JAMB UTME - Mathematics - 1986
Question 1 Report
Find the smallest number by which 252 can be multiplied to obtain a perfect square
Answer Details
Let the smallest number be x and the perfect square be y 252x = y.
By trial and error method, 252 x 9 = 1764
Check if y = 1764
y2 = 42
x = 7
Question 2 Report
Three boys shared some oranges. The first received 1/3 of the oranges and the second received 2/3 of the remaining. If the third boy received the remaining 12 oranges, how many oranges did they share
Answer Details
Let x = the number of oranges
The 1st received 1/3 of x = 1/3x
∴Remainder = x - 1/3x = 2x/3
The 2nd received 2/3 of 2x/3 = 2/3 * 2x/3 = 4x/3
The 3rd received 12 oranges
∴1/3x + 4x/9 + 12 = x
(3x + 4x + 108)/9 = x
3x + 4x + 108 = 9x
7x + 108 = 9x
9x - 7x = 108
2x = 108
x = 54 oranges
Question 3 Report
If x varies directly as y3 and x = 2 when y = 1, find x when y = 5
Answer Details
x α
y3
x = ky3
k = xy3
when x = 2, y = 1
k = 2
Thus x = 2y3 - equation of variation
= 2(5)3
= 250
Question 5 Report
Factorize (4a + 3)2 - (3a - 2)2
Answer Details
(4a + 3)2 - (3a - 2)2 = a2 - b2
= (a + b)(a - b)
= [(4a + 3) + (3a - 2)][(4a + 3) + (3a - 2)]
= [(4a + 3 + 3a - 2)][(4a + 3 - 3a + 2)]
= (7a + 1)(a + 5)
∴ (a + 5)(7a + 1)
Question 6 Report
In the diagram, PQ and RS are chords of a circle centre O which meet at T outside the circle. If TP = 24cm. TQ = 8cm and TS = 12cm, find TR.
Answer Details
PT x QT = TR x TS
24 x 8 = TR x 12
TR = 24×812
= = 16cm
Question 7 Report
make U the subject of the formula \(S = \sqrt{\frac{6}{u} - \frac{w}{2}}\)
Answer Details
S = √6u−w2
S = 12−uw2u
2us2 = 12 - uw
u(2s2 + w) = 12
u = 122s2+w
Question 8 Report
Solve the equation 3x2 + 6x - 2 = 0
Answer Details
3x2 + 6x - 2 = 0
Using almighty formula i.e. x = b±√b2−4ac2a
a = 3, b = 6, c = -2
x = −6±√62−4(3)(−2)2(3)
x = 6±√36−246
x = 6±√606
x = −6±√4×156
x = −1±√153
Question 9 Report
If U and V are two distinct fixed points and W is a variable points such that UWV is a right angle, what is the locus of W?
Question 10 Report
If 5(x + 2y) = 5 and 4(x + 3y) = 16, find 3(x + y)
Answer Details
5(x + 2y) = 5
∴ x + 2y = 1.....(i)
4(x + 3y) = 16 = 42
x + 3y = 2 .....(ii)
x + 2y = 1.....(i)
x + 3y = 2......(ii)
y = 1
Substitute y = 1 into equation (i) = x + 2y = 1
∴ x + 2(1) = 1
x + 2 = 1
∴ x = 1
∴ 3x + y = 3-1 + 1
= 3 = 1
Question 11 Report
A man kept 6 black, 5 brown and 7 purple shirts in a drawer. What is the probability of his picking a purple shirt with his eyes closed?
Question 12 Report
At what points does the straight line y = 2x + 1 intersect the curve y = 2x2 + 5x - 1?
Question 13 Report
If Musa scored 75 in biology instead of 57, his average mark in four subjects would have been 60. What was his total mark?
