JAMB UTME - Mathematics - 2008

Pergunta 1 Relatório

If p = varies inversely as the square of q and p=8 when q=4, find when p=32

+/-16

+/-8

+/-4

+/-2

Detalhes da Resposta

P ∝ 1/q
P = k/q
K = q2P
= 428
∴P = 128/q
32 = 128/q
q2 = 128/32
q2 = 4
q = √4 = +/-2

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Pergunta 2 Relatório

A student sitting on a tower 68 metres high observes his principal's car at the angle of depression of 20o. How far is the car from the bottom of the tower to the nearest metre?

184m

185m

186m

187m

Detalhes da Resposta

Tan 20o =

\frac{68 m}{x}

x tan 20o = 68
x =

\frac{68}{t a n 20}

=

\frac{68}{0.364}

x = 186.8
= 187m

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Pergunta 3 Relatório

The probability of picking a letter T from the word OBSTRUCTION is

\frac{1}{11}

\frac{3}{11}

\frac{3}{11}

\frac{4}{11}

Detalhes da Resposta

Obstruction
if a T is picked
p =

\frac{2}{11}

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Pergunta 4 Relatório

simplify $16^{-\frac{1}{2}} \times 4^{-\frac{1}{2}} \times 27^{\frac{1}{3}}$

3/8

2/3

3/4

3/2

Detalhes da Resposta

16^{\frac{? 1}{2}} \times 4^{\frac{? 1}{2}} \times 27^{\frac{1}{3}} = \frac{1}{16^{\frac{1}{2}}} \times \frac{1}{4^{\frac{1}{2}}} \times 27^{\frac{1}{3}} = \frac{1}{\sqrt{16}} \times \frac{1}{\sqrt{4}} \times s q r t [3] 27 = \frac{1}{4} \times \frac{1}{2} \times \frac{3}{1} = \frac{3}{8}

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Pergunta 5 Relatório

On a pie chart, there are six sectors of which four angles are 30o, 45o, 60o, 90o and the remaining two angles are in the ratio 2 : 1. Find the smaller angles of the remaining two angles

15o

30o

45o

60o

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Pergunta 6 Relatório

The result of rolling a fair die 150 times is as summarized in the table given.

Number	1	2	3	4	5	6
Frequency	12	18	$x$	30	$2x$	45

What is the probability of obtaining a 5?

\frac{3}{10}

\frac{1}{5}

\frac{1}{6}

\frac{1}{10}

Detalhes da Resposta

\begin{array}{cc} N u m b e r & 1 & 2 & 3 & 4 & 5 & 6 \\ F r e q u e n c y & 12 & 18 & x & 30 & 2 x & 45 \end{array}

12 + 18 + x + 30 + 2x + 45 = 150
3x + 105 = 150
3x = 150 - 105
3x = 45
x =

\frac{45}{3}

x = 15
probability of 5 =

\frac{30}{150}

=

\frac{1}{5}

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Pergunta 7 Relatório

A binary operation on the real set of numbers excluding -1 is such that for all m, n ∈ R, mΔn = m+n+mn. Find the identity element of the operation.

1

zero

-1/2

-1

Detalhes da Resposta

mΔn = m+n+mn
Let e be the identity element
∴mΔe = eΔm = m
m+e+me = m
e+me = m-m
e+me = 0
e(1+m) = 0
e = 0 / (1+m)
e = 0

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Pergunta 8 Relatório

Solve the quadratic inequalities x2 - 5x + 6 ≥0

x ≤ 2, x ≥ 3

x ≤ 3, x ≥ 2

x ≤ -2, x ≥ -3

x ≤-3, x ≥ 2

Detalhes da Resposta

x2 - 5x + 6 = 0
(X-2)(X-3) = 0
X-2 = 0 implies X = 2
X-3 = 0 implies X = 3
∴ x ≤ 2, x ≥ 3

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Pergunta 9 Relatório

Find the number of ways of selecting 6 out of 10 subjects for an examination

128

216

215

210

Detalhes da Resposta

^{10} C_{6} = \frac{10!}{(10 - 6)! 6!} = \frac{10!}{4! 6!} = \frac{10 \times 9 \times 8 \times 7 \times 6!}{4 \times 3 \times 2 \times 1 \times 6!} = 2

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Pergunta 10 Relatório

If x - 3 is directly proportional to the square of y and x = 5 when y =2, find x when y = 6.

