JAMB UTME - Mathematics - 2008
Question 1 Report
Add 11012,101112 and 1112
Question 2 Report
If X = {n2 + 1:n = 0,2,3} and Y = {n+1:n=2,3,5}, find X∩Y.
Question 3 Report
If 125x = 2010 find x
Answer Details
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
Question 4 Report
Solve the quadratic inequalities x2 - 5x + 6 ≥0
Answer Details
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
Question 5 Report
If 2x2 - kx - 12 is divisible by x-4, Find the value of k.
Answer Details
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
Question 6 Report
Find the gradient of a line which is perpendicular to the line with the equation 3x + 2y + 1 = 0
Answer Details
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
Question 7 Report
Find the minimum value of X2 - 3x + 2 for all real values of x
Answer Details
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
Question 8 Report
In the diagram above ∠OPQ is
Answer Details
a = a(base ∠s of Iss Δ)
∴ a+a+74 = 180
2a + 74 = 180
2a = 180-74
2a = 106
a = 53
∴∠OPQ = 53∘
Question 9 Report
Find the area of the figure above
[π = 22/7]
Answer Details
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
Question 11 Report
Which of the following angles is an exterior angle of a regular polygon?
Question 12 Report
If logx1/264 = 3, find the value of x
Answer Details
If logx1/264 = 3
(X 1/2)3 = 64
(X 1/2)3 = 4 3
X 1/2 = 4
X = 42
X = 16
Question 13 Report
Evaluate \( \displaystyle \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos x\,dx \)
Answer Details
∫π2−π2cosxdx=[sinx]π2−π2=sinπ2−sin−π2
= sin90 – sin-90
= sin90 – sin270
= 1 – (-1)
= 1+1
= 2
Question 14 Report
Find the capacity in liters of a cylindrical well of radius 1 meter and depth 14 meters
[π = 22/7]
Answer Details
V = πr2h
1m = 100cm
14cm = 1400cm
∴V=227×100x100x14001000=44,000liters
Question 15 Report
Differentiate sin x - x cos x
Answer Details
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
Question 16 Report
The result of rolling a fair die 150 times is ass summarized in the table above. What is the probability of obtaining a 5
Answer Details
Total possible outcome
12+18+x+30+2x+45 = 105+3x
∴105+3x = 150
3x = 150-105
3x = 45
x = 15
P(obtaining 5) =2x(105+3x)Butx=15=2(15)(105+3(15))=30(105+45)=30150=15
Question 17 Report
Find the median of 4, 1, 4, 1, 0, 4, 4, 2 and 0
Answer Details
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
Question 18 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?
Answer Details
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
Question 19 Report
If \( \tan \theta = \frac{5}{4} \) find \( \sin^2 \theta - \cos^2 \theta \)
Answer Details
(tan θ
= oppadj
)
|AB|2 = 52 + 42 →
|AB|2 = 41
→
AB = √41
sin2θ
- cos2θ
→
52√41
- (4√412
) = 2541
- 1641
= (941
)
Question 20 Report
Find the range of values of x which satisfy the inequalities \(4x - 7 \leq 3x\) and \(3x - 4 \leq 4x\)
Answer Details
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
Question 21 Report
Find the minimum value of the function y = x(1+x)
Answer Details
= 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
Question 22 Report
The cost of kerosine per liter increase from N60 to N85. What is the percentage rate of increase?
Answer Details
N85 - N60 = N25 increase
∴ percentage increase =2560×1001=1253=41.67%=42%
Question 23 Report
The probability of picking a letter T fr4om the word OBSTRUCTION is
Answer Details
OBSTRUCTION
Total possible outcome = 11
Number of chance of getting T = 2
P(picking T) = 2/11
Question 24 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
Answer Details
X * Y = 2X - 3Y + 2
2*3 = 2(2) - 3(3) + 2
=4-9+2
= -3
But -3 does not belong to positive integer
Question 25 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.
Answer Details
n(M) only = 30-10 = 20
n(E) only = 20-10 = 10
n(M∩E) = 10
∴M∪E = 20+10+10
= 40
Question 26 Report
The fifth term of an A.P is 24 and the eleventh term is 96. Find the first term.
Answer Details
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
Question 27 Report
What is the mean of the data t, 2t-1, t-2, 2t-1, 4t and 2t+2?
Answer Details
Question 28 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?
Answer Details
Ratio of least to most = 3:12
= 3/12
= 1/4
Question 29 Report
In how many ways can the letters of the word ACCEPTANCE be arranged?
Answer Details
ACCEPTANCE = 10 Letters
A = 2 letters
C = 3 letters
E = 2 letters
Can be arranged in 10! / (2!3!2!) ways
Question 30 Report
Factorize complete;y (4x+3y)2 - (3x-2y)2
Answer Details
(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)
Question 31 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
Answer Details
Question 32 Report
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?
Answer Details
| least most = | 3 | = | 1 |
| 12 | 4 |
Question 33 Report
In the diagram < OPQ is
Question 34 Report
If p = varies inversely as the square of q and p=8 when q=4, find when p=32
Answer Details
P ∝ 1/q
P = k/q
K = q2P
= 428
∴P = 128/q
32 = 128/q
q2 = 128/32
q2 = 4
q = √4 = +/-2
Question 35 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
Answer Details
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
Question 36 Report
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.
Answer Details
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
Question 37 Report
Express 1223456 to 3 significant figures
Question 38 Report
Make Q the subject of formula when \(L = \frac{4}{3}M\sqrt{PQ}\)
Answer Details
Question 39 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
Answer Details
Question 40 Report
calculate the simple interest on N6,500 for 8 years at 5% per annum.
Question 41 Report
simplify \(16^{-\frac{1}{2}} \times 4^{-\frac{1}{2}} \times 27^{\frac{1}{3}}\)
Answer Details
Question 42 Report
If x - 3 is directly proportional to the square of y and x = 5 when y =2, find x when y = 6.
Answer Details
(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
Question 43 Report
Find the number of ways of selecting 6 out of 10 subjects for an examination
Answer Details
Question 44 Report
Find the area of the figure given
Answer Details
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
Question 45 Report
Find the derivative of \(y=\frac{x^7-x^5}{x^4}\)
Answer Details
Question 46 Report
The solution of the quadratic inequality \(x^2 + x - 12 \ge 0\) is
Answer Details
(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 ,
Question 47 Report
Evaluate \( \int_{1}^{2} (6x^{2} - 2x)\,dx \)
Answer Details
∫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
Question 48 Report
In the diagram, find the size of the angle marked ao
Answer Details
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
Question 49 Report
Evaluate \( \frac{\left(\frac{3}{8} \div \frac{1}{2} + \frac{1}{2}\right)}{\left(\frac{1}{8} \times \frac{2}{3} + \frac{1}{3}\right)} \)
Answer Details
Question 50 Report
Find the mean deviation of 2, 4, 5, and 9
