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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)
Mathematics
UTME / JAMB 2019
Plane Geometry, Polygons & Circle Theorems
Solution
The locus of a point which moves so that it is equidistant from two intersecting straight lines is the
A
bisector of the two lines
B
line parallel to the two lines
C
angle bisector of the two lines
D
perpendicular bisector of the two lines
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
The locus of a points equidistant from two intersecting straight lines is a pair of bisectors that bisect the angles formed by the two intersecting lines.
Source: Myschool Past Questions Archive (Record #54974)
Mathematics
UTME / JAMB 2019
Plane Geometry, Polygons & Circle Theorems
Diagram
Solution
From the cyclic quadrilateral MNOP above, find the value of x.
Attached Figure / Diagram
A
16°
B
25°
C
42°
D
39°
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
The sum of two opposite angles of a cyclic quadrilateral = 180^\circ
\(\therefore\) (2x + 18)^\circ + 84^\circ = 180^\circ
2x + 102^\circ = 180^\circ \(\implies\) 2x = 78^\circ
x = 39^\circ
Source: Myschool Past Questions Archive (Record #54977)
Mathematics
UTME / JAMB 2019
Trigonometry, Elevation & Bearings
Solution
If \(4\sin^2 x - 3 = 0\), find the value of x, when 0^\circ \(\leq\) x \(\leq\) 90^\circ
A
90°
B
45°
C
60°
D
30°
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
\(4\sin^2 x - 3 = 0\)
\(4 \sin^2 x = 3 \implies \sin^2 x = \frac{3}{4}\)
\(\sin x = \frac{\sqrt{3}}{2}\)
\(\therefore x = \sin^{-1} (\frac{\sqrt{3}}{2})\)
x = 60^\circ
Source: Myschool Past Questions Archive (Record #54992)
Mathematics
UTME / JAMB 2019
Fractions, Decimals, Percentages & Approximations
Diagram
Solution
In the figure above, |CD| is the base of the triangle CDE. Find the area of the figure to the nearest whole number.
Attached Figure / Diagram
A
56 cm\(^2\)
B
24 cm\(^2\)
C
42 cm\(^2\)
D
34 cm\(^2\)
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
Area of rectangle ABCD = length x breadth
= 7 x 4
= 28 cm\(^2\)
Area of triangle CDE = \(\frac{1}{2}\) base x height
= \(\frac{1}{2} \times 3 \times 4\)
= 6 cm\(^2\)
Area of the figure = 28 cm\(^2\) + 6 cm\(^2\)
= 34 cm\(^2\)
Source: Myschool Past Questions Archive (Record #55004)
Mathematics
UTME / JAMB 2019
Plane Geometry, Polygons & Circle Theorems
Solution
The marks scored by 30 students in a Mathematics test are recorded in the table below:
Scores (Mark)
0
1
2
3
4
5
No of students
4
3
7
8
6
2
What is the total number of marks scored by the children?
A
82
B
15
C
63
D
75
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
Scores (Mark)
0
1
2
3
4
5
No of students
4
3
7
8
6
2
fx
0
3
14
24
24
10
75
Source: Myschool Past Questions Archive (Record #55016)