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Mathematics
UTME / JAMB 1981
Number and Numeration
Solution
simplify \(\frac{6^{2n + 1} \times 9^n \times 4^{2n}}{18^n \times 2^n \times 12^{2n}}\)
A
32n
B
3 x 23n - 1
C
2n
D
6
E
1
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
\(\frac{6^{2n + 1} \times 9^n \times 4^{2n}}{18^n \times 2^n \times 12^{2n}}\) = \(\frac{(2 \times 3^{2n + 1} \times 3^{2n}) \times 2^{4n}}{(2 \times 9)^n \times 2^n \times (6 \times 2^{2n})}\)
= \(\frac{(2^{2n + 1} \times 3^{2n + 1}) \times 3^{2n} \times 2^{4n}}{2^n \times 3^{2n} \times 2^n \times 2^{4n} \times 3^{2n}}\)
= \(\frac{2^{2n} + 1 + 4^n \times 3^{2n} + 1 + 2^n}{2^{n + n + 4n} \times 3^{2n + 4n} \times 3^{2n + 2n}}\)
= \(\frac{2^{6n + 1} \times 3^{4n + 1}}{2^{6n} x 3^{4n}}\)
= 26n + 1 - 6n x 34n + 1 - 4n
2 x 3 = 6
Source: Myschool Past Questions Archive (Record #32588)
Mathematics
UTME / JAMB 1981
Number and Numeration
Solution
The number 25 when converted from the tens and units base to the binary base (base two) is one of the following
A
10011
B
111011
C
111000
D
11001
E
110011
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
\(\begin{array}{c|c} 2 & 25\\2 & 12 R 1\\2 & 6 R 0\\2 & 3 R 0\\2 & 1 R 1\\2&0 R 1\end{array}\)
= (11001)
Source: Myschool Past Questions Archive (Record #32589)
Mathematics
UTME / JAMB 1981
Number and Numeration
Solution
Evaluate \(\frac{6.3 \times 10^5}{8.1 \times 10^3}\) to 3 significant fiqures
A
77.80
B
778.0
C
7.870
D
8.770
E
88.70
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
\(\frac{6.3 \times 10^5}{8.1 \times 10^3}\)
\(\frac{7}{9} \times 10^{2}\)
= \(0.77778 \times 10^{2}\)
= \(77.778 \approxeq 77.80\)
Source: Myschool Past Questions Archive (Record #32590)
Mathematics
UTME / JAMB 1981
Number and Numeration
Solution
The area of a circular plate is one-sixteenth the surface area of a ball of a ball, If the area of the plate is given as P cm2, then the radius of the ball is
A
\(\frac{2P}{\pi}\)
B
\(\frac{P}{\sqrt{\pi}}\)
C
\(\frac{P}{\sqrt{2\pi}}\)
D
2\(\frac{P}{\pi}\)
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
Surface area of a sphere = 4\(\pi\)r2
\(\frac{1}{16}\) of 4\(\pi\)r2
= \(\frac{\pi r^2}{4}\)
P = \(\frac{\pi r^2}{4}\)
r = 2\(\frac{P}{\pi}\)
Source: Myschool Past Questions Archive (Record #32593)
Mathematics
UTME / JAMB 1981
Number and Numeration
Solution
Simplify T = \(\frac{4R_2}{R_1^{-1} + R_2^{-1} + 4R_3^{-1}}\)
A
\(\frac{4R_1 \times R_2 R_3}{R_2R_3 + R_1R_3 + 4R_1 R_2}\)
B
\(\frac{R_1 R_2 R_3}{R_2R_3 + R_1R_2 + 4R_1 R_2}\)
C
\(\frac{16R_1 R_2 R_3}{R_2R_3 + R_1R_2 + R_1 R_2}\)
D
\(\frac{4R_1 R_2 R_3}{4R_2R_3 + R_1R_2 + 4R_1 R_2}\)
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
T = \(\frac{4R_2}{R_1^{-1} + R_2^{-1} + 4R_3^{-1}}\) = \(\frac{4R_2}{\frac{1}{R_1} + \frac{1}{R_2} + \frac{4}{R_3}}\)
= \(\frac{4R_2}{\frac{R_2R_3 + R_1R_3 + 4R_1R_2}{R_1R_2R_3}}\)
= \(\frac{4R_2 \times R_1 R_2 R_3}{R_2R_3 + R_1R_3 + 4R_1 R_2}\)
= \(\frac{4R_1 \times R_2 R_3}{R_2R_3 + R_1R_3 + 4R_1R_2}\)
T = \(\frac{4R_1 \times R_2 R_3}{R_2R_3 + R_1R_3 + 4R_1 R_2}\)
Source: Myschool Past Questions Archive (Record #32596)
Mathematics
UTME / JAMB 1981
Number and Numeration
Solution
Multiply x2 + x + 1 by x2 - x + 1
A
x4 - x + x2
B
x4 - x2 + x2
C
x4 + x2 + 1
D
x4 + x2
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
(x2 + x + 1)( x2 - x + 1)
= x2(x2 + x + 1) - x(x2 + x + 1) + (x3 + x + 1)
= x4 - x3 + x2 + x3 - x2 - x + 1
= x4 + x2 + 1
Source: Myschool Past Questions Archive (Record #32598)
Mathematics
UTME / JAMB 1981
Number and Numeration
Solution
The difference between 4\(\frac{5}{7}\) and 2\(\frac{1}{4}\) is greater than sum of \(\frac{1}{14}\) and 1\(\frac{1}{2}\) by
A
\(\frac{23}{28}\)
B
\(\frac{24}{28}\)
C
\(\frac{50}{56}\)
D
\(\frac{27}{28}\)
E
\(\frac{48}{56}\)
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
Difference between 4\(\frac{5}{7}\) and 2\(\frac{1}{4}\)
\(\frac{33}{7}\) - \(\frac{9}{4}\) = \(\frac{69}{24}\)
The sum of \(\frac{1}{14}\) and 1\(\frac{1}{14}\) + \(\frac{3}{2}\)
= \(\frac{11}{7}\)
\(\frac{69}{28}\) - \(\frac{11}{7}\) = \(\frac{25}{28}\)
= \(\frac{50}{56}\)
Source: Myschool Past Questions Archive (Record #32600)
Mathematics
UTME / JAMB 1981
Number and Numeration
Diagram
Solution
In the figure, the chords XY and ZW are produced to meet at T such that YT = WT, ZYW = 40o and YTW = 30o. What is YXW?
