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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
Indices, Logarithms & Surds
Solution
One angle of a rhombus is 60o. The shorter of the two diagonals is 8cm long. Find the length of the longer one.
A
8\(\sqrt{3}\)
B
\(\frac{16}{\sqrt{3}}\)
C
\(\sqrt{3}\)
D
\(\frac{10}{\sqrt{3}}\)
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
\(\frac{x}{4}\) = \(\frac{\tan 60}{1}\)
x = 4 tan60
= 4\(\sqrt{3}\)
BD = 2x
= 8\(\sqrt{3}\)
Source: Myschool Past Questions Archive (Record #31418)
Mathematics
UTME / JAMB 1991
Indices, Logarithms & Surds
Diagram
Solution
PMN and PQR are two secants of the circle MQTRN and PT is a tangent. If PM = 5cm, PN = 12cm and PQ = 4.8cm, calculate the respective lengths of PR and PT in centimeters
Attached Figure / Diagram
A
7.3, 5.9
B
7.7, 12.5
C
12.5, 7.7
D
5.9, 7.3
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
\(\frac{PQ}{PN} = \frac{PM}{PR} = \frac{QM}{NR}\)
\(\frac{4.8}{12} = \frac{5}{PR}\)
PR = \(\frac{5 \times 12}{4.8} = \frac{50}{4}\)
= 12.5
\(\frac{PQ}{PN} = \frac{PM}{PT} = \frac{TM}{NT}\)
\(\frac{PT}{12} = \frac{5}{PR}\)
PT2 = 60
PT = \(\sqrt{60}\)
= 7.746
= 7.7
Source: Myschool Past Questions Archive (Record #32972)
Mathematics
UTME / JAMB 1991
Indices, Logarithms & Surds
Diagram
Solution
In the figure, PQRS is a square of sides 8cm. What is the area of \(\bigtriangleup\)UVW?
Attached Figure / Diagram
A
64 sq. cm
B
40 sq.cm
C
50 sq.cm
D
10 sq.cm
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
In △ UQV
|UV|\(^2\) = 6\(^2\) + 4\(^2\) (Pythagora's theorem)
|UV| = \(\sqrt{52}\) = 7.21cm
Also, in △ WVR
|WV|\(^2\) = 2\(^2\) + 2\(^2\) = 4 + 4 = 8 (same reason as above)
|WV| = \(\sqrt{8}\) = 2.83
Therefore, the area of △ UVW = \(\frac{1}{2}\) x 2.83 x 7.21 = 10.21 ≈ 10cm\(^2\)
Source: Myschool Past Questions Archive (Record #32981)
Mathematics
WAEC / WASSCE 1991
Indices, Logarithms & Surds
Solution
(a) Using logarithm table, evaluate \(\frac{\sqrt[3]{1.376}}{\sqrt[4]{0.007}}\) correct to three significant figure.
(b) Without using Mathematical tables, find the value of \(\frac{\log 81}{\log \frac{1}{3}}\).
medium Difficulty
Verified Correct Answer: Option
Step-by-Step Mathematical Solution
(a)
No
Log
\(\sqrt[3]{1.376}\)
\(0.1386 \div 3 \) = \(0.0462\)
\(\sqrt[4]{0.007}\)
\(\bar{3}.8451 \div 4\) = \(\bar{1}.4613\)
\(\frac{\sqrt[3]{1.376}}{\sqrt[4]{0.007}}\) =
\(0.5849\)
Antilog = 3.845
\(\therefore \frac{\sqrt[3]{1.376}}{\sqrt[4]{0.007}} \approxeq 3.85\) (to 3 s.f)
(b) \(\frac{\log 81}{\log \frac{1}{3}}\)
= \(\frac{\log 3^{4}}{\log 3^{-1}}\)
= \(\frac{4 \log 3}{-1 \log 3}\)
= \(-4\)
Source: Myschool Past Questions Archive (Record #54607)
Mathematics
UTME / JAMB 1990
Indices, Logarithms & Surds
Solution
Correct 241.34(3 x 10-\(^3\))\(^2\) to 4 significant figures
A
0.0014
B
0.001448
C
0.0022
D
0.002172
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
first work out the expression and then correct the answer to 4 s.f = 241.34..............(A)
(3 x 10-\(^3\))\(^2\)............(B)
= 3\(^2\)x\(^2\)
= \(\frac{1}{10^3}\) x \(\frac{1}{10^3}\)
(Note that x\(^2\) = \(\frac{1}{x^3}\))
= 24.34 x 3\(^2\) x \(\frac{1}{10^6}\)
= \(\frac{2172.06}{10^6}\)
= 0.00217206
= 0.002172(4 s.f)
Source: WAEC / JAMB / NECO Classroom Archive (Record #31431)
Mathematics
UTME / JAMB 1990
Indices, Logarithms & Surds
Solution
Simplify \(\sqrt{27}\) + \(\frac{3}{\sqrt{3}}\)
A
4\(\sqrt{3}\)
B
\(\frac{4}{\sqrt{3}}\)
C
3\(\sqrt{3}\)
D
\(\frac{\sqrt{3}}{4}\)
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
\(\sqrt{27}\) + \(\frac{3}{\sqrt{3}}\)
= \(\sqrt{9 \times 3}\) + \(\frac{3 \times {\sqrt{3}}}{{\sqrt{3}} \times {\sqrt{3}}}\)
= 3\(\sqrt{3}\) + \(\sqrt{3}\)
= 4\(\sqrt{3}\)
Source: WAEC / JAMB / NECO Classroom Archive (Record #31437)
Mathematics
WAEC / WASSCE 1990
Indices, Logarithms & Surds
Solution
Simplify 125\(^{\frac{-1}{3}}\) x 49\(^{\frac{-1}{2}}\) x 10\(^0\)
A
350
B
35
C
1/35
D
1/350
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
125\(^{\frac{-1}{3}}\) x 49\(^{\frac{-1}{2}}\) x 10\(^0\)
= 5\(^{-1}\) x 7\(^{-1}\) x 1
= \(\frac{1}{35}\)
Source: Myschool Past Questions Archive (Record #22599)
Mathematics
WAEC / WASSCE 1990
Indices, Logarithms & Surds
Solution
If 3\(^{2x}\) = 27, what is x?
