JAMB UTME - Mathematics - 2008

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

Détails de la réponse

4X - 7 $\le$ 3X and 3X - 4 $\le$ 4X
4X - 3X $\le$ 7 and 3X - 4X $\le$ 4
X $\le$ 7 and -X $\le$ 4 = X $\ge$ -4
Range -4 $\le$ x $\le$ 7

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

Détails de la réponse

y = X2 - 3x + 2, $\frac{dy}{dx}$ = 2x - 3
at turning pt, $\frac{dy}{dx}$ = 0
∴ 2x - 3 = 0
∴ x = $\frac{3}{2}$
$\frac{d^{2}y}{d{x}^2}$ = $\frac{d}{dx}$($\frac{d}{dx}$)
= 270
∴ ymin = 2$\frac{3}{2}$ - 3$\frac{3}{2}$ + 2
= $\frac{9}{4}$ - $\frac{9}{2}$ + 2
= -$\frac{1}{4}$

If tan $\theta$ = $\frac{5}{4}$ find sin2$\theta$ - cos2$\theta$

Détails de la réponse

(tan $\theta$ = $\frac{opp}{adj}$)
|AB|2 = 52 + 42 $\to$ |AB|2 = 41
$\to$ AB = $\sqrt{41}$ sin2$\theta$ - cos2$\theta$
$\to$ $\frac{5^2}{\sqrt{41}}$ - (${\frac{4}{\sqrt{41}}}^2$) = $\frac{25}{41}$ - $\frac{16}{41}$
= ($\frac{9}{41}$)

If sinθ = 3/5. Find Tanθ

Détails de la réponse

sinθ = 3/5
x2 = 52 - 32

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?

Détails de la réponse

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

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

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
$=-1÷\frac{3}{2}=-1\times \frac{-2}{3}=\frac{2}{3}$

If 125x = 2010 find x

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

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

Détails de la réponse

In the diagram < OPQ is

Détails de la réponse

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

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)

Differentiate sin x - x cos x

Détails de la réponse

sin x - x cos x
dy/dx = cos x - [1.cos x + x -sin x]
= co x - [cos x - x sin x]
= cos x - cos x + x sin x
= x sin x

The solution of the quadratic inequality (x2 + x - 12) $\ge$ 0 is

Détails de la réponse

(x2 + x - 12) $\ge$ 0 , (x - 3)(x + 4) $\ge$ 0
For the condition to hold, each of (x - 3) and (x + 4) must be of the same sign
.i.e. x - 3 $\ge$ 0 and x + 4 $\ge$ 0
or x - 3$\le$ 0 and x + 4 $\le$ 0
when x $\ge$ 3, the condition is satisfied
when x $\ge$ -4, the condition is not satisfied.
when x $\le$ 3, the condition is not satisfied
when x $\le$ -4 , the condition is not satisfied. Thus, the solution of the inequality is x $\ge$ 3 or x $\le$ -4 ,

The fifth term of an A.P is 24 and the eleventh term is 96. Find the first term.

Détails de la réponse

U5 = 24, n = 5 and U11 = 96, n = 11
Un = a + (n-1)d
24 = a + (5-1)d imply 24 = a+4d .....eqn1
96 = a + (11-1)d imply 96 = a+10d ...eqn2
eqn1 - eqn2 -72 = -6d
d = 72/6 = 12
but 24 = a+4d
24 = a + 4(12)
24 = a + 48
a = 24-48
a = -24

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.

