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Mathematics
UTME / JAMB 2011
Number and Numeration
Solution
Find the remainder when X3 - 2X2 + 3X - 3 is divided by X2 + 1
A
2X - 1
B
X + 3
C
2X + 1
D
X - 3
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
X2 + 1 \(\frac{X - 2}{\sqrt{X^3 - 2X^2 + 3n - 3}}\)
= \(\frac {- 6X^3 + n}{-2X^2 + 2X - 3}\)
= \(\frac{(-2X^2 - 2)}{2X - 1}\)
Remainder is 2X - 1
Source: Myschool Past Questions Archive (Record #29376)
Mathematics
UTME / JAMB 2011
Number and Numeration
Solution
Factorize completely 9y2 - 16X2
A
(3y - 2x)(3y + 4x)
B
(3y + 4x)(3y + 4x)
C
(3y + 2x)(3y - 4x)
D
(3y - 4x)(3y + 4x)
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
9y2 - 16x2
= 32y2 - 42x2
= (3y - 4x)(3y +4x)
Source: Myschool Past Questions Archive (Record #29378)
Mathematics
UTME / JAMB 2011
Number and Numeration
Solution
If x varies directly as square root of y and x = 81 when y = 9, Find x when y = 1\(\frac{7}{9}\)
A
20\(\frac{1}{4}\)
B
27
C
2\(\frac{1}{4}\)
D
36
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
x \(\alpha\sqrt y\)
x = k\(\sqrt y\)
81 = k\(\sqrt9\)
k = \(\frac{81}{3}\)
= 27
therefore, x = 27\(\sqrt y\)
y = 1\(\frac{7}{9}\) = \(\frac{16}{9}\)
x = 27 x \(\sqrt{\frac{16}{9}}\)
= 27 x \(\frac{4}{3}\)
dividing 27 by 3
= 9 x 4
= 36
Source: Myschool Past Questions Archive (Record #29381)
Mathematics
UTME / JAMB 2011
Number and Numeration
Solution
T varies inversely as the cube of R. When R = 3, T = \(\frac{2}{81}\), find T when R = 2
A
\(\frac{1}{18}\)
B
\(\frac{1}{12}\)
C
\(\frac{1}{24}\)
D
\(\frac{1}{6}\)
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
T \(\alpha \frac{1}{R^3}\)
T = \(\frac{k}{R^3}\)
k = TR3
= \(\frac{2}{81}\) x 33
= \(\frac{2}{81}\) x 27
dividing 81 by 27
k = \(\frac{2}{2}\)
therefore, T = \(\frac{2}{3}\) x \(\frac{1}{R^3}\)
When R = 2
T = \(\frac{2}{3}\) x \(\frac{1}{2^3}\) = \(\frac{2}{3}\) x \(\frac{1}{8}\)
= \(\frac{1}{12}\)
Source: Myschool Past Questions Archive (Record #29385)
Mathematics
UTME / JAMB 2011
Number and Numeration
Solution
A binary operation \(\oplus\) om real numbers is defined by x \(\oplus\) y = xy + x + y for two real numbers x and y. Find the value of 3 \(\oplus\) - \(\frac{2}{3}\).
A
- \(\frac{1}{2}\)
B
\(\frac{1}{3}\)
C
-1
D
2
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
N + Y = XY + X + Y
3 + -\(\frac{2}{3}\) = 3(- \(\frac{2}{3}\)) + 3 + (- \(\frac{2}{3}\))
= -2 + 3 -\(\frac{2}{3}\)
= \(\frac{1 - 2}{1 - 3}\)
= \(\frac{1}{3}\)
Source: Myschool Past Questions Archive (Record #29421)
Mathematics
UTME / JAMB 2011
Number and Numeration
The perpendicular bisector of a line XY is the locus of a point
A
whose distance from X is always twice its distance from Y
B
whose distance from Y is always twice its distance from X.
C
which moves on the line XY
D
which is equidistant from the points X and Y
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
No expanded working has been attached for this question yet. The verified answer is Option D.
Source: Myschool Past Questions Archive (Record #29447)
Mathematics
UTME / JAMB 2011
Number and Numeration
Solution
The midpoint of P(x, y) and Q(8, 6) is (5, 8). Find x and y.
