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Mathematics
UTME / JAMB 1980
Statistics (Data Representation, Central Tendency & Dispersion)
Solution
A set of data contains a total of 130 items which are divided into six groups for display on a pie chart. If one group contains 26 items, then the sector representing this group on the pie chart contains an angle xo at the centre of the circle where x is
A
36o
B
60o
C
70o
D
72o
E
75o
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
\(\frac{26}{130}\) = \(\frac{x}{360}\)
130x = 26 x 360
= 9360
x = \(\frac{9360}{130}\)
= 72o
Source: Myschool Past Questions Archive (Record #32624)
Mathematics
UTME / JAMB 1980
Fractions, Decimals, Percentages & Approximations
Solution
In triangle FGH, < G = 90º, < H = 60º, while in triangle XYZ, < X = 60º and < Y = 30º. Form XYZ write down the ratio equal to \(\frac{FG}{FH}\)
A
\(\frac{YX}{ZX}\)
B
\(\frac{YX}{YZ}\)
C
\(\frac{ZX}{YZ}\)
D
\(\frac{YZ}{XY}\)
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
The two triangles are similar
\(\frac{FG}{YZ}\) = \(\frac{FH}{XY}\) = \(\frac{GH}{YZ}\) = \(\frac{FG}{FH}\) = \(\frac{YZ}{XZ}\)
Source: Myschool Past Questions Archive (Record #32625)
Mathematics
UTME / JAMB 1980
Plane Geometry, Polygons & Circle Theorems
Solution
A pentagon has four of its angles equal. If the size of the fifth angle is 60o, Find the size of each of the four equal angles.
A
60o
B
108o
C
120o
D
150o
E
None of the above
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
The sum of interior angle of pentagon
(5 sides) = (2n - 4) x 90o
n = 5
(2 x 5 - 4) x 90o = 6 x 90o
= 540o
540o - 60o = 480o \(\div\) 4
= 120o
Source: Myschool Past Questions Archive (Record #32626)
Mathematics
UTME / JAMB 1980
Plane Geometry, Polygons & Circle Theorems
Solution
Find the product of (2\(\sqrt{y} - 3y\)) and (3y + 2\(\sqrt{y}\))
A
4y = y2
B
4y + 9y2
C
4y - 9y2
D
-4y - 9y2
E
2y + y2
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
(2\(\sqrt{y} - 3y\)) and (3y + 2\(\sqrt{y}\))
= 6y\(\sqrt{y}\) + 4y - 9y\(^2\) - 6y\(\sqrt{y}\)
= 4y - 9y\(^2\)
Source: Myschool Past Questions Archive (Record #32627)
Mathematics
UTME / JAMB 1980
Linear, Simultaneous & Quadratic Equations
Solution
What will be the value of k so that the quadratic equation kx2 - 4x + 1 = 0 has two equal roots?
A
2
B
3
C
4
D
8
E
\(\frac{1}{4}\)
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
kx2 - 4x + 1 = 0, comparing with ax2 + bx + c = 0
a = k, b = -4, c = 1 for equal root b2 = 4ac
(-4)2 = 4k
k = \(\frac{16}{4}\)
= 4
Source: Myschool Past Questions Archive (Record #32629)
Mathematics
UTME / JAMB 1980
Plane Geometry, Polygons & Circle Theorems
Solution
Find the solution of the equation x + 2\(\sqrt{x} - 8\) = 0
A
(4, 16)
B
(2, 4)
C
(4, 1)
D
(1, 16)
E
(16, 16)
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
x + 2\(\sqrt{x} - 8 = 0, Let \sqrt{x} = y\)
x = \(y^2\)
\(y^2 + 2y\) - 8 = 0
(y + 4)(x - 2) = 0
y = -4 or 2
x = 16 or 4
Source: Myschool Past Questions Archive (Record #32630)
Mathematics
UTME / JAMB 1980
Indices, Logarithms & Surds
Solution
If it is given that \(5^{x + 1} + 5^x = 150\), then the value of x is equal to
A
3
B
4
C
1
D
2
E
\(\frac{1}{2}\)
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
\(5^{x + 1} + 5^x = 150\)
according to the laws of indices
\(5^1(5^x) + 5^x = 150\)
let \(5^x\) = y
5(y) + y = 150
6y = 150
y = \(\frac{150}{6}\)
= 25
but y = \(5^x\) = 25
\(5^x\) = 25 = \(5^2\)
equating powers since bases are the same
x = 2
Source: Myschool Past Questions Archive (Record #32631)
Mathematics
UTME / JAMB 1980
Indices, Logarithms & Surds
Solution
Solve the system of equation 2\(^{x + y}\) = 32, \(3^{3y - x}\) = 27
A
(3, 2)
B
(-3, 2)
C
(3, -2)
D
(-3, -2)
E
(2, 2)
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
\(2^{x + y} = 32, 3^{3y - x}\) = 27
\(2^{x + y} = 2^5\)
\(3^{3y - x} = 3^3\)
equating powers, we have
x + y = 5 ------------1
3y - x = 3 ------------2
solving simultaneously(any method)
