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Mathematics
UTME / JAMB 1991
Number and Numeration
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
Number and Numeration
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
Number and Numeration
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
Number and Numeration
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
Number and Numeration
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
Number and Numeration
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
Number and Numeration
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
Number and Numeration
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
Number and Numeration
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
Number and Numeration
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
Number and Numeration
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)
Mathematics
UTME / JAMB 1991
Number and Numeration
Solution
Which of the following is a factor of rs + tr - pt - ps?
A
(p - s)
B
(s - p)
C
r - p)
D
r + p)
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
rs + tr - pt - ps = rs - ps - tr - pt
= (r - p)s + (r - p)t
= (r - p)(s + t),
hence r - p is a factor
Source: Myschool Past Questions Archive (Record #31406)
Mathematics
UTME / JAMB 1991
Number and Numeration
Solution
Find the positive number n, such that thrice its square is equal to twelve times the number
A
1
B
2
C
3
D
4
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
3n2 = 12n
= 3n2 - 12n = 0
= 3n(n - 4) = 0
∴ n = 4
Source: Myschool Past Questions Archive (Record #31409)
Mathematics
UTME / JAMB 1991
Number and Numeration
Solution
Simplify \(\frac{\sqrt{1 + x} + \sqrt{x}}{\sqrt{1 + x} - \sqrt{x}}\)
A
-2x - 2\(\sqrt{x (1 + x)}\)
B
1 + 2x + 2\(\sqrt{x (1 + x)}\)
C
\(\sqrt{x (1 + x)}\)
D
1 + 2x - 2\(\sqrt{x (1 + x)}\)
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
\(\frac{\sqrt{1 + x} + \sqrt{x}}{\sqrt{1 + x} - \sqrt{x}}\)
= \((\frac{\sqrt{1 + x} + \sqrt{x}}{\sqrt{1 + x} - \sqrt{x}}) (\frac{\sqrt{1 + x} + \sqrt{x}}{\sqrt{1 + x} + \sqrt{x}})\)
= \(\frac{(1 + x) + \sqrt{x(1 + x)} + \sqrt{x(1 + x)} + x}{(1 + x) - x}\)
= \(\frac{1 + 2x + 2\sqrt{x(1 + x)}}{1}\)
= \(1 + 2x + 2\sqrt{x(1 + x)}\)
Source: Myschool Past Questions Archive (Record #31412)
Mathematics
UTME / JAMB 1991
Number and Numeration
Solution
Evaluate x2(x2 - 1)-\(\frac{1}{2}\) - (x2 - 1)\(\frac{1}{2}\)
A
(x2 - 1)-\(\frac{1}{2}\)
B
(x2 - 1)1
C
(x2 - 1)
D
(x2 - 1)-1
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
x2(x2 - 1)-\(\frac{1}{2}\) - (x2 - 1)\(\frac{1}{2}\) = \(\frac{x^2}{(x^2 - 1)^\frac{1}{2}}\) - \(\frac{(x^2 - 1)^\frac{1}{2}}{1}\)
= \(\frac{x^2 - (x^2 - 1)}{(x^2 - 1) ^\frac{1}{2}}\)
= \(\frac{x^2 - x^2 + 1}{(x^2 - 1)^\frac{1}{2}}\)
= (x2 - 1)-\(\frac{1}{2}\)
Source: Myschool Past Questions Archive (Record #31413)