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Mathematics
UTME / JAMB 2019
Algebraic Expressions, Polynomials & Factorisation
Solution
Make q the subject of the formula in the equation \(\frac{mn}{a^2} - \frac{pq}{b^2} = 1\)
A
\(q = \frac{b^2(mn - a^2)}{a^2 p}\)
B
\(q = \frac{m^2 n - a^2}{p^2}\)
C
\(q = \frac{mn - 2b^2}{a^2}\)
D
\(q = \frac{b^2 (n^2 - ma^2)}{n}\)
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
\(\frac{mn}{a^2} - \frac{pq}{b^2} = 1\)
\(\frac{mn}{a^2} - 1 = \frac{pq}{b^2}\)
\(\frac{mn - a^2}{a^2} = \frac{pq}{b^2}\)
\(pq = \frac{b^2 (mn - a^2)}{a^2}\)
\(q = \frac{b^2(mn - a^2)}{a^2 p}\)
Source: Myschool Past Questions Archive (Record #54942)
Mathematics
UTME / JAMB 2019
Trigonometry, Elevation & Bearings
Solution
The angle of elevation of the top of a tree from a point on the ground 60m away from the foot of the tree is 78^\circ. Find the height of the tree correct to the nearest whole number.
A
148m
B
382m
C
282m
D
248m
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
\(\tan 78 = \frac{h}{60}\)
\(h = 60 \tan 78\)
\(h = 60 \times 4.705 = 282.27m\)
\(\approxeq\) 282m to the nearest whole number.
Source: Myschool Past Questions Archive (Record #54944)
Mathematics
UTME / JAMB 2019
Binary Operations, Matrices & Determinants
Solution
A binary operation \(\otimes\) is defined by \(m \otimes n = mn + m - n\) on the set of real numbers, for all m, n \(\in\) R. Find the value of 3 \(\otimes\) (2 \(\otimes\) 4).
A
6
B
25
C
15
D
18
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
\(m \otimes n = mn + m - n\)
3 \(\otimes\) (2 \(\otimes\) 4)
2 \(\otimes\) 4 = 2(4) + 2 - 4 = 6
3 \(otimes\) 6 = 3(6) + 3 - 6 = 15
Source: Myschool Past Questions Archive (Record #54945)
Mathematics
UTME / JAMB 2019
Statistics (Data Representation, Central Tendency & Dispersion)
Solution
Age in years
7
8
9
10
11
No of pupils
4
13
30
44
9
The table above shows the number of pupils in a class with respect to their ages. If a pie chart is constructed to represent the age, the angle corresponding to 8 years old is
A
48.6°
B
56.3°
C
46.8°
D
13°
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
Total number of pupils : 4 + 13 + 30 + 44 + 9 = 100
The number of 8 - year olds = 13
The angle represented by the 8-year olds on the pie chart = \(\frac{13}{100} \times 360^\circ\)
= 46.8^\circ
Source: Myschool Past Questions Archive (Record #54946)
Mathematics
UTME / JAMB 2019
Plane Geometry, Polygons & Circle Theorems
Solution
In a class of 50 students, 40 students offered Physics and 30 offered Biology. How many offered both Physics and Biology?
