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Mathematics
UTME / JAMB 2019
Geometry and Trigonometry
Solution
Points X and Y are 20km North and 9km East of point O, respectively. What is the bearing of Y from X? Correct to the nearest degree.
A
24°
B
56°
C
127°
D
156°
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
\(\tan \theta = \frac{9}{20} = 0.45\)
\(\theta = \tan^{-1} (0.45) \)
= 24.23^\circ
\(\therefore\) The bearing of Y from X = 180^\circ - 24.23^\circ
= 155.77^\circ
= 156^\circ (to the nearest degree)
Source: Myschool Past Questions Archive (Record #55282)
Mathematics
UTME / JAMB 2019
Number and Numeration
Solution
If \(P = (\frac{Q(R - T)}{15})^{\frac{1}{3}}\), make T the subject of the formula.
A
\(T = \frac{15R - Q}{P^3}\)
B
\(T = R - \frac{15P^3}{Q}\)
C
\(T = \frac{R - 15P^3}{Q}\)
D
\(T = \frac{R + P^3}{15Q}\)
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
\(P = (\frac{Q(R - T)}{15})^{\frac{1}{3}}\)
\(P^3 = \frac{Q(R - T)}{15}\)
\(Q(R - T) = 15P^3\)
\(R - T = \frac{15P^3}{Q}\)
\(T = R - \frac{15P^3}{Q}\)
Source: Myschool Past Questions Archive (Record #55288)
Mathematics
UTME / JAMB 2019
Geometry and Trigonometry
Diagram
Solution
In the diagram above, O is the centre of the circle ABC, < ABO = 26^\circ and < BOC = 130^\circ. Calculate < AOC.
Attached Figure / Diagram
A
26°
B
13°
C
80°
D
102°
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
< BAC = \(\frac{130}{2}\) (angle subtended at the centre)
< BAC = 65^\circ
Also, x = 26^\circ (theorem)
y = 65^\circ - 26^\circ = 39^\circ
< AOC = 180^\circ - (39^\circ + 39^\circ)
= 102^\circ
Source: Myschool Past Questions Archive (Record #55366)
Mathematics
UTME / JAMB 2019
Geometry and Trigonometry
Solution
Each of the interior angles of a regular polygon is 140^\circ. Calculate the sum of all the interior angles of the polygon.
A
1080°
B
1260°
C
2160°
D
1800°
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
Since each interior angle = 140^\circ;
Each exterior angle = 180^\circ - 140^\circ = 40^\circ
Number of sides of the polygon = \(\frac{360^\circ}{40^\circ}\)
= 9
Sum of angles in the polygon = 140^\circ x 9
= 1260^\circ
Source: Myschool Past Questions Archive (Record #55367)
Mathematics
UTME / JAMB 2019
Number and Numeration
Solution
A man bought a car newly for ₦1,250,000. He had a crash with the car and later sold it at the rate of ₦1,085,000. What is the percentage gain or loss of the man?
A
43.7% loss
B
13.2% gain
C
13.2% loss
D
43.7% gain
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
Cost price of the car = N 1,250.00
Selling price = N 1,085.00
Loss = N (1250 - 1085)
= N 165.00
% loss = \(\frac{165}{1250} \times 100%\)
= 13.2% loss
Source: Myschool Past Questions Archive (Record #55368)
Mathematics
UTME / JAMB 2019
Number and Numeration
Solution
If the volume of a frustrum is given as \(V = \frac{\pi h}{3} (R^2 + Rr + r^2)\), find \(\frac{\mathrm d V}{\mathrm d R}\).
A
\(\frac{\pi h}{3} (2R + r)\)
B
\(2R + r + \frac{\pi h}{3}\)
C
\(\frac{\pi h}{3} (2R^2 + r + 2r)\)
D
\(\frac{2R^2}{3} \pi h\)
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
\(V = \frac{\pi h}{3} (R^2 + Rr + r^2)\)
\(V = \frac{\pi R^2 h}{3} + \frac{\pi Rr h}{3} + \frac{\pi r^2 h}{3}\)
\(\frac{\mathrm d V}{\mathrm d R} = \frac{2 \pi R h}{3} + \frac{\pi r h}{3}\)
= \(\frac{\pi}{3} (2R + r)\)
Source: Myschool Past Questions Archive (Record #55369)
Mathematics
UTME / JAMB 2019
Number and Numeration
Solution
Express \((0.0439 \div 3.62)\) as a fraction.
A
\(\frac{21}{100}\)
B
\(\frac{21}{1000}\)
C
\(\frac{12}{1000}\)
D
\(\frac{12}{100}\)
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
\((0.0439 \div 3.62)\)
= 0.01213
\(\approxeq\) 0.012
= \(\frac{12}{1000}\)
Source: Myschool Past Questions Archive (Record #55370)
Mathematics
UTME / JAMB 2019
Number and Numeration
Solution
If \(25^{1 - x} \times 5^{x + 2} \div (\frac{1}{125})^{x} = 625^{-1}\), find the value of x.
A
x = -4
B
x = 2
C
x = -2
D
x = 4
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
\(25^{1 - x} \times 5^{x + 2} \div (\frac{1}{125})^{x} = 625^{-1}\)
\((5^2)^{(1 - x)} \times 5^{(x + 2)} \div (5^{-3})^x = (5^4)^{-1}\)
\(5^{2 - 2x} \times 5^{x + 2} \div 5^{-3x} = 5^{-4}\)
\(5^{(2 - 2x) + (x + 2) - (-3x)} = 5^{-4}\)
Equating bases, we have
\(2 - 2x + x + 2 + 3x = -4\)
\(4 + 2x = -4 \implies 2x = -4 - 4\)
\(2x = -8\)
\(x = -4\)
Source: Myschool Past Questions Archive (Record #55371)
Mathematics
UTME / JAMB 2019
Geometry and Trigonometry
Solution
A bricklayer charges ₦1,500 per day for himself and ₦500 per day for his assistant. If a two bedroom flat was built for ₦95,000 and the bricklayer worked 10 days more than his assistant, how much did the assistant receive?
