JAMB UTME - Mathematics - 1983
Pregunta 1 Informe
Detalles de la respuesta
mean = 126.35
= 25.26
Pregunta 2 Informe
In the figure, find PRQ
Detalles de la respuesta
¯PQR = 1252
= 6212 ?
Pregunta 3 Informe
Detalles de la respuesta
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)
Pregunta 4 Informe
Detalles de la respuesta
w = k1v
u = k2w3
u = k2(k1v )3
= k2k21v3
k = k2k1k2
u = kv3
k = uv3
= (1)(2)3
= 8
u = 8v3
Pregunta 6 Informe
Detalles de la respuesta
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 = 5505
= 110o
Pregunta 7 Informe
If \(0.0000152 \times 0.042 = A \times 10^8\), where \(1 \le A < 10\), find A and B
Detalles de la respuesta
0.0000152 x 0.042 = A x 108
1 ≤
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
Pregunta 8 Informe
Detalles de la respuesta
672 x 360
= 30o
Similarly others give 30o, 50o, 70o, 80o and 130o respectively
Pregunta 9 Informe
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?
Detalles de la respuesta
Total sharing ratio is 9
X has 4, Y has 4 + 1
If 1 is ₦5000
Total profit = 5000 x 9
= ₦45,000
Pregunta 10 Informe
Detalles de la respuesta
Pregunta 11 Informe
Detalles de la respuesta
prob. of vowel = 613
Pregunta 12 Informe
The figure FGHK is a rhombus. What is the value of angle X?
Detalles de la respuesta
< KFG = < KHG = x(opposite angles)
x = 120?
Pregunta 13 Informe
Detalles de la respuesta
= (9a2 + 4b2)(9a2 - 4b2)
N:B 9a2 + 4b2 = (3a - 2b)(3a - 2b)
Pregunta 14 Informe
In a triangle PQT, QR = \( \sqrt{3}\text{ cm} \), PR = 3cm, PQ = \( 2\sqrt{3} \)cm and PQR = 30o. Find angles P and R
Detalles de la respuesta
Cos P = Q2+R2−p22QR
= (3)2+2(√3)2−322√3
= 3+12−912
= 612
= 12
= 0.5
Cos P = 0.5
p = cos-1 0.5 = 60∘
= < P = 60∘
If < P = 60∘ and < Q = 30
< R = 180∘
- 90∘
angle P = 60∘
and angle R is 90∘
Pregunta 15 Informe
PQRS is a desk of dimensions \(2\text{m} \times 0.8\) which is inclined at \(30^\circ\) to the horizontal. Find the inclination of the diagonal PR to the horizontal
Detalles de la respuesta
θ = tan-1(0.4)
From the diagram, the inclination of the diagonal PR to the horizontal is 10∘ 42'
Pregunta 16 Informe
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.
Detalles de la respuesta
y = kx2 + c√x
y = 2 when x = 1
2 = k + c1
k + c = 2
y = 6 when x = 4
6 = 16k + c2
12 = 32k + c
k + c = 2
32k + c = 12
= 31k + 10
k = 1031
c = 2 - 1031
= 62−1031
= 5231
y = 10x231+5231√x
Pregunta 17 Informe
Detalles de la respuesta
Pregunta 18 Informe
Simplify \( \log_{10} a^{\frac{1}{3}} + \frac{1}{4}\log_{10} a - \frac{1}{12}\log_{10} a^{7} \)
Detalles de la respuesta
= log10 a712 - log10 a712
= log10 1 = 0
Pregunta 19 Informe
Which of the following equations represents the graph above?
Detalles de la respuesta
(x + 2)(4x - 1) = 0
y = 2 - 7x - 4x2
Pregunta 20 Informe
In the figure, PQR is a straight line. Find the values of x and y.
Detalles de la respuesta
3x + 6y + 90∘ = 360∘
3x + 67 = 270.......(i) x 2, 5x + y + y = 180∘
5x + 2y = 180∘ .......(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∘
Pregunta 21 Informe
Simplify \( \frac{3\sqrt{27a^{-9}}}{8} \)
Detalles de la respuesta
= 32 x 1a3
= 32a3
Pregunta 22 Informe
Find the missing value in the following table:
