JAMB UTME - Mathematics - 1986
Pregunta 1 Informe
Simplify \( \frac{0.0324 \times 0.00064}{0.48 \times 0.012} \)
Detalles de la respuesta
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
Pregunta 2 Informe
Simplify \( \frac{9^{\frac{1}{3}} \times 27^{-\frac{1}{3}}}{3^{-\frac{1}{6}} \times 3^{\frac{2}{3}}} \)
Detalles de la respuesta
= 323×3233−16×323
= 3563−56
= 1
Pregunta 3 Informe
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.
Detalles de la respuesta
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)
Pregunta 4 Informe
Find the values of x which satisfy the equation 16x - 5 x 4x + 4 = 0
Detalles de la respuesta
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
Pregunta 5 Informe
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?
Detalles de la respuesta
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
Pregunta 6 Informe
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
Detalles de la respuesta
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∘
Pregunta 7 Informe
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
Detalles de la respuesta
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
Pregunta 8 Informe
Find all real numbers \(x\) which satisfy the inequality \(\frac{1}{3}(x + 1) - 1 > \frac{1}{5}(x + 4)\)
Detalles de la respuesta
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
Pregunta 9 Informe
make U the subject of the formula \(S = \sqrt{\frac{6}{u} - \frac{w}{2}}\)
Detalles de la respuesta
S = √6u−w2
S = 12−uw2u
2us2 = 12 - uw
u(2s2 + w) = 12
u = 122s2+w
Pregunta 10 Informe
If x varies directly as y3 and x = 2 when y = 1, find x when y = 5
Detalles de la respuesta
x α
y3
x = ky3
k = xy3
when x = 2, y = 1
k = 2
Thus x = 2y3 - equation of variation
= 2(5)3
= 250
Pregunta 11 Informe
Simplify \(\left(\frac{1}{\sqrt{5}+\sqrt{3}}-\frac{1}{\sqrt{5}-\sqrt{3}}\right)x\frac{1}{\sqrt{3}}\)
Detalles de la respuesta
(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
Pregunta 12 Informe
At what points does the straight line y = 2x + 1 intersect the curve y = 2x2 + 5x - 1?
Detalles de la respuesta
Pregunta 13 Informe
If Musa scored 75 in biology instead of 57, his average mark in four subjects would have been 60. What was his total mark?
Detalles de la respuesta
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
Pregunta 14 Informe
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?
Detalles de la respuesta
QY = 452 + 242 = 2025 + 576
= 2601
QY = √2601
= 51
Pregunta 15 Informe
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?
Detalles de la respuesta
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
Pregunta 16 Informe
Find n if log24 + log27 - log2n = 1
Detalles de la respuesta
log24 + log27 - log2n = 1
= log2(4 x 7) - log2n = 1
= log228 - log2n
= log282n
282n
= 21
= 2
2n = 28
∴ n = 14
Pregunta 17 Informe
If \( \cos \theta = \frac{a}{b} \), find \( 1 + \tan^2 \theta \)
Detalles de la respuesta
cosθ
= ab
, Sinθ
= √b2−a2a
Tanθ
= √b2−a2a2
, Tan 2 = √b2−a2a2
1 + tan2θ
= 1 + b2−a2a2
= a2+b2−a2a2
= b2a2
Pregunta 18 Informe
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?
Detalles de la respuesta
Prob. of purple shirt is 718
Pregunta 19 Informe
Find the probability that a number selected at random from 40 to 50 is a prime
Detalles de la respuesta
From 40 to 50 = 11 & number are prime i.e. 41, 43, 47
prob. of selecting a prime No. is 311
Pregunta 21 Informe
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
Detalles de la respuesta
15 - 29 years is represented by 104∘
Number of people in the group is 104360 x 0.9m
= 260000 = 26 x 104
Pregunta 22 Informe
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?
Detalles de la respuesta
.I = PTR100
R = 100PT
100×50150×6
= 223
= 713
%
Pregunta 23 Informe
Two chords QR and NP of a circle intersect inside the circle at x. If RQP = 37o, RQN = 49o and QPN = 35o, find PRQ
Detalles de la respuesta
In PNO, ONP
= 180 - (35 + 86)
= 180 - 121
= 59
PRQ = QNP = 59(angles in the same segment of a circle are equal)
Pregunta 24 Informe
Simplify \( \frac{1}{5x+5} + \frac{1}{7x+7} \)
Detalles de la respuesta
15x+5
+ 17x+7
= 15(x+1)
+ 17(x+1)
= 7+535(x+1)
= 1235(x+1)
Pregunta 25 Informe
Find the total surface area of solid cone of radius \(2\sqrt{3}\) cm and slanting side \(4\sqrt{3}\)
Detalles de la respuesta
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
Pregunta 26 Informe
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.
