2008 - JAMB Mathematics Past Questions and Answers - page 7
61
Find the range of values of x which satisfy the equalities 4x - 7 \(\leq\) 3x and 3x - 4 \(\leq\) 4x
A
-4 \(\leq\) x \(\leq\) 7
B
-7 \(\leq\) x \(\leq\) 4
C
x \(\geq\) -7
D
-7 \(\leq\) x \(\leq\) 6
correct option: a
4x - 7 \(\leq\) 3x
4x - 3x \(\leq\) 7
x \(\leq\) 7
ii. 3x - 4 \(\leq\) 4x
3x - 4x \(\leq\) 4
-x \(\leq\) -4
-4 \(\leq\) x \(\leq\) 7
Users' Answers & Comments4x - 3x \(\leq\) 7
x \(\leq\) 7
ii. 3x - 4 \(\leq\) 4x
3x - 4x \(\leq\) 4
-x \(\leq\) -4
-4 \(\leq\) x \(\leq\) 7
62
Solve the quadratic inequality x2 - 5x + 6 \(\geq\) 0
A
x \(\leq\) 2, x \(\geq\) 3
B
x \(\leq\) 3, x \(\geq\) 2
C
x \(\leq\) -2, x \(\geq\) -3
D
x \(\leq\) -3,x \(\geq\) 2
correct option: a
x2 - 5x + 6 \(\geq\) 0
x2 - 2x - 3x + 6 \(\geq\) 0
x(x - 2) - 3(x - 2) \(\geq\) 0
(x - 2) (x - 3) \(\geq\) 0
x \(\geq\) 2 or \(\geq\) 3
x \(\leq\) 2 \(\geq\)
Users' Answers & Commentsx2 - 2x - 3x + 6 \(\geq\) 0
x(x - 2) - 3(x - 2) \(\geq\) 0
(x - 2) (x - 3) \(\geq\) 0
x \(\geq\) 2 or \(\geq\) 3
x \(\leq\) 2 \(\geq\)
63
The fifth term of an A.p. is 24 and the eleventh term is 96. Find the first term
A
12
B
4
C
-12
D
-24
correct option: d
In = a + (n - 1)d
24 = a + (5 - 1)d
24 = a + 4d.......(i)
96 = a + (11 - 1)d
96 = a + 10d.......(ii)
a = 24 - 4d.......(iii)
96 = 24 - 4d + 10d
96 = 24 + 6d
96 - 24 = 6d
72 = 6d
d = \(\frac{72}{6}\)
d = 12
a = 24 - 4d
a = 24 - 4 x 12
a = 24 - 48
a = -24
Users' Answers & Comments24 = a + (5 - 1)d
24 = a + 4d.......(i)
96 = a + (11 - 1)d
96 = a + 10d.......(ii)
a = 24 - 4d.......(iii)
96 = 24 - 4d + 10d
96 = 24 + 6d
96 - 24 = 6d
72 = 6d
d = \(\frac{72}{6}\)
d = 12
a = 24 - 4d
a = 24 - 4 x 12
a = 24 - 48
a = -24
64
A binary operation defines \(\ast\) on the set of positive integers is such that x \(\ast\) y = 2x - 3y + 2 for all positive integers x and y. The binary operation is
A
commutative and closed on the set of positive integers
B
neither commutative nor closed on the set of positive integers
C
commutative but not closed on the set of positive integers
D
not commutative but closed on the set of positive integers
correct option: b
a \(\ast\) b = b \(\ast\) a
x \(\ast\) y = y \(\ast\) x
2x - 3y + 2 \(\neq\) 2y - 3y - 3x + 2
2 \(\ast\) 3 = 2(2) -3(3) + 2
= 4 - 9 + 2
= -3
1 \(\ast\) 2 = 2(1) - 3(2) + 2
= 2 - 6 + 2
= 2 - 6 + 2
= -2
Users' Answers & Commentsx \(\ast\) y = y \(\ast\) x
2x - 3y + 2 \(\neq\) 2y - 3y - 3x + 2
2 \(\ast\) 3 = 2(2) -3(3) + 2
= 4 - 9 + 2
= -3
1 \(\ast\) 2 = 2(1) - 3(2) + 2
= 2 - 6 + 2
= 2 - 6 + 2
= -2
65
a binary operation on the set of real numbers excluding -1 is such that sor all m, n \(\varepsilon\) R, m \(\Delta\) n = m + n + mn. Find the identity element of the operation.
