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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)
Mathematics
UTME / JAMB 2006
Indices, Logarithms & Surds
Solution
If (K2)\(_6\) * 3\(_6\) = 3\(_5\)(K4)\(_5\), what is the value of k?
A
1
B
4
C
3
D
2
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
(K2)\(_6\) * 3\(_6\) = 3\(_5\)(K4)\(_5\)
(K*6\(^1\) + 2*6\(^0\)) 3*6\(^0\) = 3*5\(^0\)(K*5\(^1\) + 4*5\(^0\))
3(6K + 2) = 3(5K + 4)
6K + 2 = 5K + 4
6K - 5K = 4 - 2
K = 2
Source: Myschool Past Questions Archive (Record #8753)
Mathematics
UTME / JAMB 2006
Fractions, Decimals, Percentages & Approximations
Solution
In a small village of 500 people, 350 speak the local language while 200 speak pidgin English. What percentage of the population speak both.
A
30%
B
10%
C
50%
D
14%
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
Let n(L∩P) = x
n(L) only = 350 - x
n(P) only = 200 - x
∴350 - x + x + 200 - x = 500
550 - x = 500
x = 550 - 500
x = 50 speak both
∴x% = (50/500) * (100/1)
= 10%
Source: Myschool Past Questions Archive (Record #8827)
Mathematics
UTME / JAMB 2006
Plane Geometry, Polygons & Circle Theorems
Solution
Find the tax on an income of N20,000 if no tax is paid on the first N10,000 and tax is paid at N50 in N1000 on the next N5000 and at N55 in N1000 on the remainder
A
N500
B
N552
C
N255
D
N525
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
N20,000 - N10,000 = N10,000
1st tax on N5000 at N50 per N1000
= 5*50 = N250
2nd tax on the remaining N5000 at N55 per N1000 = 5*55 = N275
Total tax paid = N250 + N275
= N525
Source: Myschool Past Questions Archive (Record #8828)
Mathematics
UTME / JAMB 2006
Indices, Logarithms & Surds
Solution
Evaluate \(\frac{2}{6-5\sqrt{3}}\)
A
\(-\left(\frac{12}{39}-\frac{10\sqrt{3}}{39}\right)\)
B
\(\frac{12}{39}-\frac{10\sqrt{3}}{39}\)
C
\(-\left(\frac{12}{39}+\frac{10\sqrt{3}}{39}\right)\)
D
\(\frac{12}{39}+\frac{10\sqrt{3}}{39}\)
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
\(\frac{2}{6-5\sqrt{3}} = \frac{2}{6-5\sqrt{3}} \times \frac{6+5\sqrt{3}}{6+5\sqrt{3}}\\
=\frac{2(6+5\sqrt{3})}{(6-5\sqrt{3})(6+5\sqrt{3})}\\
=\frac{12+10\sqrt{3}}{36-25(3)}\\
=\frac{12+10\sqrt{3}}{36-75}\\
=\frac{12+10\sqrt{3}}{-39}\\
=-\left(\frac{12}{39}+\frac{10\sqrt{3}}{39}\right)\)
Source: Myschool Past Questions Archive (Record #8829)
Mathematics
UTME / JAMB 2006
Plane Geometry, Polygons & Circle Theorems
Solution
Compute 1100112 + 111112
A
10010102
B
10100102
C
10001102
D
10001002
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
1100112 + 111112 = 10100102
Source: Myschool Past Questions Archive (Record #8830)
Mathematics
UTME / JAMB 2006
Indices, Logarithms & Surds
Solution
Simplify \((25)^{\frac{-1}{2}} \times (27)^{\frac{1}{3}} + (121)^{\frac{-1}{2}} \times (625)^{\frac{-1}{4}}\)
A
34/55
B
9/11
C
14/5
D
3/275
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
(25)\(^{\frac{-1}{2}}\) x (27)\(^{\frac{1}{3}}\) + (121)\(^{\frac{-1}{2}}\) x (625)\(^{\frac{-1}{4}}\)
5\(^{2 \times \frac{-1}{2}}\) x 3\(^{3 \times \frac{1}{3}}\) + 11\(^{2 \times \frac{-1}{2}}\) x 5\(^{4 \times \frac{-1}{4}}\)
5\(^{-1}\) x 3\(^1\) x 11\(^{-1}\) x 5\(^{-1}\)
\(\frac{1}{5} \times \frac{3}{1} + \frac{1}{11} \times \frac{1}{5}\)
\(\frac{3}{5} + \frac{1}{55} = \frac{33+1}{55}\)
= \(\frac{34}{55}\)
Source: Myschool Past Questions Archive (Record #8831)