POST UTME NILE UNIVERSITY 2018 Mathematics | Objective

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Question 1
Solve the trigonometric equation \( \sin^2 x + \cos^2 x = 1 \ \).
A. x = \frac{\pi}{2}
B. x = \frac{\pi}{4}
C. x = \frac{3\pi}{4}
D. x = \frac{5\pi}{4}
Question 2
Find the area of the paralle\logram with adjacent sides given by the vectors \mathbf{a} = \begin{pmatrix} 2 \ 3 \ 1 \end{pmatrix} and \mathbf{b} = \begin{pmatrix} 1 \ -1 \ 2 \end{pmatrix}.
A. 5
B. 6
C. 7
D. 8
Question 3
A histogram of exam scores is shown below. Find the mean of the scores.
A. 50
B. 60
C. 70
D. 80
Question 4
Let ( X ) be a random variable with probability density function ( f(x) = egin{cases} 2x & \text{if } 0 leq x leq 1 \ 0 & \text{otherwise} \end{cases} ). Find the probability that ( X ) takes a value between 0.5 and 1.
A. 0.25
B. 0.5
C. 0.75
D. 1
Question 5
A fair six-sided die is rolled. What is the probability that the number obtained is a multiple of 3?
A. 1/6
B. 1/3
C. 2/3
D. 5/6
Question 6
A polynomial function is defined as ( f(x) = ax^3 + bx^2 + cx + d ). If \( f\( -1 \ \) = 2 ), ( f(1) = 4 ), and ( f(2) = 12 ), find the value of \( a + b + c + d \).
A. ( 10 )
B. ( 12 )
C. ( 14 )
D. ( 16 )
Question 7
Let $X$ and $Y$ be indep\endent random variables with probability density functions $f_X(x) = egin{cases} 2x & 0 leq x leq 1 \ 0 & \text{otherwise} \end{cases}$ and $f_Y(y) = egin{cases} 3y^2 & 0 leq y leq 1 \ 0 & \text{otherwise} \end{cases}$. Find the probability that $X + Y leq 1$.
A. \frac{1}{4}
B. \frac{1}{2}
C. \frac{3}{4}
D. \frac{5}{8}
Question 8
In a survey of 100 students, the mean height was 175 cm with a s\tandard deviation of 5 cm. If the heights of the students are normally distributed, what is the probability that a randomly selected student will be taller than 180 cm?
A. 0.1587
B. 0.3413
C. 0.5
D. 0.8413
Question 9
A circle has a radius of 4 cm. Find its circumference.
A. 8\pi
B. 10\pi
C. 12\pi
D. 16\pi
Question 10
A set of exam scores has a mean of 80 and a s\tandard deviation of 10. If the scores are normally distributed, what is the probability that a randomly selected score will be less than 70?
A. 0.1587
B. 0.3413
C. 0.5
D. 0.8413
Question 11
Solve the inequality \( 2x^2 - 5x - 3 > 0 \).
A. \( -∞, -1 \) ∪ (3, ∞)
B. \( -∞, 1 \) ∪ (3, ∞)
C. \( -∞, -1 \) ∪ (1, ∞)
D. \( -∞, 3 \) ∪ (5, ∞)
Question 12
Solve the system of equations \begin{align*} x + y + z &= 6 \ x + 2y + 3z &= 11 \ x + 3y + 6z &= 16 \end{align*} u\sing Cramer's rule.
A. \begin{pmatrix} 1 \ 2 \ 3 \end{pmatrix}
B. \begin{pmatrix} 2 \ 3 \ 4 \end{pmatrix}
C. \begin{pmatrix} 3 \ 4 \ 5 \end{pmatrix}
D. \begin{pmatrix} 4 \ 5 \ 6 \end{pmatrix}
Question 13
Find the mean of the data set $\{ 2, 4, 6, 8, 10 \}$.
A. 4
B. 5
C. 6
D. 7
Question 14
Solve the inequality \( 2x^2 + 5x - 3 > 0 \).
A. \( -\infty, -1 \) \cup \( 3, \infty \)
B. \( -\infty, -3 \) \cup \( 1, \infty \)
C. \( -\infty, -1 \) \cup \( 1, \infty \)
D. \( -\infty, -3 \) \cup \( 3, \infty \)
Question 15
A random variable ( X ) has a probability density function (pdf) given by ( f(x) = \frac{1}{2}e^{-|x|} ). Find the probability that ( X ) is greater than 1.
A. 1/4
B. 1/2
C. 3/4
D. 1

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