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Mathematics
WAEC / WASSCE 1991
Plane Geometry, Polygons & Circle Theorems
Diagram
In the diagram above , |AD| = 10cm, |DC| = 8cm and |CF| = 15cm. Which of the following is correct?
Attached Figure / Diagram
A
Area BCF = Area DCF
B
Area ADE = Area ADF
C
Area ADE = Area DCFE
D
Area CBF = Area DABC
E
Area DABO = Area CFEO
medium Difficulty
Verified Correct Answer: Option E
Step-by-Step Mathematical Solution
No expanded working has been attached for this question yet. The verified answer is Option E.
Source: Myschool Past Questions Archive (Record #23119)
Mathematics
UTME / JAMB 1991
Plane Geometry, Polygons & Circle Theorems
Solution
Simplify 3\(\frac{1}{3}\) - 1\(\frac{1}{4}\) x \(\frac{2}{3}\) + 1\(\frac{2}{5}\)
A
2
B
3
C
4
D
5
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
3 - \(\frac{1}{3}\) - (\(\frac{5}{4}\) x \(\frac{2}{3}\)) + 1\(\frac{2}{5}\)
= \(\frac{10}{3}\) - \(\frac{5}{6}\) + \(\frac{7}{5}\)
= \(\frac{100 - 25 + 42}{30}\)
= \(\frac{117}{30}\)
= 3.9
\(\approx\) 4
Source: Myschool Past Questions Archive (Record #31367)
Mathematics
UTME / JAMB 1991
Plane Geometry, Polygons & Circle Theorems
Solution
If 2257 is the result of subtracting 4577 from 7056 in base n, find n
A
8
B
9
C
10
D
11
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
\(\begin{array}{c|c} 7056 \\ \text{- 4577} \\\hline 2257 \end{array}\)
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
Source: Myschool Past Questions Archive (Record #31370)
Mathematics
UTME / JAMB 1991
Fractions, Decimals, Percentages & Approximations
Solution
Find correct to 3 decimal places (\(\frac{1}{0.05} \div\frac{1}{5.005}\)) - (0.05 x 2.05)
A
99.998
B
98.999
C
89.899
D
9.998
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
(\(\frac{1}{0.05} \div \frac{1}{5.005}\)) - (0.05 x 2.05)
(\(\frac{1}{0.05}\) x 5.005) - (0.05 x 2.05)
\(\frac{5.005}{0.05}\) - 0.1025
100.1 - 0.1025
= 99.9975
99.998 to 3 decimal place.
Source: Myschool Past Questions Archive (Record #31371)
Mathematics
UTME / JAMB 1991
Fractions, Decimals, Percentages & Approximations
Solution
express 62 \(\div\) 3 as a decimal correct to 3 significant figures
A
20.6
B
20.667
C
20.67
D
20.7
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
\(\frac{62}{3}\) = 20.6666.....
= 20.7 to 3 sig. fig.
Source: Myschool Past Questions Archive (Record #31372)
Mathematics
UTME / JAMB 1991
Fractions, Decimals, Percentages & Approximations
Solution
Factory P produces 20,000 bags of cement per day while factory Q produces 15,000 bags per day. If P reduces production by 5% and Q increases production by 5%, determine the effective loss in the number of bags produced per day by the two factories
A
250
B
750
C
1000
D
1250
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
P produces 20,000 bags
P reduces production by 5%
i.e. reduction = \(\frac{5}{100}\) x \(\frac{20000}{1}\)
= 1000
Q produces 15000 bags
Q increases production by 5%
i.e. increase = \(\frac{5}{100}\) x \(\frac{15000}{1}\)
= 750
Effective loss by the two = 1000 - 750
= 250 bags
Source: Myschool Past Questions Archive (Record #31373)
Mathematics
UTME / JAMB 1991
Fractions, Decimals, Percentages & Approximations
Solution
If 3 gallons of spirit containing 20% water are added to 5 gallons of another spirit containing 15% water, what percentage of the mixture is water?
A
2\(\frac{4}{5}\)%
B
16\(\frac{7}{8}\)%
C
18\(\frac{1}{8}\)%
D
18\(\frac{7}{8}\)%
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
% of water in the mixture
= \(\frac{\text{Total Amount of water}}{\text{Total quantity of spirit}}\) x \(\frac{100}{1}\)
\(\frac{3(\frac{20}{100}) + 5 (\frac{15}{100})}{3 + 5}\) x \(\frac{100}{1}\)
= \(\frac{\frac{6}{10} + \frac{75}{100}}{8}\) x \(\frac{100}{1}\)
= \(\frac{0.6 + 0.75}{8}\) x \(\frac{100}{1}\)
= \(\frac{1.35}{8}\) x \(\frac{100}{1}\)
= \(\frac{33.75}{2}\)
= 16.875
= 16\(\frac{7}{8}\)
Source: Myschool Past Questions Archive (Record #31375)
Mathematics
UTME / JAMB 1991
