JAMB UTME - Mathematics - 1995

Pergunta 1 Relatório

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

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Pergunta 2 Relatório

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

1

2

3

4

Number Bases

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Pergunta 3 Relatório

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

50o

65o

85o

130o

Detalhes da Resposta

XTY = 50o = TXY =

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

= 65o

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Circles

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Pergunta 4 Relatório

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

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Pergunta 5 Relatório

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

Detalhes da Resposta

< PYU = < YPQ = 40°(Alternative) = PQ

∥

UV

< XPQ = < YPQ = 40°, Since QX = QY

therefore < XPY = 80°

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Loci

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Pergunta 6 Relatório

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

\frac{1}{3}

9

16cm

3

Number Bases

Sets

Fractions, Decimals, Approximations And Percentage

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Pergunta 7 Relatório

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

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Pergunta 8 Relatório

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

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Pergunta 9 Relatório

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}

Detalhes da Resposta

\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}

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Pergunta 10 Relatório

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

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Pergunta 11 Relatório

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

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Pergunta 12 Relatório

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

Detalhes da Resposta

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

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Ler a nota de aula em Measures Of Dispersion (JAMB)

Representation Of Data

Measures Of Dispersion

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Pergunta 13 Relatório

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

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Pergunta 14 Relatório

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

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Pergunta 15 Relatório

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

7cm

10cm

14cm

17cm

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Pergunta 16 Relatório

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

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Pergunta 17 Relatório

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

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Pergunta 18 Relatório

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

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Pergunta 19 Relatório

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

Detalhes da Resposta

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

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Pergunta 20 Relatório

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

4

{0}

\oplus

{

\oplus

}

Algebraic Fractions

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Pergunta 21 Relatório

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}

Detalhes da Resposta

\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

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Quadratic Equations

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Pergunta 22 Relatório

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

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Pergunta 23 Relatório

Determine the value of x in the figure.

134

^{\circ}

81

^{\circ}

53

^{\circ}

46

^{\circ}

Angles

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Pergunta 24 Relatório

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

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Pergunta 25 Relatório

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

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Pergunta 26 Relatório

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

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Pergunta 27 Relatório

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

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Pergunta 28 Relatório

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

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Pergunta 29 Relatório

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}

Detalhes da Resposta

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

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

⟹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

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

Ler a nota de aula em Algebraic Expressions (WAEC)

Algebraic Expressions

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Pergunta 30 Relatório

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

27

1

\frac{1}{3}

\frac{1}{8}

Number Bases

Modular Arithmetic

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Pergunta 31 Relatório

Calculate 33105 - 14425

13135

21315

43025

11035

Number Bases

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Pergunta 32 Relatório

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

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Pergunta 33 Relatório

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

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Pergunta 34 Relatório

Convert 3.1415926 to 5 decimal places

3.14160

3.14159

0.31415

3.14200

Fractions, Decimals And Approximations

Modular Arithmetic

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Pergunta 35 Relatório

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

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Pergunta 36 Relatório

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

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Pergunta 37 Relatório

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

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Pergunta 38 Relatório

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

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Pergunta 39 Relatório

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

Detalhes da Resposta

-1

\geq

x

\geq

1, 1 < x

\geq

2

Combining solutions

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

Ler a nota de aula em Linear Inequalities (WAEC)

Linear Inequalities

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Pergunta 40 Relatório

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

1.25

1.40

2.05

3.00

Measures Of Dispersion

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Pergunta 41 Relatório

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

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Pergunta 42 Relatório

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

12

9

6

2

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Pergunta 43 Relatório

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

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Pergunta 44 Relatório

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}

Detalhes da Resposta

\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

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Pergunta 45 Relatório

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

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Pergunta 46 Relatório

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

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Pergunta 47 Relatório

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

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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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