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

Add 11012,101112 and 1112

Détails de la réponse

11012 + 101112 + 1112 = 101011

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

Détails de la réponse

The probability of picking a letter T fr4om the word OBSTRUCTION is

Détails de la réponse

OBSTRUCTION
Total possible outcome = 11
Number of chance of getting T = 2
P(picking T) = 2/11

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

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

Evaluate $\frac{(\frac{3}{8}÷\frac{1}{2}+\frac{1}{2})}{(\frac{1}{8}\times \frac{2}{3}+\frac{1}{3})}$

Détails de la réponse

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

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

In the diagram above ∠OPQ is

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${}^{\circ }$

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

Détails de la réponse

X = {1,5,10}
Y = {3,4,6}
X∩Y = ∅

Evaluate ${\int }_{\frac{-\pi }{2}}^{\frac{\pi }{2}}cosxdx$

Détails de la réponse

${\int }_{\frac{-\pi }{2}}^{\frac{\pi }{2}}cosxdx=[sinx{]}_{\frac{-\pi }{2}}^{\frac{\pi }{2}}=sin\frac{\pi }{2}-sin\frac{-\pi }{2}$
= sin90 – sin-90
= sin90 – sin270
= 1 – (-1)
= 1+1
= 2

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

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

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

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

Find the derivative of $y=\frac{x^7-x^5}{x^4}$

Détails de la réponse

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.

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

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

Détails de la réponse

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

Détails de la réponse

Express 1223456 to 3 significant figures

Détails de la réponse

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

Détails de la réponse

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

Détails de la réponse

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

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

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

Find the area of the figure given

Détails de la réponse

Area of semicircle + Area of rectangle

A = $\frac{1}{2}\pi r^2$ + 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

Find the area of the figure above

[π = 22/7]

Détails de la réponse

Area of the figure = Area of rect + area of semi circle
$=L\times h+\frac{1}{2}\pi r^{2}5\times 15+\frac{1}{2}\times \frac{22}{7}\times {\left(\frac{5}{2}\right)}^2=75+\frac{(22\times 25)}{2\times 7}=75+9\frac{23}{28}=84.8cm$

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

Détails de la réponse

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

Détails de la réponse

(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

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

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

Evaluate ${\int }_1^2(6x^2-2x)dx$

Détails de la réponse

${\int }_1^2(6x^2-2x)dx=[\frac{6x^3}{3}-\frac{2x^2}{2}{]}_1^2=[2x^3-x^2{]}_1^2$
= [2(2)3 - (2)2] – [2(1)3 - (1)2]
= [16-4] – [2-1]
= 12 – 1
= 11

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

Détails de la réponse

Which of the following angles is an exterior angle of a regular polygon?

Détails de la réponse

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

Détails de la réponse

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

Détails de la réponse

X = 20/4
= 5
mean deviation = 8/4 = 2

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

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

The result of rolling a fair die 150 times is ass summarized in the table above. What is the probability of obtaining a 5

Détails de la réponse

Total possible outcome
12+18+x+30+2x+45 = 105+3x
∴105+3x = 150
3x = 150-105
3x = 45
x = 15
P(obtaining 5) =$\frac{2x}{(105+3x)}Butx=15=\frac{2\left(15\right)}{(105+3(15\left)\right)}=\frac{30}{(105+45)}=\frac{30}{150}=\frac{1}{5}$

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

Détails de la réponse

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

$\begin{array}{ccccccc}\text{Number}& 1& 2& 3& 4& 5& 6\\ \text{Frequency}& 12& 18& x& 30& 2x& 45\end{array}$

What is the probability of obtaining a 5?

Détails de la réponse

$\begin{array}{ccccccc}Number& 1& 2& 3& 4& 5& 6\\ Frequency& 12& 18& x& 30& 2x& 45\end{array}$
12 + 18 + x + 30 + 2x + 45 = 150
3x + 105 = 150
3x = 150 - 105
3x = 45
x = $\frac{45}{3}$
x = 15
probability of 5 = $\frac{30}{150}$
= $\frac{1}{5}$

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

Détails de la réponse

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

[π = 22/7]

Détails de la réponse

V = πr2h
1m = 100cm
14cm = 1400cm
$\therefore V=\frac{22}{7}\times \frac{100x100x1400}{1000}=44,000liters$

The cost of kerosine per liter increase from N60 to N85. What is the percentage rate of increase?

Détails de la réponse

N85 - N60 = N25 increase
∴ percentage increase =$\frac{25}{60}\times \frac{100}{1}=\frac{125}{3}=41.67\%=42\%$