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Mathematics
UTME / JAMB 1981
Indices, Logarithms & Surds
Solution
Solve the given equation \((\log_{3} x)^{2} - 6(\log_{3} x) + 9 = 0\)
A
27
B
9
C
\(\frac{1}{27}\)
D
18
E
81
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
\((\log_{3} x)^{2} - 6(\log_{3} x) + 9 = 0\)
Let \(\log_{3} x = a\).
\(a^{2} - 6a + 9 = 0\)
\(a^{2} - 3a - 3a + 9 = 0\)
\(a(a - 3) - 3(a - 3) = 0\)
\((a - 3)(a - 3) = 0\)
\(\implies a = 3 (twice)\)
\(\log_{3} x = 3 \implies x = 3^{3} = 27\)
Source: Myschool Past Questions Archive (Record #32603)
Mathematics
UTME / JAMB 1981
Mensuration (Areas, Perimeters & Volumes)
Solution
What is the length of an arc of a circle that subtends 2\(\frac{1}{2}\) radians at the centre when the radius of the circle is 8cm?
A
8cm
B
10cm
C
40cm
D
20cm
E
16cm
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
Given: \(\theta\) = 2\(\frac{1}{2}\) radians, r = 8cm
but 360º = 2 \(\pi\) radian.
Using: L = length of an arc = \(\frac{\theta}{2 \pi}\) x 2\(\pi\) r
= \(\frac{2.5}{2\pi}\) x 2 x \(\pi\) x 8
= 2.5 x 8 = 20cm.
Source: Myschool Past Questions Archive (Record #32604)
Mathematics
UTME / JAMB 1981
Plane Geometry, Polygons & Circle Theorems
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 1981
Indices, Logarithms & Surds
Diagram
Solution
In the figure, find the area of XYZW
Attached Figure / Diagram
A
60cm2
B
54cm2
C
27cm2
D
54\(\sqrt{2}\)cm2
E
27\(\sqrt{2}\)cm2
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
Area of trapezium XYZW = \(\frac{1}{2}\)(a + b)h
h = 6 sin 30º = 3
area = \(\frac{1}{2}\)(10 + 8) x 3
\(\frac{1}{2}\)(18) x 4
= 27cm\(^2\)
Source: Myschool Past Questions Archive (Record #32754)
Mathematics
UTME / JAMB 1981
Plane Geometry, Polygons & Circle Theorems
Diagram
Solution
In the figure, find x in terms of a, b and c.
Attached Figure / Diagram
A
a + b + c
B
180o - (a + b + c)
C
a - b - c
D
a + b
E
a + c
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
180 - x + a + b + c
= 180(sum of interior angle in triangle)
a + b + c = x.
Source: Myschool Past Questions Archive (Record #32755)
Mathematics
UTME / JAMB 1981
Mensuration (Areas, Perimeters & Volumes)
Diagram
Solution
In the diagram, XZ is the diameter of a circle's radius \(\frac{5}{2}\). If XY is 4cm, then the area of the triangle XYZ is
Attached Figure / Diagram
A
12cm2
B
28\(\frac{1}{2}\)cm2
C
16cm2
D
10cm2
E
6cm2
medium Difficulty
Verified Correct Answer: Option E
Step-by-Step Mathematical Solution
Area of the triangle XYZ = \(\frac{1}{2}\)bh = \(\frac{1}{2}\)ZY x XY
= \(\frac{1}{2}\) x 3 x 4
= 6cm2.
Source: Myschool Past Questions Archive (Record #32756)
Mathematics
UTME / JAMB 1981
Plane Geometry, Polygons & Circle Theorems
Diagram
Solution
In the figure, WU//YZ, WY//VU = 12cm, VZ = 6cm, XU = 8cm. Determine the length of WU.
Attached Figure / Diagram
A
1cm
B
6cm
C
2cm
D
4cm
E
3cm
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
From similar triangle, \(\frac{x}{6}\) = \(\frac{8}{12}\)
12x = 48
x = \(\frac{48}{12}\)
= 4
Source: Myschool Past Questions Archive (Record #32757)
Mathematics
UTME / JAMB 1980
Fractions, Decimals, Percentages & Approximations
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
Fractions, Decimals, Percentages & Approximations
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
Fractions, Decimals, Percentages & Approximations
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
Fractions, Decimals, Percentages & Approximations
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
Indices, Logarithms & Surds
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
Indices, Logarithms & Surds
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
Indices, Logarithms & Surds
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)
Mathematics
UTME / JAMB 1980
Fractions, Decimals, Percentages & Approximations
Solution
When a dealer sells a bicycle for N81 he makes a profit of 8%. What did he pay for the bicycle?
A
N73
B
N74.52
C
N75
D
N87.48
E
N75.52
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
Profit = 8%
\(\therefore S.P = N81 = (100 + 8)% = 108%\)
\(108% = N81\)
\(100% = \frac{81}{108} \times 100\)
= \(\frac{8100}{108} = N75\)
Source: Myschool Past Questions Archive (Record #32610)