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Mathematics
UTME / JAMB 1985
Mensuration (Areas, Perimeters & Volumes)
Solution
The ratio of the length of two similar rectangular blocks is 2 : 3. If the volume of the larger block is 351cm\(^3\), then the volume of the other block is?
A
234.00 cm3
B
526.50 cm3
C
166.00 cm3
D
687cm3
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
Let x represent total vol. 2 : 3 = 2 + 3 = 5
\(\frac{3}{5}\)x = 351
x = \(\frac{351 \times 5}{3}\)
= 585
Volume of smaller block = \(\frac{2}{5}\) x 585
= 234.00cm\(^3\)
Source: WAEC / JAMB / NECO Classroom Archive (Record #32260)
Mathematics
UTME / JAMB 1985
Fractions, Decimals, Percentages & Approximations
Solution
If three numbers P, Q, R are in ratio 6 : 4 : 5, find the value of \(\frac{3p - q}{4q + r}\)
A
\(\frac{3}{2}\)
B
\(\frac{2}{3}\)
C
2
D
3
E
18
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
P : Q : r = 6 : 4 : 5
5 = 6 + 4 + 5
= 15
P = \(\frac{6}{15}\), q = \(\frac{4}{15}\), r = \(\frac{5}{15}\) = \(\frac{1}{3}\)
To find \(\frac{3p - q}{4q + r}\)
3p - q = 3 x \(\frac{6}{15}\) - \(\frac{4}{15}\)
\(\frac{18}{15}\) - \(\frac{4}{15}\) = \(\frac{14}{15}\)
∴ 4q + r = 4 x \(\frac{4}{15}\) + \(\frac{5}{15}\)
\(\frac{16}{15}\) = \(\frac{16}{15}\) + \(\frac{5}{15}\)
= \(\frac{21}{15}\)
\(\frac{14}{15}\) x \(\frac{15}{21}\) = \(\frac{14}{21}\)
= \(\frac{2}{3}\)
Source: Myschool Past Questions Archive (Record #32180)
Mathematics
UTME / JAMB 1985
Fractions, Decimals, Percentages & Approximations
Solution
Arrange the following numbers in ascending order of magnitude \(\frac{6}{7}\), \(\frac{13}{15}\), 0.8650
A
\(\frac{6}{7}\) < 0.865 < \(\frac{13}{15}\)
B
\(\frac{13}{15}\) < \(\frac{6}{7}\) < 0.865
C
\(\frac{6}{7}\) < \(\frac{13}{15}\) < 0.865
D
0.865 < \(\frac{6}{7}\) < \(\frac{13}{15}\)
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
\(\frac{6}{7}\), \(\frac{13}{15}\), 0.8650
In ascending order, we have 0.8571, 0.8650, 0.8666
i.e. \(\frac{6}{7}\) < 0.8650 < \(\frac{13}{15}\)
Source: Myschool Past Questions Archive (Record #32191)
Mathematics
UTME / JAMB 1985
Fractions, Decimals, Percentages & Approximations
Solution
A sum of money was invested at 8% per annum simple interest. If after 4 years the money amounts to N330.00. Find the amount originally invested
A
N180.00
B
N165.00
C
N150.00
D
N250.00
E
N200.00
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
S.I = \(\frac{PTR}{100}\)
T = 4yrs, R = 8%, a = N330.00
330 - P = \(\frac{PTR}{100}\), A = P + I
i.e. A = P + \(\frac{PTR}{100}\)
330 = P + \(\frac{P(4) (8)}{100}\)
33000 = 32P + 100P
132P = 33000
P = N250.00
Source: Myschool Past Questions Archive (Record #32197)
Mathematics
UTME / JAMB 1985
Number Bases & Modular Arithmetic
Solution
In the equation below, Solve for x if all the numbers are in base 2: \(\frac{11}{x}\) = \(\frac{1000}{x + 101}\)
A
101
B
11
C
110
D
111
E
10
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
\(\frac{11}{x}\) = \(\frac{1000}{x + 101}\) = 11(x + 101)
1000x = 11x + 1111
1000x - 11x = 1111
101x = 1111
x = \(\frac{1111}{101}\)
x = 11
Source: Myschool Past Questions Archive (Record #32200)
Mathematics
UTME / JAMB 1985
Inequalities & Linear Programming
Solution
List all integers satisfying the inequality -2 \(\leq\) 2 x -6 < 4
A
2, 3, 4, 5
B
2, 3, 4
C
2, 5
D
3, 4, 5
E
4, 5
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
