JAMB UTME - Mathematics - 2008

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

4

5

6

7

Détails de la réponse

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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Algebraic Expressions

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Question 2 Rapport

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

Détails de la réponse

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

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Representation Of Data

Integration

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Question 3 Rapport

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

Détails de la réponse

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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Question 4 Rapport

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)

Détails de la réponse

(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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Algebraic Expressions

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Question 5 Rapport

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}

Détails de la réponse

X = {1,5,10}
Y = {3,4,6}
X∩Y = ∅

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Sets

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Question 6 Rapport

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

N3,000

N600

N300

N150

Détails de la réponse

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

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Financial Arithmetic

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Question 7 Rapport

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

Détails de la réponse

\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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Solution Of Linear Equations

Positive And Negative Integers, Rational Numbers

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Question 8 Rapport

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

√5

2√3

√20

√50

Détails de la réponse

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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Euclidean Geometry

Coordinate Geometry

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Question 9 Rapport

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

-1/4

-1/2

1/4

1/2

Détails de la réponse

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

Voir la réponse

Question 10 Rapport

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

Détails de la réponse

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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Matrices And Determinants

Binary Operations

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Question 11 Rapport

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

-

\frac{1}{8}

\frac{3}{8}

\frac{5}{8}

\frac{1}{4}

Détails de la réponse

-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}

Voir la réponse

Question 12 Rapport

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

+/-16

+/-8

+/-4

+/-2

Détails de la réponse

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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Progression

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Question 13 Rapport

Express 1223456 to 3 significant figures

123000

124000

125000

126000

Fractions, Decimals And Approximations

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Question 14 Rapport

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

4

16

32

64

Détails de la réponse

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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Logarithms

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Question 15 Rapport

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}

Détails de la réponse

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}

Voir la réponse

Question 16 Rapport

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

Détails de la réponse

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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Variation

Inequalities

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Question 17 Rapport

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

Détails de la réponse

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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Graphs Of Linear And Quadratic Functions

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Question 18 Rapport

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

1

2

5

7

Détails de la réponse

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

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Probability

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Question 19 Rapport

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

Détails de la réponse

(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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Quadratic Equations

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Question 20 Rapport

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

16

13

12

11

Détails de la réponse

\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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Modular Arithmetic

Quadratic Equations

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Question 21 Rapport

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

10! / (2!2!3!)

10! / ( 2!3!)

10! / (2!2!)

10!

Détails de la réponse

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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Permutation And Combination

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Question 22 Rapport

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

4

16

32

64

Détails de la réponse

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

Voir la réponse

Question 23 Rapport

Find the area of the figure above

[π = 22/7]

12.5 cm2

75.0 cm2

78.5 cm2

84.8 cm2

Détails de la réponse

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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Areas

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Question 24 Rapport

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!}

Détails de la réponse

Acceptance

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

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Representation Of Data

Permutation And Combination

Probability

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Question 25 Rapport

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

Détails de la réponse

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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Question 26 Rapport

If 125x = 2010 find x

2

3

4

5

Détails de la réponse

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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Question 27 Rapport

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

Détails de la réponse

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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Bearings

Coordinate Geometry

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Question 28 Rapport

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

7

1

2

3

Détails de la réponse

\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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Question 29 Rapport

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

Détails de la réponse

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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Binary Operations

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Question 30 Rapport

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

Détails de la réponse

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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Change Of Subject Of A Formula/Relation

Simple Operations On Algebraic Expressions

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Question 31 Rapport

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

\frac{1}{11}

\frac{3}{11}

\frac{3}{11}

\frac{4}{11}

Détails de la réponse

Obstruction
if a T is picked
p =

\frac{2}{11}

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Probability

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Question 32 Rapport

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

128

216

215

210

Détails de la réponse

^{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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Permutation And Combination

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Question 33 Rapport

Add 11012,101112 and 1112

1110112

1101102

1010112

1010102

Détails de la réponse

11012 + 101112 + 1112 = 101011

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Number Bases

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Question 34 Rapport

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)

Détails de la réponse

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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Sets

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Question 35 Rapport

Find the area of the figure given

12.5cm2

75.0cm2

78.5cm2

84.8cm2

Détails de la réponse

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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Areas

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Question 36 Rapport

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

60o

80o

120o

160o

Détails de la réponse

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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Circles

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Question 37 Rapport

In the diagram above ∠OPQ is

90

^{\circ}

53

^{\circ}

36

^{\circ}

26

^{\circ}

Détails de la réponse

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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Question 38 Rapport

In the diagram < OPQ is

90

^{\circ}

53

^{\circ}

36

^{\circ}

26

^{\circ}

Détails de la réponse

74 + x + x = 180

2x = 180 - 74

2x = 103

x = 53

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Angles

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Question 39 Rapport

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

Voir la réponse

Question 40 Rapport

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

Détails de la réponse

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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Probability

Sets

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Question 41 Rapport

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

zero

1

2

3

Détails de la réponse

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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Measures Of Location

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Question 42 Rapport

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

x ≤ 2, x ≥ 3

x ≤ 3, x ≥ 2

x ≤ -2, x ≥ -3

x ≤-3, x ≥ 2

Détails de la réponse

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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Progression

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Question 43 Rapport

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)

Détails de la réponse

\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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Surds (radicals)

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Question 44 Rapport

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)

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

Lisez la note de cours sur Loci (JAMB)

Lisez la note de cours sur Loci (WAEC)

Lisez la note de cours sur Euclidean Geometry (JAMB)

Loci

Euclidean Geometry

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Question 45 Rapport

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

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