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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
Algebra
Solution
Find the equation of the line through the points (-2, 1) and (-\(\frac{1}{2}\), 4)
A
y = 2x - 3
B
y = 2x + 5
C
y = 3x - 2
D
y = 2x + 1
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
\(\frac{y - y_1}{x - x_1}\) = \(\frac{y_2 - y_1}{x_2 - x_1}\)
\(\frac{y - 1}{x - -2}\) = \(\frac{4 - 1}{-\frac{1}{2} + 2}\)
= \(\frac{y - 1}{x + 2}\) = \(\frac{3}{\frac{3}{2}}\)
y = 2x + 5
Source: Myschool Past Questions Archive (Record #27339)
Mathematics
UTME / JAMB 2012
Algebra
Diagram
Solution
The shaded region above is represented by
the equation
Attached Figure / Diagram
A
y ≤ 4x + 2
B
y ≥ 4x + 2
C
y ≤ -4x + 4
D
y ≤ 4x + 4
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
Equation of the line
\(\frac{y - 4}{x - 0} = \frac{0 - 4}{1 - 0}\)
\(\frac{y - 4}{x} = \frac{-4}{1}\)
\(\therefore -4x = y - 4\)
\(y = -4x + 4\)
\(\therefore \text{The shaded portion = } y \leq -4x + 4\)
Source: Myschool Past Questions Archive (Record #33162)
Mathematics
WAEC / WASSCE 2012
Algebra
Solution
The nth term of a sequence is Tn = 5 + (n - 1)2. Evaluate T4 - T6
A
30
B
16
C
-16
D
-30
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
T4 = 5 + (4 - 1)2; where n = 4
= 5 + (3)2 = 5 + 9
= 14
T = 5 + (6 - 1)2
where n = 6
= 5 + (5)2
= 5 + 25 = 30
T4 + T6 = 14 - 30
= -16
Source: Myschool Past Questions Archive (Record #36740)
Mathematics
WAEC / WASSCE 2012
Algebra
Solution
Solve for x in the equation; \(\frac{1}{x} + \frac{2}{3x} = \frac{1}{3}\)
A
5
B
4
C
3
D
1
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
\(\frac{1}{8} + \frac{2}{3x} = \frac{1}{3}\)
= \(\frac{1}{2}\)
\(\frac{5}{3x} = \frac{1}{3}\)
3x = 15
x = \(\frac{15}{3}\)
= 5
Source: Myschool Past Questions Archive (Record #36746)
Mathematics
WAEC / WASSCE 2012
Algebra
Solution
If x + y = 2y - x + 1 = 5, find the value of x
A
3
B
2
C
1
D
-1
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
x + y = 2y - x + 1 = 5
x + y = 2y - x + 1
x + x + y - 2y = 1
2x - y = 1....(i)
2y - x + 1 = 5
-x + 2y = 5 + 1
-x = 2y = 4
x - 2y = -4 .....(ii)
solve simultaneously (i) x 2x - y = 1
(ii) x x - 2y = -4
2x - y = 1
=2x - 4y = -8
3y = 9
y = \(\frac{9}{3}\)
y = 3
substitute y = 3 into equation (i)
2x - y = 1
2x - 3 = 1
2x = 1 + 3
2x = 4
x = \(\frac{4}{2}\)
= 2
Source: Myschool Past Questions Archive (Record #36754)
Mathematics
WAEC / WASSCE 2012
Algebra
Solution
The sum of 12 and one third of n is 1 more than twice n. Express the statement in the form of an equation
A
12n - 6 = 0
B
3n - 12 = 0
C
2n - 35 = 0
D
5n - 33 = 0
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
12 = \(\frac{n}{3} - 2n = 1\), multiply through by 3
36 + n - 6n = 3
-5n = 3 - 36
-5n = -33
-5n + 33 = 0
5n - 33 = 0
Source: Myschool Past Questions Archive (Record #36755)
Mathematics
WAEC / WASSCE 2012
Algebra
Solution
Solve the inequality: \(\frac{-m}{2} - \frac{5}{4} \geq \frac{5m}{12} - \frac{7}{6}\)
A
m \(\leq \frac{5}{4}\)
B
m \(\geq \frac{5}{4}\)
C
m \(\leq - \frac{1}{11}\)
D
m \(\geq - \frac{1}{11}\)
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
\(\frac{-m}{2} - \frac{5}{4} \geq \frac{5m}{12} - \frac{7}{6}\)
= \(\frac{-2m - 5}{4} \geq \frac{5m - 14}{12}\)
12(-2m - 5) \(\geq\) 4(5m - 14)
-24m - 60 \(\geq\) 20m - 56
-24m - 20m \(\geq\) -56 + 60
44m \(\geq\) 4
m \(\leq \frac{4}{-44}\)
m \(\leq \frac{-1}{11}\)
Source: Myschool Past Questions Archive (Record #36760)
Mathematics
WAEC / WASSCE 2012
Algebra
Solution
(a) Copy and complete the table of values for \(y = 1 - 4\cos x\).
x
0^\circ
30^\circ
60^\circ
90^\circ
120^\circ
150^\circ
180^\circ
210^\circ
240^\circ
270^\circ
300^\circ
y
-3.0
1.0
4.5
-1.0
(b) Using a scale of 2cm to 30^\circ on the x- axis and 2cm to 1 unit on the y- axis, draw the graph of \(y = 1 - 4\cos x\) for \(0^\circ \leq x \leq 360^\circ\).
