JAMB UTME - Mathematics - 1983

If x is jointly proportional to the cube of y and the fourth power of z. In what ratio is x increased or decreased when y is halved and z is doubled?

Detalhes da Resposta

The scores of 16 students in a mathematics test are 65, 65, 55, 60, 60, 65, 60, 70, 75, 70, 65, 70, 60, 65, 65,
70. What is the sum of the median and modal scores?

Detalhes da Resposta

The value of $(0.303{)}^3-(0.02{)}^3$ is

Detalhes da Resposta

If 0.0000152 x 0.042 = A x 108, where 1 ≤ $\le$ A < 10, find A and B

Detalhes da Resposta

0.0000152 x 0.042 = A x 108

1 ≤ $\le$ A < 10, it means values of A includes 1 - 9

0.0000152 = 1.52 x 10-5

0.00042 = 4.2 x 10-4

1.52 x 4.2 = 6.384

10-5 x 10-4

= 10-5-4

= 10-9

= 6.38 x 10-9

A = 6.38, B = -9

Factorize completely 81a4 - 16b4

Detalhes da Resposta

81a4 - 16b4 = (9a2)2 - (4b2)2
= (9a2 + 4b2)(9a2 - 4b2)
N:B 9a2 + 4b2 = (3a - 2b)(3a - 2b)

If x + 2 and x - 1 are factors of the expression 1x3 + 2kx2 + 24, find the values of 1 and K

Detalhes da Resposta

f(x) = Lx3 + 2kx2 + 24
f(-2) = -8L + 8k = -24
4L - 4k = 12
f(1):L + 2k = -24
L - 4k = 3
3k = -27
k = -9
1 = -6

PQRS is a desk of dimensions 2m $\times$ 0.8 which is inclined at 30${}^{\circ }$ to the horizontal. Find the inclination of the diagonal PR to the horizontal

Detalhes da Resposta

tan$\theta$ = $\frac{0.8}{2}$ = (0.4)

$\theta$ = tan-1(0.4)

From the diagram, the inclination of the diagonal PR to the horizontal is 10${}^{\circ }$ 42'

y varies partly as the square of x and partly as the inverse of the square root of x. Write down the expression for y if y = 2 when x = 1 and y = 6 when x = 4.

Detalhes da Resposta

y = kx2 + $\frac{c}{\sqrt{x}}$
y = 2 when x = 1
2 = k + $\frac{c}{1}$
k + c = 2
y = 6 when x = 4
6 = 16k + $\frac{c}{2}$
12 = 32k + c
k + c = 2
32k + c = 12
= 31k + 10
k = $\frac{10}{31}$
c = 2 - $\frac{10}{31}$
= $\frac{62-10}{31}$
= $\frac{52}{31}$
y = $\frac{10x^2}{31}+\frac{52}{31\sqrt{x}}$

Simplify log10 a$\frac{1}{3}$ + $\frac{1}{4}$log10 a - $\frac{1}{12}$log10a7

Detalhes da Resposta

log10 a$\frac{1}{3}$ + $\frac{1}{4}$log10 a - $\frac{1}{12}$log10a7 = log10 a$\frac{1}{3}$ + log10$\frac{1}{4}$ - log10 a$\frac{7}{12}$
= log10 a$\frac{7}{12}$ - log10 a$\frac{7}{12}$
= log10 1 = 0

On a square paper of length 2.524375cm is inscribed square diagram of length 0.524375cm. Find the area of the paper not covered by the diagram. correct to 3 significant figures.

