JAMB UTME - Mathematics - 1988

In the figure, PS = 7cm and RY = 9cm. IF the area of parallelogram PQRS is 56cm2. Find the area of trapezium PQTS

Bayanin Amsa

From the figure, PS = QR = YT = 7cm

Area of parallelogram PQRS = 56cm

56 = base x height, where base = 7

7 x h = 56cm,

h = $\frac{56}{7}$

= 8cm

Area of trapezium $\frac{1}{2}$ (sum of two sides)x height where two sides are QT and PS but QT = QR + RY + YT = 7 +9 + 7 = 23cm

Area of trapezium PQTS = $\frac{1}{2}$(23 + 7) x 8

$\frac{1}{2}$ x 30 x 8 = 120cmsq

The solutions of x2 - 2x - 1 = 0 are the points of intersection of two graphs. if one of the graphs is y = 2 + x - x2, find the second graph

Bayanin Amsa

Simplify $\frac{1\frac{1}{2}}{\text{2 +}\frac{1}{4}\text{of 32}}$

Bayanin Amsa

$\frac{1\frac{1}{2}}{\text{2 +}\frac{1}{4}\text{of 32}}$
= $\frac{\frac{3}{2}}{2+\frac{1}{4}\times 32}$
$\frac{3}{2}$ x 4 = 6

The thickness of an 800 pages of book is 18mm. Calculate the thickness of one leaf of the book giving your answer in meters and in standard form

Bayanin Amsa

Thickness of an 800 pages book = 18mm to meter
18 x 103m = 1.8 x 10-2m
One leaf = $\frac{1.8\times {10}^{-2}}{800}$
= $\frac{1.8\times {10}^{-2}}{8\times {10}^2}$
= $\frac{-1.8}{8}$ x 10-4
= 0.225 x 10-4
= 2.25 x 10-5m

In the figure, a solid consists of a hemisphere surmounted by a right circular cone, with radius 3.0cm and height 6.0cm. Find the volume of the solid

Bayanin Amsa

The volume of the solid = vol. of cone + vol. of hemisphere

volume of cone = $\frac{1}{2}{\pi }^{2}h$

= $\frac{1\pi }{3}\times (3{)}^{2}x6=18\pi c{m}^2$

vol. of hemisphere = $\frac{4\pi r^3}{6}=\frac{2\pi r^3}{3}$

= $\frac{2\pi }{3}\times (3{)}^3=18\pi c{m}^3$

vol. of solid = 18$\pi$ + 18$\pi$

= 36$\pi$cm3

In the figure, PQ is a parallel to ST and QRS = 40${}^{\circ }$. Find the value of x.

Bayanin Amsa

From the figure, 3x + x - 40${}^{\circ }$ = 180${}^{\circ }$

4x = 180${}^{\circ }$ + 40${}^{\circ }$

4x = 220${}^{\circ }$

x = $\frac{220}{4}$

= 55${}^{\circ }$

Simplify $\frac{4a^2-49b^2}{2a^2-5ab-7b^2}$

Bayanin Amsa

$\frac{4a^2-49b^2}{2a^2-5ab-7b^2}$ = $\frac{(2a{)}^2-(7b{)}^2}{(a-b)(2a+7b)}$
= $\frac{(2a+7b)(2a-7b)}{(a-b)(2a+7b)}$
= $\frac{2a-7b}{a-b}$

Tope bought X oranges at N5.00 each and some mangoes at N4.00 each. if she bought twice as many mangoes as oranges and spent at least N65.00 and at most N130.00, find the range of values of X.