x = 3 and y = 2
Source: Myschool Past Questions Archive (Record #32632)
Mathematics
UTME / JAMB 1980
Plane Geometry, Polygons & Circle Theorems
Solution
Simplify the given expression \(\sqrt{\frac{1 - cos x}{1 + cos x}}\)
A
\(\frac{1 - cos x}{sin x}\)
B
1 - cos x
C
sin x
D
1 + cos x
E
\(\frac{1 + cos x}{sin x}\)
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
\(\sqrt{\frac{1 - cos x}{1 + cos x}}\) = a
a2 = \(\frac{1 - cosx}{1 + cosx}\)
\(\frac{1 - cosx}{1 + cosx}\) = \(\frac{1 - cosx}{1 - cosx}\)
= \(\frac{(1 - cosx)^2}{cos^2 x}\)
a2 = \(\frac{(1 - cos x)^2}{sin^2 x}\)
a = \(\frac{1 - cos x}{sin x}\)
Source: Myschool Past Questions Archive (Record #32633)
Mathematics
UTME / JAMB 1980
Mensuration (Areas, Perimeters & Volumes)
Solution
Find the area of the curved surface of a cone whose base radius is 6cm and whose height is 8cm. (take \(\pi\) = \(\frac{22}{7}\))
A
188.57cm2
B
1320cm2
C
188cm2
D
188.08cm2
E
10cm2
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
S = curved surface area = \(\pi\)rL
= \(\frac{22}{7}\) x 6 x 10
= 188.57cm2
Source: Myschool Past Questions Archive (Record #32634)
Mathematics
UTME / JAMB 1980
Linear, Simultaneous & Quadratic Equations
Solution
Solve the equation for the positive values of \(\theta\) less than 360o. 3 tan \(\theta\) + 2 = -1
A
135o or 315o
B
45o or 135o
C
315o or 180o
D
315v + 45o
E
360o or 315o
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
3 tan \(\theta\) + 2 = -1
3 tan \(\theta\) \(\frac{-3}{3}\) = -1
\(\theta\) = tan -1(-1)
\(\theta\) = 360o - 45o
= 315o
\(\theta\) = 180 - 45o = 135o
Source: Myschool Past Questions Archive (Record #32635)
Mathematics
UTME / JAMB 1980
Indices, Logarithms & Surds
Solution
Given log 2 = 0.69, log3 = 1, 10 and log7 = 1.90, all to a fixed base, find log 10.5 to the same base without using tables.
A
1.03
B
2.31
C
3.69
D
10.5
E
25
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
log 10.5 = log \(\frac{21}{2}\)
= log 21 - log 2
= log(3 x 7) - log 2
= log 3 + log 7 - log 2
= 1.10 + 1.90 - 0.69
= 3 - 0.69
= 2.31
Source: Myschool Past Questions Archive (Record #32636)
Mathematics
UTME / JAMB 1980
Plane Geometry, Polygons & Circle Theorems
Solution
Simplify 102 + log105
A
500
B
2 log10 5
C
10
D
25
E
log105 x 10100
medium Difficulty
Verified Correct Answer: Option E
Step-by-Step Mathematical Solution
102 + log105 = log10 10100 + log105
= log105 x 10100
Source: Myschool Past Questions Archive (Record #32637)
Mathematics
UTME / JAMB 1980
Plane Geometry, Polygons & Circle Theorems
Solution
Find the missing numerator \(\frac{5}{x + 1}\) - \(\frac{3}{1 - x}\) - \(\frac{7x - 1}{x^2 - 1}\) = \(\frac{?}{x + 1}\).
A
-1
B
x - 1
C
\(\frac{3(1 - 5x)}{x + 1}\)
D
1
E
3(1 - 5x)
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
\(\frac{5}{x + 1} - \frac{3}{1 - x} - \frac{7x - 1}{x^{2} - 1} = \frac{?}{x + 1}\)
\(\frac{5(x - 1) - 3(- (x + 1)) - (7x - 1)}{x^{2} - 1}\)
= \(\frac{5x - 5 + 3x + 3 - 7x + 1}{x^{2} - 1}\)
= \(\frac{x - 1}{x^{2} - 1}\)
= \(\frac{x - 1}{(x - 1)(x + 1)}\)
= \(\frac{1}{x + 1}\).
The numerator = 1.
Source: Myschool Past Questions Archive (Record #32638)
Mathematics
UTME / JAMB 1980
Linear, Simultaneous & Quadratic Equations
Solution
The expression x\(^3\) - 4x\(^2\) + cx + d is such that x + 1 is its factor, and its value is 1 when x is -2. Find c and d.
A
c = 4 and d = -9
B
c = -4 and d = 9
C
c = -20 and d = -15
D
c = 20 and d = -15
E
c = -20 and d = 15
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
F(X) = x\(^3\) - 4x\(^2\) + cx + d
= (X + 1) Q(X) + R
x = -1, R = 0,f(-1) = -1\(^3\) - 4(-1)\(^2\) + c(-1) + d = 0
-1 - 4 - c + d = 0
d - c = 5................(i)
f(-2) = -2\(^3\) - 4(-2)\(^2\) + c(-2) + d = 1
= -8 - 16 - 2c + d
= 1
-8 - 16 - 2c + d = 1
-24 - 2c + d = 1
d - 2c = 1 + 24
d - 2c = 25.................(ii)
\(\frac{d - c = 5}{-c = 20}\) d - c = 5
c = -20
d - (-20) = 5
d + 20 = 5
d = 5 - 20
= -15
c = -20, d = -15
Source: Myschool Past Questions Archive (Record #32639)