Detalhes da Resposta

Area of the paper = area of square = L x B or S2
where s = S x k
Area of the paper = (2.524)2
area of the diagram = (0.524)2
area not covered = (2.524)2 - (0.524)2
= 6.370576 - 0.274576
= 6.096
= 6.10cm2 (2 s.f)

If M represents the median and D the mode of the measurements 5, 9, 3, 5, 7, 5, 8, then (M, D) is

Detalhes da Resposta

The lengths of the sides of a right angled triangle are (3x + 1)cm, (3x - 1)cm and xcm. Find x

Detalhes da Resposta

(3x + 1)2 = (3x - 1)2 + 2 (Pythagoras's theorem)
9x2 + 6x + 1 = 9x - 6x2 + 1 + x2
x2 - 12x = 0
x(x - 12) = 0
x = 0 or 12

Simplify $\frac{3}{2x-1}$ + $\frac{2-x}{x-2}$

Detalhes da Resposta

$\frac{3}{2x-1}$ + $\frac{2-x}{x-2}$
= $\frac{3}{2x-1}$ - $\frac{x-2}{x-2}$
= $\frac{3}{2x-1}$ - 1
= $\frac{3-(2x-1)}{2x-1}$
= $\frac{3-2x+1}{2x-1}$
= $\frac{4-2x}{2x-1}$

PQRS is a cyclic quadrilateral. If ∠QPS = 75o, what is the size of ∠QRS?

Detalhes da Resposta

In a triangle PQT, QR = $\sqrt{3cm}$, PR = 3cm, PQ = $2\sqrt{3}$cm and PQR = 30o. Find angles P and R

Detalhes da Resposta

By using cosine formula, p2 = Q2 + R2 - 2QR cos p
Cos P = $\frac{Q^2+R^2-p^2}{2QR}$
= $\frac{(3{)}^2+2(\sqrt{3}{)}^2-3^2}{2\sqrt{3}}$
= $\frac{3+12-9}{12}$
= $\frac{6}{12}$
= $\frac{1}{2}$
= 0.5
Cos P = 0.5
p = cos-1 0.5 = 60${}^{\circ }$
= < P = 60${}^{\circ }$
If < P = 60${}^{\circ }$ and < Q = 30

< R = 180${}^{\circ }$ - 90${}^{\circ }$
angle P = 60${}^{\circ }$ and angle R is 90${}^{\circ }$

If a rod of length 250cm is measured as 255cm long in error, what is the percentage error in the measurement?

Detalhes da Resposta

% error = $\frac{\text{Actual error}}{\text{real value}}$ x 100
= $\frac{5}{250}$ x 100
= 2

The letters of the word MATRICULATION are cut and put into a box. One letter is drawn at random from the box. Find the probability of drawing a vowel

Detalhes da Resposta

Vowels of letters are 6 in numbers
prob. of vowel = $\frac{6}{13}$

The figure FGHK is a rhombus. What is the value of angle X?

Detalhes da Resposta

< HKF = 60${}^{?}$, < KFG = 120${}^{?}$

< KFG = < KHG = x(opposite angles)

x = 120${}^{?}$

Find the missing value in the following table:

$\begin{array}{ccccccc}x& -2& -1& 0& 1& 2& 3\\ y=x^3-x+3& & 3& 3& 3& 9& 27\end{array}$

Detalhes da Resposta

When x = -2, y = x3 - x + 3
= -23 - (-2) + 3
= -8 + 2 + 3
= -3

In a class of 60 pupils, the statistical distribution of the numbers of pupils offering Biology, History, French, Geography and Additional mathematics is as shown in the pie chart. How many pupils offer Additional Mathematics?

Detalhes da Resposta

2x - 24)${}^{\circ }$ + (3x - 18)${}^{\circ }$ + (2x + 12)${}^{\circ }$ + (x + 12)${}^{\circ }$ + x${}^{\circ }$ = 360${}^{\circ }$

9x = 360${}^{\circ }$ + 18${}^{\circ }$

x = $\frac{378}{9}$

= 42${}^{\circ }$, if x = 42${}^{\circ }$, then add maths = $\frac{2x-42}{360}$ x 60

= $\frac{2\times 42-24}{360}$ x 60

= $\frac{84-24}{6}$

= 10

If w varies inversely as V and U varies directly as w3, Find the relationship between u and v given that u = 1, when v = 2

