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JAMB UTME - Mathematics - 1995

Ibeere 1 Ìròyìn

$\begin{array}{cc} c l a s s & 1 - 3 & 4 - 6 & 7 - 9 \\ F r e q u e n c y & 5 & 8 & 5 \end{array}$

Find the standard deviation of the data using the table above

5

\(\sqrt{6}\0

\frac{5}{3}

\sqrt{5}

Awọn alaye Idahun

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

Ibeere 2 Ìròyìn

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

\frac{1}{x + 2}

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

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

x−1x+2

\frac{1}{x - 2}

Awọn alaye Idahun

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

= $\frac{1}{x + 2}$

Ibeere 7 Ìròyìn

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

Ibeere 9 Ìròyìn

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

Awọn alaye Idahun

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

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

⟹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

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

Ka ìwé ẹ̀kọ́ nípa Algebraic Expressions (WAEC)

Algebraic Expressions

Ibeere 10 Ìròyìn

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

12

9

6

2

Ibeere 11 Ìròyìn

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

Ibeere 15 Ìròyìn

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

4

{0}

\oplus

{

\oplus

}

Algebraic Fractions

Ibeere 16 Ìròyìn

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

6, -1

6, 1

1, -1

6, -6

-6, -1

Solution Of Linear Equations

Algebraic Expressions

Ibeere 18 Ìròyìn

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

Awọn alaye Idahun

< PYU = < YPQ = 40°(Alternative) = PQ $∥$ UV

< XPQ = < YPQ = 40°, Since QX = QY

therefore < XPY = 80°

Ka ìwé ẹ̀kọ́ nípa Loci (WAEC)

Ibeere 21 Ìròyìn

If x = $(\begin{matrix} 1 & 2 \\ 0 & 3 \end{matrix})$ and y = $(\begin{matrix} 2 & 1 \\ 4 & 3 \end{matrix})$ . 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

Ibeere 23 Ìròyìn

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

\frac{1100}{3}

km

\frac{2200}{3}

km

\frac{22000}{3}

km

\frac{1100}{3}

km

Coordinate Geometry

Ibeere 26 Ìròyìn

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

3

\leq

x

\leq

4

3 < x < 4

3

\leq

x < 4

3 < x

\leq

4

Linear Inequalities

Inequalities

Ibeere 27 Ìròyìn

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

50o

65o

85o

130o

Awọn alaye Idahun

XTY = 50o = TXY =

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

= 65o

Ka ìwé ẹ̀kọ́ nípa Circles (WAEC)

Ibeere 28 Ìròyìn

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

Ibeere 29 Ìròyìn

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

Probability

Ibeere 33 Ìròyìn

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

Awọn alaye Idahun

-1 $\geq$ x $\geq$ 1, 1 < x $\geq$ 2

Combining solutions

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

Ka ìwé ẹ̀kọ́ nípa Linear Inequalities (WAEC)

Linear Inequalities

Ibeere 34 Ìròyìn

$\begin{array}{cc} Class Interval & F r e q u e n c y & Class boundaries & Class Mid-point \\ 1.5 - 1.9 & 2 & 1.45 - 1.95 & 1.7 \\ 2.0 - 2.4 & 21 & 1.95 - 2.45 & 2.2 \\ 2.5 - 2.9 & 4 & 2.45 - 2.95 & 2.7 \\ 3.0 - 2.9 & 15 & 2.95 - 3.45 & 3.2 \\ 3.5 - 3.9 & 10 & 3.45 - 3.95 & 3.7 \\ 4.0 - 4.4 & 5 & 3.95 - 4.45 & 4.2 \\ 4.5 - 4.9 & 3 & 4.45 - 4.95 & 4.7 \end{array}$

The median of the distribution above is

4.0

3.4

3.2

3.0

Awọn alaye Idahun

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

Ibeere 35 Ìròyìn

$\begin{array}{cc} Age in years & 13 & 14 & 15 & 16 & 17 \\ No. of students & 3 & 10 & 30 & 42 & 15 \end{array}$

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

Ibeere 36 Ìròyìn

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.[ $π \frac{22}{7}$ ]

836cm2

308cm2

528cm2

154cm2

Ibeere 37 Ìròyìn

$| \begin{matrix} - 2 & 1 & 1 \\ 2 & 1 & k \\ 1 & 3 & - 1 \end{matrix} |$ = 23

1

2

3

4

Number Bases

Ibeere 38 Ìròyìn

Two variables x and y are such that $\frac{d y}{d x}$ = 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

Ibeere 39 Ìròyìn

Let p be a probability function on set S, where S = (a1, a2, a3, a4). Find P(a1) if P(a2) = $\frac{1}{3}$ , p(a3) = $\frac{1}{6}$ and p(a4) = $\frac{1}{5}$

\frac{7}{3}

\frac{2}{3}

\frac{1}{3}

\frac{3}{10}

Probability

Ibeere 40 Ìròyìn

Determine the value of x in the figure.

134

^{\circ}

81

^{\circ}

53

^{\circ}

46

^{\circ}

Ibeere 43 Ìròyìn

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

Ibeere 44 Ìròyìn

Find T in terms of K, Q and S if S = 2r( $π$ 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}

Awọn alaye Idahun

$\frac{s^{2}}{4 r^{2}}$ = QT $π$ + KT
$\frac{s^{2}}{4 r^{2}}$ - k $π$ = QT $π$

T = $\frac{s^{2}}{4 Q π r^{2}}$ - k

Ka ìwé ẹ̀kọ́ nípa Quadratic Equations (WAEC)

Quadratic Equations

Ibeere 45 Ìròyìn

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

Ibeere 47 Ìròyìn

$\begin{array}{cc} Class Interval & F r e q u e n c y & Class boundaries & C l a s s M i d - p o i n t \\ 1.5 - 1.9 & 2 & 1.45 - 1.95 & 1.7 \\ 2.0 - 2.4 & 21 & 1.95 - 2.45 & 2.2 \\ 2.5 - 2.9 & 4 & 2.45 - 2.95 & 2.7 \\ 3.0 - 2.9 & 15 & 2.95 - 3.45 & 3.2 \\ 3.5 - 3.9 & 10 & 3.45 - 3.95 & 3.7 \\ 4.0 - 4.4 & 5 & 3.95 - 4.45 & 4.2 \\ 4.5 - 4.9 & 3 & 4.45 - 4.95 & 4.7 \end{array}$
Find the mode of the distribution above to find the mode of the distribution.

3.2

3.3

3.7

4.2

Awọn alaye Idahun

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

Ka ìwé ẹ̀kọ́ nípa Representation Of Data (JAMB)

Ka ìwé ẹ̀kọ́ nípa Measures Of Dispersion (JAMB)

Representation Of Data

Measures Of Dispersion

Tẹ́lẹ̀

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47