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Mathematics
WAEC / WASSCE 1994
Number and Numeration
Solution
The length of an exercise book is given as 20cm correct to the nearest centimeter. In which of the following ranges of possible measurement does the actual length lie?
A
19.5 - 20.4cm
B
19.1 - 20.9cm
C
19.9 - 20.1cm
D
19.95 - 20.05cm
E
19 - 21cm
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
In mathematics, numbers ranging from 0.5 - 0.9 are approximated to 1, while 0.1 - 0.4 are taken down to 0.
Source: Myschool Past Questions Archive (Record #23584)
Mathematics
UTME / JAMB 1994
Number and Numeration
Solution
Solve for x if \(25^{x} + 3(5^{x}) = 4\)
A
1 or -4
B
0
C
1
D
-4 or 0
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
\(25^{x} + 3(5^{x}) = 4\)
Let \(5^{x}\) = y.
\((5^{2})^{x} + 3(5^{x}) - 4 = 0\)
\(y^{2} + 3y - 4 = 0\)
\(y^{2} - y + 4y - 4 = 0\)
\(y(y - 1) + 4(y - 1) = 0\)
\((y + 4)(y - 1) = 0\)
\(y = -4 ; y = 1\)
y = -4 is not possible.
y = 1 \(\implies\) x = 0.
Source: Myschool Past Questions Archive (Record #30952)
Mathematics
UTME / JAMB 1994
Number and Numeration
Solution
Evaluate \(\frac{1}{3} \div [\frac{5}{7}(\frac{9}{10} -1 + \frac{3}{4})]\)
A
\(\frac{28}{39}\)
B
\(\frac{13}{39}\)
C
\(\frac{39}{28}\)
D
\(\frac{84}{13}\)
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
\(\frac{1}{3} \div [\frac{5}{7}(\frac{9}{10} -1 + \frac{3}{4})]\)
\(\frac{1}{3} \div [\frac{5}{7}(\frac{9}{10} - \frac{10}{10} + \frac{3}{4})]\)
= \(\frac{1}{3} \div [\frac{5}{7}(\frac{-1}{10} + \frac{3}{4})]\)
= \(\frac{1}{3} \div [\frac{5}{7}(\frac{-2 + 15}{20})]\)
= \(\frac{1}{3} \div [\frac{5}{7} \times \frac{13}{20}]\)
\(\frac{1}{3} + [\frac{13}{28}]\) = \(\frac{1}{3} \times \frac{28}{13}\)
= \(\frac{28}{39}\)
Source: Myschool Past Questions Archive (Record #30957)
Mathematics
UTME / JAMB 1994
Number and Numeration
Solution
Evaluate \(\frac{0.36 \times 5.4 \times 0.63}{4.2 \times 9.0 \times 2.4}\)
A
0.013
B
0.014
C
0.14
D
0.13
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
\(\frac{0.36 \times 5.4 \times 0.63}{4.2 \times 9.0 \times 2.4}\)
= \(\frac{36}{420} \times \frac{54}{90} \times \frac{63}{240}\)
= \(\frac{6}{70} \times \frac{18}{30} \times \frac{21}{80}\)
= \(\frac{27}{2000}\)
= 0.0135
\(\approx\) = 0.014
Source: Myschool Past Questions Archive (Record #30971)
Mathematics
UTME / JAMB 1994
Number and Numeration
Solution
Evaluate \(\frac{log_5 (0.04)}{log_3 18 - log_3 2}\)
A
1
B
-1
C
\(\frac{2}{3}\)
D
-\(\frac{2}{3}\)
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
\(\frac{log_5 0.04}{log_3 18 - log_3 2}\)
= \(\frac{log_5 0.04}{log_3(\frac{18}{2})}\)
= \(\frac{log_5 0.04}{log_3 9}\)
= \(\frac{-2}{2}\)
= -1
Let log5 0.04 = x
5x = 0.04
x = \(\frac{4}{100}\) = 5\(^{-2}\)
Let log\(_3\) 9 = z
3\(^2\) = 3\(^z\)
z = 3
Source: Myschool Past Questions Archive (Record #30978)
Mathematics
UTME / JAMB 1994
Number and Numeration
Solution
Simplify \(\sqrt{48}\) - \(\frac{9}{\sqrt{3}}\) + \(\sqrt{75}\)
A
5√3
B
6√3
C
8√3
D
18√3
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
\(\sqrt{48}\) - \(\frac{9}{\sqrt{3}}\) + \(\sqrt{75}\)
Rearrange = \(\sqrt{48}\) + \(\sqrt{75}\) - \(\frac{9}{\sqrt{3}}\)
= (\(\sqrt{16}\) x \(\sqrt{3}\)) + (\(\sqrt{25}\) x \(\sqrt{3}\)) - \(\frac{9}{\sqrt{3}}\)
= 4\(\sqrt{3}\) + 5\(\sqrt{3}\) - \(\frac{9}{\sqrt{3}}\)
Rationalize \(\frac{9}{\sqrt{3}}\) \(\to\) = \(\frac{9}{\sqrt{3}}\) x \(\frac{\sqrt{3}}{\sqrt{3}}\) = 3\(\sqrt{3}\)
= 4\(\sqrt{3}\) + 5\(\sqrt{3}\) - 3\(\sqrt{3}\)
= 6\(\sqrt{3}\)
Source: Myschool Past Questions Archive (Record #30989)
Mathematics
UTME / JAMB 1994
Number and Numeration
Solution
Given that for sets A and B, in a universal set E, A \(\subseteq\) B then A \(\cap\)(A \(\cap\) B)' is
A
A
B
\(\phi\)
C
B
D
E
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
A \(\subset\) B means A is contained in B i.e. A is a subset of B(A \(\cap\) B)' = A'
A(A \(\cap\) B)' = A \(\cap\) A'
The intersection of complement of a set P and P' has no element
