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Mathematics
UTME / JAMB 1986
Indices, Logarithms & Surds
Solution
Divide the L.C.M of 48, 64, and 80 by their H.C.F
A
20
B
30
C
48
D
60
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
48 = 24 x 3, 64 = 26, 80 = 24 x 5
L.C.M = 26 x 3 x 5
H.C.F = 24
\(\frac{2^6 \times 3 \times 5}{2^4}\)
= 22 x 3 x 5
= 4 x 3 x 5
= 12 x 5
= 60
Source: Myschool Past Questions Archive (Record #32004)
Mathematics
UTME / JAMB 1986
Indices, Logarithms & Surds
Solution
Simplify \(\frac{0.0324 \times 0.00064}{0.48 \times 0.012}\)
A
3.6 x 102
B
3.6 x 10-2
C
3.6 x 10-3
D
3.6 x 10-4
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
\(\frac{0.0324 \times 0.00064}{0.48 \times 0.012}\) = \(\frac{324 \times 10^{-4} \times 10^{-5}}{48 x 10^{-2} x 12 x 10^{-3}}\)
\(\frac{324 \times 64 \times 1 0^{-9}}{48 \times 12 \times 10^{-5}}\) = 36 x 10-4
= 3.6 x 10-3
Source: Myschool Past Questions Archive (Record #32019)
Mathematics
UTME / JAMB 1986
Indices, Logarithms & Surds
Solution
Simplify \((\frac{1}{\sqrt{5} + \sqrt{3}} - \frac{1}{\sqrt{5} - \sqrt{3}}) \times \frac{1}{\sqrt{3}}\)
A
\(\frac{\sqrt{3}}{\sqrt{5}}\)
B
\(\frac{2 \sqrt{3}}{7}\)
C
-2
D
-1
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
\((\frac{1}{\sqrt{5} + \sqrt{3}} - \frac{1}{\sqrt{5} - \sqrt{3}}) \times \frac{1}{\sqrt{3}}\)
\(\frac{1}{\sqrt{5} + \sqrt{3}} - \frac{1}{\sqrt{5} - \sqrt{3}}\)
\(\frac{(\sqrt{5} - \sqrt{3}) - (\sqrt{5} + \sqrt{3})}{(\sqrt{5} + \sqrt{3})(\sqrt{5} - \sqrt{3})}\)
= \(\frac{\sqrt{5} - \sqrt{3} - \sqrt{5} - \sqrt{3}}{5 - \sqrt{15} + \sqrt{15} - 3}\)
= \(\frac{-2\sqrt{3}}{2}\)
= \(- \sqrt{3}\)
\(\therefore (\frac{1}{\sqrt{5} + \sqrt{3}} - \frac{1}{\sqrt{5} - \sqrt{3}}) \times \frac{1}{\sqrt{3}} = - \sqrt{3} \times \frac{1}{\sqrt{3}}\)
= \(-1\)
Source: Myschool Past Questions Archive (Record #32036)
Mathematics
UTME / JAMB 1986
Indices, Logarithms & Surds
Solution
Find n if lo\(g_2\) 4 + lo\(g_2\) 7 - lo\(g_2\) n = 1
A
10
B
14
C
27
D
28
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
log\(_2\) 4 + log\(_2\) 7 - log\(_2\) n = 1
= log\(_2\) (4 x 7) - log\(_2\) n = 1
\(\therefore\) log\(_2\) 28 - log\(_2\) n = 1
= \(\frac{28}{n} = 2^1\)
\(\frac{28}{n}\) = 2
2n = 28
∴ n = 14
Source: Myschool Past Questions Archive (Record #32041)
Mathematics
UTME / JAMB 1986
Indices, Logarithms & Surds
Solution
Simplify \(\frac{9^{\frac{1}{3}} \times 27^{-\frac{1}{2}}}{3^{-\frac{1}{6}} \times 3^{\frac{-2}{3}}}\)
A
\(\frac{1}{3}\)
B
1
C
3
D
9
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
\(\frac{9^{\frac{1}{3}} \times 27^{-\frac{1}{2}}}{3^{-\frac{1}{6}} \times 3^{-\frac{2}{3}}}\) = \(\frac{(3^2)^{\frac{1}{3}} \times (3^3)^{-\frac{1}{3}}}{3^{-\frac{1}{6}} \times 3^{-\frac{2}{3}}}\)
= \(\frac{3^{\frac{2}{3}} \times 3^{-\frac{3}{2}}}{3^{-\frac{1}{6}} \times 3^{-\frac{2}{3}}}\)
= \(\frac{3^{-\frac{5}{6}}}{3^{-\frac{5}{6}}}\)
= 1
Source: Myschool Past Questions Archive (Record #32057)
Mathematics
UTME / JAMB 1986
Indices, Logarithms & Surds
Solution
Factorize completely 8a + 125ax\(^3\)
A
(2a + 5x2)(4 + 26ax)
B
a(2 + 5x)(4 - 10x + 25x2)
C
(2a + 5x)(4 - 10ax + 25x2)
D
a(2 + 5x)(4 + 10ax + 25x2)
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
\(8a + 125ax^{3} = (2^{3})a + 5^{3} ax^{3}\)
= \(a(2^3 + 5^3 x^3)\)
∴\(a[2^3 + (5x)^3]\)
\(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)
∴ \(a(2^3 + (5x)^3)\)
= \(a(2 + 5x)(4 - 10x + 25x^2)\)
Source: Myschool Past Questions Archive (Record #32076)
Mathematics
UTME / JAMB 1986
Indices, Logarithms & Surds
Solution
Solve the equation 3x\(^2\) + 6x - 2 = 0
A
x = -1 \(\pm\) \(\frac{\sqrt{3}}{3}\)
B
x = -1 \(\pm\) \(\frac{\sqrt{15}}{3}\)
C
x = -2 \(\pm\) 2
D
x = 3 \(\pm\) \(\frac{\sqrt{3}}{15}\)
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
3x2 + 6x - 2 = 0
Using almighty formula i.e. x = \(\frac{b \pm \sqrt{b^2 - 4ac}}{2a}\)
a = 3, b = 6, c = -2
x = \(\frac{-6 \pm \sqrt{6^2 - 4(3)(-2)}}{2(3)}\)
x = \(\frac{-6 \pm \sqrt{36 + 24}}{6}\)
x = \(\frac{-6 \pm \sqrt{60}}{6}\)
x = \(\frac{-6 \pm \sqrt{4 \times 15}}{6}\)
x = \(-1 \pm \frac{\sqrt{15}}{3}\)
Source: Myschool Past Questions Archive (Record #32086)
Mathematics
UTME / JAMB 1986
Indices, Logarithms & Surds
Solution
If \(5^{(x + 2y)} = 5\) and \(4^{(x + 3y)} = 16\), find \(3^{(x + y)}\).
