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Mathematics
UTME / JAMB 2006
Algebraic Expressions, Polynomials & Factorisation
Solution
Find the value of k if the expression kx3 + x2 - 5x - 2 leaves a remainder 2 when it is divided by 2x + 1
A
10
B
8
C
-10
D
-8
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
F(x) = Q x D + R
Kx3 + x2 - 5x – 2 = Q(2x+1)+R
If 2x+1 = 0 implies x = -1/2
∴k(-1/2)3 + (-1/2)2 -5(-1/2) -2
= Q(2(-1/2) + 1) +2
K(-1/8) + 1/4 + 5/2 - 2 = Q(-1+1)+2
-k/8 + 1/4 + 5/2 - 2 = 0+2
(-k+2+20-16) / 8 = 2
(-k+6) / 8 = 2
-k+6 = 2*8
-k+6 = 16
-k = 16-6
-k = 10
∴k = -10
Source: Myschool Past Questions Archive (Record #9037)
Mathematics
UTME / JAMB 2006
Inequalities & Linear Programming
Solution
Solve the inequalities for which \(\frac{x+4}{3}-\frac{x-3}{2} < 4\)
A
x < 7
B
x > -7
C
x < -7
D
x > 7
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
\(\frac{x+4}{3}-\frac{x-3}{2}\) < 4
\(\frac{(2(x+4))-(3(x-3))}{6}\) < 4
2(x+4) – 3(x-3) < 4 x 6
2x + 8 – 3x+9 < 24
-x + 17 < 24
-x < 24 – 17
-x < 7
x > -7
Source: Myschool Past Questions Archive (Record #9038)
Mathematics
UTME / JAMB 2006
Plane Geometry, Polygons & Circle Theorems
Solution
The cost of renovating a 6 m square room is N540. What is the cost of renovating a 9 m square room?
A
N1215
B
N720
C
N1620
D
N810
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
Cost of removing a 6m\(^2\) room = N540
Cost of renovating a 1m\(^2\) room = 540 / 6
Cost of removing a 9m\(^2\) room
= (540 / 6) * 9
= N810
Source: Myschool Past Questions Archive (Record #9039)
Mathematics
UTME / JAMB 2006
Plane Geometry, Polygons & Circle Theorems
Solution
The sum of the first n positive integers is
A
1/2 n(n-1)
B
n(n+1)
C
n(n-1)
D
1/2 n(n+1)
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
Let the positive integers be 1, 2, ,3, 4, .....n
∴ a = 1, d = 1 and n = n
Sn = n/2(2a + (n-1)d)
= n/2 (2 + n – 1)
= 1/2n(n + 1)
Source: Myschool Past Questions Archive (Record #9040)
Mathematics
UTME / JAMB 2006
Algebraic Expressions, Polynomials & Factorisation
Solution
If \(T = 2\pi \sqrt{\frac{l}{g}}\), make g the subject of the formula
A
(4π l2) / T
B
(4π2l) / T2
C
(4π2l2) / T2
D
(2π√l) / T
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
\(T = 2\pi \sqrt{\frac{l}{g}}\)
\(\frac{T}{2\pi} = \sqrt{\frac{l}{g}}\)
\((\frac{T}{2\pi})^{2} = \frac{l}{g}\)
\(\therefore g = \frac{4 \pi^{2} l}{T^{2}}\)
Source: Myschool Past Questions Archive (Record #9041)
Mathematics
UTME / JAMB 2006
Inequalities & Linear Programming
Solution
If y = x\(^2\) - x - 12, find the range of values of x for which y \( \geq \) 0
A
x 4
B
x \( \leq \) -3 or x \( \geq \) 4
C
-3 < x \( \geq \) 4
D
-3 \( \leq \) x \( \leq \) 4
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
y = x\(^2\) - x - 12
= (x - 4)(x + 3)
∴ x = 4 or x = -3
Checking the cases for y \( \geq \) 0
We check values on the range x - 4 \(\geq\) 0; x + 3 \(\leq\) 0; x - 4 \(\leq\) 0 and x + 3 \(\geq\) 0 for the range which satisfies the inequality x\(^2\) - x - 12 \(\geq\) 0.