Bayanin Amsa

A basket contain green, black and blue balls in the ratio 5 : 2 : 1. If there are 10 blue balls. Find the corresponding new ratio when 10 green and 10 black balls are removed from the basket

Bayanin Amsa

Let x represent total number of balls in the basket.
If there are 10 blue balls, $\frac{1}{8}$ of x = 10
x = 10 x 8 = 80 balls
Green balls will be $\frac{5}{8}$ x 80 = 50 and black balls = $\frac{2}{8}$ x 80 = 20
Ratio = Green : black : blue
50 : 20 : 10
-10 : 10 : -
------------------
New Ratio 40 : 10 : 10
4 : 1 : 1

Find m such that (m + $\sqrt{3}$)(1 - $\sqrt{3}$)2 = 6 - 2$\sqrt{2}$

Bayanin Amsa

(m + $\sqrt{3}$)(1 - $\sqrt{3}$)2 = 6 - 2$\sqrt{2}$
(m + $\sqrt{3}$)(4 - 2$\sqrt{3}$) = 6 - 2$\sqrt{2}$
= 6 - 2$\sqrt{3}$
4m - 6 + 4 - 2m$\sqrt{3}$ = 6 - 2$\sqrt{3}$
comparing co-efficients,
4m - 6 = 6.......(i)
4 - 2m = -2.......(ii)
in both equations, m = 3

If $\frac{1}{p}$ = $\frac{a^2+2ab+b^2}{a-b}$ and $\frac{l}{q}$ = $\frac{a+b}{a^2-2ab+b^2}$ Find $\frac{p}{q}$

Bayanin Amsa

The solution of the quadratic equation px2 + qx + b = 0 is

Bayanin Amsa

px2 + qx + b = 0
Using almighty formula
$\frac{-b\pm \sqrt{b^2-4ac}}{2a}$.........(i)
Where a = p, b = q and c = b
substitute for this value in equation (i)
= $\frac{-q\pm \sqrt{q^2-4bp}}{2p}$

If (IPO3)4 = 11510 find P

Bayanin Amsa

1 x 43 + P x 42 + 0 x 4 + 3 = 11510
16p + 67 = 115 p = $\frac{48}{16}$
= 3

If 7 and 189 are the first and fourth terms of a geometric progression respectively find the sum for the first
three terms of the progression

Bayanin Amsa

Solve the following equation equation for x2 + $\frac{2x}{r^2}$ + $\frac{1}{r^4}$ = 0

Bayanin Amsa

x2 + $\frac{2x}{r^2}$ + $\frac{1}{r^4}$ = 0
(x + $\frac{1}{r^2}$) = 0
x + $\frac{1}{r^2}$ = 0
x = $\frac{-1}{r^2}$

Find correct to one decimal place, 0.24633 $÷$ 0.0306

Bayanin Amsa

$\frac{0.24633}{0.03060}$ multiplying throughout by 100,000
= $\frac{24633}{3060}$
= 8.05
= 8.1

Simplify $\frac{x?y}{x^{\frac{1}{3}}?x^{\frac{1}{3}}}$

Bayanin Amsa

If cos $?$ = $\frac{x}{y}$, find cosec$?$

Bayanin Amsa

Cos $\theta$ = $\frac{x}{y}$ = $\frac{\text{adj}}{\text{opp}}$
(hyp2) = opp2 + adj2
(hyp2) = x2 + y2
hyp = $\sqrt{x^2+y^2}$
Cosec$\theta$ = hyp
= x2 + y2
= $\frac{1}{y}$$\sqrt{x^2+y^2}$

For which of the following exterior angles is a regular polygon possible? i. 36o ii. 18o iii. 15o

Bayanin Amsa

for a regular polygon to be possible, it must have all sides angles equal. $\frac{360}{18}$ = 20 sides and $\frac{360}{15}$ = 24 sides
(ii) and (iii) are right

Given that 3x - 5y - 3 = 0, 2y - 6x + 5 = 0 the value of (x, y) is

Bayanin Amsa

3x - 5y = 3, 2y - 6x = -5
-5y + 3x = 3........{i} x 2
2y - 6x = -5.........{ii} x 5
Substituting for x in equation (i)
-5y + 3($\frac{19}{24}$) = 3
-5y + 3 x $\frac{19}{24}$ = 3
-5y = $\frac{3-19}{8}$
-5 = $\frac{24-19}{8}$
= $\frac{5}{8}$
y = $\frac{5}{8\times 5}$
y = $\frac{-1}{8}$
(x, y) = ($\frac{19}{24},\frac{-1}{8}$)

In triangle PQR, PQ = 1cm, QR = 2cm and PQR = 120o Find the longest side of the triangle

Bayanin Amsa

PR2 = PQ2 + QR2 - 2(QR)(PQ) COS 120o
PR2 = 12 + 22 - 2(1)(2) x - cos 60o
= 5 - 2(1)(2) x -$\frac{1}{2}$
= 5 + 2 = 7
PR = $\sqrt{7}$cm

In the figure, PQRS is a circle. If chords QR and RS are equal, calculate the value of x.

