JAMB UTME - Mathematics - 1986

Factorize (4a + 3)2 - (3a - 2)2

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(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)

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

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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${}^{\circ }$

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?

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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.

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PT x QT = TR x TS

24 x 8 = TR x 12

TR = $\frac{24\times 8}{12}$

= = 16cm

Solve the equation 3x2 + 6x - 2 = 0

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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 = $\frac{-1\pm \sqrt{15}}{3}$

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

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

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

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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

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

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

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

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

Evaluate (212)3 - (121)3 + (222)3

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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?

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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

Factorize completely 8a + 125ax3

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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)

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

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cos z = $\frac{y^2+x^2-z^2}{2yx}$

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

= $\frac{29}{36}$

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

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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

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

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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

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

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

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

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

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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?

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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

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)?

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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)

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

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

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

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

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

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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

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.

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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)

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

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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

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

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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

Factorize x2 + 2a + ax + 2x

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

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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

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

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

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?

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

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

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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

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

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

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?

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

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

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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

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?

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

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

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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 + $\frac{b^2-a^2}{a^2}$
= $\frac{a^2+b^2-a^2}{a^2}$
= $\frac{b^2}{a^2}$

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

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

Find n if log24 + log27 - log2n = 1

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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

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

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

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

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

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$\frac{a}{c}$ = $\frac{c}{d}$ = 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 = $\frac{3b^2{k}^2-b{k}^{2}d+d{k}^2}{3b^2-bd+d^2}$

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

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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

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

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

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

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

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

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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${}^{\circ }$

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

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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

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

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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

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

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

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

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$\frac{1}{3}$(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