JAMB UTME - Mathematics - 1991
Question 1 Report
Fifty boxes each of 50 bolts were inspected for the number which were defective. The following was the result:
| No. defective per box | 4 | 5 | 6 | 7 | 8 | 9 |
| No. of boxes | 2 | 7 | 17 | 10 | 8 | 6 |
The mean and the median of the distribution are respectively
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Question 2 Report
Simplify \( \frac{1}{1+\sqrt{5}} - \frac{1}{1-\sqrt{5}} \)
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= 3−√5−3−√5(3+√5)(3−√5
= −2√59−5
= −2√54
= - 12√5
Question 3 Report
Question 4 Report
In the diagram above, |PQ| = |QR|, |PS| = |RS|, ∠PSR = 30o and ∠PQR = 80o. Find ∠SPQ.
Answer Details
QRP = QPR
= 180 - 80 = 100/20 = 50o
SRP = SPR
= 180 - 30 = 150/2 = 75o
∴ SPQ = SPR - QPR
= 75 - 50 = 25o
Question 5 Report
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Question 6 Report
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Question 7 Report
Find correct to 3 decimal places \(\left(\frac{1}{0.05} + \frac{1}{5.005}\right) - (0.05 x 2.05)\)
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(10.05 x 5.005) - (0.05 x 2.05)
500.50.05
- 1025
100.1 - 0.1025
= 99.9975
99.998 to 3 sig. fiq.
Question 8 Report
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Question 9 Report
PMN and PQR are two secants of the circle MQTRN and PT is a tangent. If PM = 5cm, PN = 12cm and PQ = 4.8cm, calculate the respective lengths of PR and PT in centimeters
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Question 10 Report
Evaluate \(x^2(x^2 - 1)^{-\frac{1}{2}} - (x^2 - 1)^{\frac{1}{2}}\)
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= x2?(x2?1)(x2?1)12
= x2?x2+1(x2?1)12
= (x2 - 1)-12
Question 11 Report
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By trial and error method
Let the base to 8
i.e. Let n = 8 and it is easily verified that the subtraction holds.
The subtraction does not hold when other values of n are tried
n = 8
Question 12 Report
If w varies inversely as \( \frac{ur}{u+r} \) and is equal to 8 when u = 2 and r = 6, find a relationship between u, v, w.
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∴ w = kuvu+v
= k(u+v)uv
w = k(u+v)uv
w = 8, u = 2 and v = 6
8 = k(2+6)2(6)
= k(8)12
k = 12
Question 13 Report
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x2 - 3x - 2x + 6 = 12
x2 - 5x - 6 = 0
(x - 1)(x - 6) = 0
x = -1 or 6
Question 14 Report
Simplify cos2 x (sec2x + sec2 x tan2x)
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Question 15 Report
Rationalize \( \frac{2\sqrt{3}+3\sqrt{2}}{3\sqrt{2}-2\sqrt{3}} \)
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= 2√3+3√23√2−2√3 x 3√2+2√33√2−2√3
4(3)+9(2)+2(6)√69(2)−4(3)
12+18+12√61‘8−12
= 30+12√66
= 5 + 2√6
Question 17 Report
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Let f(m) = m3 - 2m2 - m2 + 2
= f(1)
= 1 - 2 - 2 + 2 = 0
∴ m - 1 is factor m3−2m2−m2+2m−1
= m2 - m - 2
= (m - 1)m2 - m - 2
= (m - 1)(m + 1)(m - 2)
Question 18 Report
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Question 19 Report
Fifty boxes each of 50 bolts were inspected for the number which were defective. The following was the result:
| No. defective per box | 4 | 5 | 6 | 7 | 8 | 9 |
| No. of boxes | 2 | 7 | 17 | 10 | 8 | 6 |
Find the percentage of boxes containing at least 5 defective bolts each.