| \(x\) | \(-2\) | \(-1\) | \(0\) | \(1\) | \(2\) | \(3\) |
| \(y=x^3-x+3\) | \(3\) | \(3\) | \(3\) | \(9\) | \(27\) |
Detalles de la respuesta
= -23 - (-2) + 3
= -8 + 2 + 3
= -3
Pregunta 23 Informe
Detalles de la respuesta
32 + 4 = 36
x2 = 36, x = 6
610 to base 4 = 461r2
= 124
Pregunta 24 Informe
70. What is the sum of the median and modal scores?
Detalles de la respuesta
Pregunta 25 Informe
Without using tables find the numerical value of \( \log_{7} 49 + \log_{7}\left(\frac{1}{7}\right) \)
Detalles de la respuesta
= log77
= 1
Pregunta 26 Informe
If \( \left(\frac{2}{3}\right)^m \left(\frac{3}{4}\right)^n = \frac{252}{729} \), find the values of m and n
Detalles de la respuesta
2m4n x 3n3m = 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
Pregunta 27 Informe
Detalles de la respuesta
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 720o6
= 120o
Pregunta 28 Informe
Detalles de la respuesta
The median = 47+502
972
= 4812
Pregunta 29 Informe
Detalles de la respuesta
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
Pregunta 30 Informe
Given that cos z = L , whrere z is an acute angle, find an expression for \( \frac{\cot z - \cosec z}{\sec z + \tan z} \)
Detalles de la respuesta
Given Cos z = L, z is an acute angle
cot z - cosec zsec z + tan z
= cos z
= cos zsin z
cosec z = 1sin z
cot z - cosec z = cos zsin z
- 1sin z
cot z - cosec z = L−1sin z
sec z = 1cos z
tan z = sin zcos z
sec z = 1cos z
+ sin zcos z
= 1l
+ sin zL
the original eqn. becomes
cot z - cosec zsec z + tan z
= L−1sin z1+sinzL
= L(L−1)sin z(1+sin z)
= L(L−1)sin z+1−cos2z
= sin z + 1
= 1 + √1−L2
= L(L−1)1−L+1√1−L2
Pregunta 31 Informe
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. \( \pi = \frac{22}{7} \)
Detalles de la respuesta
= 11cm
Side PT = 7cm, Side TR = 7cm
Perimeter(PTRS) = 11cm + 7cm + 7cm
= 25cm
Pregunta 32 Informe
PQRS is a cyclic quadrilateral which PQ = PS. PT is a tangent to the circle and PQ makes an angle of \(50^\circ\) with the tangent as shown in the figure. What is the size of QRS?
Detalles de la respuesta
< SPQ + < SRQ = 180(Supplementary)
80 + < QRS = 180? - 80?
= 100?
Pregunta 33 Informe
Detalles de la respuesta
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
Pregunta 34 Informe
Detalles de la respuesta
Pregunta 35 Informe
In the figure PT is a tangent to the circle with centre at O. If PQT = \(30^\circ\), find the value of PTO
Detalles de la respuesta
PTQ = 50∘ (opposite angles are supplementary)
Pregunta 36 Informe
Detalles de la respuesta
PR = 8 sin 60 = 8√32
= 4√3
Cos 45o = PRPS = 4√3PS
PS Cos45o = 4√3
PS = 4√3 x 2
= 4√6
Pregunta 37 Informe
Detalles de la respuesta
0.0746829 = 0.0747
59.8 x 0.0747 = 4.46706
= 4.48 (3s.f)
Pregunta 38 Informe
If O is the centre of the circle in the figure, find the value of x
Detalles de la respuesta
= 360 - 260
= 100∘
Pregunta 39 Informe
In a sample survey of a University Community, the following table shows the percentage distribution of the number of members per household:
| No. of members per household | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | Total |
| No. of households | 3 | 12 | 15 | 28 | 21 | 10 | 7 | 4 | 100 |
What is the median?
Detalles de la respuesta
Median is half of total frequency = 50th term
4 falls in the range = 4
Pregunta 40 Informe
Detalles de la respuesta
f(-2) = -8L + 8k = -24
4L - 4k = 12
f(1):L + 2k = -24
L - 4k = 3
3k = -27
k = -9
1 = -6
Pregunta 41 Informe
Simplify \( \frac{3}{2x-1} + \frac{2-x}{x-2} \)
Detalles de la respuesta
= 32x−1 - x−2x−2
= 32x−1 - 1
= 3−(2x−1)2x−1
= 3−2x+12x−1
= 4−2x2x−1
Pregunta 42 Informe
Make T the subject of the equation \( \frac{av}{1-v} = \sqrt{\frac{2v+T}{a+2T}} \)
Detalles de la respuesta
av1?v
= ?2v+Ta+2T
(av)3(1?v)3
= 2v+Ta+2T
a3v3(13?v)3
= 2v+Ta+2T
= 2v(1?v)3?a4v32a3v3?(1?v)3
Pregunta 43 Informe
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?
Detalles de la respuesta
Okro sector = 145270 x 360o1
= 19.33∘
= 1913 ∘
Pregunta 44 Informe
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?
Detalles de la respuesta
9x = 360∘ + 18∘
x = 3789
= 42∘ , if x = 42∘ , then add maths = 2x−42360 x 60
= 2×42−24360 x 60
= 84−246
= 10
Pregunta 45 Informe
Detalles de la respuesta
= 5250 x 100
= 2
Pregunta 46 Informe
Detalles de la respuesta
let x represent the speed, d represent distance
x = d4
d = 4x
2x = 600−d3
6x = 600 - d
6x = 600 - 4x
10x = 600
x = 60010
= 60km/hr
Pregunta 47 Informe
Detalles de la respuesta
9x2 + 6x + 1 = 9x - 6x2 + 1 + x2
x2 - 12x = 0
x(x - 12) = 0
x = 0 or 12
Pregunta 48 Informe
Detalles de la respuesta
