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Mathematics
UTME / JAMB 1983
Plane Geometry, Polygons & Circle Theorems
Solution
The lengths of the sides of a right angled triangle are (3x + 1)cm, (3x - 1)cm and xcm. Find x
A
2
B
6
C
18
D
12
E
5
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
\((3x + 1)^{2} = (3x - 1)^{2} + x^{2}\) (Pythagoras's theorem)
\(9x^{2} + 6x + 1 = 9x^{2} - 6x + 1 + x^{2}\)
x2 - 12x = 0
x(x - 12) = 0
x = 0 or 12
The sides cannot be 0 hence x = 12.
Source: Myschool Past Questions Archive (Record #32457)
Mathematics
UTME / JAMB 1983
Statistics (Data Representation, Central Tendency & Dispersion)
Solution
The scores of set of final year students in the first semester examination in a paper are 41, 29, 55, 21, 47, 70, 70, 40, 43, 56, 73, 23, 50, 50. Find the median of the scores.
A
47
B
50
C
48\(\frac{1}{2}\)
D
48
E
49
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
By re-arranging 21, 23, 29, 40, 41, 43| 47, 50| 50, 55, 56, 70, 70, 73
The median = \(\frac{47 + 50}{2}\)
\(\frac{97}{2}\)
= 48\(\frac{1}{2}\)
Source: Myschool Past Questions Archive (Record #32458)
Mathematics
UTME / JAMB 1983
Number Bases & Modular Arithmetic
Solution
Find x if (x\(_4\))\(^2\) = 100100\(_2\)
A
6
B
12
C
100
D
210
E
110
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
x\(_4\) = x \(\times\) 4\(^0\) = x in base 10.
100100\(_2\) = 1 x 2\(^5\) + 1 x 2\(^2\)
= 32 + 4
= 36 in base 10
\(\implies\) x\(^2\) = 36
x = 6 in base 10.
Convert 6 to a number in base 4.
4
6
4
1 r 2
0 r 1
= 12\(_4\)
Source: Myschool Past Questions Archive (Record #32459)
Mathematics
UTME / JAMB 1983
Indices, Logarithms & Surds
Solution
Simplify log\(_{10}\) a\(\frac{1}{3}\) + \(\frac{1}{4}\)log\(_{10}\) a - \(\frac{1}{12}\)log\(_{10}\)a7
A
1
B
\(\frac{7}{6}\)log10 a
C
zero
D
10
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
log10 a\(\frac{1}{3}\) + \(\frac{1}{4}\)log10 a - \(\frac{1}{12}\)log10a7 = log10 a\(\frac{1}{3}\) + log10\(\frac{1}{4}\) - log10 a\(\frac{7}{12}\)
= log\(_{10}\) a\(\frac{7}{12}\) - log\(_{10}\) a\(\frac{7}{12}\)
= log\(_{10}\) 1 = 0
Source: Myschool Past Questions Archive (Record #32460)
Mathematics
UTME / JAMB 1983
Variations
Solution
If w varies inversely as V and U varies directly as w3, Find the relationship between u and v given that u = 1, when v = 2
A
u = \(\frac{8}{v^3}\)
B
v = \(\frac{8}{u^2v^3}\)
C
u = 8v3
D
v = 8u2
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
W \(\alpha\) \(\frac{1}{v}\)u \(\alpha\) w3
w = \(\frac{k1}{v}\)
u = k2w3
u = k2(\(\frac{k1}{v}\))3
= \(\frac{k_2k_1^2}{v^3}\)
k = k2k1k2
u = \(\frac{k}{v^3}\)
k = uv3
= (1)(2)3
= 8
u = \(\frac{8}{v^3}\)
Source: Myschool Past Questions Archive (Record #32461)
Mathematics
UTME / JAMB 1983
Linear, Simultaneous & Quadratic Equations
Solution
Solve the simultaneous equations for x in x2 + y - 8 = 0, y + 5x - 2 = 0
A
-28, 7
B
6, -28
C
6, -1
D
-1, 7
E
3, 2
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
x2 + y - 8 = 0, y + 5x - 2 = 0
Rearranging, x2 + y = 8.....(i)
5x + y = 2.......(ii)
Subtract eqn(ii) from eqn(i)
x2 - 5x - 6 = 0
(x - 6)(x + 1) = 0
x = 6, -1
Source: Myschool Past Questions Archive (Record #32462)
Mathematics
UTME / JAMB 1983
Plane Geometry, Polygons & Circle Theorems
Solution
Find the missing value in the following table \(\begin{array}{c|c} x & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline Y = x^3 - x + 3 & & 3 & 3 & 3 & 9 & 27\end{array}\)
A
-3
B
3
C
-9
D
13
E
9
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
When x = -2, y = x3 - x + 3
= -23 - (-2) + 3
= -8 + 2 + 3
= -3
Source: Myschool Past Questions Archive (Record #32463)
Mathematics
UTME / JAMB 1983
Statistics (Data Representation, Central Tendency & Dispersion)
Solution
Find the angle of the sectors representing each item in pie chart of the following data 6, 10, 14, 16, 26
A
15o, 25o, 35o, 40o, 65o
B
60o, 100o, 140o, 160o, 260o
C
6o, 10o, 14o, 16o, 26o
D
30o, 50o, 70o, 80o, 130o
E
none of the above
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
6 + 10 + 14 + 16 + 26 = 72
\(\frac{6}{72}\) x 360
= 30o
Similarly others give 30o, 50o, 70o, 80o and 130o respectively
Source: Myschool Past Questions Archive (Record #32464)
Mathematics
UTME / JAMB 1983
Statistics (Data Representation, Central Tendency & Dispersion)
Solution
The scores of 16 students in a Mathematics test are 65, 65, 55, 60, 60, 65, 60, 70, 75, 70, 65, 70, 60, 65, 65, 70. What is the sum of the median and modal scores?