Detalles de la respuesta
PT x QT = TR x TS
24 x 8 = TR x 12
TR = 24×812
= = 16cm
Pregunta 27 Informe
If \( \frac{a}{c} = \frac{c}{d} = k \), find the value of \( \frac{3a^2-ac+c^2}{3b^2-bd+d^2} \)
Detalles de la respuesta
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
Pregunta 28 Informe
Factorize completely 8a + 125ax3
Detalles de la respuesta
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)
Pregunta 29 Informe
Divide the L.C.M of 48, 64, and 80 by their H.C.F
Detalles de la respuesta
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
Pregunta 30 Informe
Factorize (4a + 3)2 - (3a - 2)2
Detalles de la respuesta
(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)
Pregunta 31 Informe
The figure is a solid with the trapezium PQRS as its uniform cross-section. Find its volume
Detalles de la respuesta
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
Pregunta 32 Informe
Simplify \( \frac{1}{x-2} + \frac{1}{x+2} + \frac{2x}{x^2-4} \)
Detalles de la respuesta
1x−2
+ 1x+2
+ 2xx2−4
= (x+2)+(x−2)+2x(x+2)(x−2)
= 4xx2−4
Pregunta 33 Informe
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
Detalles de la respuesta
Anambra = 185900
x 3601
= 74o
Pregunta 34 Informe
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)?
Detalles de la respuesta
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)
Pregunta 35 Informe
If 5(x + 2y) = 5 and 4(x + 3y) = 16, find 3(x + y)
Detalles de la respuesta
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
Pregunta 36 Informe
If P = 18, Q = 21, R = -6 and S = -4, Calculate \( \frac{(P-Q)^3+S^2}{R^3} + S^2 \)
Detalles de la respuesta
(P−Q)3+S2R3
= (18−21)3+(−4)2(−6)3
= −27+16R3
= −11−216
= 11216
Pregunta 37 Informe
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?
Detalles de la respuesta
Pregunta 38 Informe
Solve the equation 3x2 + 6x - 2 = 0
Detalles de la respuesta
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
Pregunta 39 Informe
A regular polygon of n sides has 160o as the size of each interior angle. Find n
Detalles de la respuesta
Interior + exterior = 360
160 + exterior = 360
Exterior = 360 - 160
Exterior = 20
n = 360/exterior
n = 360/20
n = 18
Pregunta 40 Informe
In \( \triangle \)XYZ, determine the cosine of angle Z.
Detalles de la respuesta
cos z = y2+x2−z22yx
= 9+36−162(3)(6)
= 2936
Pregunta 41 Informe
Find the reciprocal of \( \frac{\frac{2}{3}}{\frac{1}{2}+\frac{1}{3}} \)
Detalles de la respuesta
23
= 23
= 23
x 65
= 45
reciprocal of 45
= 145
= 54
Pregunta 42 Informe
Find the median of the numbers 89, 141, 130, 161, 120, 131, 131, 100, 108 and 119
Detalles de la respuesta
Arrange in ascending order
89, 100, 108, 119, |120, 130|, 131, 131, 141, 161
Median = 120+1302
= 125
Pregunta 43 Informe
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
Detalles de la respuesta
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
Pregunta 44 Informe
An arc of a circle of radius 6cm is 8cm long. Find the area of the sector
Detalles de la respuesta
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
Pregunta 45 Informe
Find the smallest number by which 252 can be multiplied to obtain a perfect square
Detalles de la respuesta
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
Pregunta 46 Informe
In the figure, \( \triangle \)PQT is isosceles. PQ = QT, SRQ = \(35^\circ\), TPQ = \(20^\circ\) and PQR is a straight line.Calculate TSR
Detalles de la respuesta
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∘
Pregunta 48 Informe
If \( y = \frac{x}{x - 3} + \frac{x}{x + 4} \) find y when \( x = -2 \)
Detalles de la respuesta
y = xx−3
+ xx+4
when x = -2
y = −2−5
+ (−2)−2+4
= −25
+ −22
= 4+−1010
= −1410
= -75