We find that the inequality is satisfied on the range x \(\leq\) -3 and x \(\geq\) 4.
Source: Myschool Past Questions Archive (Record #9042)
Mathematics
UTME / JAMB 2006
Indices, Logarithms & Surds
Solution
How many terms of the series 3, -6, +12, - 24, + ..... are needed to make a total of 1 - 2\(^8\)?
A
12
B
10
C
9
D
8
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
3, - 6, +12, -24
a = 3, r = -2
\(Sn = \frac{a(1-r^n)}{1-r}\\
∴1-2^8 = \frac{3(1-(-2^{n}))}{1-(-2)}\\
1-2^8 = \frac{3(1-(-2^{n}))}{3}\)
1 - 2\(^8\) = 1 - (-2)\(^n\)
Since 1 - 2\(^8\) = 1 - (-2)\(^n\)
Source: Myschool Past Questions Archive (Record #9043)
Mathematics
UTME / JAMB 2006
Binary Operations, Matrices & Determinants
Solution
Find p, q for which \(\begin{pmatrix} 2p & 8 \\ 3 & -5q \end{pmatrix}\)\(\begin{pmatrix} 1 \\ 2\end{pmatrix}\) = \(\begin{pmatrix}24 \\ -17\end{pmatrix}\)
A
-4
B
45
C
4,2
D
2
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
2p x 1 + 8 x 2 = 24
\(\to\) 4p = 24 - 8 = 16,
p = 4
3 x 1 + -5q x 2 = -17
\(\to\) -10q = -17 - 3
-10q = -20
q = 2
Source: WAEC / JAMB / NECO Classroom Archive (Record #30481)
Mathematics
UTME / JAMB 2006
Binary Operations, Matrices & Determinants
Solution
If x = \(\begin{pmatrix} 1 & 0 & 1 \\ 2 & -1 & 0 \\ -1 & 0 & 1\end{pmatrix}\) and y = \(\begin{pmatrix} -1 & 1 & 2 \\ 0 & -1 & -1 \\ 2 & -1 & 1\end{pmatrix}\)
find 2x - y
A
\(\begin{pmatrix} 3 & -1 & 0 \\ 4 & -3 & -1 \\ -4 & 1 & 1\end{pmatrix}\)
B
\(\begin{pmatrix} 3 & -1 & 0 \\ 4 & -1 & 1 \\ -4 & 1 & 1\end{pmatrix}\)
C
\(\begin{pmatrix} 3 & -1 & 0 \\ 4 & -3 & 1 \\ -4 & 1 & 1\end{pmatrix}\)
D
\(\begin{pmatrix} 3 & -1 & 0 \\ 4 & 1 & 1 \\ -4 & -1 & 1\end{pmatrix}\)
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
2(1) - (-1) = 3 2(2) - (0) = 4 2(-1) - (2) = -4
2(0) - (1) = -1 2(-1) - (-1) = -1 2(0) - (-1) = 1
2(1) - 2 = 0 2(0) - (-1) = 1 2(1) - (1) = 1
\(\begin{pmatrix} 3 & -1 & 0 \\ 4 & -1 & 1 \\ -4 & 1 & 1\end{pmatrix}\)
Source: WAEC / JAMB / NECO Classroom Archive (Record #30522)
Mathematics
UTME / JAMB 2006
Linear, Simultaneous & Quadratic Equations
Solution
Find the roots of x\(^3\) - 2x\(^2\) - 5x + 6 = 0
A
1, -2, 3
B
1, 2, -3,
C
-1, -2, 3
D
-1, 2, -3
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
Equation: x\(^3\) - 2x\(^2\) - 5x + 6 = 0.
First, bring out a\(_n\) which is the coefficient of x\(^3\) = 1.
Then, a\(_0\) which is the coefficient void of x = 6.
The factors of a\(_n\) = 1; The factors of a\(_0\) = 1, 2, 3 and 6.
