JAMB UTME - Mathematics - 1986

Factorize completely 8a + 125ax3

Awọn alaye Idahun

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)

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.

Awọn alaye Idahun

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)

Find n if log24 + log27 - log2n = 1

Awọn alaye Idahun

log24 + log27 - log2n = 1
= log2(4 x 7) - log2n = 1
= log228 - log2n
= log$\frac{28}{2n}$
$\frac{28}{2n}$ = 21
= 2
2n = 28
∴ n = 14

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

Awọn alaye Idahun

From the circle centre 0, if PQ & PR are tangents from P and QRP = 34${}^{\circ }$

Then the angle marked x i.e. QOP

34${}^{\circ }$ x 2 = 68${}^{\circ }$

Simplify $\frac{9^{\frac{1}{3}}\times {27}^{-\frac{1}{3}}}{3^{-\frac{1}{6}}\times 3^{\frac{2}{3}}}$

Awọn alaye Idahun

Simplify $\frac{1}{x-2}$ + $\frac{1}{x+2}$ + $\frac{2x}{x^2-4}$

Awọn alaye Idahun

$\frac{1}{x-2}$ + $\frac{1}{x+2}$ + $\frac{2x}{x^2-4}$
= $\frac{(x+2)+(x-2)+2x}{(x+2)(x-2)}$
= $\frac{4x}{x^2-4}$

Find the total surface area of solid cone of radius 2$\sqrt{3}$cm and slanting side 4$\sqrt{3}$

Awọn alaye Idahun

Total surface area of a solid cone
r = 2$\sqrt{3}$
= $\pi r^2$ + $\pi$rH
H = 4$\sqrt{3}$, $\pi$r(r + H)
∴ Area = $\pi$2$\sqrt{3}$ [2$\sqrt{3}$ + 4$\sqrt{3}$]
= $\pi$2$\sqrt{3}$(6$\sqrt{3}$)
= 12$\pi$ x 3
= 36$\pi$cm2

Simplify $\frac{0.0324\times 0.00064}{0.48\times 0.012}$

Awọn alaye Idahun

$\frac{0.0324\times 0.00064}{0.48\times 0.012}$ = $\frac{324\times {10}^{-4}\times {10}^{-5}}{48x{10}^{-2}x12x{10}^{-3}}$
$\frac{324\times 64\times 10^{-9}}{48\times 12\times {10}^{-5}}$ = 36 x 10-4
= 3.6 x 10-3

Find the median of the numbers 89, 141, 130, 161, 120, 131, 131, 100, 108 and 119

Awọn alaye Idahun

Arrange in ascending order
89, 100, 108, 119, |120, 130|, 131, 131, 141, 161
Median = $\frac{120+130}{2}$
= 125

Solve the equation 3x2 + 6x - 2 = 0

Awọn alaye Idahun

3x2 + 6x - 2 = 0
Using almighty formula i.e. x = $\frac{b\pm \sqrt{b^2-4ac}}{2a}$
a = 3, b = 6, c = -2
x = $\frac{-6\pm \sqrt{6^2-4\left(3\right)(-2)}}{2\left(3\right)}$
x = $\frac{6\pm \sqrt{36-24}}{6}$
x = $\frac{6\pm \sqrt{60}}{6}$
x = $\frac{-6\pm \sqrt{4\times 15}}{6}$
x = $\frac{-1\pm \sqrt{15}}{3}$

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

Awọn alaye Idahun

Anambra = $\frac{185}{900}$ x $\frac{360}{1}$
= 74o

In $△$XYZ, determine the cosine of angle Z.

Awọn alaye Idahun

cos z = $\frac{y^2+x^2-z^2}{2yx}$

= $\frac{9+36-16}{2\left(3\right)\left(6\right)}$

= $\frac{29}{36}$

At what points does the straight line y = 2x + 1 intersect the curve y = 2x2 + 5x - 1?

(-2, -3) and($\frac{1}{2}$, 0)

Awọn alaye Idahun

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?

Awọn alaye Idahun

QY = 452 + 242 = 2025 + 576
= 2601
QY = $\sqrt{2601}$
= 51

In the figure, $△$PQT is isosceles. PQ = QT, SRQ = 35${}^{\circ }$, TPQ = 20${}^{\circ }$ and PQR is a straight line.Calculate TSR

Awọn alaye Idahun

Given $△$ isosceles PQ = QT, SRQ = 35${}^{\circ }$

TPQ = 20${}^{\circ }$

PQR = is a straight line

Since PQ = QT, angle P = angle T = 20${}^{\circ }$

Angle PQR = 180${}^{\circ }$ - (20 + 20) = 140${}^{\circ }$

TQR = 180${}^{\circ }$ - 140${}^{\circ }$ = 40${}^{\circ }$ < on a straight line

QSR = 180${}^{\circ }$ - (40 + 35)${}^{\circ }$ = 105${}^{\circ }$

TSR = 180${}^{\circ }$ - 105${}^{\circ }$

= 75${}^{\circ }$

Simplify ($\frac{1}{\sqrt{5}+\sqrt{3}}-\frac{1}{\sqrt{5}-\sqrt{3}}$ x $\frac{1}{\sqrt{3}}$)