Answer Details
Let x represent Musa's total mark when he scores 57 in biology and Let Y represent Musa's total mark when he now scored 75 in biology, if he scored 75 in biology his new total mark will be Y4
= 60, y = 4 x 60 = 240
To get his total mark when he scored 57, subtract 57 from 75 to give 18, then subtract this 18 from the new total mark(ie. 240)
= 240 - 18
= 222
Question 14 Report
In the figure, \( \triangle \)PQT is isosceles. PQ = QT, SRQ = \(35^\circ\), TPQ = \(20^\circ\) and PQR is a straight line.Calculate TSR
Answer Details
Given △ isosceles PQ = QT, SRQ = 35∘
TPQ = 20∘
PQR = is a straight line
Since PQ = QT, angle P = angle T = 20∘
Angle PQR = 180∘ - (20 + 20) = 140∘
TQR = 180∘ - 140∘ = 40∘ < on a straight line
QSR = 180∘ - (40 + 35)∘ = 105∘
TSR = 180∘ - 105∘
= 75∘
Question 15 Report
A girl walks 45 meters in the direction 050o from a point Q to a point X. She then walks 24 meters in the direction 140o from X to a point Y. How far is she then from Q?
Answer Details
QY = 452 + 242 = 2025 + 576
= 2601
QY = √2601
= 51
Question 16 Report
Simplify \( \frac{0.0324 \times 0.00064}{0.48 \times 0.012} \)
Answer Details
0.0324×0.000640.48×0.012
= 324×10−4×10−548x10−2x12x10−3
324×64×10−948×12×10−5
= 36 x 10-4
= 3.6 x 10-3
Question 17 Report
Find the probability that a number selected at random from 40 to 50 is a prime
Answer Details
From 40 to 50 = 11 & number are prime i.e. 41, 43, 47
prob. of selecting a prime No. is 311
Question 18 Report
Find the reciprocal of \( \frac{\frac{2}{3}}{\frac{1}{2}+\frac{1}{3}} \)
Answer Details
23
= 23
= 23
x 65
= 45
reciprocal of 45
= 145
= 54
Question 19 Report
Two chords QR and NP of a circle intersect inside the circle at x. If RQP = 37o, RQN = 49o and QPN = 35o, find PRQ
Answer Details
In PNO, ONP
= 180 - (35 + 86)
= 180 - 121
= 59
PRQ = QNP = 59(angles in the same segment of a circle are equal)
Question 20 Report
Simplify \( \frac{1}{x-2} + \frac{1}{x+2} + \frac{2x}{x^2-4} \)
Answer Details
1x−2
+ 1x+2
+ 2xx2−4
= (x+2)+(x−2)+2x(x+2)(x−2)
= 4xx2−4
Question 22 Report
The table below gives the scores of a group of students in a Mathematical test.
| Scores | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| Frequency | 2 | 4 | 7 | 14 | 12 | 6 | 4 | 1 |
If the mode in m and the number of students who scored 4 or less is s. What is (s, m)?
Answer Details
M = mode = the number having the highest frequency = 4
S = Number of students with 4 or less marks
= 14 + 7 + 4 + 2
= 27
∴ (M,S) = (27, 4)
Question 23 Report
A regular polygon of n sides has 160o as the size of each interior angle. Find n
Answer Details
Interior + exterior = 360
160 + exterior = 360
Exterior = 360 - 160
Exterior = 20
n = 360/exterior
n = 360/20
n = 18
Question 24 Report
Find the values of x which satisfy the equation 16x - 5 x 4x + 4 = 0
Answer Details
16x - 5 x 4x + 4 = 0
(4x)2 - 5(4x) + 4 = 0
let 4x = y
y2 - 5y + 4 = 0
(y - 4)(y - 1) = 0
y = 4 or 1