Detalhes da Resposta

W $\alpha$ $\frac{1}{v}$u $\alpha$ w3
w = $\frac{k_1}{v}$
u = k2w3
u = k2($\frac{k_1}{v}$)3
= $\frac{k_2{k}_1^2}{v^3}$
k = k2k1k2
u = $\frac{k}{v^3}$
k = uv3
= (1)(2)3
= 8
u = $\frac{8}{v^3}$

Find x if (xbase4)2 = 100100base2

Detalhes da Resposta

(x2)4 to base10 gives (1 x 25) + (1 x 25) + (1 x 2)
32 + 4 = 36
x2 = 36, x = 6
610 to base 4 = $\begin{array}{cc}4& 6\\ 1r2& \end{array}$
= 124

In the figure, find PRQ

Detalhes da Resposta

Angle subtended at any part of the circumference of the circle $\frac{{125}^o}{2}$ at centre = 360${}^{?}$ - 235${}^{?}$ = 125${}^{?}$

$\overline{PQR}$ = $\frac{125}{2}$

= 62$\frac{1}{2}$${}^{?}$

The farm yield of four crops on a piece of land in Ondo are represented on the pie chart. what is the angle of the sector occupies by Okro in the chart?

Detalhes da Resposta

Adding the values of all the items together, it gives 70

Okro sector = $\frac{145}{270}$ x $\frac{{360}^o}{1}$

= 19.33${}^{\circ }$

= 19$\frac{1}{3}$${}^{\circ }$

A ship H leaves a port P and sails 30 km due south. Then it sails 50km due west. What is the bearing of H from P

Detalhes da Resposta

If O is the centre of the circle in the figure, find the value of x

Detalhes da Resposta

From the diagram; The value of x = 360${}^{\circ }$ - 2(130${}^{\circ }$)

= 360 - 260

= 100${}^{\circ }$

PQRS is a cyclic quadrilateral which PQ = PS. PT is a tangent to the circle and PQ makes an angle of 50${}^{?}$ with the tangent as shown in the figure. What is the size of QRS?

Detalhes da Resposta

< SPQ = 80${}^{?}$

< SPQ + < SRQ = 180(Supplementary)

80 + < QRS = 180${}^{?}$ - 80${}^{?}$

= 100${}^{?}$

Simplify $\frac{3\sqrt{27a^{-9}}}{8}$

Detalhes da Resposta

$\frac{3\sqrt{27a^{-9}}}{8}$ = $\frac{3a^{-3}}{2}$
= $\frac{3}{2}$ x $\frac{1}{a^3}$
= $\frac{3}{2a^3}$

One interior angle of a convex hexagon is 170o and each of the remaining interior angles is equal to xo. Find x

Detalhes da Resposta

An hexagon polygon is a six sided polygon
n = 6
sum of all interior angles of hexagon polygon will be (2n - 4) x 90o
= [2 x 6 - 4] x 90
= 720o
If one angle is 170o and each of the remaining five angles is 5xo
5xo + 170o = 720o
= 5x + 550
x = $\frac{550}{5}$
= 110o

Given a regular hexagon, calculate each interior angle of the hexagon

Detalhes da Resposta

Sum of interior angles of polygon = (2n - 4)rt < s

sum of interior angles of an hexagon
(2 x 6 - 4) x 90o = (12 - 4) x 90o
= 8 x 90o
= 720o
each interior angle will have $\frac{{720}^o}{6}$
= 120o

PQR is the diagram of a semicircle RSP with centre at Q and radius of length 3.5cm. If QPT = 60o. Find the perimeter of the figure PTRS. $\pi$ = $\frac{22}{7}$

Detalhes da Resposta

Circumference of PRS = $\frac{\pi }{2}$ = $\frac{22}{7}$ x $\frac{7}{1}$ x $\frac{1}{2}$
= 11cm
Side PT = 7cm, Side TR = 7cm
Perimeter(PTRS) = 11cm + 7cm + 7cm
= 25cm

In the figure PT is a tangent to the circle with centre at O. If PQT = 30${}^{?}$, find the value of PTO