Awọn alaye Idahun

($\frac{1}{\sqrt{5}+\sqrt{3}}-\frac{1}{\sqrt{5}-\sqrt{3}}$ x $\frac{1}{\sqrt{3}}$)
$\frac{1}{\sqrt{5}+\sqrt{3}}-\frac{1}{\sqrt{5}-\sqrt{3}}$ = $\frac{\sqrt{5}-\sqrt{3}-(\sqrt{5}+\sqrt{3})}{(\sqrt{5}+\sqrt{3})\left(5\sqrt{3}\right)}$
= $\frac{-2\sqrt{3}}{5-3}$
= $\frac{-2\sqrt{3}}{7}$

Evaluate (212)3 - (121)3 + (222)3

Awọn alaye Idahun

If cos$\theta$ = $\frac{a}{b}$, find 1 + tan2$\theta$

Awọn alaye Idahun

cos$\theta$ = $\frac{a}{b}$, Sin$\theta$ = $\sqrt{\frac{b^2-a^2}{a}}$
Tan$\theta$ = $\sqrt{\frac{b^2-a^2}{a^2}}$, Tan 2 = $\sqrt{\frac{b^2-a^2}{a^2}}$
1 + tan2$\theta$ = 1 + $\frac{b^2-a^2}{a^2}$
= $\frac{a^2+b^2-a^2}{a^2}$
= $\frac{b^2}{a^2}$

The table below gives the scores of a group of students in a Mathematical test.

$\begin{array}{ccccccccc}Scores& 1& 2& 3& 4& 5& 6& 7& 8\\ Frequency& 2& 4& 7& 14& 12& 6& 4& 1\end{array}$

If the mode in m and the number of students who scored 4 or less is s. What is (s, m)?

Awọn alaye Idahun

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)

Factorize x2 + 2a + ax + 2x

Awọn alaye Idahun

An arc of a circle of radius 6cm is 8cm long. Find the area of the sector

Awọn alaye Idahun

Radius of the circle r = 6cm, Length of the arc = 8cm
Area of sector = $\frac{\theta }{360}$ x 2$\pi$r2........(i)
Length of arc = $\frac{\theta }{360}$ x 2$\pi$r........(ii)
from eqn. (ii) $\theta$ = $\frac{240}{\pi }$, subt. for $\theta$ in eqn (i)
Area x $\frac{240}{1}$ x $\frac{1}{360}$ x $\frac{\pi 6}{1}$
= 24cm2

If P = 18, Q = 21, R = -6 and S = -4, Calculate $\frac{(P-Q{)}^3+S^2}{R^3}$ + S2

Awọn alaye Idahun

$\frac{(P-Q{)}^3+S^2}{R^3}$ = $\frac{(18-21{)}^3+(-4{)}^2}{(-6{)}^3}$
= $\frac{-27+16}{R^3}$
= $\frac{-11}{-216}$
= $\frac{11}{216}$

If Musa scored 75 in biology instead of 57, his average mark in four subjects would have been 60. What was his total mark?

Awọn alaye Idahun

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 $\frac{Y}{4}$ = 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

Find the smallest number by which 252 can be multiplied to obtain a perfect square

Awọn alaye Idahun

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

The figure is a solid with the trapezium PQRS as its uniform cross-section. Find its volume

Awọn alaye Idahun

Volume of solid = cross section x H

Since the cross section is a trapezium

= $\frac{1}{2}(6+11)\times 12\times 8$

= 6 x 17 x 8 = 816m3

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

Awọn alaye Idahun

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

Find the probability that a number selected at random from 40 to 50 is a prime

Awọn alaye Idahun

From 40 to 50 = 11 & number are prime i.e. 41, 43, 47
prob. of selecting a prime No. is $\frac{3}{11}$

Factorize (4a + 3)2 - (3a - 2)2

Awọn alaye Idahun

(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)

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

Awọn alaye Idahun

15 - 29 years is represented by 104${}^{\circ }$

Number of people in the group is $\frac{104}{360}$ x 0.9m

= 260000 = 26 x 104

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?

Awọn alaye Idahun

Let x r3epresent total number of pencils shared

B : S : T = 2 + 3 + 5 = 10

2 : 3 : 5

= 210 $\frac{2}{10}$ x y

= 5

2y =5

2y = 50

∴ y = 502 $\frac{50}{2}$

= 25

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?

Awọn alaye Idahun

Find the values of x which satisfy the equation 16x - 5 x 4x + 4 = 0

Awọn alaye Idahun

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

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?

