JAMB UTME - Mathematics - 1990
Question 1 Rapport
Below are the scores of a group of students in a test:
| Scores | 1 | 2 | 3 | 4 | 5 | 6 |
| No. of students | 1 | 4 | 5 | 6 | \(x\) | 2 |
If the average scores is 3.5, find the value of x.
Détails de la réponse
Average = 3.5
3.5 = (1×1)+(2×4)+(3×5)+(4×6)+5x+(6×2)1+4+5+6+x+2
3.51 = 1+8+15+24+5x+1218+x
3.51 = 60+5x18+x
60 + 5x = 3.5(18 ÷ x)
60 + 5x = = 63 + 1.5x
5x - 1.5x = 63 - 60
1.5x = 3
x = 31.5
3015 = 2
Question 2 Rapport
In the figure, YXZ = \(30^\circ\), XYZ = \(105^\circ\) and XY = 8cm. Calculate YZ
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Question 3 Rapport
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Question 4 Rapport
What is the equation of the quadratic function represented by the graph?
Détails de la réponse
i.e. B where the graph touches the graph touches the x-axis y = 0
x2 - x - 2 = 0 = (x + 1)(x - 2) = 0
Hence roots of the equation are -1 and 2 as shown in the graph
Question 5 Rapport
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Question 6 Rapport
Simplify and express in standard form \( \frac{0.00275 \times 0.0064}{0.025 \times 0.08} \)
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Question 7 Rapport
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Question 8 Rapport
In the figure, PQR is a semicircle. Calculate the area of the shaded region.
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Question 9 Rapport
In triangles XYZ and XQP, XP = 4cm, XQ = 5cm and PQ = QY = 3cm. Find ZY
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Question 10 Rapport
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Question 11 Rapport
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Question 12 Rapport
Simplify \( \sqrt{160r^2} + \sqrt{71r^4} + \sqrt{100r^2} \)
Détails de la réponse
Simplifying from the innermost radical and progressing outwards we have the given expression
√160r2+71r4+100r8 = √160r2+81r4
√160r2+9r2 = √169r2
= 13r
Question 13 Rapport
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Question 14 Rapport
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Question 15 Rapport
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Question 16 Rapport
Find the solution of the equation \(x - 8\sqrt{x} + 15 = 0\)
Détails de la réponse
x + 15 = 8√x
square both sides = (x + 15)2 = (8 √x 2
x2 + 225 + 30x = 64x
x2 + 225 + 30x - 64x = 0
x2 - 34x + 225 = 0
(x - 9)(x - 25) = 0
x = 9 or 25
Question 17 Rapport
Détails de la réponse
1st term = 3, common ratio 63 = 2
n = 18, sum og GP is given by Sn = a(rn−1)r−1
s18 = 3(218−1)2−1
= 3(217 - 1)
Question 18 Rapport
If the heights of two circular cylinder are in the ratio 2 : 3 and their volumes?
Détails de la réponse
h1h2
= 23
h2 = 2h13
r1r2
= 98
r2 = 9r18
v1 = π
(9r18
)2(2h13
)
= π
r1 2h1 x 2732
v = πr12h1×2732πr12h1
= 2732
v2 : v1 = 27 : 32
Question 19 Rapport
Détails de la réponse
tan θ= 2.0000
θ = 63o
Question 20 Rapport
Détails de la réponse
To find f(2) = f(x - 4)
= f(2)
x - 4 = 2
x = 6
f(2) = 62 + 2(6) + 3
= 36 + 12 + 3
= 51
Question 21 Rapport
If \(a = 2\), \(b = -2\) and \(c = -\frac{1}{2}\), evaluate \((ab^2 - bc^2)(a^2c - abc)\)
Détails de la réponse
[2(2)2 - (- 2x12 )] [22(-12 ) - 2(-2)(-12 )]
[8 = 12 ][-2 - 2] = 172 x 42
= -34
Question 22 Rapport
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Adding up = 120
nos ≥ 16 are 16 + 30 + 20 + 26 = 92
The requires sum of angles = 92120 x 360o1
= 276o
Question 23 Rapport
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Question 24 Rapport