A
42
B
20
C
70
D
54
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
n(Total) = 50
n(Physics) = 40
n(Biology) = 30
Let n(Physics and Biology) = x
n(Physics only) = 40 -x
n(Biology only) = 30 - x
40 - x + 30 - x + x = 50
70 - x = 50
x = 20
Source: Myschool Past Questions Archive (Record #54947)
Mathematics
UTME / JAMB 2019
Indices, Logarithms & Surds
Solution
Rationalize \(\frac{\sqrt{2} + \sqrt{3}}{\sqrt{2} - \sqrt{3}}\)
A
\(-5 - 2\sqrt{6}\)
B
\(-5 + 3\sqrt{2}\)
C
\(5 - 2\sqrt{3}\)
D
\(5 + 2\sqrt{6}\)
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
\(\frac{\sqrt{2} + \sqrt{3}}{\sqrt{2} - \sqrt{3}}\)
= \((\frac{\sqrt{2} + \sqrt{3}}{\sqrt{2} - \sqrt{3}})(\frac{\sqrt{2} + \sqrt{3}}{\sqrt{2} + \sqrt{3}})\)
= \(\frac{2 + \sqrt{6} + \sqrt{6} + 3}{2 - \sqrt{6} + \sqrt{6} - 3}\)
= \(\frac{5 + 2\sqrt{6}}{-1}\)
= \(- 5 - 2\sqrt{6}\)
Source: Myschool Past Questions Archive (Record #54951)
Mathematics
UTME / JAMB 2019
Trigonometry, Elevation & Bearings
Diagram
Solution
Find the length of the chord |AB| in the diagram shown above.
Attached Figure / Diagram
A
4.2 cm
B
4.3 cm
C
3.2 cm
D
3.4 cm
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
Length of chord = \(2r \sin (\frac{\theta}{2})\)
= \(2(3) \sin (\frac{68}{2})\)
= \(6 \sin 34\)
= \(6 \times 0.559\)
= 3.354 cm \(\approxeq\) 3.4 cm
Source: Myschool Past Questions Archive (Record #54955)
Mathematics
UTME / JAMB 2019
Trigonometry, Elevation & Bearings
Solution
Given \(\sin 58^\circ = \cos p^\circ\), find p.
A
48°
B
58°
C
32°
D
52°
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
\(\sin \theta = \cos (90 - \theta)\)
\(\sin \theta = \cos (90 - 58)\)
= \(\cos 32\)
Source: Myschool Past Questions Archive (Record #54956)
Mathematics
UTME / JAMB 2019
Plane Geometry, Polygons & Circle Theorems
Solution
\(\frac{\frac{2}{3} \div \frac{4}{5}}{\frac{1}{4} + \frac{3}{5} - \frac{1}{3}}\)
A
\(\frac{31}{50}\)
B
\(\frac{20}{31}\)
C
\(\frac{31}{20}\)
D
\(\frac{50}{31}\)
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
\(\frac{\frac{2}{3} \div \frac{4}{5}}{\frac{1}{4} + \frac{3}{5} - \frac{1}{3}}\)
\(\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4}\)
= \(\frac{5}{6}\)
\(\frac{1}{4} + \frac{3}{5} - \frac{1}{3} = \frac{15 + 36 - 20}{60}\)
= \(\frac{31}{60}\)
\(\therefore \frac{\frac{2}{3} \div \frac{4}{5}}{\frac{1}{4} + \frac{3}{5} - \frac{1}{3}} = \frac{5}{6} \div \frac{31}{60}\)
= \(\frac{5}{6} \times \frac{60}{31}\)
= \(\frac{50}{31}\)
Source: Myschool Past Questions Archive (Record #54958)
Mathematics
UTME / JAMB 2019
Linear, Simultaneous & Quadratic Equations
If \(6x^3 + 2x^2 - 5x + 1\) divides \(x^2 - x - 1\), find the remainder.
A
9x + 9
B
2x + 6
C
6x + 8
D
5x - 3
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
No expanded working has been attached for this question yet. The verified answer is Option A.
Source: Myschool Past Questions Archive (Record #54960)
Mathematics
UTME / JAMB 2019
Permutations, Combinations & Probability
Solution
If a fair coin is tossed 3 times, what is the probability of getting at least two heads?
A
\(\frac{2}{3}\)
B
\(\frac{4}{5}\)
C
\(\frac{2}{5}\)
D
\(\frac{1}{2}\)
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
The outcomes are {HHH, HHT, HTT, HTH, THH, THT, TTH, TTT}
P(at least two heads) = \(\frac{4}{8}\)
= \(\frac{1}{2}\)
Source: Myschool Past Questions Archive (Record #54961)
Mathematics
UTME / JAMB 2019
Permutations, Combinations & Probability
Solution
In how many ways can the word MATHEMATICIAN be arranged?
