 3 Ripoti

Finds a positive value of p if the expression 2x2 - px + p leaves a remainder 6 when divided by x - p and q

1

2

3

4

Modular Arithmetic

Tazama Jibu

Swali 4 Ripoti

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

50o

65o

85o

130o

Maelezo ya Majibu

XTY = 50o = TXY =

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

= 65o

Soma maelezo ya somo kwenye Circles (WAEC)

Circles

Tazama Jibu

Swali 5 Ripoti

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}

Maelezo ya Majibu

\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

Tazama Jibu

Swali 6 Ripoti

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

Tazama Jibu

Swali 7 Ripoti

If $x = \begin{pmatrix}1 & 2 \\ 0 & 3\end{pmatrix}$ and $y = \begin{pmatrix}2 & 1 \\ 4 & 3\end{pmatrix}$. Find $xy$.

(\begin{matrix} 10 & 7 \\ 12 & 9 \end{matrix})

(\begin{matrix} 2 & 7 \\ 4 & 17 \end{matrix})

(\begin{matrix} 10 & 4 \\ 4 & 6 \end{matrix})

(\begin{matrix} 4 & 3 \\ 10 & 9 \end{matrix})

Matrices And Determinants

Modular Arithmetic

Tazama Jibu

Swali 8 Ripoti

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

27

1

\frac{1}{3}

\frac{1}{8}

Number Bases

Modular Arithmetic

Tazama Jibu

Swali 9 Ripoti

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

Tazama Jibu

Swali 10 Ripoti

Convert 3.1415926 to 5 decimal places

3.14160

3.14159

0.31415

3.14200

Fractions, Decimals And Approximations

Modular Arithmetic

Tazama Jibu

Swali 11 Ripoti

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

Tazama Jibu

Swali 12 Ripoti

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

Tazama Jibu

Swali 13 Ripoti

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

Tazama Jibu

Swali 14 Ripoti

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

Tazama Jibu

Swali 15 Ripoti

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

Tazama Jibu

Swali 16 Ripoti

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

12

9

6

2

Tazama Jibu

Swali 17 Ripoti

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

1

2

3

4

Number Bases

Tazama Jibu

Swali 18 Ripoti

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

Tazama Jibu

Swali 19 Ripoti

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

Tazama Jibu

Swali 20 Ripoti

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}

Maelezo ya Majibu

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

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

⟹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

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

Soma maelezo ya somo kwenye Algebraic Expressions (WAEC)

Algebraic Expressions

Tazama Jibu

Swali 21 Ripoti

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 me and r

1.11

1.65

1.85

2.45

Tazama Jibu

Swali 22 Ripoti

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

Tazama Jibu

Swali 23 Ripoti

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

Tazama Jibu

Swali 24 Ripoti

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

Tazama Jibu

Swali 25 Ripoti

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

7cm

10cm

14cm

17cm

Tazama Jibu

Swali 26 Ripoti

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

Tazama Jibu

Swali 27 Ripoti

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

Tazama Jibu

Swali 28 Ripoti

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

Tazama Jibu

Swali 29 Ripoti

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

Tazama Jibu

Swali 30 Ripoti

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

Tazama Jibu

Swali 31 Ripoti

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

Maelezo ya Majibu

-1

\geq

x

\geq

1, 1 < x

\geq

2

Combining solutions

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

Soma maelezo ya somo kwenye Linear Inequalities (WAEC)

Linear Inequalities

Tazama Jibu

Swali 32 Ripoti

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

Tazama Jibu

Swali 33 Ripoti

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

Tazama Jibu

Swali 34 Ripoti

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}

Maelezo ya Majibu

\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

Soma maelezo ya somo kwenye Quadratic Equations (WAEC)

Quadratic Equations

Tazama Jibu

Swali 35 Ripoti

If $x - 1$ and $x + 1$ are both factors of the equation $x^3 + px^2 + qx + 6 = 0$, evaluate $p$ and $q$

6, -1

6, 1

1, -1

6, -6

-6, -1

Solution Of Linear Equations

Algebraic Expressions

Tazama Jibu

Swali 36 Ripoti

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

Maelezo ya Majibu

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

Soma maelezo ya somo kwenye Representation Of Data (JAMB)

Soma maelezo ya somo kwenye Measures Of Dispersion (JAMB)

Representation Of Data

Measures Of Dispersion

Tazama Jibu

Swali 37 Ripoti

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

Tazama Jibu

Swali 38 Ripoti

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

Tazama Jibu

Swali 39 Ripoti

Calculate 33105 - 14425

13135

21315

43025

11035

Number Bases

Tazama Jibu

Swali 40 Ripoti

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

Tazama Jibu

Swali 41 Ripoti

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

Tazama Jibu

Swali 42 Ripoti

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}

Maelezo ya Majibu

\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}

Tazama Jibu

Swali 43 Ripoti

Determine the value of x in the figure.

134

^{\circ}

81

^{\circ}

53

^{\circ}

46

^{\circ}

Angles

Tazama Jibu

Swali 44 Ripoti

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

4

{0}

\oplus

{

\oplus

}

Algebraic Fractions

Tazama Jibu

Swali 45 Ripoti

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

\frac{1}{3}

9

16cm

3

Number Bases

Sets

Fractions, Decimals, Approximations And Percentage

Tazama Jibu

Swali 46 Ripoti

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

Maelezo ya Majibu

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

Tazama Jibu

Swali 47 Ripoti

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

Tazama Jibu

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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