Simplify \( \frac{1}{x-2} + \frac{1}{x+2} + \frac{2x}{x^2-4} \)
Détails de la réponse
= (x+2)+(x−2)+2x(x+2)(x−2)
= 4xx2−4
Question 25 Rapport
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Question 26 Rapport
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prob. of selecting a prime No. is 311
Question 27 Rapport
Détails de la réponse
= log693 + log612 + log664 - log672
log6729 + log612 + log664 - log672
log6(729 x 12 x 64) = log6776
= log665 = 5 log66 = 5
N.B: log66 = 1
Question 28 Rapport
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Question 29 Rapport
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Question 30 Rapport
Given that \(x^2 + y^2 + z^2 = 194\), calculate z if \(x = 7\) and \(\sqrt{y} = 3\)
Détails de la réponse
Given that x2 + y2 + z2 = 194, calculate z if x = 7 and √y
= 3
x = 7
∴ x2 = 49
√y
= 3
∴ y2 = 81 = x2 + y2 + z2 = 194
49 + 81 + z2 = 194
130 + z2 = 194
z2 = 194 - 130
= 64
z = √64
= 8
Question 31 Rapport
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Question 32 Rapport
Détails de la réponse
Using difference of two squares which says
a2 - b2 = (a + b)(a - b) = 9(x + y)2 - 4(x - y)2
= [3(x + y)]2 - [2(x - y)]-2
= [3(x + y) + 2(x - y) + 2(x - y)][3(x + y) - 2(x - y)]
= [3x +3y + 2x - 2y][3x + 3y - 2x + 2y]
= (5x + y)(x + 5y)
Question 33 Rapport
Three children shared a basket of mangoes in such a way that the first child took \( \frac{1}{4} \) of the mangoes and the second \( \frac{3}{4} \) of the remainder. What fraction of the mangoes did the third child take?
Détails de la réponse
If the first child takes 14 it will remain 1 - 14 = 34
Next, the second child takes 34 of the remainder
which is 34 i.e. find 34 of 34
= 34 x 34
= 916
the fraction remaining now = 34 - 916
= 12?916
= 316
Question 35 Rapport
If \( \cos \theta = \frac{12}{13} \). Find \( \theta + \cos^2 \theta \)
Détails de la réponse
x2 + 122 = 132
x2 = 169- 144 = 25
x = 25
= 5
Hence, tanθ = 512 and cosθ = 1213
If cos2θ = 1 + 1tan2θ
= 1 + 1(5)212
= 1 + 125144
= 1 + 14425
= 25+14425
= 16925
Question 36 Rapport
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Question 37 Rapport
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sample space = 16
sum of nos. removed are (2), 3, (4), 5
3, (4), 5, (6)
(4), 5, (6), 7
(5), 6, 7, (8)
Even nos. = 8 of them
Pr(even sum) = 816
= 12
Question 38 Rapport
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Question 40 Rapport
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Question 41 Rapport
If \(x + \frac{1}{x} = 4\), find \(x^2 + \frac{1}{x^2}\)
Détails de la réponse
= (x + 1x )2 = x2 + 1x2 + 2
x2 + 1x2 = ( x + 1x2 )2 - 2
= (4)2 - 2
= 16 - 2
= 14
Question 42 Rapport
Simplify \(4\frac{3}{4} - 6\frac{1}{4}\)
Détails de la réponse
194 - 254 ............(A)
215 - 54 .............(B)
Now work out the value of A and the value of B and then find the value AB
A = 194 - 254
= −64
B = 215 x 520
= 10520
= 214
But then AB = −64
214 = −64 ÷ 214
= −64 x 421
= −2484
= −27
Question 43 Rapport
Simplify \( \left(\frac{1}{x^{-1}} + \frac{1}{y^{-1}}\right)^{-1} \)
Détails de la réponse
= (x + y)-1 = (x)y
= xy
Question 44 Rapport
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Question 45 Rapport
Find the curved surface area of the frustrum in the figure.
Détails de la réponse
6x = 4(6 + x) = 24 + 4x
x = 12 = c = πRL−πL
= π(6)√182+62−π×4×√122+42
= 6π√360−4π√160
= 36π√10−16π√10
= 20π√10 cm2
Question 46 Rapport
Détails de la réponse
To find all possible values = (a - b)2 + b2 - 2ab
Substituting the given values = (a - b)2
= 16 - 7
= 9
(a - b) = ± 9
= ± 3
OR a - b = 3, -3