-2 \(\leq\) 2x - 6 < 4 = 2x - 6 < 4
= 2x < 10
= x < 5
2x \(\geq\) -2 + 6 \(\geq\)
= x \(\geq\) 2
∴ 2 \(\leq\) x < 5 [2, 3, 4]
Source: Myschool Past Questions Archive (Record #32203)
Mathematics
UTME / JAMB 1985
Plane Geometry, Polygons & Circle Theorems
Solution
Find correct to two decimals places 100 + \(\frac{1}{100}\) + \(\frac{3}{1000}\) + \(\frac{27}{10000}\)
A
100.02
B
1000.02
C
100.22
D
100.01
E
100.51
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
100 + \(\frac{1}{100}\) + \(\frac{3}{1000}\) + \(\frac{27}{10000}\)
\(\frac{1000,000 + 100 + 30 + 27}{10000}\) = \(\frac{1,000.157}{10000}\)
= 100.02
Source: Myschool Past Questions Archive (Record #32204)
Mathematics
UTME / JAMB 1985
Fractions, Decimals, Percentages & Approximations
Solution
John gives one-third of his money to Janet who has N105.00. He then finds that his money is reduced to one-fourth of what Janet now has. Find how much money john has at first
A
N45.00
B
N48.00
C
N52.00
D
N60.00
E
N58.00
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
Let x be John's money, Janet already had N105, \(\frac{1}{3}\) of x was given to Janet
Janet now has \(\frac{1}{3}\)x + 105 = \(\frac{x + 315}{3}\)
John's money left = \(\frac{2}{3}\)x = \(\frac{1}{4} \times\frac{ x + 315}{3}\)
cross multiplying
24x = 3x + 945
∴ x = 45
Source: Myschool Past Questions Archive (Record #32209)
Mathematics
UTME / JAMB 1985
Plane Geometry, Polygons & Circle Theorems
Solution
Write h in terms of a, b, c, d if a = \(\frac{b(1 - ch)}{1 - dh}\)
A
h = \(\frac{a - b}{ad}\)
B
h = \(\frac{1 - b}{ad - bc}\)
C
h = \(\frac{(a - b)^2}{ad - bc}\)
D
h = \(\frac{a - b}{ad - bc}\)
E
h = \(\frac{b - a}{ab - dc}\)
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
a = \(\frac{b(1 - ch)}{1 - dh}\)
a = \(\frac{b - bch}{1 - dh}\)
a - adh = b - bch
a - b = -bch + adh
a - b = adh - bch
a - b = h(ad - bc)
h = \(\frac{a - b}{ad - bc}\)
Source: Myschool Past Questions Archive (Record #32216)
Mathematics
UTME / JAMB 1985
Fractions, Decimals, Percentages & Approximations
Solution
22\(\frac{1}{2}\)% of the Nigerian Naira is equal to 17\(\frac{1}{10}\)% of a foreign currency M. What is the conversion rate of the M to the Naira?
A
1M = 1\(\frac{15}{57}\)N
B
1M = 38\(\frac{1}{4}\)N
C
1M = 1\(\frac{18}{57}\)N
D
1M = 384\(\frac{3}{4}\)N
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
N = 22\(\frac{1}{2}\)%, M = 17\(\frac{1}{10}\)%
M = \(\frac{171}{10}\)%, N = \(\frac{45}{2}\)
\(\frac{45}{2}\) x \(\frac{10}{171}\)
= \(\frac{225}{171}\)
= 1 \(\frac{54}{171}\)
= 1 \(\frac{18}{57}\)
Source: Myschool Past Questions Archive (Record #32218)
Mathematics
UTME / JAMB 1985
Linear, Simultaneous & Quadratic Equations
Solution
Find the values of p for which the equation x\(^2\) - (p - 2)x + 2p + 1 = 0
A
(21, 0)
B
(0, 12)
C
(1, 2)
D
(3, 4)
E
(4, 5)
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
Equal roots implies b\(^2\) - 4ac = 0
a = 1, b = - (p - 2), c = 2p + 1
[-(p - 2)]\(^2\) - 4 x 1 x (2p + 1) = 0
p\(^2\) - 4p + 4 - 4(2p + 1) = 0
p\(^2\) - 4p = 4 - 8p - 4 = 0
p\(^2\) - 12p = 0
p(p - 12) = 0
p = 0 or 12
Source: Myschool Past Questions Archive (Record #32219)
Mathematics
UTME / JAMB 1985
Indices, Logarithms & Surds
Solution
If \(e^{x} = 1 + x + \frac{x^{2}}{1.2} + \frac{x^{3}}{1.2.3} + ... \), find \(\frac{1}{e^{\frac{1}{2}}}\)
A
1 - \(\frac{x}{2}\) + \(\frac{x^2}{1.2^3}\) + \(\frac{x^3}{2^4.3}\) + .........