A
1
B
o
C
-\(\frac{1}{2}\)
D
-1
correct option: b
m \(\Delta\) n = m + n + mn
m \(\Delta\) e = e \(\Delta\) m = m
m \(\Delta\) e = m
m + e + me = m
e + me = m - m
e + me = 0
e(1 + m) = 0
e = \(\frac{0}{1 + m}\) = 0
Users' Answers & Commentsm \(\Delta\) e = e \(\Delta\) m = m
m \(\Delta\) e = m
m + e + me = m
e + me = m - m
e + me = 0
e(1 + m) = 0
e = \(\frac{0}{1 + m}\) = 0
66
Find the values of x and y respectively if
\(\begin{pmatrix} 1 & 0 \ -1 & -1\ 2 & 2 \end{pmatrix}\) + \(\begin{pmatrix} x & 1 \ -1 & 0\ y & -2 \end{pmatrix}\) = \(\begin{pmatrix} -2 & 1 \ -2 & -1\ -30 & 0 \end{pmatrix}\)
\(\begin{pmatrix} 1 & 0 \ -1 & -1\ 2 & 2 \end{pmatrix}\) + \(\begin{pmatrix} x & 1 \ -1 & 0\ y & -2 \end{pmatrix}\) = \(\begin{pmatrix} -2 & 1 \ -2 & -1\ -30 & 0 \end{pmatrix}\)
A
-3, -2
B
-5, -3
C
-2, -5
D
-3, -5
correct option: d
\(\begin{pmatrix} 1 & 0 \ -1 & -1\ 2 & 2 \end{pmatrix}\) + \(\begin{pmatrix} x & 1 \ -1 & 0\ y & -2 \end{pmatrix}\) = \(\begin{pmatrix} -2 & 1 \ -2 & -1\ -30 & 0 \end{pmatrix}\)
therefore, (x, y) = (-3, -5) respectively
Users' Answers & Commentstherefore, (x, y) = (-3, -5) respectively
67
\(\begin{pmatrix} -2 & 1 \ 2 & 3 \end{pmatrix}\) + \(\begin{pmatrix}p & q \ r & s\end{pmatrix}\) = \(\begin{pmatrix} 1 & 0 \0 & 1 \end{pmatrix}\). What is the value of r?
A
-\(\frac{1}{8}\)
B
\(\frac{3}{8}\)
C
\(\frac{5}{8}\)
D
\(\frac{1}{4}\)
correct option: d
-2p + r = 1.......(i)
2p + 3r = 0.......(ii)
r - 1 + 2p ........(iii)
2p + 3(1 + 2p) = 0
2p + 3(1 + 2p) = 0
2p + 3 + 6p = 0
3 - 8p = 0 \(\to\) 8p = 3
p = \(\frac{3}{8}\)
6 = 1 - 2 \(\frac{3}{8}\)
= 1 - \(\frac{6}{8}\)
\(\frac{8 - 6}{8}\) = \(\frac{2}{8}\)
= \(\frac{1}{4}\)
Users' Answers & Comments2p + 3r = 0.......(ii)
r - 1 + 2p ........(iii)
2p + 3(1 + 2p) = 0
2p + 3(1 + 2p) = 0
2p + 3 + 6p = 0
3 - 8p = 0 \(\to\) 8p = 3
p = \(\frac{3}{8}\)
6 = 1 - 2 \(\frac{3}{8}\)
= 1 - \(\frac{6}{8}\)
\(\frac{8 - 6}{8}\) = \(\frac{2}{8}\)
= \(\frac{1}{4}\)
68
Find the angle subtended at the centre of a circle by a chord which is equal in length to the radius of the circle
A
30o
B
45o
C
60o
D
90o
correct option: c
Equilateral are equal in sides and angle
180o = 60o + 60o + 60o
\(\theta\) = 60o
Users' Answers & Comments180o = 60o + 60o + 60o
\(\theta\) = 60o
69
Find the capacity in litres of a cylindrical well of radius 1 meter and depth 14 metres.
A
44 000 litres
B
4400 litres
C
440 litres
D
44 litres
correct option: c
r = 1
h = 14
\(\pi\) = \(\frac{22}{7}\)
A = \(\pi\)r2h
A = \(\frac{22}{7}\) x 12 x 14
A = 44m2
A = 44 x 10 litres
A = 440 litres
Users' Answers & Commentsh = 14
\(\pi\) = \(\frac{22}{7}\)
A = \(\pi\)r2h
A = \(\frac{22}{7}\) x 12 x 14
A = 44m2
A = 44 x 10 litres
A = 440 litres
70
The locus of a point equidistant from two points P(6,2) and R(4,2) is a perpendicular bisector of PR passing through
A
(2, 5)
B
(5, 2)
C
(1, 0)
D
(0,1)
correct option: b
let (6, 2) be represented as (x1, y1) and (4, 2) be (x2, y2)
p(6, 2) R(4, 2)
m.p = (\(\frac{x_1 + x_2}{2}\)) (\(\frac{y_1 + y_2}{2}\))
= (\(\frac{6 + 4}{2}\) , \(\frac{2 + 2}{2}\))
= (\(\frac{10}{2}\),\(\frac{4}{2}\))
= (5, 2)
Users' Answers & Commentsp(6, 2) R(4, 2)
m.p = (\(\frac{x_1 + x_2}{2}\)) (\(\frac{y_1 + y_2}{2}\))
= (\(\frac{6 + 4}{2}\) , \(\frac{2 + 2}{2}\))
= (\(\frac{10}{2}\),\(\frac{4}{2}\))
= (5, 2)