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Mathematics
UTME / JAMB 2014
Sets, Logic & Venn Diagrams
Solution
If P = {1,2,3,4,5} and P \(\cup\) Q = {1,2,3,4,5,6,7}, list the elements in Q
A
{6}
B
{7}
C
{6,7}
D
{5,6}
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
Q = (P \(\cup\) Q) - P
{6,7}
Source: Myschool Past Questions Archive (Record #27004)
Mathematics
UTME / JAMB 2014
Algebraic Expressions, Polynomials & Factorisation
Solution
If gt2 - k - w = 0, make g the subject of the formula
A
\(\frac{k + w}{t^2}\)
B
\(\frac{k - w}{t^2}\)
C
\(\frac{k + w}{t}\)
D
\(\frac{k - w}{t}\)
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
gt2 - k - w = 0
gt2 = k + w
\(g = \frac{k + w}{t^2}\)
Source: Myschool Past Questions Archive (Record #27005)
Mathematics
UTME / JAMB 2014
Algebraic Expressions, Polynomials & Factorisation
Solution
Factorize 2y2 - 15xy + 18x2
A
(2y - 3x) (y + 6x)
B
(2y - 3x) (y - 6x)
C
(2y + 3x) (y - 6x)
D
(3y + 2x) (y - 6x)
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
2y2 - 15xy + 18x2
2y2 - 12xy - 3xy + 18x2
2y(y - 6x) - 3x(y - 6x)
(2y - 3x) (y - 6x)
Source: Myschool Past Questions Archive (Record #27006)
Mathematics
UTME / JAMB 2014
Algebraic Expressions, Polynomials & Factorisation
Solution
Find the value of k if y - 1 is a factor of y3 + 4y2 + ky - 6
A
-6
B
-4
C
O
D
1
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
if y - 1 is a factor of y3 + 4y2 + ky - 6, then
f(1) = (1)3 + 4(1)2 + k(1) - 6 = 0 (factor theorem)
1 + 4 + k - 6 = 0
5 - 6 + k = 0
-1 + k = 0
k = 1
Source: Myschool Past Questions Archive (Record #27007)
Mathematics
UTME / JAMB 2014
Variations
Solution
y varies directly as w2. When y = 8, w = 2. Find y when w = 3
A
18
B
12
C
9
D
6
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
\(y \propto w^2\)
y = kw2
8 = k(2)2
8 = k(4)
k = 8/4
k = 2
Thus y = 2w2
When w = 3, y = 2(3)2
y = 2 x 9 = 18
Source: Myschool Past Questions Archive (Record #27008)
Mathematics
UTME / JAMB 2014
Variations
Solution
P varies directly as Q and inversely as R. When Q = 36 and R = 16, P = 27. Find the relation between P, Q and R.
A
\(P = \frac{Q}{12R}\)
B
\(P = \frac{12Q}{R}\)
C
\(P = 12QR\)
D
\(P = \frac{12}{QR}\)
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
\(P \propto \frac{Q}{R}\)
\(P = K \frac{Q}{R}\)
When Q = 36, R = 16, P = 27
Then substitute into the equation
\(27 = K \frac{36}{16}\)
\(K = \frac{27 \times 16}{36}\)
\(K = 12\)
So the equation connecting P, Q and R is
\(P = \frac{12Q}{R}\)
Source: Myschool Past Questions Archive (Record #27009)
Mathematics
UTME / JAMB 2014
Plane Geometry, Polygons & Circle Theorems
Solution
What is the solution of \(\frac{x - 5}{x + 3} < -1\)?
A
-3 < x < 1
B
x 1
C
-3 < x < 5
D
x 5
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
Consider the range -3 < x < -1
= { -2, -1, 0}, for instance
When x = -2,
\(\frac{-2 - 5}{-2 + 3} < -1\)
\(\frac{-7}{1} < -1\)
When x = -1,
\(\frac{-1 - 5}{-1 + 3} < -1\)
\(\frac{-6}{2} < -1\)
= -3 < -1
When x = 0,
\(\frac{0 - 5}{0 + 3} < -1\)
\(\frac{- 5}{3} < -1\)
Hence -3 < x < 1
Source: Myschool Past Questions Archive (Record #27010)
Mathematics
UTME / JAMB 2014
Inequalities & Linear Programming
Solution
Evaluate the inequality \(\frac{x}{2} + \frac{3}{4} \leq \frac{5x}{6} - \frac{7}{12}\)
A
\(x \geq 4\)
B
\(x \leq 3\)
C
\(x \geq -3\)
D
\(x \leq -4\)
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
\(\frac{x}{2} + \frac{3}{4} \leq \frac{5x}{6} - \frac{7}{12}\)
\(12\frac{x}{2} + 12\frac{3}{4} \leq 12\frac{5x}{6} - 12\frac{7}{12}\)
6x + 9 \(\leq\) 10x - 7
6x - 10x \(\leq\) - 7 - 9
-4x \(\leq\) -16
-4x/-4 \(\geq\) -16/-4
x \(\geq\) 4
Source: Myschool Past Questions Archive (Record #27011)
Mathematics
UTME / JAMB 2014
Sequences & Series (AP & GP)
Solution
The 4th term of an A.P. is 13 while the 10th term is 31. Find the 24th term.