A
(2, 10)
B
(2, 8)
C
(2, 12)
D
(2, 6)
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
P(x, y) Q(8, 6)
midpoint = (5, 8)
x + 8 = 5
\(\frac{y + 6}{2}\) = 8
x + 8 = 10
x = 10 - 8 = 2
y + 6 = 16
y + 16 - 6 = 10
therefore, P(2, 10)
Source: Myschool Past Questions Archive (Record #29452)
Mathematics
UTME / JAMB 2011
Number and Numeration
Solution
Find the value of x at the minimum point of the curve y = x3 + x2 - x + 1
A
\(\frac{1}{3}\)
B
-\(\frac{1}{3}\)
C
1
D
-1
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
y = x3 + x2 - x + 1
\(\frac{dy}{dx}\) = \(\frac{d(x^3)}{dx}\) + \(\frac{d(x^2)}{dx}\) - \(\frac{d(x)}{dx}\) + \(\frac{d(1)}{dx}\)
\(\frac{dy}{dx}\) = 3x2 + 2x - 1 = 0
\(\frac{dy}{dx}\) = 3x2 + 2x - 1
At the maximum point \(\frac{dy}{dx}\) = 0
3x2 + 2x - 1 = 0
(3x2 + 3x) - (x - 1) = 0
3x(x + 1) -1(x + 1) = 0
(3x - 1)(x + 1) = 0
therefore x = \(\frac{1}{3}\) or -1
For the maximum point
\(\frac{d^2y}{dx^2}\) < 0
\(\frac{d^2y}{dx^2}\) 6x + 2
when x = \(\frac{1}{3}\)
\(\frac{dx^2}{dx^2}\) = 6(\(\frac{1}{3}\)) + 2
= 2 + 2 = 4
\(\frac{d^2y}{dx^2}\) > o which is the minimum point
when x = -1
\(\frac{d^2y}{dx^2}\) = 6(-1) + 2
= -6 + 2 = -4
-4 < 0
therefore, \(\frac{d^2y}{dx^2}\) < 0
the minimum point is 1/3
Source: Myschool Past Questions Archive (Record #29499)
Mathematics
UTME / JAMB 2011
Number and Numeration
Solution
Evaluate \(\int^{1}_{0}\)(3 - 2x)dx
A
33m
B
5
C
2
D
6
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
\(\int^{1}_{0}\)(3 - 2x)dx
[3x - x\(^2\)]\(_{0} ^{1}\)
[3(1) - (1)\(^2\)] - [3(0) - (0)\(^2\)]
(3 - 1) - (0 - 0) = 2 - 0
= 2
Source: Myschool Past Questions Archive (Record #29500)
Mathematics
UTME / JAMB 2011
Number and Numeration
Solution
The sum of four consecutive integers is 34. Find the least of these numbers
A
7
B
6
C
8
D
5
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
Let the numbers be a, a + 1, a + 2, a + 3
a + a + 1 + a + 2 + a + 3 = 34
4a = 34 - 6
4a = 28
a = \(\frac{28}{4}\)
= 7
The least of these numbers is a = 7
Source: Myschool Past Questions Archive (Record #29503)
Mathematics
UTME / JAMB 2011
Number and Numeration
Solution
In how many ways can five people sit round a circular table?
A
24
B
60
C
12
D
120
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
The first person will sit down and the remaining will join.
i.e. (n - 1)!
= (5 - 1)! = 4!
= 24 ways
Source: Myschool Past Questions Archive (Record #29532)
Mathematics
UTME / JAMB 2011
Number and Numeration
Solution
Find the probability that a number picked at random from the set(43, 44, 45, ..., 60) is a prime number.
A
\(\frac{2}{3}\)
B
\(\frac{1}{3}\)
C
\(\frac{2}{9}\)
D
\(\frac{7}{9}\)
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
Prime numbers = (43,47,53,59)
N = (43, 44, 45,..., 60)
The universal set contains 18 numbers.
The prime numbers between 43 and 60 are 4
Probability of picking a prime number = \(\frac{4}{18}\)
= \(\frac{2}{9}\)
Source: Myschool Past Questions Archive (Record #29534)
Mathematics
UTME / JAMB 2011
Number and Numeration
Diagram
From the venn diagram above, the complement of the set P\(\cap\)Q is given by
Attached Figure / Diagram
A
{a, b, d, e}
B
{b, d}
C
{a, e}
D
{c}
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
No expanded working has been attached for this question yet. The verified answer is Option A.
Source: Myschool Past Questions Archive (Record #33148)
Mathematics
UTME / JAMB 2011
Number and Numeration
Diagram
Solution
The pie chart shows the distribution of courses offered by students. What percentage of the students offer English?
Attached Figure / Diagram
A
30%
B
25%
C
35%
D
20%
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
\(\frac{90}{360} \times 100 = \frac{100}{4}\)
= 25
Source: Myschool Past Questions Archive (Record #33149)
Mathematics
UTME / JAMB 2011
Number and Numeration
Diagram
Solution
The bar chart above shows the distribution of SS2 students in a school.
Find the total number of students
Attached Figure / Diagram
A
180
B
135
C
210
D
105
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
45 + 60 + 30 + 45 = 180 students.
Source: Myschool Past Questions Archive (Record #33150)