Indices, Logarithms & Surds
Solution
Simplify 2log \(\frac{2}{5}\) - log\(\frac{72}{125}\) + log 9
A
1 - 4 log3
B
-1 + 2 log 3
C
-1 + 5 log2
D
1 - 2log 2
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
2log \(\frac{2}{5}\) - log\(\frac{72}{125}\) + log 9
[\(\frac{2}{5}\))2 x 9] = log \(\frac{4}{25}\) x \(\frac{9}{1}\) x \(\frac{125}{72}\)
= log \(\frac{72}{125}\)
= log \(\frac{5}{2}\)
= log \(\frac{10}{4}\)
= log 10 - log 4
= log10 10 - log10 22
= 1 - 2 log2
Source: Myschool Past Questions Archive (Record #31377)
Mathematics
UTME / JAMB 1991
Indices, Logarithms & Surds
Solution
Simplify \(\frac{1}{1 + \sqrt{5}}\) - \(\frac{1}{1 - \sqrt{5}}\)
A
- \(\frac{1}{2}\sqrt{5}\)
B
\(\frac{1}{2}\sqrt{5}\)
C
-- \(\frac{1}{4}\sqrt{5}\)
D
5
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
\(\frac{1}{1 + \sqrt{5}}\) - \(\frac{1}{1 - \sqrt{5}}\)
= \(\frac{1 - \sqrt{5} - 1 - \sqrt{5}}{(1 + \sqrt{5}) (1 - \sqrt{5}}\)
= \(\frac{-2\sqrt{5}}{1 - 5}\)
= \(\frac{-2\sqrt{5}}{- 4}\)
= \(\frac{1}{2}\sqrt{5}\)
Source: Myschool Past Questions Archive (Record #31378)
Mathematics
UTME / JAMB 1991
Indices, Logarithms & Surds
Solution
Rationalize \(\frac{2\sqrt{3} + 3 \sqrt{2}}{3\sqrt{2} - 2 \sqrt{3}}\)
A
5 - 2\(\sqrt{6}\)
B
5 + 2\(\sqrt{6}\)
C
5\(\sqrt{6}\)
D
5
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
\(\frac{2\sqrt{3} + 3 \sqrt{2}}{3\sqrt{2} - 2 \sqrt{3}}\)
= \(\frac{2\sqrt{3} + 3 \sqrt{2}}{3\sqrt{2} - 2 \sqrt{3}}\) x \(\frac{3\sqrt{2} + 2 \sqrt{3}}{3\sqrt{2} - 2 \sqrt{3}}\)
\(\frac{4(3) + 9(2) + 2(6) \sqrt{6}}{9(2) - 4(3)}\)
\(\frac{12 + 18 + 12\sqrt{6}}{1`8 - 12}\)
= \(\frac{30 + 12\sqrt{6}}{6}\)
= 5 + 2\(\sqrt{6}\)
Source: Myschool Past Questions Archive (Record #31382)
Mathematics
UTME / JAMB 1991
Plane Geometry, Polygons & Circle Theorems
Solution
Multiply (x2 - 3x + 1) by (x - a)
A
x3 - (3 + a) x2 + (1 + 3a)x - a
B
x3 - (3 - a)x2 + 3ax - a
C
x3 - (3 - a)x2 - (1 = 3a) - a
D
x3 + (3 - a)x2 + (1 + 3a) - a
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
(x2 - 3x + 1)(x - a) = x3 - 3x2 + x - ax2 + 3ax - a
= x3 - (3 + a) x2 + (1 + 3a)x - a
Source: Myschool Past Questions Archive (Record #31384)
Mathematics
UTME / JAMB 1991
Linear, Simultaneous & Quadratic Equations
Solution
Evaluate \(\frac{xy^2 - x^2y}{x^2 - xy}\) When x = -2 and y = 3
A
3
B
-\(\frac{3}{5}\)
C
\(\frac{3}{5}\)
D
-3
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
\(\frac{xy^2 - x^2y}{x^2 - xy^1}\)
= \(\frac{(-2)(3)^2 - (-2)^2(3)}{(-2)^2 - (-2)(3)}\)
= \(\frac{-30}{10}\)
= -3
Source: Myschool Past Questions Archive (Record #31387)
Mathematics
UTME / JAMB 1991
Plane Geometry, Polygons & Circle Theorems
Solution
If g(x) = x2 + 3x + 4, find g(x + 1) - g(x)
A
(x + 2)
B
2(x + 2)
C
(2x + 1)
D
(x2 + 4)
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
g(x) = x2 + 3x + 4
= g(x + 1) = (x + 1)^2 + 3(x + 1) + 4
= x2 + 1 + 2x + 3x + 3 + 4
= x2 + 5x + 8
g(x + 1) - g(x) = x2 + 5x + 8 - (x2 + 3x + 4)
= x2 + 5x + 8 - x2 + 3x + 4
= 2x + 4
= 2(x + 2)
Source: Myschool Past Questions Archive (Record #31403)
Mathematics
UTME / JAMB 1991
Algebraic Expressions, Polynomials & Factorisation
Solution
Factorize \(m^3 - 2m^2\) - m + 2
A
(m2 + 1)(m - 2)
B
(m - 1)(m + 1)(m + 2)
C
(m - 2)(m + 1)(m - 1)
D
(m2 + 2)(m - 1)
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
\(m^3 - 2m^2\) - m + 2
Let f(m) = \(m^3 - 2m^2\) - m + 2
= f(1)
= 1 - 2 - 1 + 2 = 0
∴ m - 1 is factor \(\frac{m^3 - 2m^2 - m + 2}{m - 1}\)
= \(m^2\) - m - 2
= (\(m^2\) - 2m + m - 2)
= m(m - 2) + 1(m - 2)
= ( m + 1)( m - 2)
= (m - 1)(m + 1)(m - 2)
Source: Myschool Past Questions Archive (Record #31404)
Mathematics
UTME / JAMB 1991
Algebraic Expressions, Polynomials & Factorisation
Solution
Factorize 1 - (a - b)2
A
(1 - a - b)(1 - a + b)
B
(1 + a - b)(1 - a + b)
C
(1 - a + b)(1 - a + b)
D
(1 + a + b)(1 + a + b)
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
1 - (a - b)2 = [1 + (a - b)][1 - a + b]
= (1 + a - b)(1 - a + b)
Source: Myschool Past Questions Archive (Record #31405)