30

21

16

12

Detalhes da Resposta

(x – 3) ∝ y2
X-3 = Ky2
K = X-3 / y2
= 5-2/22
= 2/4
= 1/2
∴X-3 = 1/2y2
X-3 = 1/2(6)2
X-3 = 1/2 x 30/1
X-3 = 18
X = 21

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Pergunta 11 Relatório

In the diagram < OPQ is

90

^{\circ}

53

^{\circ}

36

^{\circ}

26

^{\circ}

Detalhes da Resposta

74 + x + x = 180

2x = 180 - 74

2x = 103

x = 53

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Pergunta 12 Relatório

Find the mean deviation of 2, 4, 5, and 9

1

2

5

7

Detalhes da Resposta

X = 20/4
= 5
mean deviation = 8/4 = 2

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Pergunta 13 Relatório

Find the range of values of x which satisfy the inequalities $4x - 7 \leq 3x$ and $3x - 4 \leq 4x$

-4

\leq

x

\leq

7

-7

\leq

x

\leq

4

x

\geq

-7

\leq

x

\leq

6

Detalhes da Resposta

4X - 7

\leq

3X and 3X - 4

\leq

4X
4X - 3X

\leq

7 and 3X - 4X

\leq

4
X

\leq

7 and -X

\leq

4 = X

\geq

-4
Range -4

\leq

x

\leq

7

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Pergunta 14 Relatório

If $ \log X^{\frac{1}{2}} $ 64 = 3, find the value of x.

4

16

32

64

Detalhes da Resposta

l o g X^{\frac{1}{2}}

64 = 3, find the value of X
recall that X = 10p
log10X = P
64 =

X^{\frac{1}{2}}

43 =

X^{\frac{1}{2}}

4^{3 \times \frac{1}{2}}

=

X^{\frac{1}{2} \times \frac{1}{3}}

4 =

X^{\frac{1}{2}}

(4)2 =

X^{\frac{1}{2} \times 2}

x = 16

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Pergunta 15 Relatório

In how many ways can the letters of the word ACCEPTANCE be arranged?

\frac{10!}{2! 2! 3!}

\frac{10!}{2! 3!}

\frac{10!}{2! 2!}

\frac{10!}{3! 2! 2!}

Detalhes da Resposta

Acceptance

\frac{10!}{3! 2! 2!}

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Pergunta 16 Relatório

Express 1223456 to 3 significant figures

123000

124000

125000

126000

Fractions, Decimals And Approximations

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Pergunta 17 Relatório

A binary operation * is defined on the set of positive integers is such x*y = 2x-3y+2 for all positive integers x and y. The binary operation is

commutative and close on the set of positive integers

neither commutative nor closed on the set of ppositive integers

commutative but not closed on the set of positive integers

not cummutative but closed on the set of positive integers

Detalhes da Resposta

X * Y = 2X - 3Y + 2
2*3 = 2(2) - 3(3) + 2
=4-9+2
= -3
But -3 does not belong to positive integer

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Pergunta 18 Relatório

If x > 0, find the range of number x-3, 3x+2,x-1, 4x, 2x-1, x-2, 2x-2, 3x and 3x+1

3x+3

3x+1

2x+5

2x+1

Detalhes da Resposta

x-3, 3x+2,x-1, 4x, 2x-1, x-2, 2x-2, 3x, 3x+1
Range = 3x+2 - (x-3)
= 3x+2 - x - 3
= 2x + 5

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Pergunta 19 Relatório

In the diagram, find the size of the angle marked ao

60o

80o

120o

160o

Detalhes da Resposta

2 x s = 280o(Angle at centre = 2 x < at circum)

S = $\frac{280^{o}}{2}$

= 140

< O = 360 - 280 = 80o

60 + 80 + 140 + a = 360o

(< in a quad); 280 = a = 360

a = 360 - 280

a = 80o

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Pergunta 20 Relatório

The locus of a point equidistant from two points p(6,2) and R(4,2) is a perpendicular bisector of PR passing through

(2,5)