Attached Figure / Diagram
A
50o
B
75o
C
35o
D
115o
E
30o
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
< WYT = \(\frac{180 - 30}{2}\)
= \(\frac{150}{2}\)
= 75o(base angles of isosceles D)
YXW = 75 - 40 = 35o(Exterior angle is equal to sum of interior angles)
Source: Myschool Past Questions Archive (Record #32753)
Mathematics
UTME / JAMB 1980
Number and Numeration
Solution
Three numbers, x, y and z are connected by the relationships y = \(\frac{4x}{9}\) + 1. If x = 99, Find z = \(\frac{4y}{9}\) + 1
A
6\(\frac{1}{3}\)
B
20
C
21
D
176\(\frac{4}{9}\)
E
None of the above
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
y = \(\frac{4x}{9}\) + 1, z = \(\frac{4y}{9}\) + 1, z = \(\frac{4}{9}\)(\(\frac{4}{9}\) x + 1) + 1
= [\(\frac{4}{9}\)(\(\frac{4}{9}\)(99) + 1]
= \(\frac{4}{9}\)(45) + 1
= 21
Source: WAEC / JAMB / NECO Classroom Archive (Record #32622)
Mathematics
UTME / JAMB 1980
Number and Numeration
Solution
(\(\frac{x^a}{x^b}\))a + b = (\(\frac{x^{a + b}}{x^{a - b}}\))\(\frac{a^2}{b}\)
A
x-a2
B
xb2
C
xa2 - b2
D
\(\frac{1}{x^{a2 + b2}}\)
E
xb2 - a2
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
(\(\frac{x^a}{x^b}\))a + b = (\(\frac{x^{a2 + ab}}{x^{b2 + ab}}\))
= xa2 - b2
{\(\frac{xa + b}{xa - b}\)} = xa + b - a + b
= x2b
= x2a
= xa2 - b2
= xb2 \(\div \) a2 = \(\frac{1}{x^\text a2 + b2}\)
Source: WAEC / JAMB / NECO Classroom Archive (Record #32628)
Mathematics
UTME / JAMB 1980
Number and Numeration
Solution
Find, correct to three significant figures, the value of \(\sqrt{41830}\).
A
205
B
647
C
2050
D
6470
E
64.7
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
\(\sqrt{41830} = 204.5238\)
\(\approxeq 205\) (to three significant figures)
Source: Myschool Past Questions Archive (Record #32605)
Mathematics
UTME / JAMB 1980
Number and Numeration
Solution
Write down the number 0.0052048 correct to three significant figures
A
0.005
B
0.0052
C
0.00520
D
5.2048
E
5204
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
0.0052048 = 0.00520
Source: Myschool Past Questions Archive (Record #32606)
Mathematics
UTME / JAMB 1980
Number and Numeration
Solution
Evaluate \((2^{0} + 4^{-\frac{1}{2}})^{2}\)
A
\(\frac{1}{4}\)
B
\(\frac{5}{4}\)
C
\(\frac{9}{4}\)
D
4
E
9
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
\((2^{0} + 4^{-\frac{1}{2}})^{2}\)
= \((1 + (\frac{1}{4})^{\frac{1}{2}})^{2}\)
= \((1 + \frac{1}{2})^{2}\)
= \((\frac{3}{2})^{2} = \frac{9}{4}\)
Source: Myschool Past Questions Archive (Record #32607)
Mathematics
UTME / JAMB 1980
Number and Numeration
Solution
Which of the following is NOT a factor of 12\(^{4}\) - 5\(^{4}\)?
A
7
B
13
C
17
D
49
E
169
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
124 - 54 = (122)2 - (52)2
= (122 + 52) x (122 - 52)
(144 + 25)(144 - 25)
= (169)(119)
Factors of above 169, 17, 13, 7
Source: Myschool Past Questions Archive (Record #32608)
Mathematics
UTME / JAMB 1980
Number and Numeration
Solution
Rationalize the denominator of the given expression \(\frac{\sqrt{1 + a} - \sqrt{a}}{\sqrt{1 + a} + \sqrt{a}}\)
A
1 + 2a - 2\(\sqrt{a(1 + a)}\)
B
\(\sqrt{1(1 + a)}\)
C
2a - 2\(\sqrt{a(1 + a)}\)
D
1 + 2a - 2\(\sqrt{a + b}\)
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
\(\frac{\sqrt{1 + a} - \sqrt{a}}{\sqrt{1 + a} + \sqrt{a}}\) = \(\frac{\sqrt{1 + a} - \sqrt{a}}{\sqrt{1 + a} + \sqrt{a}}\) x \(\frac{\sqrt{1 + a} - \sqrt{a}}{\sqrt{1 + a} - \sqrt{a}}\)
= \(\frac{\sqrt{1 + a + a}}{1 + a - a}\)
= 2a + a(1 + a)
= 1 + 2a - 2\(\sqrt{a(1 + a)}\)
Source: Myschool Past Questions Archive (Record #32609)