Detalhes da Resposta

FROM the diagram, PQT = 50${}^{\circ }$

PTQ = 50${}^{\circ }$(opposite angles are supplementary)

Find the mean of the following 24.57, 25.63, 24.32, 26.01, 25.77

Detalhes da Resposta

$\frac{24.57+25.63+24.32+26.01+25.77}{5}$
mean = $\frac{126.3}{5}$
= 25.26

Make T the subject of the equation $\frac{av}{1?v}$ = $\sqrt{\frac{2v+T}{a+2T}}$

Detalhes da Resposta

$\frac{av}{1?v}$ = $\sqrt{\frac{2v+T}{a+2T}}$
$\frac{(av{)}^3}{(1?v{)}^3}$ = $\frac{2v+T}{a+2T}$
$\frac{a^3{v}^3}{(1^3?v{)}^3}$ = $\frac{2v+T}{a+2T}$
= $\frac{2v(1?v{)}^3?a^4{v}^3}{2a^3{v}^3?(1?v{)}^3}$

Which of the following equations represents the graph above?

Detalhes da Resposta

x = -2 or x = $\frac{1}{4}$

(x + 2)(4x - 1) = 0

y = 2 - 7x - 4x2

Without using tables find the numerical value of log749 + log7($\frac{1}{7}$)

Detalhes da Resposta

log749 + log7$\frac{1}{7}$
= log77
= 1

The scores of set of final year students in the first semester examination in a paper are 41, 29, 55, 21, 47, 70, 70, 40, 43, 56, 73, 23, 50, 50. Find the median of the scores.

Detalhes da Resposta

By re-arranging 21, 23, 29, 40, 41, 43| 47, 50| 50, 55, 56, 70, 70, 73
The median = $\frac{47+50}{2}$
$\frac{97}{2}$
= 48$\frac{1}{2}$

Given that cos z = L , whrere z is an acute angle, find an expression for $\frac{\text{cot z - cosec z}}{\text{sec z + tan z}}$

Detalhes da Resposta

Given Cos z = L, z is an acute angle
$\frac{\text{cot z - cosec z}}{\text{sec z + tan z}}$ = cos z
= $\frac{\text{cos z}}{\text{sin z}}$
cosec z = $\frac{1}{\text{sin z}}$
cot z - cosec z = $\frac{\text{cos z}}{\text{sin z}}$ - $\frac{1}{\text{sin z}}$
cot z - cosec z = $\frac{L-1}{\text{sin z}}$
sec z = $\frac{1}{\text{cos z}}$
tan z = $\frac{\text{sin z}}{\text{cos z}}$
sec z = $\frac{1}{\text{cos z}}$ + $\frac{\text{sin z}}{\text{cos z}}$
= $\frac{1}{l}$ + $\frac{\text{sin z}}{L}$
the original eqn. becomes
$\frac{\text{cot z - cosec z}}{\text{sec z + tan z}}$ = $\frac{L-\frac{1}{\text{sin z}}}{1+sin\frac{z}{L}}$
= $\frac{L(L-1)}{\text{sin z}(1+\text{sin z})}$
= $\frac{L(L-1)}{\text{sin z}+1-co{s}^{2}z}$
= sin z + 1
= 1 + $\sqrt{1-L^2}$
= $\frac{L(L-1)}{1-L+1\sqrt{1-L^2}}$

Solve the following equations 4x - 3 = 3x + y = x - y = 3, 3x + y = 2y + 5x - 12

Detalhes da Resposta

4x - 3 = 3x + y = x - y = 3.......(i)
3x + y = 2y + 5x - 12.........(ii)
eqn(ii) + eqn(i)
3x = 15
x = 5
substitute for x in equation (i)
5 - y = 3
y = 2

If ($\frac{2}{3}$)m ($\frac{3}{4}$)n = $\frac{252}{729}$, find the values of m and n

Detalhes da Resposta

($\frac{2}{3}$)m ($\frac{3}{4}$)n = $\frac{252}{729}$
$\frac{2^m}{4^n}$ x $\frac{3^n}{3^m}$ = 283-6
2m - 2n x 3n - m = 28 x 3-6
m - 2n = 8........(i)
-m + n = -6........(ii)
Solving the equations simultaneously
m = 4, n = -2

A man drove for 4 hours at a certain speed, he then doubled his speed and drove for another 3 hours. Although he covered 600 kilometers. At what speed did he drive for the last 3 hours?