(5,2)

(1,0)

(0,1)

Detalhes da Resposta

X = \frac{6 + 2}{2} = \frac{10}{2} = 5 Y = \frac{2 + 2}{2} = \frac{4}{2} = 2 ∴ X, Y = (5, 2)

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Pergunta 21 Relatório

Make Q the subject of formula when $L = \frac{4}{3}M\sqrt{PQ}$

\frac{9 L^{2}}{16 M^{2}} P

\frac{3 L}{4 M \sqrt{P}}

\frac{\sqrt{3 L}}{4 M P}

\frac{3 L^{2}}{16 M^{2}} P

Detalhes da Resposta

L = \frac{4}{3} M \sqrt{P Q} = \frac{3}{4 M} \times L = \sqrt{P Q} = {(\frac{3 L}{4 M})}^{2} = (\sqrt{P Q})^{2} = \frac{9 L^{2}}{16 M^{2}} = P Q = Q = \frac{9 L^{2}}{16 M^{2} P}

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Pergunta 22 Relatório

A book seller sells Mathematics and English books. If 30 customers buy Mathematics books, 20 customers buy English books and 10 customers buy the two books, How many customers has he altogether.

30

40

50

60

Detalhes da Resposta

n(M) only = 30-10 = 20
n(E) only = 20-10 = 10
n(M∩E) = 10
∴M∪E = 20+10+10
= 40

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Pergunta 23 Relatório

Calculate the distance between points L(-1, -6) and M(-3, -5)

√5

2√3

√20

√50

Detalhes da Resposta

L

(\begin{matrix} x_{1} & y_{1} \\ - 1 & - 6 \end{matrix})

m L

(\begin{matrix} x_{2} & y_{2} \\ - 3 & - 5 \end{matrix})

D =

\sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}

D =

\sqrt{(- 3 - (- 1)^{2} + (- 5 - (- 6)^{2}}

D =

\sqrt{(- 3 + 1)^{2} + (- 5 + 6)^{2}}

D =

\sqrt{(- 2)^{2} + 1^{2}}

D =

\sqrt{4 + 1}

D =

\sqrt{5}

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Pergunta 24 Relatório

Find the median of 4, 1, 4, 1, 0, 4, 4, 2 and 0

zero

1

2

3

Detalhes da Resposta

4, 4, 1, 0, 4, 4, 2, 0
Arrange in ascending order:
0, 0, 1, 1, 2, 4, 4, 4, 4
Position of median = N+1 / 2 = 9+1 / 2 = 5th
Median = 2

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Pergunta 25 Relatório

The bar chart above shows the number of times the word a, and , in, it, the , to appear in a paragraph in a book. What is the ratio of the least frequent word?

1/4

1/3

2/3

3/4

Detalhes da Resposta

Ratio of least to most = 3:12
= 3/12
= 1/4

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Pergunta 26 Relatório

Find the area of the figure above

[π = 22/7]

12.5 cm2

75.0 cm2

78.5 cm2

84.8 cm2

Detalhes da Resposta

Area of the figure = Area of rect + area of semi circle

= L \times h + \frac{1}{2} π r^{2} 5 \times 15 + \frac{1}{2} \times \frac{22}{7} \times {(\frac{5}{2})}^{2} = 75 + \frac{(22 \times 25)}{2 \times 7} = 75 + 9 \frac{23}{28} = 84.8 c m

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Pergunta 27 Relatório

Find the gradient of a line which is perpendicular to the line with the equation 3x + 2y + 1 = 0

3/2

2/3

-2/3

-3/2

Detalhes da Resposta

3X + 2Y + 1 = 0
2Y = -3X - 1

\frac{- 3}{2} X - \frac{1}{2}

Gradient of 3X + 2Y +1 = 0 is -3/2
Gradient of a line perpendicular to 3X + 2Y + 1 = 0

= - 1 \div \frac{3}{2} = - 1 \times \frac{- 2}{3} = \frac{2}{3}

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Pergunta 28 Relatório

The solution of the quadratic inequality $x^2 + x - 12 \ge 0$ is

x

\geq

3 or x

\geq

-4

x

\leq

3 or x

\leq

-4

x

\geq

3 or x

\leq

-4

x

\geq

-3 or x

\leq

4

Detalhes da Resposta

(x2 + x - 12)

