JAMB UTME - Mathematics - 2008
Pregunta 1 Informe
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.
Detalles de la respuesta
n(M) only = 30-10 = 20
n(E) only = 20-10 = 10
n(M∩E) = 10
∴M∪E = 20+10+10
= 40
Pregunta 2 Informe
Find the median of 4, 1, 4, 1, 0, 4, 4, 2 and 0
Detalles de la respuesta
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
Pregunta 3 Informe
If p = varies inversely as the square of q and p=8 when q=4, find when p=32
Detalles de la respuesta
P ∝ 1/q
P = k/q
K = q2P
= 428
∴P = 128/q
32 = 128/q
q2 = 128/32
q2 = 4
q = √4 = +/-2
Pregunta 4 Informe
Find the capacity in liters of a cylindrical well of radius 1 meter and depth 14 meters
[π = 22/7]
Detalles de la respuesta
V = πr2h
1m = 100cm
14cm = 1400cm
∴V=227×100x100x14001000=44,000liters
Pregunta 5 Informe
calculate the simple interest on N6,500 for 8 years at 5% per annum.
Detalles de la respuesta
Pregunta 6 Informe
If logx1/264 = 3, find the value of x
Detalles de la respuesta
If logx1/264 = 3
(X 1/2)3 = 64
(X 1/2)3 = 4 3
X 1/2 = 4
X = 42
X = 16
Pregunta 7 Informe
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?
Detalles de la respuesta
Ratio of least to most = 3:12
= 3/12
= 1/4
Pregunta 8 Informe
In the diagram, find the size of the angle marked ao
Detalles de la respuesta
2 x s = 280o(Angle at centre = 2 x < at circum)
S = 280o2
= 140
< O = 360 - 280 = 80o
60 + 80 + 140 + a = 360o
(< in a quad); 280 = a = 360
a = 360 - 280
a = 80o
Pregunta 10 Informe
If 2x2 - kx - 12 is divisible by x-4, Find the value of k.
Detalles de la respuesta
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
Pregunta 11 Informe
If \( \log X^{\frac{1}{2}} \) 64 = 3, find the value of x.
Detalles de la respuesta
logX12
64 = 3, find the value of X
recall that X = 10p
log10X = P
64 = X12
43 = X12
43×12
= X12×13
4 = X12
(4)2 = X12×2
x = 16
Pregunta 12 Informe
Solve the quadratic inequalities x2 - 5x + 6 ≥0
Detalles de la respuesta
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
Pregunta 13 Informe
Find the minimum value of X2 - 3x + 2 for all real values of x
Detalles de la respuesta
y = X2 - 3x + 2, dydx
= 2x - 3
at turning pt, dydx
= 0
∴ 2x - 3 = 0
∴ x = 32
d2ydx2
= ddx
(ddx
)
= 270
∴ ymin = 232
- 332
+ 2
= 94
- 92
+ 2
= -14
Pregunta 14 Informe
if X = {n2 + 1 : n = 0, 2, 3} and y = {n + 1 : n = 2, 3, 4, 5}, find x ∩ y
Detalles de la respuesta
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
Pregunta 15 Informe
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.
Detalles de la respuesta
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 = 01+m
= 0
Pregunta 16 Informe
simplify \(16^{-\frac{1}{2}} \times 4^{-\frac{1}{2}} \times 27^{\frac{1}{3}}\)
Detalles de la respuesta
Pregunta 17 Informe
If x - 3 is directly proportional to the square of y and x = 5 when y =2, find x when y = 6.
Detalles de la respuesta
(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
Pregunta 18 Informe
Find the area of the figure above
[π = 22/7]
Detalles de la respuesta
Area of the figure = Area of rect + area of semi circle
=L×h+12πr25×15+12×227×(52)2=75+(22×25)2×7=75+92328=84.8cm
Pregunta 19 Informe
Factorize complete;y (4x+3y)2 - (3x-2y)2
Detalles de la respuesta
(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)
Pregunta 20 Informe
In the diagram above ∠OPQ is
Detalles de la respuesta
a = a(base ∠s of Iss Δ)
∴ a+a+74 = 180
2a + 74 = 180
2a = 180-74
2a = 106
a = 53
∴∠OPQ = 53∘
Pregunta 21 Informe
Find the values of x and y respectively if
\[ \begin{pmatrix} 1 & 0\\ -1 & -1\\ 2 & 2 \end{pmatrix} + \begin{pmatrix} x & 1\\ -1 & 0\\ y & -2 \end{pmatrix} = \begin{pmatrix} -2 & 1\\ -2 & -1\\ -30 & 0 \end{pmatrix} \]Detalles de la respuesta
⎛⎜⎝10−1−122⎞⎟⎠
+ ⎛⎜⎝x1−10y−2⎞⎟⎠
= ⎛⎜⎝−21−2−1−300⎞⎟⎠
therefore, (x, y) = (-3, -5) respectively
Pregunta 22 Informe
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
Detalles de la respuesta
Pregunta 23 Informe
\[ \begin{pmatrix} -2 & 1 \\ 2 & 3 \end{pmatrix} + \begin{pmatrix} p & q \\ r & s \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}. \] What is the value of r?
Detalles de la respuesta
-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 →
8p = 3
p = 38
6 = 1 - 2 38
= 1 - 68
8−68
= 28
= 14
Pregunta 24 Informe
What is the mean of the data t, 2t-1, t-2, 2t-1, 4t and 2t+2?
Detalles de la respuesta
Pregunta 25 Informe
Find the number of ways of selecting 6 out of 10 subjects for an examination
Detalles de la respuesta
Pregunta 26 Informe
\[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos x\, dx \]
Detalles de la respuesta
∫π2−π2
cosx
[sinx]π2π2
= sinπ2
- (-sin π2
)
= sin π2
+ sin π2
= 2 sin π2
= 2 sin 1.5714
= 2(0.2704)
= 0.5
= 1
Pregunta 27 Informe
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?