i.e. n(A \(\cap\) A') = \(\phi\)
Source: Myschool Past Questions Archive (Record #31000)
Mathematics
UTME / JAMB 1994
Number and Numeration
Solution
Simplify \(\frac{(2m - u)^2 - (m - 2u)^2}{5m^2 - 5u^2}\)
A
\(\frac{3}{5}\)
B
\(\frac{2}{5}\)
C
\(\frac{2m - u}{5m + u}\)
D
\(\frac{m - 2u}{m + 5u}\)
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
\(\frac{(2m - u)^2 - (m - 2u)^2}{5m^2 - 5u^2}\)
= \(\frac{2m - u + m - 2u)(2m - u - m + 2u)}{5(m + u)(m - u)}\)
= \(\frac{3(m - u)(m + u)}{5(m + u)(m - u)}\)
= \(\frac{3}{5}\)
Source: Myschool Past Questions Archive (Record #31006)
Mathematics
UTME / JAMB 1994
Number and Numeration
Solution
Find P if \(\frac{x - 3}{(1 - x)(x + 2)}\) = \(\frac{p}{1 - x}\) + \(\frac{Q}{x + 2}\)
A
\(\frac{-2}{3}\)
B
\(\frac{-5}{3}\)
C
\(\frac{5}{3}\)
D
\(\frac{2}{3}\)
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
\(\frac{x - 3}{(1 - x)(x + 2)}\) = \(\frac{p}{1 - x}\) + \(\frac{Q}{x + 2}\)
Multiply both sides by LCM i.e. (1 - x(x + 2))
∴ x - 3 = p(x + 2) + Q(1 - x)
When x = +1
(+1) - 3 = p(+1 + 2) + Q(1 - 1)
-2 = 3p + 0(Q)
3p = -2
∴ p = \(\frac{-2}{3}\)
Source: Myschool Past Questions Archive (Record #31011)
Mathematics
UTME / JAMB 1994
Number and Numeration
Solution
Find the range of values of x for which \(\frac{1}{x}\) > 2 is true
A
x < \(\frac{1}{2}\)
B
x < 0 or x < \(\frac{1}{2}\)
C
0 < x < \(\frac{1}{2}\)
D
1 < x < 2
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
\(\frac{1}{x}\) > 2 = \(\frac{x}{x^2}\) > 2
x > 2x2
= 2x2 2 - x 0, 2x - 1 0, x 0
x < 0, x , \(\frac{1}{2}\) = 0
Source: Myschool Past Questions Archive (Record #31012)
Mathematics
UTME / JAMB 1994
Number and Numeration
Solution
Find the value of log10 r + log10 r2 + log10 r4 + log10 r8 + log10 r16 + log10 r32 = 63
A
10-1
B
10o
C
10
D
102
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
log10 r + log10 r2 + log10 r4 + log10 r8 + log10 r16 + log10 r32 = 63
log10r63 = 63
63 = 1063
∴ r = 10
Source: Myschool Past Questions Archive (Record #31014)
Mathematics
UTME / JAMB 1994
Number and Numeration
Solution
A binary operation \(\oplus\) is defines on the set of all positive integers by a \(\oplus\) b = ab for all positive integers a, b. Which of the following properties does NOT hold?
A
closure
B
identity
C
positive
D
inverse
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
a \(\oplus\) b = ab
The set of all national rules Q, is closed under the operations, additions, subtraction, multiplication and division. Since a \(\oplus\) b = ab; b \(\oplus\) a = ba = ab
The number 1 is the identity element under multiplication
Source: Myschool Past Questions Archive (Record #31016)
Mathematics
UTME / JAMB 1994
Number and Numeration
Solution
\(\begin{array}{c|c} \oplus mod 10 & 2 & 4 & 6 & 8 \\ \hline 2 & 4 & 8 & 2 & 6 \\4 & 8 & 6 & 4 & 2\\ 4 & 8 & 6 & 4 & 2\\ 6 & 2 & 4 & 6 & 8\\ 8 & 6 & 2 & 8 & 4\end{array}\)
The multiplication table above has modulo 10 on the set S = (2, 4, 6, 8). Find the inverse of 2
A
2
B
4
C
6
D
8
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
The inverse of 2 is 6 since 2 x 6 = 12; under mod 10
12 = 2 which is also the value required
Source: Myschool Past Questions Archive (Record #31017)
Mathematics
UTME / JAMB 1994
Number and Numeration
Solution
If M(4, q) is the mid-point of the line joining L(p, -2) and N(q, p). Find the values of p and q
A
P = 2, q = 4
B
p = 3, q = 1
C
p = 5, q = 3
D
p = 6, q = 2
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
\(\frac{p + q}{2}\) = 4
p + q = 8 ....(i)
\(\frac{p - 2}{2}\) = q
p - 2q 2 ....(ii)
q = 2, p = 6
3q = 6
Source: Myschool Past Questions Archive (Record #31041)
Mathematics
UTME / JAMB 1994
Number and Numeration
Solution
If y = 3t3 + 2t2 - 7t + 3, find \(\frac{dy}{dt}\) at t = -1
A
-1
B
1
C
-2
D
2
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
y = 3t3 + 2t2 - 7t + 3
\(\frac{dy}{dt}\) = 9t2 + 4t - 7
When t = -1
\(\frac{dy}{dt}\) = 9(-1)2 + 4(-1) - 7
= 9 - 4 -7
= 9 - 11
= -2
Source: Myschool Past Questions Archive (Record #31047)