POST UTME NILE UNIVERSITY 2024 Mathematics | Objective

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Question 1
A random variable $X$ has a probability distribution given by $P\( X = x \) = \frac{1}{2} cdot \frac{1}{2} = \frac{1}{4}$ for $x = 1, 2, 3, 4$. Find the expected value of $X$.
A. 1.5
B. 2
C. 2.5
D. 3
Question 2
Determine the sum of the infinite geometric series - 3 + 6 + 12 + … with first term - 3 and common ratio 2.
A. 9
B. -3
C. 12
D. -9
Question 3
A random sample of 25 students from a university had a mean height of 175.5 cm with a s\tandard deviation of 5.2 cm. If the population s\tandard deviation is 6.1 cm, calculate the 95% confidence interval for the population mean.
A. 168.1 cm, 182.9 cm
B. 170.5 cm, 180.5 cm
C. 172.1 cm, 178.9 cm
D. 174.5 cm, 176.5 cm
Question 4
A sequence is defined as: \[ a_n = \frac{1}{n} + \frac{1}{n+1} \] Calculate the sum of the first 10 terms of the sequence.
A. 2.928968253
B. 2.928968253
C. 2.928968253
D. 2.928968253
Question 5
If the sum of the first n terms of an arithmetic progression is 5n^2 + 4n, find the first term and the common difference.
A. a = 2, d = 3
B. a = 3, d = 2
C. a = 4, d = 1
D. a = 1, d = 4
Question 6
A curve is defined by the equation y = 2x^2 + 3x - 1. Find the area under the curve between x = 0 and x = 2.
A. \frac{13}{3}
B. \frac{13}{2}
C. \frac{13}{4}
D. \frac{13}{5}
Question 7
A fair six-sided die is rolled. What is the probability that the number rolled is greater than 4?
A. 1/6
B. 1/3
C. 2/3
D. 5/6
Question 8
A vector $\vec{a}$ has a magnitude of 5 and points in the direction of $\frac{1}{\sqrt{2}} \hat{i} + \frac{1}{\sqrt{2}} \hat{j}$. Find the magnitude of the vector $\vec{a} + \vec{b}$, where $\vec{b}$ is a vector with a magnitude of 3 and points in the opposite direction of $\vec{a}$.
A. 2
B. 4
C. 6
D. 8
Question 9
Find the equation of the circle with center \( -2, 3 \) and radius 4.
A. \( x + 2 \)^2 + \( y - 3 \)^2 = 16
B. \( x - 2 \)^2 + \( y + 3 \)^2 = 16
C. \( x + 2 \)^2 + \( y + 3 \)^2 = 16
D. \( x - 2 \)^2 + \( y - 3 \)^2 = 16
Question 10
Solve the inequality $|x - 2| > 3$.
A. \( -∞, -1 \) ∪ (4, ∞)
B. \( -∞, 1 \) ∪ (2, ∞)
C. \( -∞, -1 \) ∪ (1, ∞)
D. \( -∞, 1 \) ∪ (4, ∞)
Question 11
Solve for ( x ) in the equation \( \log_{10} \( x^2 \ \) = 4 ).
A. \( x = 10^2 \)
B. \( x = 10^4 \)
C. \( x = 10^{-2} \)
D. \( x = 10^{-4} \)
Question 12
Determine the value of $\int_{0}^{\pi} \frac{\sin^2 x}{1 + \cos^2 x} dx$.
A. \frac{\pi}{2}
B. \frac{\pi}{4}
C. \frac{\pi}{8}
D. \frac{\pi}{16}
Question 13
Find the value of $\int_0^1 \frac{1}{x^2 + 2x + 2} dx$.
A. 0.5
B. 1
C. 1.5
D. 2
Question 14
Find the derivative of $f(x) = \frac{1}{x^2 + 1}$ u\sing the chain rule.
A. \frac{-2x}{\( x^2 + 1 \)^2}
B. \frac{2x}{\( x^2 + 1 \)^2}
C. \frac{-x}{\( x^2 + 1 \)^2}
D. \frac{x}{\( x^2 + 1 \)^2}
Question 15
Solve the equation $\sin^2 x + \cos^2 x = 1$ for $x$ in the interval $[0, 2\pi]$.
A. \frac{\pi}{4}
B. \frac{\pi}{2}
C. \frac{3\pi}{4}
D. \pi

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