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Mathematics
UTME / JAMB 2012
Number and Numeration
Solution
Find the remainder when 2x3 - 11x2 + 8x - 1 is divided by x + 3
A
-871
B
-781
C
-187
D
-178
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
Hence f(x) = 2x3 - 11x2 + 8x - 1
f(-3) = 2(-3)3 - 11(-3)2 + 8(-3) - 1
= 2(-27) - 11(9) + 8(-3) - 1
= -54 - 99 - 24 - 1
= -178
Source: Myschool Past Questions Archive (Record #27220)
Mathematics
UTME / JAMB 2012
Algebra
Solution
Solve for x and y in the equations below
x2 - y2 = 4
x + y = 2
A
x = 0, y = -2
B
x = 0, y = 2
C
x = 2, y = 0
D
x = -2, y = 0
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
x2 - y2 = 4 .... (1)
x + y = 2 .... (2)
Simplify eqn (1)
(x + y)(x - y) = 4
From eqn (2)
x + y = 2 so substitute it into simplified eqn (1), we have
2 (x - y) = 4
therefore,
x - y = 2 ... (1)
x + y = 2
---------
2x = 4
---------
x = 2, when y = 0
Source: Myschool Past Questions Archive (Record #27221)
Mathematics
UTME / JAMB 2012
Number and Numeration
Solution
If y varies directly as \(\sqrt{n}\) and y = 4 when n = 4, find y when n = 1\(\frac{7}{9}\)
A
\(\sqrt{17}\)
B
\(\frac{4}{3}\)
C
\(\frac{8}{3}\)
D
\(\frac{2}{3}\)
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
y \(\propto \sqrt{n}\)
y = k\(\sqrt{n}\)
when y = 4, n = 4
4 = k\(\sqrt{4}\)
4 = 2k
k = 2
Therefore,
y = 2\(\sqrt{n}\)
y = 2\(\sqrt{\frac{16}{9}}\)
y = 2\((\frac{4}{3})\)
y = \(\frac{8}{3}\)
Source: Myschool Past Questions Archive (Record #27222)
Mathematics
UTME / JAMB 2012
Number and Numeration
Solution
U is inversely proportional to the cube of V and U = 81 when V = 2. Find U when V = 3
A
24
B
27
C
32
D
36
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
U \(\propto \frac{1}{V^3}\)
U = \(\frac{k}{V^3}\)
k = UV\(^3\)
k = 81 x 2\(^3\) = 81 x 8
When V = 3,
U = \(\frac{k}{V^3}\)
U = \(\frac{81 \times 8}{3^3}\)
U = \(\frac{81 \times 8}{27}\) = 24
Source: Myschool Past Questions Archive (Record #27223)
Mathematics
UTME / JAMB 2012
Number and Numeration
Solution
The value of y for which \(\frac{1}{5}y + \frac{1}{5} < \frac{1}{2}y + \frac{2}{5}\) is
A
\(y > \frac{2}{3}\)
B
\(y < \frac{2}{3}\)
C
\(y > -\frac{2}{3}\)
D
\(y < -\frac{2}{3}\)
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
\(\frac{1}{5}y + \frac{1}{5} < \frac{1}{2}y + \frac{2}{5}\)
Collect like terms
\(\frac{y}{5} - \frac{y}{2} < \frac{2}{5} - \frac{1}{5}\)
\(\frac{2y - 5y}{10} < \frac{2 - 1}{5}\)
\(\frac{-3y}{10} < \frac{1}{5}\)
\(y > \frac{-2}{3}\)
Source: Myschool Past Questions Archive (Record #27326)
Mathematics
UTME / JAMB 2012
Number and Numeration
Solution
Find the range of values of m which satisfy (m - 3)(m - 4) < 0
A
2 < m < 5
B
-3 < m < 4
C
3 < m < 4
D
-4 < m < 3
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
(m - 3)(m - 4) < 0
(m - 3) < 0 ; (m - 4) < 0
m < 3 ; m < 4
3 < m < 4
Source: Myschool Past Questions Archive (Record #27327)
Mathematics
UTME / JAMB 2012
Algebra
Solution
The nth term of a sequence is n2 - 6n - 4. Find the sum of the 3rd and 4th terms.
A
24
B
23
C
-24
D
-25
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
n2 - 6n - 4
For the 3rd term,
32 - 6(3) - 4
9 - 18 -4 = -13
For the 4th term,
42 - 6(4) - 4
16 - 24 - 4 = -12
Sum of both terms
-13 - 12 = -25
Source: Myschool Past Questions Archive (Record #27328)
Mathematics
UTME / JAMB 2012
Number and Numeration
Solution
The sum to infinity of a geometric progression is \(-\frac{1}{10}\) and the first term is \(-\frac{1}{8}\). Find the common ratio of the progression.