B
1 - \(\frac{x}{2}\) + \(\frac{x^2}{1.2^3}\) + \(\frac{x^4}{2.4.3}\) + ..
C
1 + \(\frac{x}{2}\) + \(\frac{x^2}{1.2}\) + \(\frac{x^3}{1.2.3}\) + \(\frac{x^4}{1.23.4}\) + .........
D
1 - x + \(\frac{x^2}{1.2^3}\) + \(\frac{x^3}{2^4.3}\) + .........
E
1 + \(\frac{x}{2}\) + \(\frac{x^2}{1.2^3}\) + \(\frac{x^4}{1.2.6}\) + .........
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
\(e^{x} = 1 + x + \frac{x^{2}}{1.2} + \frac{x^{3}}{1.2.3} + ...\)
\(\frac{1}{e^{\frac{1}{2}}} = e^{-\frac{1}{2}}\)
\(e^{-\frac{1}{2}} = 1 - \frac{x}{2} + \frac{x^{2}}{1.2^{3}} - \frac{x^{3}}{1.2^{4}.3} + ... \)
Source: Myschool Past Questions Archive (Record #32220)
Mathematics
UTME / JAMB 1985
Indices, Logarithms & Surds
Solution
\((\sqrt[4]{3} + \sqrt[4]{2})(\sqrt[4]{3} - \sqrt[4]{2})(\sqrt{3} + \sqrt{2})\) is equal to
A
1
B
(\(\sqrt{2} + 4\sqrt{2}\))
C
(6\(\sqrt{2}\)
D
8
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
\((\sqrt[4]{3} + \sqrt[4]{2})(\sqrt[4]{3} - \sqrt[4]{2})(\sqrt{3} + \sqrt{2})\)
\((\sqrt[4]{3} + \sqrt[4]{2})(\sqrt[4]{3} - \sqrt[4]{2}) = \sqrt{3} - \sqrt[4]{6} + \sqrt[4]{6} - \sqrt{2}\)
= \(\sqrt{3} - \sqrt{2}\)
\((\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{2}) = 3 + \sqrt{6} - \sqrt{6} - 2\)
= \(3 - 2 = 1\)
Source: Myschool Past Questions Archive (Record #32221)
Mathematics
UTME / JAMB 1985
Variations
Solution
In a restaurant, the cost of providing a particular type of food is partly constant and partially inversely proportional to the number of people. If cost per head for 100 people is 30k and the cost for 40 people is 60k, Find the cost for 50 people?
A
15k
B
20k
C
50k
D
40k
E
45k
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
C = a + k
\(\frac{1}{N}\) = c
= \(\frac{aN + k}{N}\)
CN = aN + K
30(100) = a(100) + k
3000 = 100a + k.......(i)
60(40) = a(40) + k
2400 = 40a + k.......(ii)
eqn (i) - eqn (ii)
600 = 60a
a = 10
subt. for a in eqn (i) 3000 = 100(10) + K
3000 - 1000 = k
k = 2000
CN = 10N + 2000. when N = 50,
50C = 10(50) + 2000
50C = 500 + 2000
C = \(\frac{2500}{50}\)
= 50k
Source: Myschool Past Questions Archive (Record #32222)
Mathematics
UTME / JAMB 1985
Plane Geometry, Polygons & Circle Theorems
Solution
The factors of 9 - (x2 - 3x - 1)2 are
A
-(x - 4)(x + 1) (x - 1)(x - 2)
B
(x - 4)(x - 2) (x - 1)(x + 1)
C
-(x - 2)(x + 1) (x - 2) (x - 1)
D
(x - 2)(x + 2) (x - 1)(x + 1)
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
9 - (x2 - 3x - 1)2 = [3 - (x2 - 3x - 1)] [3 + (x2 - 3x - 1)]
= (3 - x2 + 3x + 1)(3 + x2 - 3x - 1)
= (4 + 3x - x2)(x2 - 3x + 2)
= (4 - x)(1 + x)(x - 1)(x - 2)
= -(x - 4)(x + 1) (x - 1)(x - 2)
Source: Myschool Past Questions Archive (Record #32223)