Bayanin Amsa

SRT is a straight line, where QRT = 120

SRQ = 180${}^{\circ }$ - 120${}^{\circ }$ = 60${}^{\circ }$ - (angle on a straight line)

also angle QRS = 180${}^{\circ }$ - 100${}^{\circ }$ (angle on a straight line) . In angles where QR = SR and angle SRQ = 60${}^{\circ }$

x = 100 - 60 = 40${}^{\circ }$

Evaluate $\frac{8^{\frac{1}{3}}\times 5^{\frac{2}{3}}}{{10}^{\frac{2}{3}}}$ = $\frac{8^{\frac{1}{3}}\times 5^{\frac{2}{3}}}{{10}^{\frac{2}{3}}}$

Bayanin Amsa

$\frac{8^{\frac{1}{3}}\times 5^{\frac{3}{2}}}{\frac{2}{{10}^3}}$ = $\frac{(2^3{)}^{\frac{1}{3}}\times 5^{\frac{3}{2}}}{(2\times 5{)}^{\frac{2}{3}}}$
= $\frac{2\times 5}{2^{\frac{2}{3}}\times 5^{\frac{3}{2}}}$
= 21 - $\frac{2}{3}$
= 2$\frac{1}{3}$
= 3$\sqrt{2}$

If log102 = 0.3010 and log103 = 0.4771, evaluate; without using logarithm tables, log104.5

Bayanin Amsa

If log102 = 0.3010 and log103 = 0.4771,
log104.5 = log10 $\frac{(3\times 3)}{2}$
log103 + log103 - log102 = 0.4771 + 0.4771 - 0.0310
= 0.6532

If x and y represent the mean and the median respectively of the following set of numbers 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, find the $\frac{x}{y}$ correct to one decimal place

Bayanin Amsa

Mean $\overline{x}$ = $\frac{156}{10}$ = 15.6
Median = $\overline{y}$ = $\frac{15+16}{2}$
$\frac{31}{2}$ = 15.5
$\frac{x}{y}$ = $\frac{15.6}{15.5}$ = 1.0065
1.0(1 d.p)

If cos 60o = 1/2, which of the following angle has cosine of -1/2?

Bayanin Amsa

cos60o = 1/2, cos(180o/60o) = -1/2

cos120o = -1/2

if x is the addition of the prime numbers between 1 and 6; and y the H.C.F. of 6, 9, 15. Find the product of x and y

Bayanin Amsa

Prime numbers between 1 and 6 are 2, 3 and 5
x = 2 + 3
= 5 = 10
H.C.F. of 6, 9, 15 = 3
∴ y = 3
X x y = 10 x 3
= 30

If two dice are thrown together, what is the probability of obtaining at least a score of 10?

Bayanin Amsa

The total sample space when two dice are thrown together is 6 x 6 = 36
$\begin{array}{ccccccc}& 1& 2& 3& 4& 5& 6\\ 1.& 1.1& 1.2& 1.3& 1.4& 1.5& 1.6\\ 2& 2.1& 2.2& 2.3& 2.4& 2.5& 2.6\\ 3& 3.1& 3.2& 3.3& 3.4& 3.5& 3.6\\ 4& 4.1& 4.2& 4.3& 4.4& 4.5& 4.6\\ 5& 5.1& 5.2& 5.3& 5.4& 5.5& 5.6\\ 6& 6.1& 6.2& 6.3& 6.4& 6.5& 6.6\end{array}$
At least 10 means 10 and above
P(at least 10) = $\frac{6}{36}$
= $\frac{1}{6}$