A
1
B
1.5
C
4.5
D
18
E
40.5
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
3\(^{2x}\) = 27
3\(^{2x}\) = 3\(^3\)
2x = 3
x = 1.5
Source: Myschool Past Questions Archive (Record #22600)
Mathematics
WAEC / WASSCE 1990
Indices, Logarithms & Surds
Solution
Simplify \(\frac{\log \sqrt{8}}{\log 8}\)
A
1/3
B
1/2
C
1/3log√2
D
1/3log√8
E
1/2log√2
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
\(\frac{\log \sqrt{8}}{\log 8}\)
= \(\frac{\log 8^{\frac{1}{2}}}{log 8}\)
= \(\frac{\frac{1}{2} \log 8}{\log 8}\)
= \(\frac{1}{2}\)
Source: Myschool Past Questions Archive (Record #22603)
Mathematics
WAEC / WASSCE 1990
Indices, Logarithms & Surds
Solution
Evaluate using the logarithm table, log(0.65)2
A
1.6258
B
0.6272
C
0.6258
D
\(\overline{1}\).6258
E
1.6272
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
log(0.65)\(^2\) = 2log(0.65) but log0.65 = \(\overline{1}\).8129
∴2 x \(\overline{1}\).8129 = \(\overline{1}\).6258
Source: Myschool Past Questions Archive (Record #22604)
Mathematics
WAEC / WASSCE 1990
Indices, Logarithms & Surds
Solution
If log x = \(\bar{2}.3675\) and log y = 0.9750, what is the value of x + y? Correct to three significant figures
A
1.18
B
1.31
C
9.03
D
9.44
E
9.46
medium Difficulty
Verified Correct Answer: Option E
Step-by-Step Mathematical Solution
log x = \(\bar{2}.3675\) ; log y = 0.9750
\(x = 10^{\bar{2}.3675} = 0.02331 \)
\(y = 10^{0.9750} = 9.441 \)
\(x + y = 9.4641 \approxeq 9.46\)
Source: Myschool Past Questions Archive (Record #22605)
Mathematics
UTME / JAMB 1990
Indices, Logarithms & Surds
Solution
Simplify 3log\(_6\)9 + log\(_6\)12 + log\(_6\)64 - log\(_6\)72
A
5
B
7776
C
log631
D
(7776)6
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
3 log\(_6\)9 + log\(_6\)12 + log\(_6\)64 - log\(_6\)72
= log\(_6\)9\(^3\) + log\(_6\)12 + log\(_6\)64 - log\(_6\)72
log\(_6\)729 + log\(_6\)12 + log\(_6\)64 - log\(_6\)72
log\(_6\)[\(\frac{(729 \times 12 \times 64)}{72}\)] = log\(_6\)7776
= log\(_6\)6\(^5\) = 5log\(_6\)6 = 5
N.B: log\(_6\)6 = 1
Source: Myschool Past Questions Archive (Record #31441)
Mathematics
UTME / JAMB 1990
Indices, Logarithms & Surds
Solution
Given that x2 + y2 + z2 = 194, calculate z if x = 7 and \(\sqrt{y}\) = 3
A
\(\sqrt{10}\)
B
8
C
12.2
D
13.4
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
Given that x2 + y2 + z2 = 194, calculate z if x = 7 and \(\sqrt{y}\) = 3
x = 7
∴ x2 = 49
\(\sqrt{y}\) = 3
∴ y2 = 81 = x2 + y2 + z2 = 194
49 + 81 + z2 = 194
130 + z2 = 194
z2 = 194 - 130
= 64
z = \(\sqrt{64}\)
= 8
Source: Myschool Past Questions Archive (Record #31470)
Mathematics
UTME / JAMB 1990
Indices, Logarithms & Surds
Find the sum of the first twenty terms of the progression log a, log a2, log a3.....
A
log a20
B
log a21
C
log a200
D
log a210
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
No expanded working has been attached for this question yet. The verified answer is Option D.
Source: Myschool Past Questions Archive (Record #31471)