A
N20,000
B
N28,000
C
N31,200
D
N41,000
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
Let the number of days worked by the assistant = t
\(\therefore\) The bricklayer worked (t + 10) days.
1500(t + 10) + 500(t) = N 95,000
1500t + 15,000 + 500t = N 95,000
2000t = N 95,000 - N 15,000
2000t = N 80,000
t = 40 days
\(\therefore\) The assistant worked for 40 days and received N (500 x 40)
= N 20,000
Source: Myschool Past Questions Archive (Record #55372)
Mathematics
UTME / JAMB 2019
Algebra
Solution
Find the equation of the locus of a point A(x, y) which is equidistant from B(0, 2) and C(2, 1)
A
4x + 2y = 3
B
4x - 3y = 1
C
4x - 2y = 1
D
4x + 2y = -1
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
Since A(x, y) is the point of equidistance between B and C, then
AB = AC
(AB)\(^2\) = (AC)\(^2\)
Using the distance formula,
(x - 0)\(^2\) + (y - 2)\(^2\) = (x - 2)\(^2\) + (y - 1)\(^2\)
x\(^2\) + y\(^2\) - 4y + 4 = x\(^2\) - 4x + 4 + y\(^2\) - 2y + 1
x\(^2\) - x\(^2\) + y\(^2\) - y\(^2\) + 4x - 4y + 2y = 5 - 4
4x - 2y = 1
Source: Myschool Past Questions Archive (Record #55373)
Mathematics
UTME / JAMB 2019
Number and Numeration
Solution
A factory worker earns ₦50,000 per month out of which he spends 15% on his children's education, ₦13,600 on Food, 3% on electricity and uses the rest for his personal purpose. How much does he have left?
A
N21,850
B
N18,780
C
N27,400
D
N32,500
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
Education = \(\frac{15}{100} \times N 50,000\)
= N 7,500
Food = N 13,600
Electricity = \(\frac{3}{100} \times N 50,000\)
= N 1,500
Leftover : N (50,000 - (7,500 + 13,600 + 1,500))
= N 27,400
Source: Myschool Past Questions Archive (Record #55374)
Mathematics
UTME / JAMB 2019
Number and Numeration
Solution
A binary operation \Delta is defined by a \Delta b = a + 3b + 2.
Find (3 \Delta 2) \Delta 5
A
35
B
59
C
28
D
87
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
a \Delta b = a + 3b + 2
(3 \Delta 2) \Delta 5 = (3 + 3(2) + 2) \Delta 5
= 11 \Delta 5
= 11 + 3(5) + 2
= 28
Source: Myschool Past Questions Archive (Record #55375)
Mathematics
UTME / JAMB 2019
Number and Numeration
Solution
If M varies directly as N and inversely as the root of P. Given that M = 3, N = 5 and P = 25. Find the value of P when M = 2 and N = 6.
A
36
B
63
C
47
D
81
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
\(M \propto N \) ; \(M \propto \frac{1}{\sqrt{P}}\).
\(\therefore M \propto \frac{N}{\sqrt{P}}\)
\(M = \frac{k N}{\sqrt{P}}\)
when M = 3, N = 5 and P = 25;
\(3 = \frac{5k}{\sqrt{25}}\)
\(k = 3\)
\(M = \frac{3N}{\sqrt{P}}\)
when M = 2 and N = 6,
\(2 = \frac{3(6)}{\sqrt{P}} \implies \sqrt{P} = \frac{18}{2}\)
\(\sqrt{P} = 9 \implies P = 9^2\)
P = 81
Source: Myschool Past Questions Archive (Record #55376)
Mathematics
UTME / JAMB 2019
Statistics and Probability
Solution
This table below gives the scores of a group of students in a Further Mathematics Test.
Score
1
2
3
4
5
6
7
Frequency
4
6
8
4
10
6
2
Find the mode of the distribution.
A
7
B
10
C
5
D
4
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
Mode = Score with the highest frequency
= 5
Source: Myschool Past Questions Archive (Record #55377)
Mathematics
UTME / JAMB 2019
Statistics and Probability
Solution
This table below gives the scores of a group of students in a Further Mathematics Test.
Score
1
2
3
4
5
6
7
Frequency
4
6
8
4
10
6
2
Calculate the mean deviation for the distribution
A
4.32
B
2.81
C
1.51
D
3.90
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
Score(x)
1
2
3
4
5
6
7
Total
Frequency (f)
4
6
8
4
10
6
2
40
fx
4
12
24
16
50
36
14
156
x - \(\bar{x}\)
-2.9
-1.9
-0.9
0.1
1.1
2.1
3.1
|x - \(\bar{x}\)|
2.9
1.9
0.9
0.1
1.1
2.1
3.1
f|x - \(\bar{x}\)|
11.6
11.4
7.2
0.4
11
12.6
6.2
60.4
Mean = \(\frac{\sum fx}{\sum f}\)
= \(\frac{156}{40}\)
= 3.9
M.D = \(\frac{\sum f|x - \bar{x}|}{\sum f}\)
= \(\frac{60.4}{40}\)
= 1.51
Source: Myschool Past Questions Archive (Record #55378)