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Mathematics
UTME / JAMB 2007
Integration & Applications
Solution
Integrate \(\frac{x^2 -\sqrt{x}}{x}\) with respect to x
A
\(\frac{x^2}{2}-2\sqrt{x}+K\)
B
\(\frac{2(x^2 - x)}{3x}+K\)
C
\(\frac{x^2}{2}-\sqrt{x}+K\)
D
\(\frac{(x^2 - x)}{3x}+K\)
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
\(\int \frac{x^2 -\sqrt{x}}{x} = \int \frac{x^2}{x} - \frac{x^{\frac{1}{2}}}{x}\\
\int x - x^{\frac{-1}{2}}\\
=\left(\frac{1}{2}\right)x^2 - \frac{x^{\frac{1}{2}}}{\frac{1}{2}}+K\\
=\frac{x^2}{2}-2x^{\frac{1}{2}}+K\\
=\frac{x^2}{2}-2\sqrt{x}+K\)
Source: Myschool Past Questions Archive (Record #9221)
Mathematics
UTME / JAMB 2007
Differentiation & Applications
Solution
If y = x cos x, find dy/dx
A
sin x - x cos x
B
sin x + x cos x
C
cos x + x sin x
D
cos x - x sin x
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
y = x cos x
dy/dx = 1. cos x + x (-sin x)
= cos x - x sin x
Source: Myschool Past Questions Archive (Record #9222)
Mathematics
UTME / JAMB 2007
Linear, Simultaneous & Quadratic Equations
Solution
Find the value of x for which the function f(x) = 2x\(^3\) - x\(^2\) - 4x + 4 has a maximum value
A
2/3
B
1
C
- 2/3
D
- 1
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
f(x) = 2x\(^3\) - x\(^2\) - 4x – 4
f’(x) = 6x\(^2\) - 2x – 4
As f’(x) = 0
Implies 6x\(^2\) - 2x – 4 = 0
3x – x – 2 = 0 (By dividing by 2)
(3x + 2)(x - 1) = 0
3x + 2 = 0 implies x = \(\frac{-2}{3}\)
Or x - 1 = 0 implies x = 1
f’(x) = 6x\(^2\) - 2x – 4
f’’(x) = 12x – 2
At max point f’’(x) < 0
∴f’’(x) = 12x – 2 at x = 1
= 12(1) – 2
= 12 – 2 = 10
∴ Max at x = \(\frac{-2}{3}\)
Source: Myschool Past Questions Archive (Record #9223)
Mathematics
UTME / JAMB 2007
Integration & Applications
Solution
Determine the value of \(\int_0 ^{\frac{\pi}{2}
}(-2cos x)dx\)
A
-2
B
-1/2
C
-3
D
-3/2
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
\(\int_0 ^{\frac{\pi}{2}}(-2cos x)dx = [-2sin x + c]_0 ^{\frac{\pi}{2}}\)
=(-2sin\(\frac{\pi}{2}+c+2sin0-c)\)
=-2sin90+c+2sin0-c
=-2(1)+2(0)
=-2
Source: Myschool Past Questions Archive (Record #9224)
Mathematics
UTME / JAMB 2007
Binary Operations, Matrices & Determinants
Solution
A binary operation ⊕ on real numbers is defined by x⊕y = xy + x + y for any two real numbers x and y. The value of (-3/4)⊕6 is
A
3/4
B
-9/2
C
45/4
D
-3/4
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
x⊕y = xy + x + y
= -3/4 (6) + (-3/4) + 6
= -9/2 - 3/4 + 6
= (-18-3+3+24) / 4
= 3/4
Source: Myschool Past Questions Archive (Record #9225)
Mathematics
UTME / JAMB 2007
Linear, Simultaneous & Quadratic Equations
Diagram
Solution
The graph above is represented by
Attached Figure / Diagram
A
y = x3 - 3x - 2
B
y = x3 + 2x2 - x - 2
C
y = x3 - 4x2 + 5x - 2
D
y = x3 - 4x + 2
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
x = -2, x = -1 and x = 1
then the factors; x+2, x+1 and x-1
Product of the factors; (x+2)(x+1)(x-1)
= y = (x + 2)(x2 - x + x - 1)
= y = (x+2)(x2-1)
x3 - x + 2x2 - 2 = y
x3 + 2x2 - x - 2 = y
Source: Myschool Past Questions Archive (Record #9226)
Mathematics
UTME / JAMB 2007
Algebraic Expressions, Polynomials & Factorisation
Solution
Make L the subjects of the formula if d = \(\sqrt{\frac{42w}{5l}}\)
A
\(\sqrt{\frac{42w}{5d}}\)
B
\(\frac{42W}{5d^2}\)
C
\(\frac{42}{5dW}\)
D
\(\frac{1}{d}\sqrt{\frac{42w}{5}}\)
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
d = \(\sqrt{\frac{42w}{5l}}\)
square both side of the equation
\(d^2 = \left(\sqrt{\frac{42W}{5l}}\right)^2\\
d^2 = \frac{42W}{5l}\\
5ld^2=42W\\
l = \frac{42W}{5d^2}\)
Source: Myschool Past Questions Archive (Record #9227)
Mathematics
UTME / JAMB 2007
Inequalities & Linear Programming
Solution