A
6794800 ways
B
2664910 ways
C
6227020800 ways
D
129729600 ways
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
MATHEMATICIAN = 13 letters with 2M, 3A, 2T, 2I.
Hence, the word MATHEMATICIAN can be arranged in \(\frac{13!}{2! 3! 2! 2!}\)
= 129729600 ways
Source: Myschool Past Questions Archive (Record #54962)
Mathematics
UTME / JAMB 2019
Binary Operations, Matrices & Determinants
Solution
Given matrix M = \(\begin{vmatrix} -2 & 0 & 4 \\ 0 & -1 & 6 \\ 5 & 6 & 3 \end{vmatrix}\), find \(M^{T} + 2M\)
A
\(\begin{vmatrix} -4 & 2 & 1\\ 6 & 0 & 5 \\ 0 & 6 & 2 \end{vmatrix}\)
B
\(\begin{vmatrix} -6 & 0 & 13\\ 0 & -3 & 18 \\ 14 & 18 & 9 \end{vmatrix}\)
C
\(\begin{vmatrix} 5 & 2 & 6 \\ 0 & 1 & 1\\ 3 & 4 & -7 \end{vmatrix}\)
D
\(\begin{vmatrix} -4 & 0 & 8 \\ 0 & -2 & -16 \\ 10 & 12 & 6 \end{vmatrix}\)
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
M = \(\begin{vmatrix} -2 & 0 & 4 \\ 0 & -1 & 6 \\ 5 & 6 & 3 \end{vmatrix}\)
M\(^{T}\) = \(\begin{vmatrix} -2 & 0 & 5 \\ 0 & -1 & 6\\ 4 & 6 & 3 \end{vmatrix}\)
2M = \(\begin{vmatrix} -4 & 0 & 8\\ 0 & -2 & 12\\ 10 & 12 & 6\end{vmatrix}\)
M\(^T\) + 2M = \(\begin{vmatrix} -6 & 0 & 13 \\ 0 & -3 & 18 \\ 14 & 18 & 9 \end{vmatrix}\)
Source: Myschool Past Questions Archive (Record #54967)
Mathematics
UTME / JAMB 2019
Statistics (Data Representation, Central Tendency & Dispersion)
Solution
Score (x)
0
1
2
3
4
5
6
Freq (f)
5
7
3
7
11
6
7
Find the mean of the data.
A
3.26
B
4.91
C
6.57
D
3.0
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
Score (x)
0
1
2
3
4
5
6
Freq (f)
5
7
3
7
11
6
7
46
fx
0
7
6
21
44
30
42
150
Mean = \(\frac{\sum fx}{\sum f}\)
= \(\frac{150}{46}\)
= 3.26
Source: Myschool Past Questions Archive (Record #54970)
Mathematics
UTME / JAMB 2019
Statistics (Data Representation, Central Tendency & Dispersion)
Solution
Score (x)
0
1
2
3
4
5
6
Freq (f)
5
7
3
7
11
6
7
Find the variance
A
3.42
B
4.69
C
4.85
D
3.72
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
x
f
fx
(x - \(\bar{x}\))
(x - \(\bar{x}\))\(^2\)
f(x - \(\bar{x}\))\(^2\)
0
5
0
-3.26
10.628
53.14
1
7
7
-2.26
5.108
35.756
2
3
6
-1.26
1.588
4.764
3
7
21
-0.26
0.068
0.476
4
11
44
0.74
0.548
6.028
5
6
30
1.74
3.028
18.168
6
7
42
2.74
7.508
52.556
46
150
170.888
Variance = \(\frac{\sum f(x - \bar{x})}{\sum f}\)
= \(\frac{170.888}{46}\)
= 3.72
Source: Myschool Past Questions Archive (Record #54971)