JAMB UTME - Mathematics - 1983
Vraag 1 Verslag
Find the missing value in the following table:
| \(x\) | \(-2\) | \(-1\) | \(0\) | \(1\) | \(2\) | \(3\) |
| \(y=x^3-x+3\) | \(3\) | \(3\) | \(3\) | \(9\) | \(27\) |
Antwoorddetails
= -23 - (-2) + 3
= -8 + 2 + 3
= -3
Vraag 2 Verslag
The figure FGHK is a rhombus. What is the value of angle X?
Antwoorddetails
< KFG = < KHG = x(opposite angles)
x = 120?
Vraag 3 Verslag
Antwoorddetails
= 5250 x 100
= 2
Vraag 4 Verslag
Without using tables find the numerical value of \( \log_{7} 49 + \log_{7}\left(\frac{1}{7}\right) \)
Antwoorddetails
= log77
= 1
Vraag 5 Verslag
Simplify \( \frac{3\sqrt{27a^{-9}}}{8} \)
Antwoorddetails
= 32 x 1a3
= 32a3
Vraag 6 Verslag
In the figure, find PRQ
Antwoorddetails
¯PQR = 1252
= 6212 ?
Vraag 7 Verslag
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. \( \pi = \frac{22}{7} \)
Antwoorddetails
= 11cm
Side PT = 7cm, Side TR = 7cm
Perimeter(PTRS) = 11cm + 7cm + 7cm
= 25cm
Vraag 8 Verslag
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Vraag 9 Verslag
If \( \left(\frac{2}{3}\right)^m \left(\frac{3}{4}\right)^n = \frac{252}{729} \), find the values of m and n
Antwoorddetails
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
Vraag 10 Verslag
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Vraag 11 Verslag
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?
Antwoorddetails
Median is half of total frequency = 50th term
4 falls in the range = 4
Vraag 12 Verslag
In a triangle PQT, QR = \( \sqrt{3}\text{ cm} \), PR = 3cm, PQ = \( 2\sqrt{3} \)cm and PQR = 30o. Find angles P and R
Antwoorddetails
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∘
Vraag 13 Verslag
Antwoorddetails
0.0746829 = 0.0747
59.8 x 0.0747 = 4.46706
= 4.48 (3s.f)
Vraag 14 Verslag
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?
Antwoorddetails
Okro sector = 145270 x 360o1
= 19.33∘
= 1913 ∘
Vraag 15 Verslag
Vraag 16 Verslag
Antwoorddetails
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)
Vraag 17 Verslag
Antwoorddetails
mean = 126.35
= 25.26
Vraag 18 Verslag
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.
Antwoorddetails
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
Vraag 19 Verslag
In the figure, PQR is a straight line. Find the values of x and y.
Antwoorddetails
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∘
Vraag 20 Verslag
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?
Antwoorddetails
< SPQ + < SRQ = 180(Supplementary)
80 + < QRS = 180? - 80?
= 100?
Vraag 21 Verslag
Simplify \( \frac{3}{2x-1} + \frac{2-x}{x-2} \)
Antwoorddetails
= 32x−1 - x−2x−2
= 32x−1 - 1
= 3−(2x−1)2x−1
= 3−2x+12x−1
= 4−2x2x−1
Vraag 22 Verslag
Antwoorddetails
f(-2) = -8L + 8k = -24
4L - 4k = 12
f(1):L + 2k = -24
L - 4k = 3
3k = -27
k = -9
1 = -6
Vraag 23 Verslag
In the figure PT is a tangent to the circle with centre at O. If PQT = \(30^\circ\), find the value of PTO
Antwoorddetails
PTQ = 50∘ (opposite angles are supplementary)
Vraag 24 Verslag
Which of the following equations represents the graph above?
Antwoorddetails
(x + 2)(4x - 1) = 0
y = 2 - 7x - 4x2
Vraag 25 Verslag
Antwoorddetails
The median = 47+502
972
= 4812
Vraag 26 Verslag
Antwoorddetails
prob. of vowel = 613
Vraag 27 Verslag
Antwoorddetails
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
Vraag 28 Verslag
Antwoorddetails
= (9a2 + 4b2)(9a2 - 4b2)
N:B 9a2 + 4b2 = (3a - 2b)(3a - 2b)
Vraag 29 Verslag
Simplify \( \log_{10} a^{\frac{1}{3}} + \frac{1}{4}\log_{10} a - \frac{1}{12}\log_{10} a^{7} \)
Antwoorddetails
= log10 a712 - log10 a712
= log10 1 = 0
Vraag 30 Verslag
Antwoorddetails
PR = 8 sin 60 = 8√32
= 4√3
Cos 45o = PRPS = 4√3PS
PS Cos45o = 4√3
PS = 4√3 x 2
= 4√6
Vraag 31 Verslag
If O is the centre of the circle in the figure, find the value of x
Antwoorddetails
= 360 - 260
= 100∘
Vraag 32 Verslag
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?
Antwoorddetails
Total sharing ratio is 9
X has 4, Y has 4 + 1
If 1 is ₦5000
Total profit = 5000 x 9
= ₦45,000
Vraag 34 Verslag
Make T the subject of the equation \( \frac{av}{1-v} = \sqrt{\frac{2v+T}{a+2T}} \)
Antwoorddetails
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
Vraag 35 Verslag
Antwoorddetails
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
Vraag 36 Verslag
Antwoorddetails
672 x 360
= 30o
Similarly others give 30o, 50o, 70o, 80o and 130o respectively
Vraag 37 Verslag
Antwoorddetails
w = k1v
u = k2w3
u = k2(k1v )3
= k2k21v3
k = k2k1k2
u = kv3
k = uv3
= (1)(2)3
= 8
u = 8v3
Vraag 38 Verslag
Vraag 39 Verslag
Given that cos z = L , whrere z is an acute angle, find an expression for \( \frac{\cot z - \cosec z}{\sec z + \tan z} \)
Antwoorddetails
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
Vraag 40 Verslag
Antwoorddetails
9x2 + 6x + 1 = 9x - 6x2 + 1 + x2
x2 - 12x = 0
x(x - 12) = 0
x = 0 or 12
Vraag 41 Verslag
Antwoorddetails
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
Vraag 42 Verslag
Antwoorddetails
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
Vraag 43 Verslag
Antwoorddetails
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
Vraag 44 Verslag
If \(0.0000152 \times 0.042 = A \times 10^8\), where \(1 \le A < 10\), find A and B
Antwoorddetails
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
Vraag 45 Verslag
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?
Antwoorddetails
9x = 360∘ + 18∘
x = 3789
= 42∘ , if x = 42∘ , then add maths = 2x−42360 x 60
= 2×42−24360 x 60
= 84−246
= 10
Vraag 46 Verslag
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
Antwoorddetails
θ = tan-1(0.4)
From the diagram, the inclination of the diagonal PR to the horizontal is 10∘ 42'
Vraag 47 Verslag
Antwoorddetails
32 + 4 = 36
x2 = 36, x = 6
610 to base 4 = 461r2
= 124
Vraag 48 Verslag
70. What is the sum of the median and modal scores?
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