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Mathematics
UTME / JAMB 2011
Permutations, Combinations & Probability
Solution
Raial has 7 different posters to be hanged in her bedroom, living room and kitchen. Assuming she has plans to place at least a poster in each of the 3 rooms, how many choices does she have?
A
49
B
170
C
21
D
210
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
The first poster has 7 ways to be arranges, the second poster can be arranged in 6 ways and the third poster in 5 ways.
= 7 x 6 x 5
= 210 ways
or \(\frac{7}{P_3}\) = \(\frac{7!}{(7 - 3)!}\) = \(\frac{7!}{4!}\)
= \(\frac{7 \times 6 \times 5 \times 4!}{4!}\)
= 210 ways
Source: Myschool Past Questions Archive (Record #29369)
Mathematics
UTME / JAMB 2011
Algebraic Expressions, Polynomials & Factorisation
Solution
Make R the subject of the formula if T = \(\frac {KR^2 + M}{3}\)
A
\(\sqrt\frac{3T - K}{M}\)
B
\(\sqrt\frac{3T - M}{K}\)
C
\(\sqrt\frac{3T + K}{M}\)
D
\(\sqrt\frac{3T - K}{M}\)
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
T = \(\frac{KR^2 + M}{3}\)
3T = KR2 + M
KR2 = 3T - M
R2 = \(\frac{3T - M}{K}\)
R = \(\sqrt\frac{3T - M}{K}\)
Source: Myschool Past Questions Archive (Record #29375)
Mathematics
UTME / JAMB 2011
Linear, Simultaneous & Quadratic Equations
Solution
Find the remainder when X3 - 2X2 + 3X - 3 is divided by X2 + 1
A
2X - 1
B
X + 3
C
2X + 1
D
X - 3
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
X2 + 1 \(\frac{X - 2}{\sqrt{X^3 - 2X^2 + 3n - 3}}\)
= \(\frac {- 6X^3 + n}{-2X^2 + 2X - 3}\)
= \(\frac{(-2X^2 - 2)}{2X - 1}\)
Remainder is 2X - 1
Source: Myschool Past Questions Archive (Record #29376)
Mathematics
UTME / JAMB 2011
Algebraic Expressions, Polynomials & Factorisation
Solution
Factorize completely 9y2 - 16X2
A
(3y - 2x)(3y + 4x)
B
(3y + 4x)(3y + 4x)
C
(3y + 2x)(3y - 4x)
D
(3y - 4x)(3y + 4x)
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
9y2 - 16x2
= 32y2 - 42x2
= (3y - 4x)(3y +4x)
Source: Myschool Past Questions Archive (Record #29378)
Mathematics
UTME / JAMB 2011
Linear, Simultaneous & Quadratic Equations
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
Variations
Solution
If x varies directly as square root of y and x = 81 when y = 9, Find x when y = 1\(\frac{7}{9}\)
A
20\(\frac{1}{4}\)
B
27
C
2\(\frac{1}{4}\)
D
36
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
x \(\alpha\sqrt y\)
x = k\(\sqrt y\)
81 = k\(\sqrt9\)
k = \(\frac{81}{3}\)
= 27
therefore, x = 27\(\sqrt y\)
y = 1\(\frac{7}{9}\) = \(\frac{16}{9}\)
x = 27 x \(\sqrt{\frac{16}{9}}\)
= 27 x \(\frac{4}{3}\)
dividing 27 by 3
= 9 x 4
= 36
Source: Myschool Past Questions Archive (Record #29381)
Mathematics
UTME / JAMB 2011
Variations
Solution
T varies inversely as the cube of R. When R = 3, T = \(\frac{2}{81}\), find T when R = 2
A
\(\frac{1}{18}\)
B
\(\frac{1}{12}\)
C
\(\frac{1}{24}\)
D
\(\frac{1}{6}\)
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
T \(\alpha \frac{1}{R^3}\)
T = \(\frac{k}{R^3}\)
k = TR3
= \(\frac{2}{81}\) x 33
= \(\frac{2}{81}\) x 27
dividing 81 by 27
k = \(\frac{2}{2}\)
therefore, T = \(\frac{2}{3}\) x \(\frac{1}{R^3}\)
When R = 2
T = \(\frac{2}{3}\) x \(\frac{1}{2^3}\) = \(\frac{2}{3}\) x \(\frac{1}{8}\)
= \(\frac{1}{12}\)
Source: Myschool Past Questions Archive (Record #29385)
Mathematics
UTME / JAMB 2011
Inequalities & Linear Programming
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
Sequences & Series (AP & GP)
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
Sequences & Series (AP & GP)
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)
Mathematics
UTME / JAMB 2011
Binary Operations, Matrices & Determinants
Solution
A binary operation \(\oplus\) om real numbers is defined by x \(\oplus\) y = xy + x + y for two real numbers x and y. Find the value of 3 \(\oplus\) - \(\frac{2}{3}\).