Simplify $\frac{x-7}{x^2-9}$ x $\frac{x^2-3x}{x^2-49}$

Bayanin Amsa

$\frac{x-7}{x^2-9}$ x $\frac{x^2-3x}{x^2-49}$
= $\frac{x-7}{(x-3)(x+3)}$ x $\frac{x(x-3)}{(x-7)(x+7)}$
= $\frac{x}{(x+3)(x+7)}$

If a metal pipe 10cm long has an external diameter of 12cm and a thickness of 1cm find the volume of the metal used in making the pipe

Bayanin Amsa

The volume of the pipe is equal to the area of the cross section and length.
let outer and inner radii be R and r respectively.
Area of the cross section = (R2 - r2)
where R = 6 and r = 6 - 1
= 5cm
Area of the cross section = (62 - 52)$\pi$ = (36 - 25)$\pi$cm sq
vol. of the pipe = $\pi$ (R2 - r2)L where length (L) = 10
volume = 11$\pi$ x 10
= 110$\pi$cm3

PQR is a triangle in which PQ = 10cm and QPR = 60oS is a point equidistant from P and Q. Also S is a point equidistant from PQ and PR. If U is the foot of the perpendicular from S on PR, find the length SU in cm to one decimal place

Bayanin Amsa

$△$PUS is right angled
$\frac{US}{5}$ = sin60o
US = 5 x $\frac{\sqrt{3}}{2}$
= 2.5$\sqrt{3}$
= 4.33cm

A 5.0g of salt was weighted by Tunde as 5.1g. What is the percentage error?

Bayanin Amsa

% error = $\frac{\text{actual error}}{\text{true value}}$ x 100
Where actual error = 5.1 - 5.0 = 0.1
true value = 5.0g
% error = $\frac{0.1}{5.0}$ x 100
= $\frac{10}{5}$
= 2

Four interior angles of a pentagon are 90o - xo, 90o + xo, 110o - 2xo, 110o + 2xo. Find the fifth interior angle

Bayanin Amsa

Let the fifth interior angle be y: sum of interior angle of a pentagon
= (2 x 5 - 4) x 90o
= 6 x 90o
= 540o
(90 - x) + (90 + x) + (110 - 2x) + (110 + 2x) + y = 540o
400o + y = 540o
y = 540 - 400o
y = 140o

In the figure, XR and YQ are tangents to the circle YZXP if ZXR = 45o and YZX = 55o, Find ZYQ

Bayanin Amsa

< RXZ = < ZYX = 45O(Alternate segment)

< ZYQ = 90 + 45

= 135O

What is the solution of the equation x2 - x - 1 + 0?

Bayanin Amsa

In the figure, STQ = SRP, PT = TQ = 6cm and QS = 5cm. Find SR

Bayanin Amsa

From similar triangle, $\frac{QS}{QP}=\frac{TQ}{QR}=\frac{5}{12}=\frac{6}{QR}$

QT = $\frac{6\times 12}{5}=\frac{72}{5}=SR=QR-QS$

= $\frac{72}{5}-5=\frac{72-25}{5}$

= $\frac{47}{5}$

For what values of x is the curve y = $\frac{x^2+3}{x+4}$ decreasing?

Bayanin Amsa

A tax player is allowed $\frac{1}{8}$th of his income tax-free, and pays 20% on the remainder. If he pays ₦490.00 tax, what is his income?

Bayanin Amsa

He pays tax on 1 - $\frac{7}{8}$ = $\frac{1}{7}$th of his income
20% is 490, 100% is $\frac{1000}{20}$ x 490, ₦2,450.00
= $\frac{7}{8}$ of his income = ₦2,450.00

$\frac{1}{\frac{7}{8}}$ x 2450
= $\frac{8\times 2450}{7}$
= $\frac{19600}{7}$
= ₦2800.00