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Question 20 Report
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= (1 + a - b)(1 - a + b)
Question 21 Report
If the exterior angles of a pentagon are xo, (x + 5)o, (x + 10), (x + 15)o and (x + 20)o, find x
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Question 22 Report
What is the product of \( \frac{27}{5^1}(3)^{-3} \) and \( \frac{(1)^{-1}}{5} \)?
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= 275 x 127 x 51
= 1
Question 23 Report
In the figure, PQRS is a square of sides 8cm. What is the area of \( \triangle UVW \)?
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Question 24 Report
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Question 25 Report
If \( \cos x = \sqrt{\frac{a}{b}} \), find coses x
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y2 + √(a)2 = √(b)2 by pythagoras
y2 = b - a
∴ y = b - a
cosec x = 1sinx = 1y
by = √b√b−a
= √bb−a
Question 26 Report
Simplify \(3\frac{1}{3} - 1\frac{1}{4} x \frac{2}{3} + 1\frac{2}{5}\)
Answer Details
3 - 13
- (54
x 23
) + 125
= 103
- 56
+ 75
= 100−25+4230
= 11730
= 3.9
≈
4
Question 28 Report
Simplify \(2\log \frac{2}{5} - \log \frac{72}{125} + \log 9\)
Answer Details
[25 )2 x 9] = log 425 x 91 x 12572
= log 72125
= log 52
= log 104
= log 10 - log 4
= log10 10 - log10 22
= 1 - 2 log2
Question 30 Report
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Question 31 Report
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Question 33 Report
Question 34 Report
A car travels from calabar to Enugu, a distance of p km with an average speed of u km per hour and continues to benin, a distance of q km, with an average speed of w km per hour. Find its average speed from Calabar to Benin
Answer Details
Average speed = total DistanceTotal Time
from Calabar to Enugu in time t1, hence
t1 = pu
also from Enugu to Benin
t2 = qw
Average speed = p+qt1+t2
= p+qpu+qw
= p + q × uwpw+qu
= uw(p+q)pw+qu
Question 35 Report
Evaluate \( \frac{xy^2-x^2y}{x^2-xy^1} \), when x = -2 and y = 2
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xy2−x2yx2−xy1
= (−2)(3)2−(−2)2(3)(−2)2−(−2)(3)
= -3
Question 36 Report
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Question 37 Report
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Question 38 Report
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BC = 60 tan 62
AC60 = tan621
AC = 60 tan 64
AB = AC - BC
= 60(tan 64o - tan 62o)
Question 40 Report
In this figure, \(PQ = PR = PS\) and \(\angle SRT = 68^\circ\). Find \(\angle QPS\).
Answer Details
i.e. 2(y + x) + QPS = 360∘
QPS = 360∘ - 2(y + x)
But x + y + 68∘ = 180∘
x + y = 180∘ - 68∘ = 180∘
x + y = 180∘ - 68∘
= 112∘
QPS = 360∘ - 2(112∘ )
360∘ - 224∘ = 136∘
Question 41 Report
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Question 42 Report
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Question 43 Report
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= Total Amount of waterTotal quantity of spirit x 1001
3(20100)+5(15100)3+5 x 1001
= 610+751008 x 1001
= 0.6+0.758 x 1001
= 1.358
x 1001
= 33.752
= 16.875
= 1678
Question 44 Report
Musa borrows N10.00 at 2% per month simple interest and repays N8.00 after 4 months. How much does he still owe?
Answer Details
I = PRT100
= 10×2×4100
= 45
= 0.8
Total amount = N10.80
He pays N8.00
Remainder = 10.80 - 8.00
= N2.80
Question 46 Report
Question 47 Report
A crate of soft drinks contains 10 bottles of Coca-cola, 8 of Fanta and 6 of sprite. If one bottle is selected at random, what is the probability that it is NOT a Cocacola bottle?
Answer Details
Coca-cola = 10 bottles, Fanta = 8 bottles
Sprite = 6 bottles, Total = 24
P(cola-cola) = 1024
P(not coca-cola) = 1 - 1024
24?1024
= 1424
= 724
Question 48 Report
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