2014 - JAMB Mathematics Past Questions and Answers - page 2
11
Factorize 2y2 - 15xy + 18x2
A
(2y - 3x) (y + 6x)
B
(2y - 3x) (y - 6x)
C
(2y + 3x) (y - 6x)
D
(3y + 2x) (y - 6x)
correct option: b
2y2 - 15xy + 18x2
2y2 - 12xy - 3xy + 18x2
2y(y - 6x) - 3x(y - 6x)
(2y - 3x) (y - 6x)
Users' Answers & Comments2y2 - 12xy - 3xy + 18x2
2y(y - 6x) - 3x(y - 6x)
(2y - 3x) (y - 6x)
12
Find the value of k if y - 1 is a factor of y3 + 4y2 + ky - 6
A
-6
B
-4
C
O
D
1
correct option: d
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
Users' Answers & Commentsf(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
13
y varies directly as w2. When y = 8, w = 2. Find y when w = 3
A
18
B
12
C
9
D
6
correct option: a
\(y \propto w^2\)
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
Users' Answers & Commentsy = 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
14
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
\(P = \frac{12}{QR}\)
correct option: b
\(P \propto \frac{Q}{R}\)
\(P = K \frac{Q}{R}\)
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
\(P = \frac{12Q}{R}\)
Users' Answers & Comments\(P = K \frac{Q}{R}\)
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
\(P = \frac{12Q}{R}\)
15
What is the solution of \(\frac{x - 5}{x + 3} < -1\)?
A
-3 < x < 1
B
x < -3 or x > 1
C
-3 < x < 5
D
x < -3 or x > 5
correct option: a
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
Users' Answers & Comments= { -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
16
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
\(x \leq -4\)
correct option: a
\(\frac{x}{2} + \frac{3}{4} \leq \frac{5x}{6} - \frac{7}{12}\)
\(12\frac{x}{2} + 12\frac{3}{4} \leq 12\frac{5x}{6} - 12\frac{7}{12}\)
6x + 9 \(\leq\) 10x - 7
6x - 10x \(\leq\) - 7 - 9
-4x \(\leq\) -16
-4x/-4 \(\geq\) -16/-4
x \(\geq\) 4
Users' Answers & Comments\(12\frac{x}{2} + 12\frac{3}{4} \leq 12\frac{5x}{6} - 12\frac{7}{12}\)
6x + 9 \(\leq\) 10x - 7
6x - 10x \(\leq\) - 7 - 9
-4x \(\leq\) -16
-4x/-4 \(\geq\) -16/-4
x \(\geq\) 4
17
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
correct option: c
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
Users' Answers & Commentsa + 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
18
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
correct option: a
Common ratio r of the G.P is
\(r = \frac{T_n + 1}{T_n} = \frac{T_2}{T_1}\)
\(r = \frac{\sqrt{10} + 2\sqrt{5}}{\sqrt{10} + \sqrt{5}}\)
\(r = \frac{\sqrt{10} + 2\sqrt{5}}{\sqrt{10} + \sqrt{5}} \times \frac{\sqrt{10} - \sqrt{5}}{\sqrt{10} - \sqrt{5}} \)
\( = \frac{(\sqrt{10})(\sqrt{10}) + (\sqrt{10})(-\sqrt{5}) + (2\sqrt{5})(\sqrt{10}) + (2\sqrt{5})(-\sqrt{5})}{(\sqrt{10})^2 - (\sqrt{5})^2}\)
\(\frac{10 - \sqrt{50} + 2\sqrt{50} - 10}{10 - 5}\)
\(\frac{\sqrt{50}}{5}\)
\(\frac{\sqrt{25 \times 2}}{5}\)
\(\frac{5\sqrt{2}}{5}\)
\(\sqrt{2}\)
Users' Answers & Comments\(r = \frac{T_n + 1}{T_n} = \frac{T_2}{T_1}\)
\(r = \frac{\sqrt{10} + 2\sqrt{5}}{\sqrt{10} + \sqrt{5}}\)
\(r = \frac{\sqrt{10} + 2\sqrt{5}}{\sqrt{10} + \sqrt{5}} \times \frac{\sqrt{10} - \sqrt{5}}{\sqrt{10} - \sqrt{5}} \)
\( = \frac{(\sqrt{10})(\sqrt{10}) + (\sqrt{10})(-\sqrt{5}) + (2\sqrt{5})(\sqrt{10}) + (2\sqrt{5})(-\sqrt{5})}{(\sqrt{10})^2 - (\sqrt{5})^2}\)
\(\frac{10 - \sqrt{50} + 2\sqrt{50} - 10}{10 - 5}\)
\(\frac{\sqrt{50}}{5}\)
\(\frac{\sqrt{25 \times 2}}{5}\)
\(\frac{5\sqrt{2}}{5}\)
\(\sqrt{2}\)
19
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
correct option: b
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
Users' Answers & Commentsx * 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
20
Find y, if \(\begin{pmatrix}
5 & -6 \\ 2 & -7
\end{pmatrix}\begin{pmatrix}
5 \\ 2
\end{pmatrix} = \begin{pmatrix}
7 \\ -11
\end{pmatrix}\)
5 & -6 \\ 2 & -7
\end{pmatrix}\begin{pmatrix}
5 \\ 2
\end{pmatrix} = \begin{pmatrix}
7 \\ -11
\end{pmatrix}\)
A
8
B
5
C
3
D
2
correct option: c
\(\begin{pmatrix}
5 & -6 \\ 2 & -7
\end{pmatrix}\begin{pmatrix}
5 \\ 2
\end{pmatrix} = \begin{pmatrix}
7 \\ -11
\end{pmatrix}\)
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
Users' Answers & Comments5 & -6 \\ 2 & -7
\end{pmatrix}\begin{pmatrix}
5 \\ 2
\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