A
7
B
1
C
3
D
27
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
\(5^{(x + 2y)} = 5\)
∴ x + 2y = 1.....(i)
\(4^{(x + 3y)} = 16 = 4^2\)
x + 3y = 2 .....(ii)
x + 2y = 1.....(i)
x + 3y = 2......(ii)
y = 1
Substitute y = 1 into equation (i)
\(x + 2y = 1 \implies x + 2(1) = 1\)
\(x + 2 = 1 \implies x = -1\)
\(\therefore 3^{(x + y)} = 3^{(-1 + 1)}\)
\(3^{0} = 1\)
Source: Myschool Past Questions Archive (Record #32111)
Mathematics
UTME / JAMB 1986
Indices, Logarithms & Surds
Solution
A girl walks 45 meters in the direction 050o from a point Q to a point X. She then walks 24 meters in the direction 140o from X to a point Y. How far is she then from Q?
A
69m
B
57m
C
51m
D
21m
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
QY = 452 + 242 = 2025 + 576
= 2601
QY = \(\sqrt{2601}\)
= 51
Source: Myschool Past Questions Archive (Record #32159)
Mathematics
UTME / JAMB 1986
Indices, Logarithms & Surds
Solution
Find the total surface area of solid cone of radius 2\(\sqrt{3}\)cm and slanting side 4\(\sqrt{3}\)
A
8\(\sqrt{3}\pi \)cm2
B
24\(\pi \)cm2
C
15\(\sqrt{3}\pi \)cm2
D
36\(\pi \)cm2
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
Total surface area of a solid cone
r = 2\(\sqrt{3}\)
= \(\pi r^2\) + \(\pi\)rH
H = 4\(\sqrt{3}\), \(\pi\)r(r + H)
∴ Area = \(\pi\)2\(\sqrt{3}\) [2\(\sqrt{3}\) + 4\(\sqrt{3}\)]
= \(\pi\)2\(\sqrt{3}\)(6\(\sqrt{3}\))
= 12\(\pi\) x 3
= 36\(\pi \)cm2
Source: Myschool Past Questions Archive (Record #32165)
Mathematics
UTME / JAMB 1985
Indices, Logarithms & Surds
Solution
Find x if log\(_9\)x = 1.5
A
72.0
B
27.0
C
36.0
D
3.5
E
24.5
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
If log\(_9\)x = 1.5,
9\(^1.5\) = x
9^\(\frac{3}{2}\) = x
(\sqrt9)\(^3\) = 3
∴ x = 27
Source: WAEC / JAMB / NECO Classroom Archive (Record #32213)
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
Indices, Logarithms & Surds
Solution
In \(\bigtriangleup\) XYZ, XY = 13cm, YZ = 9cm, XZ = 11cm and XYZ = \(\theta\). Find cos\(\theta\)o
A
\(\frac{4}{39}\)
B
\(\frac{43}{39}\)
C
\(\frac{209}{3}\)
D
\(\frac{43}{78}\)
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
cos\(\theta\) = \(\frac{13^2 + 9^2 - 11^2}{2(13)(9)}\)
= \(\frac{169 + 81 - 21}{26 \times 9}\)
cos\(\theta\) = \(\frac{129}{26 \times 9}\)
= \(\frac{43}{78}\)
Source: Myschool Past Questions Archive (Record #32244)
Mathematics
UTME / JAMB 1985
Indices, Logarithms & Surds
Solution
If the hypotenuse of right angled isosceles triangle is 2, what is the length of each of the other sides?
A
\(\sqrt{2}\)
B
\(\frac{1}{2}\)
C
22
D
1
E
2 - 1
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
45o = \(\frac{x}{2}\), Since 45o = \(\frac{1}{\sqrt{2}}\)
x = 2 x \(\frac{1}{\sqrt{2}}\)
= \(\frac{2\sqrt{2}}{2}\)
= \(\sqrt{2}\)
Source: Myschool Past Questions Archive (Record #32254)