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Mathematics
WAEC / WASSCE 1993
Permutations, Combinations & Probability
Diagram
Solution
A die is rolled 200 times the outcomes obtained are shown in the table below.
Find the probability of obtaining 2.
Attached Figure / Diagram
A
0.002
B
0.015
C
0.15
D
16
E
0.2
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
Prob(obtaining a 2) = \(\frac{30}{200}\)
= \(0.15\)
Source: Myschool Past Questions Archive (Record #23234)
Mathematics
WAEC / WASSCE 1993
Permutations, Combinations & Probability
Solution
Number
1
2
3
4
5
6
No of times
25
30
45
28
40
32
A die rolled 200 times. The outcome obtained are shown in the table above.
What is the probability of obtaining a number less than 3 ?
A
0.125
B
0.150
C
0.275
D
0.500
E
0.725
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
Prob(less than 3) = \(\frac{25 + 30}{200}\)
= \(\frac{11}{40}\)
= 0.275
Source: Myschool Past Questions Archive (Record #23235)
Mathematics
UTME / JAMB 1993
Integration & Applications
Solution
Integrate \(\frac{1 - x}{x^3}\) with respect to x
A
\(\frac{x - x^2}{x^4}\) + k
B
\(\frac{4}{x^4} - \frac{3 + k}{x^3}\)
C
\(\frac{1}{x} - \frac{1}{2x^2}\) + k
D
\(\frac{1}{3x^2} - \frac{1}{2x}\) + k
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
\(\int \frac{1 - x}{x^3}\)
= \(\int^{1}_{x^3} - \int^{x}_{x^3}\)
= x-3 dx - x-2dx
= \(\frac{1}{2x^2} + \frac{1}{x}\)
Source: Myschool Past Questions Archive (Record #31059)
Mathematics
UTME / JAMB 1993
Number Bases & Modular Arithmetic
Solution
change 7110 to base 8
A
1078
B
1068
C
718
D
178
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
\(\begin{array}{c|c} 8 & 71 \\ 8 & 8 \text{rem} 7\\ 8 & 1 \text{rem} 0\end{array}\)
= 1078
Source: Myschool Past Questions Archive (Record #31127)
Mathematics
UTME / JAMB 1993
Fractions, Decimals, Percentages & Approximations
Solution
Evaluate \(\frac{3524}{0.05}\) correct to 3 significant figures
A
705
B
70,000
C
70, 480
D
70, 500
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
\(\frac{3524}{0.05}\) = 70480
\(\approx\) 70500(3 s.g)
Source: Myschool Past Questions Archive (Record #31128)
Mathematics
UTME / JAMB 1993
Indices, Logarithms & Surds
Solution
If 9\(^{(x - \frac{1}{2})} = 3^{x^2}\) Find the value of x
A
\(\frac{1}{2}\)
B
1
C
2
D
3
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
9(x - \(\frac{1}{2}\)) 3x2 = 3\(^{2(x - \frac{1}{2})}\) = 3\(^x{^2}\)
∴ 2(x - \(\frac{1}{2}\)) = x2
2x - 1 = x\(^2\)
hence x\(^2\) - 2x + 1 = 0
(x - 1)(x - 1) = 0
x = 1
Source: Myschool Past Questions Archive (Record #31130)
Mathematics
UTME / JAMB 1993
Indices, Logarithms & Surds
Solution
Solve for y in the equation 10\(^y\) x 5\(^{(2y - 2)}\) x 4\(^{(y - 1)}\) = 1
A
\(\frac{3}{4}\)
B
\(\frac{5}{4}\)
C
\(\frac{2}{3}\)
D
5
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
10\(^y\) x 5\(^{(2y - 2)}\) x 4\(^{(y - 1)}\) = 1
but 10\(^y\) = (5 x 2)\(^y\) = 5\(^y\) x 2\(^y\)
= (Law of indices)
5\(^y\) x 2\(^y\) x 5\(^{(2y - 2)}\) x 4\(^{(y - 1)}\) = 1
but 4\(^{(y - 1)}\) = 2\(^{2(y - 1)}\)
= 2\(^{2y - 2}\) (Law of indices)
5\(^y\) x 5\(^{(2y -2)}\) x 2\(^{(2y- 2)}\) x 2\(^y\) = 1
5\(^{(3y -2)}\) x 2\(^y\) x 2\(^{(2y -2)}\) = 1
= 5\(^{(3y -2)}\) x 2\(^{(3y -2)}\) = 1
But a\(^o\) = 1
10\(^{(3y -2)}\) = 10\(^o\)
3y - 2 = 0
∴ y = \(\frac{2}{3}\)
Source: Myschool Past Questions Archive (Record #31135)
Mathematics
UTME / JAMB 1993
Plane Geometry, Polygons & Circle Theorems
Solution
simplify \(\frac{1}{√3 - 2}\) - \(\frac{1}{√3 + 2}\)
A
3
B
\(\frac{2}{3}\)
C
7
D
-4
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
\(\frac{1}{√3 - 2}\) - \(\frac{1}{√3 + 2}\)
L.C.M = (3- 2) (3 + 2)
∴ \(\frac{1}{\sqrt{3 - 2}}\) - \(\frac{1}{\sqrt{3 - 2}}\) = \(\frac{\sqrt{3 + 2} - \sqrt{3 - 2}}{\sqrt{3 - 2} + \sqrt{3 - 2}}\)
\(\frac{√3 + 2 - √3 + 2}{3 - 2√3 + 2√3 - 4}\) = \(\frac{4}{3 - 2}\)
= \(\frac{4}{-1}\)
= -4