Detalhes da Resposta

Speed = $\frac{distance}{time}$
let x represent the speed, d represent distance
x = $\frac{d}{4}$
d = 4x
2x = $\frac{600-d}{3}$
6x = 600 - d
6x = 600 - 4x
10x = 600
x = $\frac{600}{10}$
= 60km/hr

In a figure, PQR = 60o, PRS = 90o, RPS = 45o, QR = 8cm. Determine PS

Detalhes da Resposta

From the diagram, sin 60o = $\frac{PR}{8}$
PR = 8 sin 60 = $\frac{8\sqrt{3}}{2}$
= 4$\sqrt{3}$
Cos 45o = $\frac{PR}{PS}$ = $\frac{4\sqrt{3}}{PS}$
PS Cos45o = 4$\sqrt{3}$
PS = 4$\sqrt{3}$ x 2
= 4$\sqrt{6}$

Solve the simultaneous equations for x in x2 + y - 8 = 0, y + 5x - 2 = 0

Detalhes da Resposta

x2 + y - 8 = 0, y + 5x - 2 = 0
Rearranging, x2 + y = 8.....(i)
5x + y = 2.......(ii)
Subtract eqn(ii) from eqn(i)
x2 - 5x - 6 = 0
(x - 6)(x + 1) = 0
x = 6, -1

Correct each of the numbers 59.81798 and 0.0746829 to three significant figures and multiply them, giving your answer to three significant figures

Detalhes da Resposta

59.81798 = 59.8 (3 s.f)
0.0746829 = 0.0747
59.8 x 0.0747 = 4.46706
= 4.48 (3s.f)

A construction company is owned by two partners X and Y and it is agreed that their profit will be divided in the ratio 4:5. At the end of the year, Y received ₦5,000 more than X. What is the total profit of the company for the year?

Detalhes da Resposta

Total sharing ratio is 9
X has 4, Y has 4 + 1
If 1 is ₦5000
Total profit = 5000 x 9
= ₦45,000

Find the angle of the sectors representing each item in pie chart of the following data 6, 10, 14, 16, 26

Detalhes da Resposta

6 + 10 + 14 + 16 + 26 = 72
$\frac{6}{72}$ x 360
= 30o
Similarly others give 30o, 50o, 70o, 80o and 130o respectively

In a sample survey of a University Community, the following table shows the percentage distribution of the number of members per household:

$\begin{array}{cccccccccc}\text{No. of members per household}& 1& 2& 3& 4& 5& 6& 7& 8& \text{Total}\\ \text{No. of households}& 3& 12& 15& 28& 21& 10& 7& 4& 100\end{array}$

What is the median?

Detalhes da Resposta

$\begin{array}{cc}x& f\\ 1& 3\\ 2& 13\\ 3& 15\\ 4& 28\\ 5& 21\\ 6& 10\\ 7& 7\\ 8& 4\end{array}$
Median is half of total frequency = 50th term
4 falls in the range = 4

In the figure, PQR is a straight line. Find the values of x and y.

Detalhes da Resposta

$\frac{2}{3}$ + 3y + 45${}^{\circ }$ = 180${}^{\circ }$

3x + 6y + 90${}^{\circ }$ = 360${}^{\circ }$

3x + 67 = 270.......(i) x 2, 5x + y + y = 180${}^{\circ }$

5x + 2y = 180${}^{\circ }$.......(ii) x 6

6 x 12y = 540..........(iii)

30x + 12y = 1080.........(iv)

egn(iv) - eqn(iii)

24x = 540

x = 22.5 and y = 33.75${}^{\circ }$