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Mathematics
UTME / JAMB 1983
Plane Geometry, Polygons & Circle Theorems
Diagram
Solution
In the figure, PQR is a straight line. Find the values of x and y
Attached Figure / Diagram
A
x = 22.5o, y = 33.75o
B
x = 15o, y = 52.5o
C
x = 22.5o, y = 45.0o
D
x = 56.25o, y = 11.5o
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
\(\frac{3}{2}\)x + 3y + 45^\circ = 180^\circ
3x + 6y + 90^\circ = 360^\circ
3x + 6y = 270.......(i) x 2,
5x + y + y = 180^\circ
5x + 2y = 180^\circ ... (ii) x 6
6x + 12y = 540 ... (iii)
30x + 12y = 1080 ... (iv)
eqn(iv) - eqn(iii)
24x = 540
x = 22.5^\circ and y = 33.75^\circ
Source: Myschool Past Questions Archive (Record #32773)
Mathematics
UTME / JAMB 1983
Plane Geometry, Polygons & Circle Theorems
Diagram
Solution
In the diagram, if PS = SR and PQ||SQ, what is the size of PQR?
Attached Figure / Diagram
A
25o
B
50o
C
55o
D
65o
E
75o
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
o = 180
o
Source: Myschool Past Questions Archive (Record #32774)
Mathematics
UTME / JAMB 1982
Fractions, Decimals, Percentages & Approximations
Diagram
Solution
The diagram is the distance time graph of a vehicle. Find its average speed in kilometers per hour during the journey
Attached Figure / Diagram
A
155km/hr
B
50km/hr
C
40km/hr
D
124km/hr
E
84km/hr
medium Difficulty
Verified Correct Answer: Option E
Step-by-Step Mathematical Solution
Distance = 155 - 50 = 105km
Time = 75mins
= \(\frac{75}{60}\)hr = \(\frac{5}{4}\)hr
Average speed = \(\frac{Distance}{time}\) = \(\frac{105}{\frac{5}{4}}\)
= \(\frac{105 \times 4}{5}\)
= 84km\h
Source: WAEC / JAMB / NECO Classroom Archive (Record #32759)
Mathematics
UTME / JAMB 1982
Indices, Logarithms & Surds
Solution
Rationalize the expression \(\frac{1}{\sqrt{2} + \sqrt{5}}\)
A
\(\frac{1}{3}\)(\(\sqrt{5} - \sqrt{2}\)
B
\(\frac{\sqrt{2}}{3}\) + \(\frac{\sqrt{5}}{5}\)
C
\(\sqrt{2} - \sqrt{5}\)
D
5(\(\sqrt{2} - \sqrt{5}\)
E
\(\frac{1}{3}\)(\(\sqrt{2} - \sqrt{5}\)
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
\(\frac{1}{\sqrt{2} + \sqrt{5}}\)
\(\frac{1}{\sqrt{2} + \sqrt{5}} \times \frac{(\sqrt{2} - \sqrt{5})}{(\sqrt{2} - \sqrt{5})}\)
= \(\frac{\sqrt{2} - \sqrt{5}}{2 - 5}\)
= \(\frac{\sqrt{2} - \sqrt{5}}{-3}\)
= \(\frac{1}{3} (\sqrt{5} - \sqrt{2})\)
Source: Myschool Past Questions Archive (Record #32481)
Mathematics
UTME / JAMB 1982
Plane Geometry, Polygons & Circle Theorems
Solution
Simplify 3 - 2 \(\div\) \(\frac{4}{5}\) + \(\frac{1}{2}\)
A
1\(\frac{3}{4}\)
B
-1
C
1\(\frac{3}{10}\)
D
1
E
1\(\frac{9}{10}\)
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
3 - 2 \(\div\) (\(\frac{4}{5}\)) + \(\frac{1}{2}\)
3 - (2 x \(\frac{5}{4}\)) + \(\frac{1}{2}\) = 3 - \(\frac{10}{4}\) + \(\frac{1}{2}\)
= 3 - \(\frac{5}{2}\) + \(\frac{1}{2}\)
= \(\frac{6 - 5 + 1}{2}\)
= \(\frac{2}{2}\)
= 1
Source: Myschool Past Questions Archive (Record #32482)
Mathematics
UTME / JAMB 1982
Fractions, Decimals, Percentages & Approximations
Solution
If N560.70 is shared in the ratio 7 : 2 : 1, what is the smallest share?