\geq

0 , (x - 3)(x + 4)

\geq

0
For the condition to hold, each of (x - 3) and (x + 4) must be of the same sign
.i.e. x - 3

\geq

0 and x + 4

\geq

0
or x - 3

\leq

0 and x + 4

\leq

0
when x

\geq

3, the condition is satisfied
when x

\geq

-4, the condition is not satisfied.
when x

\leq

3, the condition is not satisfied
when x

\leq

-4 , the condition is not satisfied. Thus, the solution of the inequality is x

\geq

3 or x

\leq

-4 ,

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Pergunta 29 Relatório

If X = {n2 + 1:n = 0,2,3} and Y = {n+1:n=2,3,5}, find X∩Y.

{1,3}

{5,10}

∅

{4,6}

Detalhes da Resposta

X = {1,5,10}
Y = {3,4,6}
X∩Y = ∅

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Pergunta 30 Relatório

calculate the simple interest on N6,500 for 8 years at 5% per annum.

N3,000

N600

N300

N150

Detalhes da Resposta

I = \frac{P \times T \times R}{100} = \frac{7500 \times 8 \times 5}{100} = N 3000

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Pergunta 31 Relatório

If $ \frac{1+\sqrt{2}}{1-\sqrt{2}} $ is expressed in the form of x+y√2 find the values of x and y

(-3, -2)

(-2, 3)

(3,2)

(2,-3)

Detalhes da Resposta

\frac{1 + \sqrt{2}}{1 - \sqrt{2}} \times \frac{1 + \sqrt{2}}{1 + \sqrt{2}} = \frac{1 + (1 + \sqrt{2}) + \sqrt{2} (1 + \sqrt{2})}{1^{2} - (\sqrt{2})^{2}} = \frac{(1 + \sqrt{2} + \sqrt{2} + 2)}{1 - 2} = \frac{3 + 2 \sqrt{2}}{- 1} = - 3 - 2 \sqrt{2} ∴ X a n d Y = - 3 a n d - 2

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Pergunta 32 Relatório

\[ \begin{pmatrix} -2 & 1 \\ 2 & 3 \end{pmatrix} + \begin{pmatrix} p & q \\ r & s \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}. \] What is the value of r?

-

\frac{1}{8}

\frac{3}{8}

\frac{5}{8}

\frac{1}{4}

Detalhes da Resposta

-2p + r = 1.......(i)
2p + 3r = 0.......(ii)
r - 1 + 2p ........(iii)
2p + 3(1 + 2p) = 0
2p + 3(1 + 2p) = 0
2p + 3 + 6p = 0
3 - 8p = 0

\to

8p = 3
p =

\frac{3}{8}

6 = 1 - 2

\frac{3}{8}

= 1 -

\frac{6}{8}

\frac{8 - 6}{8}

=

\frac{2}{8}

=

\frac{1}{4}

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Pergunta 33 Relatório

In how many ways can the letters of the word ACCEPTANCE be arranged?

10! / (2!2!3!)

10! / ( 2!3!)

10! / (2!2!)

10!

Detalhes da Resposta

ACCEPTANCE = 10 Letters
A = 2 letters
C = 3 letters
E = 2 letters
Can be arranged in 10! / (2!3!2!) ways

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Pergunta 34 Relatório

Evaluate $ \frac{\left(\frac{3}{8} \div \frac{1}{2} + \frac{1}{2}\right)}{\left(\frac{1}{8} \times \frac{2}{3} + \frac{1}{3}\right)} $

1/4

1/3

1/2

1

Detalhes da Resposta

\frac{(\frac{3}{8} \div \frac{1}{2} + \frac{1}{2})}{(\frac{1}{8} \times \frac{2}{3} + \frac{1}{3})} = \frac{(\frac{3}{8} \times \frac{2}{1} + \frac{1}{2})}{(\frac{1}{8} \times \frac{2}{3} + \frac{1}{3})} = \frac{(\frac{3}{4} - \frac{1}{3})}{(\frac{1}{12} + \frac{1}{3})} = (\frac{9 - 4}{12}) \div (\frac{1 + 4}{12}) = (\frac{5}{12}) \div (\frac{5}{12}) = \frac{5}{12} \times \frac{12}{5} = 1

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Pergunta 35 Relatório

If 2x2 - kx - 12 is divisible by x-4, Find the value of k.

