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Mathematics
UTME / JAMB 2006
Statistics (Data Representation, Central Tendency & Dispersion)
Solution
If the mean of five consecutive integers is 30, find the largest of the numbers
A
28
B
30
C
32
D
34
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
Let the consecutive numbers be a, a+1, a+2, a+3, a+4
Mean = \(\frac{(a+a+1+a+2+a+3+a+4)}{5}\)
30 = \(\frac{5a+10}{5}\)
30\( \times 5 = 5a + 10\)
5a = 150 - 10
5a = 140
a = 28
∴ a + 4 = 28 + 4
= 32
Source: Myschool Past Questions Archive (Record #8639)
Mathematics
UTME / JAMB 2006
Permutations, Combinations & Probability
Solution
A bag contains 5 black, 4 white and x red marbles. If the probability of picking a red marble is 2/5, find the value of x
A
8
B
10
C
4
D
6
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
Black = 5
White = 4
Red = x
Total = 9+x
P(red) = 2/5
x / (9+x) = 2/5
5x = 2(9+x)
5x = 18+2x
5x-2 = 18
3x = 18
x = 6
Source: Myschool Past Questions Archive (Record #8656)
Mathematics
UTME / JAMB 2006
Statistics (Data Representation, Central Tendency & Dispersion)
Solution
Find the variance of 2x, 2x-1 and 2x+1
A
2/3
B
2
C
√(2/3)
D
1
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
\(\sum x = 6x\)
\(\sum(x-\bar{x})^2\) = 2
\(\bar{x} = \frac{\sum x}{n}\)
= \(\frac{6x}{3}\)
= 2x
Variance = \(\frac{\sum(x-\bar{x})^2}{n}\)
= \(\frac{2}{3}\)
Source: Myschool Past Questions Archive (Record #8663)
Mathematics
UTME / JAMB 2006
Permutations, Combinations & Probability
Diagram
Solution
The table above shows the distribution of recharge cards of four major GSM operators. What is the probability that a recharge card selected at random will be GTN or Qtel?
Attached Figure / Diagram
A
3/20
B
1/4
C
2/5
D
3/4
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
P(GTN) = 5/20
P(Qtel) = 3/20
∴P(GTN or Qtel) = (5/20) + (3/20)
= 8/20
= 2/5
Source: Myschool Past Questions Archive (Record #8671)
Mathematics
UTME / JAMB 2006
Statistics (Data Representation, Central Tendency & Dispersion)
Diagram
Solution
The pie chart above shows the expenditure of a family whose income is N30,000. If the expenditure on food is twice that on housing and that on school fees is twice that on transport, how much does the family spend on food?
Attached Figure / Diagram
A
N 28 000
B
N 25 500
C
N15 000
D
N 12 500
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
Expenditure on housing = x
Expenditure on food = 2x
Expenditure on school fees = 90o
Expenditure on transport = 45o
x + 2x + 90 + 45 = 360o
3x + 135 = 360o
3x = 360 - 135
3x = 225
x = 225/3 = 75o
∴∠ for food = 2x = 2 * 75
= 150
360o = N30,000
1o = ?
1o = 30,000/360
150o = (30000/360) * (150/1)
= N12500
Source: Myschool Past Questions Archive (Record #8682)
Mathematics
UTME / JAMB 2006
Plane Geometry, Polygons & Circle Theorems
Solution
For what value of n is \(^{n+1}C_3\) = 4(\(^nC_3\))?
A
6
B
5
C
4
D
3
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
\(^{n+1}C_3 = 4(^nC_3)\\\frac{(n+1)!}{(n+1-3)!3!} = 4\left(\frac{n!}{(n-3)!3!}\right)\\\frac{(n+1)n!}{(n-2)(n-3)!}=4\left(\frac{n!}{n-3!}\right)\\=\frac{n+1}{n-2}=\frac{4}{1}\\n+1 = 4(n-2)\\n+1 = 4n-8\\-3n = -9\\\frac{-9}{-3}\\n=3\)
Source: Myschool Past Questions Archive (Record #8692)
Mathematics
UTME / JAMB 2006
Coordinate Geometry
Solution
The gradient of a curve is 2x + 7 and the curve passes through point (2, 0). find the equation of the curve.