Detalles de la respuesta
Number123456Frequency1218x302x45
12 + 18 + x + 30 + 2x + 45 = 150
3x + 105 = 150
3x = 150 - 105
3x = 45
x = 453
x = 15
probability of 5 = 30150
= 15
Pregunta 28 Informe
If 125x = 2010 find x
Detalles de la respuesta
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
Pregunta 29 Informe
Calculate the distance between points L(-1, -6) and M(-3, -5)
Detalles de la respuesta
D = √(x2−x1)2+(y2−y1)2
D = √(−3−(−1)2+(−5−(−6)2
D = √(−3+1)2+(−5+6)2
D = √(−2)2+12
D = √4+1
D = √5
Pregunta 30 Informe
The probability of picking a letter T from the word OBSTRUCTION is
Detalles de la respuesta
Obstruction
if a T is picked
p = 211
Pregunta 31 Informe
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.
Detalles de la respuesta
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
Pregunta 32 Informe
Find the range of values of x which satisfy the inequalities \(4x - 7 \leq 3x\) and \(3x - 4 \leq 4x\)
Detalles de la respuesta
4X - 7 ≤
3X and 3X - 4 ≤
4X
4X - 3X ≤
7 and 3X - 4X ≤
4
X ≤
7 and -X ≤
4 = X ≥
-4
Range -4 ≤
x ≤
7
Pregunta 33 Informe
Find the area of the figure given
Detalles de la respuesta
Area of semicircle + Area of rectangle
A = 12πr2 + LB
A = 12×2277×(52)2+(15×5)
= 12×227×254+75
A = 27528+751
275+210028=237528
A = 84.8cm2
Pregunta 34 Informe
The locus of a point equidistant from two points p(6,2) and R(4,2) is a perpendicular bisector of PR passing through
Detalles de la respuesta
Pregunta 35 Informe
If X = {n2 + 1:n = 0,2,3} and Y = {n+1:n=2,3,5}, find X∩Y.
Pregunta 36 Informe
Evaluate \( \int_{1}^{2} (6x^{2} - 2x)\,dx \)
Detalles de la respuesta
∫21(6x2−2x)dx=[6x33−2x22]21=[2x3−x2]21
= [2(2)3 - (2)2] – [2(1)3 - (1)2]
= [16-4] – [2-1]
= 12 – 1
= 11
Pregunta 37 Informe
Evaluate \( \frac{\left(\frac{3}{8} \div \frac{1}{2} + \frac{1}{2}\right)}{\left(\frac{1}{8} \times \frac{2}{3} + \frac{1}{3}\right)} \)
Detalles de la respuesta
Pregunta 38 Informe
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?
Detalles de la respuesta
Tan 20o = 68mx
x tan 20o = 68
x = 68tan20
= 680.364
x = 186.8
= 187m
Pregunta 39 Informe
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
Detalles de la respuesta
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
Pregunta 40 Informe
In how many ways can the letters of the word ACCEPTANCE be arranged?
Detalles de la respuesta
Acceptance
10!3!2!2!
Pregunta 41 Informe
In the diagram < OPQ is
Detalles de la respuesta
2x = 180 - 74
2x = 103
x = 53
Pregunta 42 Informe
Find the mean deviation of 2, 4, 5, and 9
Detalles de la respuesta
X = 20/4
= 5
mean deviation = 8/4 = 2
Pregunta 43 Informe
Find the gradient of a line which is perpendicular to the line with the equation 3x + 2y + 1 = 0
Detalles de la respuesta
3X + 2Y + 1 = 0
2Y = -3X - 1
−32X−12
Gradient of 3X + 2Y +1 = 0 is -3/2
Gradient of a line perpendicular to 3X + 2Y + 1 = 0
=−1÷32=−1×−23=23
Pregunta 44 Informe
Find the minimum value of the function y = x(1+x)
Detalles de la respuesta
= 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
Pregunta 45 Informe
The solution of the quadratic inequality \(x^2 + x - 12 \ge 0\) is
Detalles de la respuesta
(x2 + x - 12) ≥
0 , (x - 3)(x + 4) ≥
0
For the condition to hold, each of (x - 3) and (x + 4) must be of the same sign
.i.e. x - 3 ≥
0 and x + 4 ≥
0
or x - 3≤
0 and x + 4 ≤
0
when x ≥
3, the condition is satisfied
when x ≥
-4, the condition is not satisfied.
when x ≤
3, the condition is not satisfied
when x ≤
-4 , the condition is not satisfied. Thus, the solution of the inequality is x ≥
3 or x ≤
-4 ,
Pregunta 46 Informe
In how many ways can the letters of the word ACCEPTANCE be arranged?
Detalles de la respuesta
ACCEPTANCE = 10 Letters
A = 2 letters
C = 3 letters
E = 2 letters
Can be arranged in 10! / (2!3!2!) ways
Pregunta 47 Informe
Make Q the subject of formula when \(L = \frac{4}{3}M\sqrt{PQ}\)
Detalles de la respuesta
Pregunta 48 Informe
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
Detalles de la respuesta
X * Y = 2X - 3Y + 2
2*3 = 2(2) - 3(3) + 2
=4-9+2
= -3
But -3 does not belong to positive integer
Pregunta 49 Informe
Add 11012,101112 and 1112
Detalles de la respuesta
11012 + 101112 + 1112 = 101011
Pregunta 50 Informe
If \( \frac{1+\sqrt{2}}{1-\sqrt{2}} \) is expressed in the form of x+y√2 find the values of x and y
Detalles de la respuesta