4

5

6

7

Detalhes da Resposta

2x2 - kx - 12 is divisible by x-4
implies x is a factor ∴ x = 4
f(4) implies 2(4)2 - k(4) - 12 = 0
32 - 4k - 12 = 0
-4k + 20 = 0
-4k = -20
k = 5

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Pergunta 36 Relatório

Find the minimum value of the function y = x(1+x)

-1/4

-1/2

1/4

1/2

Detalhes da Resposta

y = x(1+x)
= x + x2
dy/dx = 1 + 2x
As dy/dy → 0
1 + 2x = 0
2x = -1
X = -1/2
Y = x(1+x)
= -1/2(1 - 1/2) at x = -1/2
= -1/2(1/2)
= -1/4

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Pergunta 37 Relatório

Find the area of the figure given

12.5cm2

75.0cm2

78.5cm2

84.8cm2

Detalhes da Resposta

Area of semicircle + Area of rectangle

A = $\frac{1}{2} π r^{2}$ + LB

A = $\frac{1}{2} \times \frac{22}{77} \times (\frac{5}{2})^{2} + (15 \times 5)$

= $\frac{1}{2} \times \frac{22}{7} \times \frac{25}{4} + 75$

A = $\frac{275}{28} + \frac{75}{1}$

$\frac{275 + 2100}{28} = \frac{2375}{28}$

A = 84.8cm2

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Pergunta 38 Relatório

if X = {n2 + 1 : n = 0, 2, 3} and y = {n + 1 : n = 2, 3, 4, 5}, find x ∩ y

{1, 3}

{5,10}

Φ

(4, 6)

Detalhes da Resposta

X = {n2 + 1 : n = 0, 2, 3}
y = {n + 1 : n = 2, 3, 4, 5}
find x ∩ y
x = (n2 + 1)(22 + 1)(32 + 1)
y = (2 + 1) (3 + 1)(5 + 1)
x = {1, 5, 10}
y = {3, 4, 6}
x ∩ y =

Φ

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Pergunta 39 Relatório

Evaluate $ \int_{1}^{2} (6x^{2} - 2x)\,dx $

16

13

12

11

Detalhes da Resposta

\int_{1}^{2} (6 x^{2} - 2 x) d x = [\frac{6 x^{3}}{3} - \frac{2 x^{2}}{2}]_{1}^{2} = [2 x^{3} - x^{2}]_{1}^{2}

= [2(2)3 - (2)2] – [2(1)3 - (1)2]
= [16-4] – [2-1]
= 12 – 1
= 11

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Pergunta 40 Relatório

Add 11012,101112 and 1112

1110112

1101102

1010112

1010102

Detalhes da Resposta

11012 + 101112 + 1112 = 101011

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Number Bases

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Pergunta 41 Relatório

Find the values of x and y respectively if

\[ \begin{pmatrix} 1 & 0\\ -1 & -1\\ 2 & 2 \end{pmatrix} + \begin{pmatrix} x & 1\\ -1 & 0\\ y & -2 \end{pmatrix} = \begin{pmatrix} -2 & 1\\ -2 & -1\\ -30 & 0 \end{pmatrix} \]

-3, -2

-5, -3

-2, -5

-3, -5

Detalhes da Resposta

(\begin{matrix} 1 & 0 \\ - 1 & - 1 \\ 2 & 2 \end{matrix})

+

(\begin{matrix} x & 1 \\ - 1 & 0 \\ y & - 2 \end{matrix})

=

(\begin{matrix} - 2 & 1 \\ - 2 & - 1 \\ - 30 & 0 \end{matrix})

therefore, (x, y) = (-3, -5) respectively

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Pergunta 42 Relatório

If logx1/264 = 3, find the value of x

4

16

32

64

Detalhes da Resposta

If logx1/264 = 3
(X 1/2)3 = 64
(X 1/2)3 = 4 3
X 1/2 = 4
X = 42
X = 16

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Pergunta 43 Relatório

Factorize complete;y (4x+3y)2 - (3x-2y)2

(x+5y)(7x+y)