A
N392.49
B
N56.70
C
N113.40
D
N112.14
E
N56.07
medium Difficulty
Verified Correct Answer: Option E
Step-by-Step Mathematical Solution
7 + 2 + 1 = 10
\(\frac{1}{10}\) x 560.70
= N56.07
Source: Myschool Past Questions Archive (Record #32483)
Mathematics
UTME / JAMB 1982
Plane Geometry, Polygons & Circle Theorems
Solution
Seven years ago, the age of a father was three times that of his son, but in six years time the age of the son will be half that of his father, representing the present ages of the father and son by x and y, respectively, the two equations relating x and y are
A
3y - x = 0; 2y - x = 0
B
3y - x = 14; x - 2y = 6
C
3y - x =7; x - 2y = 6
D
3y - x = 14; y - 2x = 6
E
x + 3y = 7; x = 2y = 12
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
7 years ago, Father(x - 7) years old, Son (y - 7) years
x - 7 = 3(y - 7)
x - 7 = 3y - 21
3y - x = -7 + 21 = 14
3y - x = 14 ... (1)
In six years time,
x + 6 = 2(y + 6)
x + 6 = 2y + 12
2y + 12 = x + 6
12 - 6 = x - 2y
6 = x - 2y ... (2)
Source: Myschool Past Questions Archive (Record #32484)
Mathematics
UTME / JAMB 1982
Plane Geometry, Polygons & Circle Theorems
Solution
The factors of 6x - 5 - x2 are
A
-(x + 3)(x + 2)
B
(x + 5)(x + 1)
C
(x - 5)(1 - x)
D
(x + 1)(x + 5)
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
6x - 5 - x2 = (-1)(-x2 - 5 + 6x)
= x2 - 6x + 5
= (x - 5)(x - 1)
-(x - 1) = 1 - x
= (x - 5)(1 - x)
Source: Myschool Past Questions Archive (Record #32485)
Mathematics
UTME / JAMB 1982
Linear, Simultaneous & Quadratic Equations
Solution
The solution of the quadratic equation bx\(^2\) + cx + a = 0 is given by
A
x = b \(\pm\) \(\frac{\sqrt{b^2 - 4ac}}{2a}\)
B
x = c \(\pm\) \(\frac{\sqrt{b^2 - 4ab}}{2b}\)
C
x = -c \(\pm\) \(\frac{\sqrt{c^2 - 4ab}}{2b}\)
D
x = -b \(\pm\) \(\frac{\sqrt{b^2 - 4ac}}{2b}\)
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
bx\(^2\) + cx + a = 0
a = b; b = c; c = a
x = -b \(\pm\) \(\frac{\sqrt{b^2 - 4ac}}{2a}\)
x = -c \(\pm\) \(\frac{\sqrt{c^2 - 4ab}}{2b}\)
Source: Myschool Past Questions Archive (Record #32486)
Mathematics
UTME / JAMB 1982
Linear, Simultaneous & Quadratic Equations
Solution
The graphical method of solving the equation x3 + 3x2 + 4x - 28 = 0 is by drawing the graphs of the curves
A
y = x3 and y = 3x2 + x - 28
B
y = x3 + 3x2 + 4x + 4 and the line y = \(\frac{28}{x}\)
C
y = x3 + 3x2 + 4x and y
D
y = x2 + 3x + 4 and y = \(\frac{28}{x}\)
E
y = x2 + 3x + 4 and line y = 28x
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
The graphical method of solving the equation x3 + 3x2 + 4x - 28 = 0 is by drawing the graphs of the curves
y = x2 + 3x + 4 and y = \(\frac{28}{x}\)`.