Source: Myschool Past Questions Archive (Record #31138)
Mathematics
UTME / JAMB 1993
Indices, Logarithms & Surds
Solution
If 2log3 y + log3 x2 = 4, then y is
A
4 - log3
B
\(\frac{4}{log_3 x}\)
C
\(\frac{4}{x}\)
D
\(\pm\) \(\frac{9}{x}\)
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
2log3y + log3x2 = 4
log3y2 + log3x2 = 4
∴ log3 (x2y2) = log381(correct all to base 4)
x2y2 = 81
∴ xy = \(\pm\)9
∴ y = \(\pm\)\(\frac{9}{x}\)
Source: Myschool Past Questions Archive (Record #31142)
Mathematics
UTME / JAMB 1993
Plane Geometry, Polygons & Circle Theorems
Solution
Solve without using tables log5(62.5) - log5(\(\frac{1}{2}\))
A
3
B
4
C
5
D
8
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
log5(62.5) - log5(\(\frac{1}{2}\))
= log5\(\frac{(62.5)}{\frac{1}{2}}\) - log5(2 x 62.5)
= log5(125)
= log553 - 3log55
= 3
Source: Myschool Past Questions Archive (Record #31147)
Mathematics
UTME / JAMB 1993
Fractions, Decimals, Percentages & Approximations
Solution
If N225.00 yields N27.00 in x years simple interest at the rate of 4% per annum, find x
A
3
B
4
C
12
D
17
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
Principal = N255.00, Interest = 27.00
year = x Rate: 4%
∴ 1 = \(\frac{PRT}{100}\)
27 = \(\frac{225 \times 4 \times T}{100}\)
2700 = 900T
T = 3 years
Source: Myschool Past Questions Archive (Record #31152)
Mathematics
UTME / JAMB 1993
Linear, Simultaneous & Quadratic Equations
Solution
If \(\sqrt{x^2 + 9}\) = x + 1, solve for x
A
5
B
4
C
3
D
2
E
1
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
\(\sqrt{x^2 + 9}\) = x + 1
x\(^2\) + 9 = (x + 1)\(^2\)
0 = x\(^2\) + 2x + 1 - x\(^2\) - 9
= 2x - 8 = 0
2(x - 4) = 0
x = 4
Source: Myschool Past Questions Archive (Record #31158)
Mathematics
UTME / JAMB 1993
Algebraic Expressions, Polynomials & Factorisation
Solution
Make x the subject of the relation \(\frac{1 + ax}{1 - ax}\) = \(\frac{p}{q}\)
A
\(\frac{p + q}{a(p - q)}\)
B
\(\frac{p - q}{a(p + q)}\)
C
\(\frac{p - q}{apq}\)
D
\(\frac{pq}{a(p - q)}\)
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
\(\frac{1 + ax}{1 - ax}\) = \(\frac{p}{q}\) by cross multiplication,
q(1 + ax) = p(1 - ax)
q + qax = p - pax
qax + pax = p - q
∴ x = \(\frac{p - q}{a(p + q)}\)
Source: Myschool Past Questions Archive (Record #31159)
Mathematics
UTME / JAMB 1993
Algebraic Expressions, Polynomials & Factorisation
Solution
Which of the following is a factor of 15 + 7x - 2x2
A
x + 3
B
x - 3
C
x - 5
D
x + 5
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
Factorize 15 + 7x - 2x2
(5 - x)(3 + 2x); suppose 15 + 7x = 2x2 = 0
∴ (5 - x)(3 + 2x) = 0
x = 5 or x = -\(\frac{3}{2}\)
Since 5 is a root, then (x - 5) is a factor
Source: Myschool Past Questions Archive (Record #31166)
Mathematics
UTME / JAMB 1993
Plane Geometry, Polygons & Circle Theorems
Solution
Evaluate (x + \(\frac{1}{x}\) + 1)\(^2\) - (x - \(\frac{1}{x}\) - 1)\(^2\)
A
4x2
B
(\(\frac{2}{x}\) + 2)2
C
4
D
4(1 + x)
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
\(\left(x + \frac{1}{x} + 1\right)^2 - \left(x - \frac{1}{x} - 1\right)^2,\)
We can use the difference of squares formula:
\(a^2 - b^2 = (a - b)(a + b),\)
where
\(a = x + \frac{1}{x} + 1 \quad \text{and} \quad b = x - \frac{1}{x} - 1.\)
Calculating \( a - b \):
\(a - b = \left(x + \frac{1}{x} + 1\right) - \left(x - \frac{1}{x} - 1\right) = x + \frac{1}{x} + 1 - x + \frac{1}{x} + 1 = 2 \cdot \frac{1}{x} + 2 = 2\left(\frac{1}{x} + 1\right).\)
Calculating \( a + b \):
\(a + b = \left(x + \frac{1}{x} + 1\right) + \left(x - \frac{1}{x} - 1\right) = x + \frac{1}{x} + 1 + x - \frac{1}{x} - 1 = 2x.\)
Substituting back into the difference of squares formula:
\(a^2 - b^2 = (a - b)(a + b) = \left(2\left(\frac{1}{x} + 1\right)\right)(2x) = 4x\left(\frac{1}{x} + 1\right).\)
Distributing \(4x\):
\(= 4x \cdot \frac{1}{x} + 4x \cdot 1 = 4 + 4x.\)
Thus, the evaluated expression is: 4 + 4x = 4(1 + x)
Source: Myschool Past Questions Archive (Record #31167)