(x+5y)(7x-y)

(x-5y)(7x+y)

(x-5y)(7x-y)

Detalhes da Resposta

(4x+3y)2 - (3x-2y)2
(4x+3y+3x-2y)(4x+3y-(3x-2y))
(4x+3y+3x-2y)(4x+3y-3x+2y)
(x+5y)(7x+y)

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Pergunta 44 Relatório

A binary operation on the set of real numbers excluding -1 is such that sor all $m, n \in R$, $m \Delta n = m + n + mn$. Find the identity element of the operation.

1

0

-

\frac{1}{2}

-1

Detalhes da Resposta

m

Δ

n = m + n + mn
m

Δ

e = e

Δ

m = m
m

Δ

e = m
m + e + me = m
e + me = m - m
e + me = 0
e(1 + m) = 0
e =

\frac{0}{1 + m}

= 0

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Binary Operations

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Pergunta 45 Relatório

\[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos x\, dx \]

7

1

2

3

Detalhes da Resposta

\int_{- \frac{π}{2}}^{\frac{π}{2}}

cosx

[s i n x]_{\frac{π}{2}}^{\frac{π}{2}}

= sin

\frac{π}{2}

- (-sin

\frac{π}{2}

)
= sin

\frac{π}{2}

+ sin

\frac{π}{2}

= 2 sin

\frac{π}{2}

= 2 sin 1.5714
= 2(0.2704)
= 0.5
= 1

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Integration

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Pergunta 46 Relatório

In the diagram above ∠OPQ is

90

^{\circ}

53

^{\circ}

36

^{\circ}

26

^{\circ}

Detalhes da Resposta

a = a(base ∠s of Iss Δ)
∴ a+a+74 = 180
2a + 74 = 180
2a = 180-74
2a = 106
a = 53
∴∠OPQ = 53

^{\circ}

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Circles

Angles

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Pergunta 47 Relatório

What is the mean of the data t, 2t-1, t-2, 2t-1, 4t and 2t+2?

2t

2t-1

2/3t+1

2t-1/3

Detalhes da Resposta

M e a n = \frac{t + 2 t - 1 + t - 2 + 2 t - 14 t + 2 t + 2}{6} M e a n = \frac{12 t - 2}{6} = \frac{2 (6 t - 1)}{6} = \frac{6 t - 1}{3} = \frac{6 t}{3} - \frac{1}{3} = 2 t - \frac{1}{3}

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Representation Of Data

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Pergunta 48 Relatório

Find the capacity in liters of a cylindrical well of radius 1 meter and depth 14 meters

[π = 22/7]

44,000 liters

4,400 liters

440 liters

44 liters

Detalhes da Resposta

V = πr2h
1m = 100cm
14cm = 1400cm

∴ V = \frac{22}{7} \times \frac{100 x 100 x 1400}{1000} = 44, 000 l i t e r s

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Volumes

Inequalities

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Pergunta 49 Relatório

If 125x = 2010 find x

2

3

4

5

Detalhes da Resposta

125x = 20
1xX2 + 2xX1 + 5xX0 = 20
X2 + 2X + 5 = 20
X2 + 2X - 15 = 0
(X + 5)( X - 3) = 0
X + 5 implies X = -5
X - 3 implies X = 3
But X cannot be negative
∴X = 3

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Number Bases

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Pergunta 50 Relatório

Find the minimum value of X2 - 3x + 2 for all real values of x

-

\frac{1}{4}

-

\frac{1}{2}

\frac{1}{4}

\frac{1}{2}

Detalhes da Resposta

y = X2 - 3x + 2,

\frac{d y}{d x}

= 2x - 3
at turning pt,

\frac{d y}{d x}

= 0
∴ 2x - 3 = 0
∴ x =

\frac{3}{2}

\frac{d^{2} y}{d x^{2}}

=

\frac{d}{d x}

(

\frac{d}{d x}

)
= 270
∴ ymin = 2

\frac{3}{2}

- 3

\frac{3}{2}

+ 2
=

\frac{9}{4}

-

\frac{9}{2}

+ 2
= -

\frac{1}{4}

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