If cos2$\theta$ + $\frac{1}{8}$ = sin2$\theta$, find tan$\theta$

Bayanin Amsa

cos2$\theta$ + $\frac{1}{8}$ = sin2$\theta$..........(i)
from trigometric ratios for an acute angle, where cos$\theta$ + sin2$\theta$ = 1 - cos$\theta$ ........(ii)
Substitute for equation (i) in (i) = cos2$\theta$ + $\frac{1}{8}$ = 1 - cos2$\theta$
= cos2$\theta$ + cos2$\theta$ = 1 - $\frac{1}{8}$
2 cos2$\theta$ = $\frac{7}{8}$
cos2$\theta$ = $\frac{7}{2\times 3}$
$\frac{7}{16}$ = cos$\theta$
$\sqrt{\frac{7}{16}}$ = $\frac{\sqrt{7}}{4}$
but cos $\theta$ = $\frac{\text{adj}}{\text{hyp}}$
opp2 = hyp2 - adj2
opp2 = 42 ($\sqrt{7}$)2
= 16 - 7
opp = $\sqrt{9}$ = 3
than $\theta$ = $\frac{\text{opp}}{\text{hyp}}$
= $\frac{3}{\sqrt{7}}$
$\frac{3}{\sqrt{7}}$ x $\frac{7}{\sqrt{7}}$ = $\frac{3\sqrt{7}}{7}$

$\begin{array}{ccccccccc}Scores\left(x\right)& 0& 1& 2& 3& 4& 5& 6& 7\\ Frequency\left(f\right)& 7& 11& 6& 7& 7& 5& 3& \end{array}$

In the distribution above, the mode and median respectively are

Bayanin Amsa

From the distribution, Mode = 1 and
Median = $\frac{2+2}{2}$
= 2
= 1, 2

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

Bayanin Amsa

$\frac{x+2}{x+1}$ - $\frac{x-2}{x+2}$ = $\frac{(x+2)(x+2)-(x-2)-(x-2)(x+1)}{(x+1)(x+2)}$
= $\frac{(x^2+4x+4)-(x^2-x-2)}{(x+1)(x+2)}$ = $\frac{x^2+4x+4-x^2+x+2}{(x+1)(x+2)}$
= $\frac{5x+6}{(x+1)(x+2)}$

Two sisters, Taiwo and Keyinde, own a store. The ratio of Taiwo's share to Kehinde's is 11:9. Later, Keyinde sells $\frac{2}{3}$ of her share to Taiwo for ₦720.00. Find the value of the store.

Bayanin Amsa

Let value of store = X
Ratio of Taiwo's share to kehine's is 11:9 Keyinde sells $\frac{2}{3}$ of her share to Taiwo for ₦720
$\frac{2}{3}$ of 9 = 6
∴ Sum of the ratio = 11 + 9 = 20
$\frac{6}{20}$ of x = ₦720
$\frac{6x}{20}$ = 720
∴ x = $\frac{720\times 20}{6}$
x = ₦24,000

In a class of 150 students, the sector in a pie chart representing the students offering Physics has angle 12o. How many students are offering Physics?

Bayanin Amsa

No of students offering Physics are $\frac{12}{360}$ x 150
= 5

If the sum of the 8th and 9th terms of an arithmetic progression is 72 and the 4th term is -6, find the common difference

Bayanin Amsa

Factorize completely (x2 + x)2 - (2x + 2)2

Bayanin Amsa

If (g(y)) = $\frac{y-3}{11}$ + $\frac{11}{y^2-9}$. what is g(y + 3)?

Bayanin Amsa

If x varies inversely as the cube root of y and x = 1 when y = 8, find y when x = 3

Bayanin Amsa

if a = -3, b = 2, c = 4, evaluate $\frac{a^3-b^3-c^{\frac{1}{2}}}{b-a-c}$

Bayanin Amsa

In the figure, PS = RS = QS and QRS = 50o. Find QPR

Bayanin Amsa

In the figure PS = RS = QS, they will have equal base QR = RP

In angle SQR, angle S = 50O

In angle QRP, 65 + 65 = 130O

Since RQP = angle RPQ = $\frac{180-130}{2}$

= $\frac{50}{2}={25}^o$

QPR = 25O