4x = 4
x = 1
4x = 1
i.e. 4x = 4o, x = 0
∴ x = 1 or 0
Question 25 Report
Find all real numbers \(x\) which satisfy the inequality \(\frac{1}{3}(x + 1) - 1 > \frac{1}{5}(x + 4)\)
Answer Details
13
(x + 1) - 1 > 15
(x + 4)
= x+13
- 1 > x+45
x+13
- x+45
- 1 > 0
= 5x+5−3x−1215
= 2x - 7 > 15
= 2x > 12
= x > 11
Question 26 Report
If \( \frac{a}{c} = \frac{c}{d} = k \), find the value of \( \frac{3a^2-ac+c^2}{3b^2-bd+d^2} \)
Answer Details
ac
= cd
= k
∴ ab
= bk
cd
= k
∴ c = dk
= 3a2−ac+c23b2−bd+d2
= 3(bk)2−(bk)(dk)+dk23b2−bd+a2
= 3b2k2−bk2d+dk23b2−bd+d2
k = 3b2k2−bk2d+dk23b2−bd+d2
Question 27 Report
Divide the L.C.M of 48, 64, and 80 by their H.C.F
Answer Details
48 = 24 x 3, 64 = 26, 80 = 24 x 5
L.C.M = 26 x 3 x 5
H.C.F = 24
26×3×524
= 22 x 3 x 5
= 4 x 3 x 5
= 12 x 5
= 60
Question 28 Report
Find the median of the numbers 89, 141, 130, 161, 120, 131, 131, 100, 108 and 119
Answer Details
Arrange in ascending order
89, 100, 108, 119, |120, 130|, 131, 131, 141, 161
Median = 120+1302
= 125
Question 29 Report
Simplify \( \frac{1}{5x+5} + \frac{1}{7x+7} \)
Answer Details
15x+5
+ 17x+7
= 15(x+1)
+ 17(x+1)
= 7+535(x+1)
= 1235(x+1)
Question 30 Report
Udoh deposited ₦150.00 in the bank. At the end of 5 years the simple interest on the principal was ₦55.00. At what rate per annum was the interest paid?
Answer Details
.I = PTR100
R = 100PT
100×50150×6
= 223
= 713
%
Question 31 Report
An open rectangular box externally measures 4m x 3m x 4m. Find the total cost of painting the box externally if it costs ₦2.00 to paint one square meter
Answer Details
Total surface area(s) = 2(4 x 3) + 2(4 x 4)
= 2(12) + 2(16)
= 24 + 32
= 56cm2
1m2 costs ₦2.00
∴ 56m∴ will cost 56 x ₦2.00
= ₦112.00
Question 32 Report
In 1984, Ike was 24 yrs old and his father was 45 yrs old. In what year was Ike exactly half his father's age?
Answer Details
Let the no. of years be y
24 - y = 12
(45 - y)
45 - y = 2(24 - y)
45 - y = 48 - 2y
2y - y = 48 - 45
∴ y = 3
The exact year = 1984
1984 - 3 = 1981
Question 33 Report
If P = 18, Q = 21, R = -6 and S = -4, Calculate \( \frac{(P-Q)^3+S^2}{R^3} + S^2 \)
Answer Details
(P−Q)3+S2R3
= (18−21)3+(−4)2(−6)3
= −27+16R3
= −11−216
= 11216
Question 34 Report
An arc of a circle of radius 6cm is 8cm long. Find the area of the sector
Answer Details
Radius of the circle r = 6cm, Length of the arc = 8cm
Area of sector = θ360
x 2π
r2........(i)
Length of arc = θ360
x 2π
r........(ii)
from eqn. (ii) θ
= 240π
, subt. for θ
in eqn (i)
Area x 2401
x 1360
x π61
= 24cm2
Question 35 Report
PQ and PR are tangents from P to a circle centre O as shown in the figure. If QRP = \(34^\circ\), find the angle marked x
Answer Details
From the circle centre 0, if PQ & PR are tangents from P and QRP = 34∘
Then the angle marked x i.e. QOP
34∘ x 2 = 68∘
Question 36 Report
The ages of Tosan and Isa differ by 6 and the product of their ages is 187. Write their ages in the form (x, y), where x > y.