A
y = x2 + 7x + 9
B
y = x2 + 7x - 18
C
y = x2 + 7x + 18
D
y = x2 + 14x + 11
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
dy/dx = 2x + 7
y = ∫2x + 7
y = x2 + 7x + C at (2,0)
0 = 22 + 7(2) + C
0 = 4 + 14 + C
0 = 18 + C
C = -18
∴ The equation is y = x2 + 7x - 18
Source: Myschool Past Questions Archive (Record #8694)
Mathematics
UTME / JAMB 2006
Differentiation & Applications
Solution
Differentiate (x\(^2\) - \(\frac{1}{x}\))\(^2\) with respect to x
A
4x3 - 4x - 2/x
B
4x3 - 2 + 2/x3
C
4x3 - 2 - 2/x3
D
4x3 - 3x + 2/x
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
y = (x\(^2\) - \(\frac{1}{x}\))\(^2\)
y = (x\(^2\) - \(\frac{1}{x}\))(x\(^2\) - \(\frac{1}{x}\))
y = x\(^4\) - x - x + 1/x2
y = x\(^4\) - 2x + 1/x2
y = x\(^4\) - 2x + x-2
dy/dx = 4x\(^3\) - 2 - 2x\(^-3\)
= 4x\(^3\) - 2 - \(\frac{2}{x^3}\)
Source: Myschool Past Questions Archive (Record #8696)
Mathematics
UTME / JAMB 2006
Differentiation & Applications
Solution
Find the value of x for which the function 3x\(^3\) - 9x\(^2\) is minimum
A
zero
B
2
C
3
D
5
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
y = 3x\(^3\) - 9x\(^2\)
\(\frac{dy}{dx}\) = 9x\(^2\) - 18x
As \(\frac{dy}{dx}\) = 0
9x\(^2\) - 18x = 0
9x(x-2) = 0
9x = 0 which implies x = 0
x - 2 = 0 implies x = 2
\(\frac{d^2y}{dx^2}\) = 18x - 18
when x = 0
\(\frac{d^2y}{dx^2}\) > 0 ∴ x is minimum
when x = 2, \(\frac{d^2y}{dx^2}\) = 18
∴ the value > 0 x is minimum
Source: Myschool Past Questions Archive (Record #8697)
Mathematics
UTME / JAMB 2006
Differentiation & Applications
Solution
If dy/dx = x + cos x, find y
A
x2 - sin x + c
B
x2 + sin x + c
C
x2/2 - sin x + c
D
x2/2 + sin x + c
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
dy/dx = x + cos x
y = ∫(x + cos x)dx
y = 1/2x2 + sin x + C
Source: Myschool Past Questions Archive (Record #8713)
Mathematics
UTME / JAMB 2006
Trigonometry, Elevation & Bearings
Solution
Differentiate (cos \theta - sin \theta)\(^2\)
A
-2 cos 2θ
B
-2 sin2θ
C
1 - 2 cos 2θ
D
1 - 2 sin 2θ
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
y = (cos\theta - sin\theta)
dy/dx = 2(cos\theta - sin\theta)(-sin\theta - cos\theta)
dy/dx = 2(-cos\thetasin\theta - cos2\theta + sin2\theta)
dy/dx = 2(- cos\(^2\)\theta + sin\(^2\)\theta)
= -2(cos\(^2\)\theta - sin\(^2\)\theta)
= -2(1 - 2sin\(^2\)\theta)
= -2cos2\theta
Source: Myschool Past Questions Archive (Record #8727)
Mathematics
UTME / JAMB 2006
Integration & Applications
Solution
Evaluate \(\int_{-4}^0(1 - 2x)dx\)
A
-16
B
-20
C
20
D
10
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
\(\int_{-4}^0(1 - 2x)\)dx = \([x - x^2]_{-4}^0\) = (0 - 0 + C) - (-4 - (-4)\(^2\) + C)
= C - ( - 4 - 16 +C )
= C - ( - 20 + C)
= C + 20 - C
= 20
Source: Myschool Past Questions Archive (Record #8732)
Mathematics
UTME / JAMB 2006
Plane Geometry, Polygons & Circle Theorems
Solution
simplify \(\frac{\frac{7}{9}-\frac{2}{3}}{\frac{1}{3}+\frac{\frac{2}{5}}{\frac{4}{5}}}\)
A
\(\frac{11}{12}\)
B
\(\frac{5}{6}\)
C
\(\frac{1}{5}\)
D
\(\frac{2}{15}\)
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
\(\frac{\frac{7}{9}-\frac{2}{3}}{\frac{1}{3}+\frac{\frac{2}{5}}{\frac{4}{5}}}\\
=\frac{\frac{7-6}{9}}{\frac{1}{3}+\frac{\frac{2}{5}}{\frac{4}{5}}}\\
=\frac{\frac{1}{9}}{\frac{1}{3}+\left(\frac{2}{5}\times \frac{5}{4}\right)}\\
=\frac{\frac{1}{9}}{\frac{1}{3}+\frac{1}{2}}\
=\frac{\frac{1}{9}}{\frac{2+3}{6}}\\
=\frac{\frac{1}{9}}{\frac{5}{6}}\\
=\frac{1}{9}\times\frac{6}{5}\\
=\frac{2}{15}\)
Source: Myschool Past Questions Archive (Record #8741)
Mathematics
UTME / JAMB 2006
Indices, Logarithms & Surds
Solution
If m:n = 13:11, find m\(^2\) - n\(^2\) : (m + n)\(^2\)
A
1:11
B
1:13
C
1:10
D
1:12
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
m : n = 13 : 11
m\(^2\) - n\(^2\) : (m + n)2
= 13\(^2\) - 11\(^2\) : (13 + 11)\(^2\)
= 169 - 121 : 24\(^2\)
= 48 : 576
= 1 : 12
Source: Myschool Past Questions Archive (Record #8742)
Mathematics
UTME / JAMB 2006
Indices, Logarithms & Surds
Solution
Calculate the logarithm to base 9 of 3\(^{-4}\) x 9\(^2\) x (81)\(^{-1}\)
A
2
B
zero
C
-2
D
-4
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
\(3^{-4}\times 9^2 \times 81^{-1}\)
=\(log_9 (3^{-4}\times 9^2 \times 81^{-1})\)
= \(log_9 \left(\frac{1}{3^4}\times 9^2 \times \frac{1}{81}\right)\)
= \(log_9 \left(\frac{1}{81}\times \frac{81}{1}\times \frac{1}{81}\right)\)
= \(log_9 \frac{1}{81}\)
= \(log_9 \frac{1}{9^2}\)
= \(log_9 9^{-2}\)
= -2\(log_9 9\)
- 2 \(\times 1\)
= -2
Source: Myschool Past Questions Archive (Record #8747)