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Mathematics
UTME / JAMB 2007
Plane Geometry, Polygons & Circle Theorems
Solution
A senatorial candidate had planned to visit seven cities prior to a primary election. However, he could only visit four of the cities. How many different itineraries could be considered?
A
640
B
840
C
520
D
920
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
Number of itineraries = 7P4
= \(\frac{7!}{(7-4)!}\)
=\(\frac{7!}{3!}\)
=\(\frac{7 \times 6 \times 5 \times 4 \times 3!}{3!}\)
= 840
Source: Myschool Past Questions Archive (Record #9144)
Mathematics
UTME / JAMB 2007
Statistics (Data Representation, Central Tendency & Dispersion)
Diagram
Solution
The pie chart above illustrate the amount of private time a student spends in a week studying various subjects. Find the value of k
Attached Figure / Diagram
A
90o
B
60o
C
30o
D
40o
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
K + 2K + 3K + 75 + 105 = 360 (at a point)
6K + 180 = 360
6K = 360 - 180
6K = 180
k = 180/6 = 30º
Source: Myschool Past Questions Archive (Record #9150)
Mathematics
UTME / JAMB 2007
Permutations, Combinations & Probability
Diagram
Solution
The table above shows the number of pupils in each age group in a class. What is the probability that a pupil chosen at random is at least 11 years old?
Attached Figure / Diagram
A
27/40
B
17/20
C
3/30
D
33/40
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
P(At least 11 yrs) = P(11yrs) + P(12yrs)
= \(\frac{27}{40}\) + \(\frac{7}{40}\)
= \(\frac{34}{40}\)
= \(\frac{17}{20}\)
Source: Myschool Past Questions Archive (Record #9152)
Mathematics
UTME / JAMB 2007
Statistics (Data Representation, Central Tendency & Dispersion)
Solution
What is the mean deviation of 3, 5, 8, 11, 12 and 21?
A
4.7
B
60
C
3.7
D
10
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
Mean \(= \bar{x} = \frac{60}{6} = 10\)
mean deviation \(= \frac{\sum|x-\bar{x}|}{n} = \frac{28}{6}\)
= 4.7
Source: Myschool Past Questions Archive (Record #9218)
Mathematics
UTME / JAMB 2007
Statistics (Data Representation, Central Tendency & Dispersion)
Solution
Marks
3
4
5
6
7
8
Frequency
5
(y - 1)
y
9
4
1
The table above gives the frequency distribution of marks obtained by a group of students in a test. If the total mark scored is 200, calculate the value of y
A
15
B
13
C
11
D
8
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
Total mark scored = 200
∴200 = 15 + 4y - 4 + 5y + 54 + 28 + 8
200 = 9y + 101
200 - 101 = 9y
99 = 9y
∴y = 11
Source: Myschool Past Questions Archive (Record #9219)
Mathematics
UTME / JAMB 2007
Permutations, Combinations & Probability
Solution
In how many ways can 6 subjects be selected from 10 subjects for an examination
A
218
B
216
C
215
D
210
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
\(^{10}C_6 = \frac{10!}{(10-6)!6!}=\frac{10!}{4!6!}\\
=\frac{(10\times 9\times 8\times 7 \times 6!)}{4\times 3\times 2\times 1\times 6!}\\
=210\)
Source: Myschool Past Questions Archive (Record #9220)
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)