The numbers to test for the roots are \(\pm (\frac{a_0}{a_n})\).
= \(\pm (1, 2, 3, 6)\).
Test for +1: 1\(^3\) - 2(1\(^2\)) - 5(1) + 6 = 1 - 2 - 5 + 6 = 0.
Therefore x = 1 is a root of the equation.
Using long division method, \(\frac{x^3 - 2x^2 - 5x + 6}{x - 1}\) = x\(^2\) - x - 6.
x\(^2\) - x - 6 = (x - 3)(x + 2).
x = -2, 3.
\(\therefore\) The roots of the equation = 1, -2 and 3.
Source: Myschool Past Questions Archive (Record #30534)
Mathematics
UTME / JAMB 2006
Fractions, Decimals, Percentages & Approximations
Diagram
Solution
The response of 160 pupils in a school asked to indicate their favourite subjects is given in the bar chart above. What percentage of the pupils have English and Health education as the their favourite subjects?
Attached Figure / Diagram
A
55%
B
52%
C
36%
D
22%
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
English = 52 students
Health education = 36 students
Total = 88 students
= \(\frac{88}{160} \times 100%\)
= 55%
Source: Myschool Past Questions Archive (Record #33132)
Mathematics
WAEC / WASSCE 2006
Fractions, Decimals, Percentages & Approximations
Solution
Evaluate (0.13)\(^3\)correct to three significant figures
A
0.00219
B
0.00220
C
0.00300
D
0.00390
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
(0.13)\(^3\) = 0.13 x 0.13 x 0.13 = 0.002197
= 0.00220 (3 s.f)
Source: Myschool Past Questions Archive (Record #36156)
Mathematics
WAEC / WASSCE 2006
Plane Geometry, Polygons & Circle Theorems
Solution
Simplify: 11011\(_2\) - 1101\(_2\)
A
101000\(_2\)
B
1100\(_2\)
C
1110\(_2\)
D
1011\(_2\)
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
11011\(_2\) - 1101\(_2\) = 1110\(_2\)
Source: Myschool Past Questions Archive (Record #36157)
Mathematics
WAEC / WASSCE 2006
Indices, Logarithms & Surds
Solution
Simplify \(\frac{25 ^\frac{2}{3} \div 25 ^\frac{1}{6}}{( \frac{1}{5})^{-\frac{7}{6}} \times ( \frac{1}{5})^{\frac{1}{6}}}\)
A
25
B
\(\frac{1}{5}\)
C
1
D
\(\frac{1}{25}\)
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
\(\frac{25 \frac{2}{3} \div 25 \frac{1}{6}}{( \frac{1}{5})^{-\frac{7}{6}} \times ( \frac{1}{5})^{\frac{1}{6}}}\) = \(\frac{25^{4 - \frac{1}{6}}}{(\frac{1}{5})^{-7 + \frac{1}{6}}}\)
= \(\frac{25^{\frac{1}{2}}}{(\frac{1}{5})^{-1}}\)
= \(\frac{(5^2)^{\frac{1}{2}}}{(5^{-1})^{-1}}\)
= \(\frac{5}{5}\)
= 1
Source: Myschool Past Questions Archive (Record #36158)
Mathematics
WAEC / WASSCE 2006
Plane Geometry, Polygons & Circle Theorems
Solution
Simplify \(\frac{x - 4}{4} - \frac{x - 3}{6}\)
A
\(\frac{x - 18}{12}\)
B
\(\frac{x - 6}{12}\)
C
\(\frac{x - 18}{24}\)
D
\(\frac{x - 6}{24}\)
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
\(\frac{x - 4}{4} - \frac{x - 3}{6}\) = \(\frac{3(x - 4) - 2(x - 3)}{12}\)
= \(\frac{3x -12 - 2x + 6}{12}\)
= \(\frac{3x - 2x - 12 + 6}{12}\)
= \(\frac{x - 6}{12}\)
Source: Myschool Past Questions Archive (Record #36159)