A
89
B
75
C
73
D
69
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
a + 3d = 13 .......... (1)
a + 9d = 31 .......... (2)
(2) - (1): 6d = 18
d = 18/6 = 3
From (1), a + 3(3) = 13
a + 9 = 13
a = 13 - 9 = 4
Hence,
T24 = a + 23d
T24 = 4 + 23(3)
T24 = 4 + 69
T24 = 73
Source: Myschool Past Questions Archive (Record #27012)
Mathematics
UTME / JAMB 2014
Indices, Logarithms & Surds
Solution
What is the common ratio of the G.P. \((\sqrt{10} + \sqrt{5}) + (\sqrt{10} + 2\sqrt{5}) + ... \)?
A
\(\sqrt{2}\)
B
\(\sqrt{5}\)
C
3
D
5
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
Common ratio r of the G.P is
\(r = \frac{T_n + 1}{T_n} = \frac{T_2}{T_1}\)
\(r = \frac{\sqrt{10} + 2\sqrt{5}}{\sqrt{10} + \sqrt{5}}\)
\(r = \frac{\sqrt{10} + 2\sqrt{5}}{\sqrt{10} + \sqrt{5}} \times \frac{\sqrt{10} - \sqrt{5}}{\sqrt{10} - \sqrt{5}} \)
\( = \frac{(\sqrt{10})(\sqrt{10}) + (\sqrt{10})(-\sqrt{5}) + (2\sqrt{5})(\sqrt{10}) + (2\sqrt{5})(-\sqrt{5})}{(\sqrt{10})^2 - (\sqrt{5})^2}\)
\(\frac{10 - \sqrt{50} + 2\sqrt{50} - 10}{10 - 5}\)
\(\frac{\sqrt{50}}{5}\)
\(\frac{\sqrt{25 \times 2}}{5}\)
\(\frac{5\sqrt{2}}{5}\)
\(\sqrt{2}\)
Source: Myschool Past Questions Archive (Record #27013)
Mathematics
UTME / JAMB 2014
Binary Operations, Matrices & Determinants
Solution
A binary operation * is defined by x * y = xy. If x * 2 = 12 - x, find the possible values of x
A
3,4
B
3,-4
C
-3,4
D
-3,-4
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
x * y = xy
x * 2 = 12 - x
Thus by comparison,
x = x, y = 2
But x * y = x * 2
xy = 12 - x
x2 = 12 - x
x2 + x - 12 = 0
x2 + 4x - 3x - 12 = 0
x(x + 4) - 3(x + 4) = 0
(x - 3)(x + 4) = 0
x - 3 = 0 or x + 4 = 0
So x = 3 or x = -4
Source: Myschool Past Questions Archive (Record #27014)
Mathematics
UTME / JAMB 2014
Binary Operations, Matrices & Determinants
Solution
Find y, if \(\begin{pmatrix}5 & -6 \\2 & -7\end{pmatrix}\begin{pmatrix}x \\ y \end{pmatrix} = \begin{pmatrix}7 \\-11\end{pmatrix}\)
A
8
B
5
C
3
D
2
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
\(\begin{pmatrix}5 & -6 \\2 & -7\end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix}7 \\-11\end{pmatrix}\)
By matrices multiplication;
5x - 6y = 7 ........(1)
2x - 7y = -11 ......(2)
2 x (1): 10x - 12y = 14 .......(3)
5 x (2): 10x - 35y = -55 ......(4)
(3) - (4): 23y = 69
y = 69/23 = 3
Source: Myschool Past Questions Archive (Record #27015)
Mathematics
UTME / JAMB 2014
Plane Geometry, Polygons & Circle Theorems
Solution
If \(\begin{vmatrix}-x & 12 \\-1 & 4 \end{vmatrix} = - 12\), find x.
A
-6
B
-2
C
3
D
6
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
\(\begin{vmatrix}-x & 12 \\-1 & 4\end{vmatrix} = - 12\)
-4x - (-1)12 = -12
-4x + 12 = -12
-4x = -12 - 12
-4x = - 24
x = 6
Source: Myschool Past Questions Archive (Record #27016)
Mathematics
UTME / JAMB 2014
Plane Geometry, Polygons & Circle Theorems
Solution
Find the value of \(\begin{vmatrix}0 & 3 & 2 \\1 & 7 & 8 \\0 & 5 & 4\end{vmatrix}\).
A
12
B
10
C
-1
D
-2
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
\(0 \begin{vmatrix}7 & 8 \\5 & 4\end{vmatrix} -3 \begin{vmatrix}1 & 8 \\0 & 4\end{vmatrix} +2 \begin{vmatrix}1 & 7 \\0 & 5\end{vmatrix}\)
= 0(28 - 40) - 3(4 - 0) + 2(5 - 0)
= 0(-12) - 3(4) + 2(5)
= 0 - 12 + 10
= -2
Source: Myschool Past Questions Archive (Record #27017)
Mathematics
UTME / JAMB 2014
Plane Geometry, Polygons & Circle Theorems
Solution
How many sides has a regular polygon whose interior angle is 135^\circ?
A
12
B
10
C
9
D
8
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
If each interior angle of the polygon is 135^\circ, then each exterior angle is 180^\circ - 135^\circ = 45^\circ.
Hence, number of sides =
\(\frac{360^\circ}{\text{one exterior angle}}\)
\(\frac{360^\circ}{45^\circ}\)
= 8
Source: Myschool Past Questions Archive (Record #27018)