A
\(-\frac{1}{5}\)
B
\(-\frac{1}{4}\)
C
\(-\frac{1}{3}\)
D
\(-\frac{1}{2}\)
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
Sr = \(\frac{a}{1 - r}\)
\(-\frac{1}{10}\) = \(\frac{1}{8} \times \frac{1}{1 - r}\)
\(-\frac{1}{10}\) = \(\frac{1}{8(1 - r)}\)
\(-\frac{1}{10}\) = \(\frac{1}{8 - 8r}\)
cross multiply...
-1(8 - 8r) = -10
-8 + 8r = -10
8r = -2
r = -1/4
Source: Myschool Past Questions Archive (Record #27329)
Mathematics
UTME / JAMB 2012
Number and Numeration
Solution
The binary operation * is defined on the set of integers such that p * q = pq + p - q. Find 2 * (3 * 4)
A
11
B
13
C
15
D
22
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
p * q = pq + p - q
First execute for 3 * 4
==> 3(4) + 3 - 4 = 12 + 3 - 4 = 11
Now we execute for 2 * 11
==> 2(11) + 2 - 11 = 22 + 2 - 11 = 13
Source: Myschool Past Questions Archive (Record #27330)
Mathematics
UTME / JAMB 2012
Number and Numeration
Solution
The binary operation on the set of real numbers is defined by m*n = \(\frac{mn}{2}\) for all m, n \(\in\) R. If the identity element is 2, find the inverse of -5
A
\(-\frac{4}{5}\)
B
\(-\frac{2}{5}\)
C
4
D
5
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
m * n = \(\frac{mn}{2}\)
Identify, e = 2
Let a \(\in\) R, then
a * a\(^{-1}\) = e
a * a\(^{-1}\) = 2
-5 * a\(^{-1}\) = 2
\(\frac{-5 \times a^{-1}}{2} = 2\)
\(a^{-1} = \frac{2 \times 2}{-5}\)
\(a^{-1} = -\frac{4}{5}\)
Source: Myschool Past Questions Archive (Record #27331)
Mathematics
UTME / JAMB 2012
Algebra
Solution
If \(\begin{vmatrix} 5 & 3 \\ x & 2 \end{vmatrix}\) = \(\begin{vmatrix} 3 & 5 \\ 4 & 5 \end{vmatrix}\), find the value of x
A
3
B
4
C
5
D
7
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
\(\begin{vmatrix} 5 & 3 \\ x & 2 \end{vmatrix}\) = \(\begin{vmatrix} 3 & 5 \\ 4 & 5 \end{vmatrix}\)
10 - 3x = 15 - 20
-3x = 15 - 20 - 10
-3x = -15
x = 5
Source: Myschool Past Questions Archive (Record #27332)
Mathematics
UTME / JAMB 2012
Algebra
Solution
Given that I3 is a unit matrix of order 3, find |I3|
A
-1
B
O
C
1
D
2
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
Recall that a unit matrice is a diagonal matrix in which the elements in the leading diagonal is unity. Therefore,
I3 = \(\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\)
I3 = \(+1\begin{vmatrix} 1 & 0 \\ 0 & 1 \end{vmatrix} - 0\begin{vmatrix} 0 & 0 \\ 0 & 1 \end{vmatrix} + 0 \begin{vmatrix} 0 & 1 \\ 0 & 0 \end{vmatrix} \)
I3 = +1(1 - 0) - 0(0 - 0) + 0(0 - 0)
= 1(1)
= 1
Source: Myschool Past Questions Archive (Record #27333)
Mathematics
UTME / JAMB 2012
Geometry and Trigonometry
Solution
The angles of a polygon are given by x, 2x, 3x, 4x and 5x respectively. Find the value of x.
A
24o
B
30o
C
33o
D
36o
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
Since there are 5 angles given, the polygon is a pentagon.
Sum of interior angles of a pentagon = (2(5) - 4) x 90^\circ = 540^\circ
\(\therefore\) x + 2x + 3x + 4x + 5x = 15x
15x = 540^\circ
\(x = \frac{540}{15} = 36^\circ\)
Source: Myschool Past Questions Archive (Record #27334)
Mathematics
UTME / JAMB 2012
Number and Numeration
Solution
Calculate the volume of a cuboid of length 0.76cm, breadth 2.6cm and height 0.82cm.
A
3.92cm3
B
2.13cm3
C
1.97cm3
D
1.62cm3
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
Volume of cuboid = L x b x h
= 0.76cm x 2.6cm x 0.82cm
= 1.62cm3
Source: Myschool Past Questions Archive (Record #27335)
Mathematics
UTME / JAMB 2012
Geometry and Trigonometry
Solution
The gradient of the straight line joining the points P(5, -7) and Q(-2, -3) is
A
\(\frac{1}{2}\)
B
\(\frac{2}{5}\)
C
\(-\frac{4}{7}\)
D
\(-\frac{2}{3}\)
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
PQ = \(\frac{y_1 - y_0}{x_1 - x_0}\) = \(\frac{-3 - (-7)}{-2 - 5}\) = \(\frac{-3 + 7}{-2 - 5}\) = \(\frac{4}{-7}\)
Source: Myschool Past Questions Archive (Record #27337)