(c) Use the graph to : (i) solve the equation \(1 - 4\cos x = 0\) ; (ii) find the value of y when x = 105^\circ ; (iii) find x when y = 1.5.
medium Difficulty
Verified Correct Answer: Option
Step-by-Step Mathematical Solution
(a)
x
0^\circ
30^\circ
60^\circ
90^\circ
120^\circ
150^\circ
180^\circ
210^\circ
240^\circ
270^\circ
300^\circ
y
-3.0
-2.5
-1.0
1.0
3.0
4.5
5.0
4.5
3.0
1.0
-1.0
(b)
(c)(i) x = 75^\circ or x = 285^\circ
(ii) When x = 105^\circ, y = 2.
(iii) When y = 1.5, x = 96^\circ or x = 264^\circ.
Source: Myschool Past Questions Archive (Record #52730)
Mathematics
UTME / JAMB 2011
Algebra
Solution
Solve the inequality x2 + 2x > 15.
A
x 5
B
-5 < x < 3
C
x 5
D
x > 3 or x < -5
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
x2 + 2x > 15
x2 + 2x - 15 > 0
(x2 + 5x) - (3x - 15) > 0
x(x + 5) - 3(x + 5) >0
(x - 3)(x + 5) > 0
therefore, x = 3 or -5
i.e. x< 3 or x > -5
Source: WAEC / JAMB / NECO Classroom Archive (Record #29387)
Mathematics
UTME / JAMB 2011
Algebra
Solution
Solve for x and y respectively in the simultaneous equations -2x - 5y = 3. x + 3y = 0
A
-3, -9
B
9, -3
C
-9,3
D
3, -9
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
-2x -5y = 3
x + 3y = 0
x = -3y
-2 (-3y) - 5y = -3
6y - 5y = 3
y = 3
but, x = -3y
x = -3(3)
x = -9
therefore, x = -9, y = 3
Source: Myschool Past Questions Archive (Record #29379)
Mathematics
UTME / JAMB 2011
Algebra
Solution
Solve the inequality -6(x + 3) \(\leq\) 4(x - 2)
A
x \(\leq\) 2
B
x \(\geq\) -1
C
x \(\geq\) -2
D
x \(\leq\) -1
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
-6(x + 3) \(\leq\) 4(x - 2)
-6(x +3) \(\leq\) 4(x - 2)
-6x -18 \(\leq\) 4x - 8
-18 + 8 \(\leq\) 4x +6x
-10 \(\leq\) 10x
10x \(\geq\) -10
x \(\geq\) -1
Source: Myschool Past Questions Archive (Record #29386)
Mathematics
UTME / JAMB 2011
Algebra
Solution
Find the sum of the first 18 terms of the series 3, 6, 9,..., 36.
A
505
B
513
C
433
D
635
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
3, 6, 9,..., 36.
a = 3, d = 3, i = 36, n = 18
Sn = \(\frac{n}{2}\) [2a + (n - 1)d
S18 = \(\frac{18}{2}\) [2 x 3 + (18 - 1)3]
= 9[6 + (17 x 3)]
= 9 [6 + 51] = 9(57)
= 513
Source: Myschool Past Questions Archive (Record #29400)
Mathematics
UTME / JAMB 2011
Algebra
Solution
The seconds term of a geometric series is 4 while the fourth term is 16. Find the sum of the first five terms
A
60
B
62
C
54
D
64
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
T2 = 4, T4 = 16
Tx = arn-1
T2 = ar2-1 = 4 i.e. ar3 = 16, i.e. ar = 4
T4 = ar4-1
therefore, \(\frac{T_4}{T_r}\) = \(\frac{ar^3}{ar}\) = \(\frac{16}{4}\)
r2 = 4 and r = 2
but ar = 4
a = \(\frac{4}{r}\) = \(\frac{4}{2}\)
a = 2
Sn = \(\frac{a(r^n - 1)}{r - 1}\)
S5 = \(\frac{2(2^5 - 1)}{2 - 1}\)
= \(\frac{2(32 - 1)}{2 - 1}\)
= 2(31)
= 62
Source: Myschool Past Questions Archive (Record #29413)