The solution of the quadratic inequality (x\(^2\) + x - 12) \ge 0 is
A
x ≥ -3 or x ≤ 4
B
x ≥ 3 or x ≥ -4
C
x ≤ 3 or x \(\leq\) -4
D
x ≥ 3 or x ≤ -4
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
(x\(^2\) + x - 12) \ge 0
(x + 4)(x - 3) \ge 0
Either x + 4 \ge 0 implies x \ge - 4
Or x - 3 \ge 0 implies x \ge 3
∴ x \ge 3 or x \(\leq\) -4
Source: Myschool Past Questions Archive (Record #9228)
Mathematics
UTME / JAMB 2007
Algebraic Expressions, Polynomials & Factorisation
Solution
Factorize 2t2 + t - 15
A
(2t - 3)(t + 5)
B
(t + 3)(2t - 5)
C
(t + 3)(t - 5)
D
(2t + 3)(t - 5)
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
2t2 + t - 15 = (2t-5)(t+3)
Source: Myschool Past Questions Archive (Record #9229)
Mathematics
UTME / JAMB 2007
Inequalities & Linear Programming
Solution
Solve the inequalities -3(x - 2) < -2(x + 3)
A
x > 12
B
x < 12
C
x > -12
D
x < - 12
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
-3(x-3) -12/-1
x > 12
Source: Myschool Past Questions Archive (Record #9230)
Mathematics
UTME / JAMB 2007
Plane Geometry, Polygons & Circle Theorems
Solution
W ∝ L2 and W = 6 when L = 4. If L = √17 find W
A
67/8
B
65/8
C
63/8
D
61/8
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
W ∝ L2
W = KL2
K = W/L2
K = 6/42
K = 6/16 = 3/8
W = 3/8 L2
W = 3/8(√17)2
W = 3/8 x 17
W = 51/8 = 63/8
Source: Myschool Past Questions Archive (Record #9231)
Mathematics
UTME / JAMB 2007
Binary Operations, Matrices & Determinants
Solution
A binary operation Δ is defined by aΔb = a + b + 1 for any numbers a and b. Find the inverse of the real number 7 under the operation Δ, if the identity element is -1
A
-7
B
-9
C
5
D
9
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
a*e = a + e + 1 = a
implies e+ 1 = 0
∴ e = -1
7 * e = -1
∴ a + 7 + 1 = -1
a + 8 = -1
a+8 = -1
a = -1-8
a = -9
Source: Myschool Past Questions Archive (Record #9232)
Mathematics
UTME / JAMB 2007
Sequences & Series (AP & GP)
Solution
The nth term of the sequence \(\frac{3}{2}\), 3, 7, 16, 35, 74 ..... is
A
5 . 2n-2 - n
B
5 . 2n-2 - \(\frac{(n+1)}{2}\)
C
3 . 2n-2
D
\(\frac{3n}{2}\)
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
3/2, 3, 7, 16, 35, 74, ....
Using the method of substitution
When n = 1, 5 . 2n-2 - 1 = 5 . 21-2 - 1
= 5 x 2-1 - 1
= 5 x 1/2 - 1
= 5/2 - 1 = 3/2
When n = 1, 5 . 2n-2 - n
= 5 . 22-2 - 2
= 5 x 20 - 2
= 5 x 1 – 2 = 3
When n = 3, 5 . 2n-2 - n
= 5 . 23-2 - 3
= 5 x 21 - 3
= 5 x 2 – 3
= 10 -3 = 7
When n = 4, 5 . 2n-2 - n
= 5 . 24-2 - 4
= 5 x 22 - 4
= 5 x 4 – 4
= 20 – 4 = 16
When n = 5, 5 . 2n-2 - n
= 5 . 25-2 - 5
= 5 x 23 - 5
= 5 x 8 – 5
= 40 – 5
= 35
When n = 6, 5 . 2n-2 - n
= 5 . 26-2 - 6
= 5 x 24 - 6
= 5 x 16 – 6
= 80 – 6 = 74
∴ 5 . 2n-2 - 1 = 3/2, 3, 7, 16, 35, 74
Source: Myschool Past Questions Archive (Record #9233)
Mathematics
UTME / JAMB 2007
Sequences & Series (AP & GP)
Solution
Find the sum to infinity of the series \(2+\frac{3}{2}+\frac{9}{8}+\frac{27}{32}+......\)
A
1
B
2
C
8
D
4
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
a = 2
r = \(\frac{3}{4}\)
S = \(\frac{a}{1-r}\)
S = \(\frac{2}{1-\frac{3}{4}}\)
= \(\frac{2}{\frac{1}{4}}\)
S = \(\frac{2}{1}\times \frac{4}{1}\)
= 8.
Source: Myschool Past Questions Archive (Record #9234)
Mathematics
UTME / JAMB 2007
Indices, Logarithms & Surds
Solution
Find y, if \(\sqrt{12}-\sqrt{147}+y\sqrt{3} = 0\)
A
5
B
1
C
7
D
3
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
\(\sqrt{12}\) -\(\sqrt{147}\) + y\(\sqrt{3}\) = 0
\(\sqrt{4\times 3}\) - \(\sqrt{49\times 3}\) + y\(\sqrt{3}\) = 0
2\(\sqrt{3}\) - 7\(\sqrt{3}\) + y\(\sqrt{3}\) = 0
y\(\sqrt{3}\) = 7\(\sqrt{3}\) - 2\(\sqrt{3}\)
y = \(\frac{5\sqrt{3}}{\sqrt{3}}\)
y = 5
Source: Myschool Past Questions Archive (Record #9235)