A
- \(\frac{1}{2}\)
B
\(\frac{1}{3}\)
C
-1
D
2
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
N + Y = XY + X + Y
3 + -\(\frac{2}{3}\) = 3(- \(\frac{2}{3}\)) + 3 + (- \(\frac{2}{3}\))
= -2 + 3 -\(\frac{2}{3}\)
= \(\frac{1 - 2}{1 - 3}\)
= \(\frac{1}{3}\)
Source: Myschool Past Questions Archive (Record #29421)
Mathematics
UTME / JAMB 2011
Plane Geometry, Polygons & Circle Theorems
Solution
If \(\begin{vmatrix} 2 & 3 \\ 5 & 3x \end{vmatrix}\) = \(\begin{vmatrix} 4 & 1 \\ 3 & 2x \end{vmatrix}\), find the value of x.
A
-6
B
6
C
-12
D
12
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
\(\begin{vmatrix} 2 & 3 \\ 5 & 3x \end{vmatrix}\) = \(\begin{vmatrix} 4 & 1 \\ 3 & 2x \end{vmatrix}\)
(2 x 3x) - (5 x 3) = (4 x 2x) - (3 x 1)
6x - 15 = 8x - 3
6x - 8x = 15 - 3
-2x = 12
x = \(\frac{12}{-2}\)
= -6
Source: Myschool Past Questions Archive (Record #29429)
Mathematics
UTME / JAMB 2011
Plane Geometry, Polygons & Circle Theorems
Solution
Evaluate \(\begin{vmatrix} 4 & 2 & -1 \\ 2 & 3 & -1 \\ -1 & 1 & 3 \end{vmatrix}\)
A
25
B
45
C
15
D
55
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
\(\begin{vmatrix} 4 & 2 & -1 \\ 2 & 3 & -1 \\ -1 & 1 & 3 \end{vmatrix}\)
4 \(\begin{vmatrix} 3 & -1 \\ 1 & 3 \end{vmatrix}\) -2 \(\begin{vmatrix} 2 & -1 \\ -1 & 3\end{vmatrix}\) -1 \(\begin{vmatrix} 2 & 3 \\ -1 & 1 \end{vmatrix}\)
4[(3 x 3) - (-1 x 1)] -2 [(2x 3) - (-1 x -1)] -1 [(2 x 1) - (-1 x 3)]
= 4[9 + 1] -2 [6 - 1] -1 [2 + 3]
= 4(10) - 2(5) - 1(5)
= 40 - 10 - 5
= 25
Source: Myschool Past Questions Archive (Record #29434)
Mathematics
UTME / JAMB 2011
Binary Operations, Matrices & Determinants
Solution
The inverse of matrix N = \(\begin{vmatrix} 2 & 3 \\
1 & 4\end{vmatrix}\) is
A
\(\frac{1}{5}\) \(\begin{vmatrix} 2 & 1 \\ 3 & 4\end{vmatrix}\)
B
\(\frac{1}{5}\) \(\begin{vmatrix} 4 & -3 \\ -1 & 2\end{vmatrix}\)
C
\(\frac{1}{5}\) \(\begin{vmatrix} 2 & -1 \\ -3 & 4\end{vmatrix}\)
D
\(\frac{1}{5}\) \(\begin{vmatrix} 4 & 3 \\ 1 & 2\end{vmatrix}\)
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
N = [2 3]
N-1 = \(\frac{adj N}{|N|}\)
adj N = \(\begin{vmatrix} 4 & -3 \\ -1 & 2 \end{vmatrix}\)
|N| = (2 x4) - (1 x 3)
= 8 - 3
=5
N-1 = \(\frac {1}{5}\) \(\begin{vmatrix} 4 & -3 \\ -1 & 2 \end{vmatrix}\)
Source: Myschool Past Questions Archive (Record #29438)
Mathematics
UTME / JAMB 2011
Plane Geometry, Polygons & Circle Theorems
Solution
What is the size of each interior angle of a 12-sided regular polygon?
A
120o
B
150o
C
30o
D
180o
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
Interior angle = (n - 2)180
but, n = 12
= (12 -2)180
= 10 x 180
= 1800
let each interior angle = x
x = \(\frac{(n - 2)180}{n}\)
x = \(\frac{1800}{12}\)
= 150o
Source: Myschool Past Questions Archive (Record #29442)