JAMB UTME - Mathematics - 2008

Question 1 Report

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}

Sets

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Question 2 Report

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

x ≤ 2, x ≥ 3

x ≤ 3, x ≥ 2

x ≤ -2, x ≥ -3

x ≤-3, x ≥ 2

Progression

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Question 3 Report

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

Answer Details

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}

Read lesson note on Graphs Of Linear And Quadratic Functions (WAEC)

Graphs Of Linear And Quadratic Functions

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Question 4 Report

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

Representation Of Data

Integration

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Question 5 Report

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

Answer Details

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

Read lesson note on Binary Operations (JAMB)

Binary Operations

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Question 6 Report

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

4

5

6

7

Algebraic Expressions

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Question 7 Report

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

60o

80o

120o

160o

Answer Details

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

Read lesson note on Circles (WAEC)

Circles

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Question 8 Report

Find the area of the figure above

[π = 22/7]

12.5 cm2

75.0 cm2

78.5 cm2

84.8 cm2

Areas

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Question 9 Report

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

Answer Details

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

Read lesson note on Bearings (WAEC)

Read lesson note on Coordinate Geometry (WAEC)

Bearings

Coordinate Geometry

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Question 10 Report

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

Answer Details

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

Read lesson note on Variation (JAMB)

Read lesson note on Inequalities (JAMB)

Variation

Inequalities

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Question 11 Report

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

Answer Details

(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 ,

Read lesson note on Quadratic Equations (WAEC)

Quadratic Equations

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Question 12 Report

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)

Algebraic Expressions

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Question 13 Report

If 125x = 2010 find x

2

3

4

5

Number Bases

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Question 14 Report

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)

Surds (radicals)

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Question 15 Report

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

Change Of Subject Of A Formula/Relation

Simple Operations On Algebraic Expressions

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Question 16 Report

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

16

13

12

11

Answer Details

\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

Read lesson note on Modular Arithmetic (WAEC)

Read lesson note on Quadratic Equations (WAEC)

Modular Arithmetic

Quadratic Equations

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Question 17 Report

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

10! / (2!2!3!)

10! / ( 2!3!)

10! / (2!2!)

10!

Permutation And Combination

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Question 18 Report

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

4

16

32

64

Answer Details

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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Question 19 Report

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

-1/4

-1/2

1/4

1/2

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Question 20 Report

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!}

Representation Of Data

Permutation And Combination

Probability

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Question 21 Report

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)

Sets

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Question 22 Report

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

3/8

2/3

3/4

3/2

Number Bases

Simple Operations On Algebraic Expressions

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Question 23 Report

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

128

216

215

210

Answer Details

^{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

Read lesson note on Permutation And Combination (JAMB)

Permutation And Combination

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Question 24 Report

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

\frac{1}{11}

\frac{3}{11}

\frac{3}{11}

\frac{4}{11}

Probability

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Question 25 Report

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

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Question 26 Report

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}

Answer Details

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}

View Answer

Question 27 Report

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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Question 28 Report

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

Algebraic Expressions

Quadratic Equations

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Question 29 Report

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

zero

1

2

3

Measures Of Location

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Question 30 Report

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

Solution Of Linear Equations

Positive And Negative Integers, Rational Numbers

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Question 31 Report

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

Probability

Sets

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Question 32 Report

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

Matrices And Determinants

Binary Operations

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Question 33 Report

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

7

1

2

3

Answer Details

\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

Read lesson note on Integration (JAMB)

Integration

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Question 34 Report

In the diagram < OPQ is

90

^{\circ}

53

^{\circ}

36

^{\circ}

26

^{\circ}

Angles

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Question 35 Report

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

4

16

32

64

Logarithms

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Question 36 Report

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}

Answer Details

\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}

View Answer

Question 37 Report

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)

Loci

Euclidean Geometry

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Question 38 Report

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

Answer Details

(\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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Question 39 Report

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

√5

2√3

√20

√50

Answer Details

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}

Read lesson note on Euclidean Geometry (JAMB)

Read lesson note on Coordinate Geometry (JAMB)

Euclidean Geometry

Coordinate Geometry

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Question 40 Report

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

-