Source: Myschool Past Questions Archive (Record #32487)
Mathematics
UTME / JAMB 1982
Indices, Logarithms & Surds
Solution
Write the equation 2 log2x - x log2(1 + y) = 3 in a form not involving logarithms
A
2x(1 + y) = 3
B
2x - x(1 + y) = 8
C
x2 = 8(1 + y)x
D
x2 - x(1 + y) = 8
E
x2 - (1 + y)2 = 8
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
2log2 x - x log2 (1 + y) = 3
log2 \(\frac{x^2}{(1 + y)^x}\) = 3
= \(\frac{x^2}{(1 + y)^x}\)
= 23
= 8
= x2 = 8(1 + y)x
Source: Myschool Past Questions Archive (Record #32488)
Mathematics
UTME / JAMB 1982
Linear, Simultaneous & Quadratic Equations
Solution
Find \(\alpha\) and \(\beta\) such that x\(\frac{3}{8}\) x y\(\frac{-6}{7}\) x (\(\frac{y^{\frac{9}{7}}}{x^{\frac{45}{8}}}\))\(\frac{1}{9}\) = \(\frac{x^{\alpha}}{y^{\beta}}\)
A
\(\alpha\) = 1, \(\beta\) = \(\frac{5}{7}\)
B
\(\alpha\)= 1, \(\beta\) = -\(\frac{5}{7}\)
C
\(\alpha\)= \(\frac{3}{5}\), \(\beta\) = -6
D
\(\alpha\)= 1, \(\beta\) = -\(\frac{3}{5}\)
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
x\(\frac{3}{8}\) x y\(\frac{-6}{7}\) x (\(\frac{y^{\frac{9}{7}}}{x^{\frac{45}{8}}}\))\(\frac{1}{9}\) = \(\frac{y^{\alpha}}{y^{\beta}}\)
x\(\frac{3}{8}\) x y\(\frac{-6}{7}\) x y\(\frac{1}{7}\) = x\(\alpha\)
= x\(\frac{3}{8}\) + \(\frac{5}{8}\) + y\(\frac{6}{7}\) + \(\frac{1}{7}\)
= x\(\alpha\)y\(\beta\)
x1y\(\frac{-5}{7}\) = x\(\alpha\)y\(\beta\)
\(\alpha\) = 1, \(\beta\) = \(\frac{5}{7}\)
Source: Myschool Past Questions Archive (Record #32489)
Mathematics
UTME / JAMB 1982
Plane Geometry, Polygons & Circle Theorems
Solution
Which of the following lines is not parallel to the line 3y + 2x + 7 = 0?
A
3y + 2x - 7 = 0
B
9y + 6x + 17 = 0
C
24y + 16x + 19 = 0
D
3y - 2x + 7 = 0
E
15y + 10x - 13 = 0
medium Difficulty
Verified Correct Answer: Option E
Step-by-Step Mathematical Solution
Two lines are said to be parallel if the slope of the two lines are equal.
The equation : \(3y + 2x + 7 = 0\)
\(3y = -2x - 7\)
\(y = \frac{-2}{3} x - \frac{7}{3}\)
\(\frac{\mathrm d y}{\mathrm d x} = - \frac{2}{3}\)
All the options have the same slope except 15y + 10x - 13 = 0.
Source: Myschool Past Questions Archive (Record #32490)
Mathematics
UTME / JAMB 1982
Inequalities & Linear Programming
Solution
What is the least possible value of \(\frac{9}{1 + 2x^2}\) if 0 \(\geq\) x \(\geq\) 2?
A
9
B
5
C
1
D
2
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
0 \(\geq\) x \(\geq\) 2 \(\to\) 0, 1, 2
If x = 0, \(\frac{9}{1 + 2x^2}\)
\(\frac{9}{1 + 2(0)^2}\) = \(\frac{9}{1}\)
= 3
If x = 2, \(\frac{9}{1 + 2(1)^2}\)
= \(\frac{9}{3}\)
= 3
If x = 2, \(\frac{9}{1 + 2(2)^2}\)
= \(\frac{9}{9}\)
= 1
The least value of \(\frac{9}{1 + 2x^2}\) is 1 when x = 2
Source: Myschool Past Questions Archive (Record #32491)
Mathematics
UTME / JAMB 1982
Fractions, Decimals, Percentages & Approximations
Solution
A man bought wrist watch for N150 but was only able to sell it for N120. Find the loss per cent on the transaction
A
25%
B
11%
C
20%
D
80%
E
30%
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
Loss% = \(\frac{\text{actual loss}}{\text{cost price}}\) x 100%
\(\frac{N150 - N120}{N150} \times 100%\)
\(\frac{N30}{N150} \times 100%\)
= 20%
Source: Myschool Past Questions Archive (Record #32492)