A
125
B
130
C
140
D
150
E
137.5
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
\(\begin{array}{c|c} Scores & Frequency \\\hline 55 & 1\\ 60 & 4\\ 65 & 6\\ 70 & 4\\ 75 & 1 \end{array}\)
Median = 65(middle number)
Mode = 65(most common)
Median + Mode = 65 + 65 = 130
Source: Myschool Past Questions Archive (Record #32465)
Mathematics
UTME / JAMB 1983
Permutations, Combinations & Probability
Solution
The letters of the word MATRICULATION are cut and put into a box. One letter is drawn at random from the box. Find the probability of drawing a vowel
A
\(\frac{7}{13}\)
B
\(\frac{5}{13}\)
C
\(\frac{6}{13}\)
D
\(\frac{8}{13}\)
E
\(\frac{4}{13}\)
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
Vowels of letters are 6 in numbers
prob. of vowel = \(\frac{6}{13}\)
Source: Myschool Past Questions Archive (Record #32466)
Mathematics
UTME / JAMB 1983
Fractions, Decimals, Percentages & Approximations
Solution
Correct each of the numbers 59.81798 and 0.0746829 to three significant figures and multiply them, giving your answer to three significant figures
A
4.46
B
4.48
C
4.47
D
4.49
E
4.50
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
59.81798 = 59.8(3 s.f)
0.0746829 = 0.0747
59.8 x 0.0747 = 4.46706
= 4.47(3s.f)
Source: Myschool Past Questions Archive (Record #32467)
Mathematics
UTME / JAMB 1983
Fractions, Decimals, Percentages & Approximations
Solution
If a rod of length 250cm is measured as 255cm long in error, what is the percentage error in the measurement?
A
55
B
10
C
5
D
4
E
2
medium Difficulty
Verified Correct Answer: Option E
Step-by-Step Mathematical Solution
% error = \(\frac{\text{Actual error}}{\text{real value}}\) x 100
= \(\frac{5}{250}\) x 100
= 2%
Source: Myschool Past Questions Archive (Record #32468)
Mathematics
UTME / JAMB 1983
Indices, Logarithms & Surds
Solution
If \((\frac{2}{3})^{m} (\frac{3}{4})^{n} = \frac{256}{729}\), find the values of m and n.
A
m = 4, n = 2
B
m = -4, n = -2
C
m = -4, n =2
D
m = 4, n = -2
E
m = -2, n = 4
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
(\(\frac{2}{3}\))m (\(\frac{3}{4}\))n = \(\frac{256}{729}\)
\(\frac{2^m}{4^n}\) x \(\frac{3^n}{3^m}\) = \(\frac{2^{8}}{3^{6}}\)
\(2^{m} \div 2^{2n} = 2^{8}; 3^{n} \div 3^{m} = 3^{-6}\)
m - 2n = 8........(i)
-m + n = -6........(ii)
Solving the equations simultaneously
m = 4, n = -2
Source: Myschool Past Questions Archive (Record #32469)
Mathematics
UTME / JAMB 1983
Indices, Logarithms & Surds
Solution
Without using tables find the numerical value of log\(_7\)49 + log\(_7\)(\(\frac{1}{7}\))
A
1
B
2
C
3
D
7
E
9
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
log\(_7\)49 + log\(_7\)\(\frac{1}{7}\)
= log\(_7\)7
= 1
Source: Myschool Past Questions Archive (Record #32470)
Mathematics
UTME / JAMB 1983
Plane Geometry, Polygons & Circle Theorems
Solution
One interior angle of a convex hexagon is 170o and each of the remaining interior angles is equal to xo. Find x
A
120o
B
110o
C
105o
D
102o
E
100o
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
An hexagon polygon is a six sided polygon
n = 6
sum of all interior angles of hexagon polygon will be (2n - 4) x 90o
= [2 x 6 - 4] x 90
= 720o
If one angle is 170o and each of the remaining five angles is 5xo
5xo + 170o = 720o
= 5x + 550
x = \(\frac{550}{5}\)
= 110o
Source: Myschool Past Questions Archive (Record #32472)