Awọn alaye Idahun

.I = $\frac{PTR}{100}$
R = $\frac{100}{PT}$
$\frac{100\times 50}{150\times 6}$ = $\frac{22}{3}$
= 7$\frac{1}{3}$%

Divide the L.C.M of 48, 64, and 80 by their H.C.F

Awọn alaye Idahun

48 = 24 x 3, 64 = 26, 80 = 24 x 5
L.C.M = 26 x 3 x 5
H.C.F = 24
$\frac{2^6\times 3\times 5}{2^4}$
= 22 x 3 x 5
= 4 x 3 x 5
= 12 x 5
= 60

If y = $\frac{x}{x-3}$ + $\frac{x}{x+4}$ find y when x = -2

Awọn alaye Idahun

y = $\frac{x}{x-3}$ + $\frac{x}{x+4}$ when x = -2
y = $\frac{-2}{-5}$ + $\frac{(-2)}{-2+4}$
= $\frac{-2}{5}$ + $\frac{-2}{2}$
= $\frac{4+-10}{10}$
= $\frac{-14}{10}$
= -$\frac{7}{5}$

Find the reciprocal of $\frac{\frac{2}{3}}{\frac{1}{2}+\frac{1}{3}}$

Awọn alaye Idahun

$\frac{2}{3}$ = $\frac{2}{3}$
= $\frac{2}{3}$ x $\frac{6}{5}$
= $\frac{4}{5}$
reciprocal of $\frac{4}{5}$ = $\frac{1}{\frac{4}{5}}$
= $\frac{5}{4}$

If $\frac{a}{c}$ = $\frac{c}{d}$ = k, find the value of $\frac{3a^2?ac+c^2}{3b^2?bd+d^2}$

Awọn alaye Idahun

$\frac{a}{c}$ = $\frac{c}{d}$ = k
∴ $\frac{a}{b}$ = bk
$\frac{c}{d}$ = k
∴ c = dk
= $\frac{3a^2-ac+c^2}{3b^2-bd+d^2}$
= $\frac{3(bk{)}^2-(bk\left)\right(dk)+d{k}^2}{3b^2-bd+a^2}$
= $\frac{3b^2{k}^2-b{k}^{2}d+d{k}^2}{3b^2-bd+d^2}$
k = $\frac{3b^2{k}^2-b{k}^{2}d+d{k}^2}{3b^2-bd+d^2}$

Simplify $\frac{1}{5x+5}$ + $\frac{1}{7x+7}$

Awọn alaye Idahun

$\frac{1}{5x+5}$ + $\frac{1}{7x+7}$ = $\frac{1}{5(x+1)}$ + $\frac{1}{7(x+1)}$
= $\frac{7+5}{35(x+1)}$
= $\frac{12}{35(x+1)}$

Find all real numbers x which satisfy the inequality $\frac{1}{3}$(x + 1) - 1 > $\frac{1}{5}$(x + 4)

Awọn alaye Idahun

$\frac{1}{3}$(x + 1) - 1 > $\frac{1}{5}$(x + 4)
= $\frac{x+1}{3}$ - 1 > $\frac{x+4}{5}$
$\frac{x+1}{3}$ - $\frac{x+4}{5}$ - 1 > 0
= $\frac{5x+5-3x-12}{15}$
= 2x - 7 > 15
= 2x > 12
= x > 11

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?

Awọn alaye Idahun

Prob. of purple shirt is $\frac{7}{18}$

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

Awọn alaye Idahun

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

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.

Awọn alaye Idahun

PT x QT = TR x TS

24 x 8 = TR x 12

TR = $\frac{24\times 8}{12}$

= = 16cm

Two chords QR and NP of a circle intersect inside the circle at x. If RQP = 37o, RQN = 49o and QPN = 35o, find PRQ

Awọn alaye Idahun

In PNO, ONP
= 180 - (35 + 86)
= 180 - 121
= 59
PRQ = QNP = 59(angles in the same segment of a circle are equal)

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?

Awọn alaye Idahun

Let the no. of years be y
24 - y = $\frac{1}{2}$(45 - y)
45 - y = 2(24 - y)
45 - y = 48 - 2y
2y - y = 48 - 45
∴ y = 3
The exact year = 1984
1984 - 3 = 1981

If x varies directly as y3 and x = 2 when y = 1, find x when y = 5

Awọn alaye Idahun

x $\alpha$ y3
x = ky3
k = $\frac{x}{y^3}$
when x = 2, y = 1
k = 2
Thus x = 2y3 - equation of variation
= 2(5)3
= 250

A regular polygon of n sides has 160o as the size of each interior angle. Find n

Awọn alaye Idahun

Interior + exterior = 360
160 + exterior = 360
Exterior = 360 - 160
Exterior = 20
n = 360/exterior
n = 360/20
n = 18

If 5(x + 2y) = 5 and 4(x + 3y) = 16, find 3(x + y)

Awọn alaye Idahun

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

make U the subject of the formula S = $\sqrt{\frac{6}{u}-\frac{w}{2}}$

Awọn alaye Idahun

S = $\sqrt{\frac{6}{u}-\frac{w}{2}}$
S = $\frac{12-uw}{2u}$
2us2 = 12 - uw
u(2s2 + w) = 12
u = $\frac{12}{2s^2+w}$