Answer Details
x - y = 6.......(i)
xy = 187.......(ii)
From equation (i), x(6 + y)
sub. for x in equation (ii) = y(6 + y)
= 187
y2 + 6y = 187
y2 + 6y - 187 = 0
(y + 17)(y - 11) = 0
y = -17 or y = 11
y cannot be negative, y = 11
Sub. for y in equation(i) = x - 11
= 16
x = 6 + 11
= 17
∴(x, y) = (17, 11)
Question 37 Report
Find the total surface area of solid cone of radius \(2\sqrt{3}\) cm and slanting side \(4\sqrt{3}\)
Answer Details
Total surface area of a solid cone
r = 2√3
= πr2
+ π
rH
H = 4√3
, π
r(r + H)
∴ Area = π
2√3
[2√3
+ 4√3
]
= π
2√3
(6√3
)
= 12π
x 3
= 36π
cm2
Question 38 Report
If \( \cos \theta = \frac{a}{b} \), find \( 1 + \tan^2 \theta \)
Answer Details
cosθ
= ab
, Sinθ
= √b2−a2a
Tanθ
= √b2−a2a2
, Tan 2 = √b2−a2a2
1 + tan2θ
= 1 + b2−a2a2
= a2+b2−a2a2
= b2a2
Question 39 Report
In \( \triangle \)XYZ, determine the cosine of angle Z.
Question 40 Report
The people in a city with a population of 0.9 million were grouped according to their ages. Use the diagram to determine the number of people in the 15 - 29 years group
Answer Details
15 - 29 years is represented by 104∘
Number of people in the group is 104360 x 0.9m
= 260000 = 26 x 104
Question 41 Report
Find n if log24 + log27 - log2n = 1
Answer Details
log24 + log27 - log2n = 1
= log2(4 x 7) - log2n = 1
= log228 - log2n
= log282n
282n
= 21
= 2
2n = 28
∴ n = 14
Question 42 Report
Factorize completely 8a + 125ax3
Answer Details
8a + 125ax3 = 23a + 53ax3
= a(23 + 53x3)
∴a[23 + (5x)3]
a3 + b3 = (a + b)(a2 - ab + b2)
∴ a(23 + (5x)3)
= a(2 + 5x)(4 - 10x + 25x2)
Question 43 Report
A number of pencils were shared out among Bisi, Sola and Tunde in the ratio of 2 : 3 : 5 respectively. If Bisi got 5, how many were share out?
Answer Details
Let x r3epresent total number of pencils shared
B : S : T = 2 + 3 + 5 = 10
2 : 3 : 5
= 210
x y
= 5
2y =5
2y = 50
∴ y = 502
= 25
Question 44 Report
If \( y = \frac{x}{x - 3} + \frac{x}{x + 4} \) find y when \( x = -2 \)
Answer Details
y = xx−3
+ xx+4
when x = -2
y = −2−5
+ (−2)−2+4
= −25
+ −22
= 4+−1010
= −1410
= -75
Question 45 Report
Simplify \(\left(\frac{1}{\sqrt{5}+\sqrt{3}}-\frac{1}{\sqrt{5}-\sqrt{3}}\right)x\frac{1}{\sqrt{3}}\)
Answer Details
(1√5+√3−1√5−√3
x 1√3
)
1√5+√3−1√5−√3
= √5−√3−(√5+√3)(√5+√3)(5√3)
= −2√35−3
= −2√37
Question 46 Report
Of the nine hundred students admitted in a university in 1979, the following was the distribution by state: Anambrs 185, Imo 135, Kaduna 90, Kwara 110, Ondo 155, Oyo 225. In a pie chart drawn to represent this distribution, The angle subtended at the centre by anambra is
Answer Details
Anambra = 185900
x 3601
= 74o
Question 47 Report
The figure is a solid with the trapezium PQRS as its uniform cross-section. Find its volume
Answer Details
Volume of solid = cross section x H
Since the cross section is a trapezium
= 12(6+11)×12×8
= 6 x 17 x 8 = 816m3
Question 48 Report
Simplify \( \frac{9^{\frac{1}{3}} \times 27^{-\frac{1}{3}}}{3^{-\frac{1}{6}} \times 3^{\frac{2}{3}}} \)
Answer Details
= 323×3233−16×323
= 3563−56
= 1
