POST UTME MOUNTAIN TOP UNIVERSITY 2023 Mathematics | Objective

Practice these randomly selected questions to test your readiness.

Question 1
A rec\tangular prism has a length of 5 cm, a width of 3 cm, and a height of 2 cm. Find its volume.
A. 5 \cdot 3 \cdot 2
B. 5 \cdot 3 + 2
C. 5 \cdot 2 + 3
D. 3 \cdot 2 + 5
Question 2
Solve the equation \( \frac{1}{2} \sin^2 x + \frac{1}{4} \cos^2 x = \frac{1}{4} \)
A. \frac{\pi}{6}
B. \frac{\pi}{4}
C. \frac{\pi}{3}
D. \frac{\pi}{2}
Question 3
Let \( mathbf{a} = egin{pmatrix} 2 \ 3 \end{pmatrix} \) and \( mathbf{b} = egin{pmatrix} 4 \ 5 \end{pmatrix} \). Find the unit vector in the direction of \( mathbf{a} + mathbf{b} \).
A. \( egin{pmatrix} 0.6 \ 0.8 \end{pmatrix} \)
B. \( egin{pmatrix} 0.8 \ 0.6 \end{pmatrix} \)
C. \( egin{pmatrix} 0.4 \ 0.6 \end{pmatrix} \)
D. \( egin{pmatrix} 0.6 \ 0.4 \end{pmatrix} \)
Question 4
Solve the inequality \( x^2 - 4x + 4 \geq 0 \)
A. \\left( -\\infty, 2 \\right]
B. \\left( 2, \\infty \\right]
C. \left\( -\infty, 2 \right \) \cup \left\( 2, \infty \right \)
D. \\left( -\\infty, 2 \\right]
Question 5
Find the determinant of the 3x3 matrix \begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9 \end{bmatrix}.
A. 0
B. 1
C. 2
D. 3
Question 6
A set of data has a mean of 25 and a s\tandard deviation of 5. Find the probability that a randomly selected value from the set is greater than 30.
A. \( \frac{1}{4} \)
B. \( \frac{1}{2} \)
C. \( \frac{3}{4} \)
D. \( \frac{1}{3} \)
Question 7
The polynomial $P(x) = x^3 - 6x^2 + 11x - 6$ has a root at $x = 1$. Find the value of $P(2)$.
A. 0
B. 2
C. 4
D. 6
Question 8
Find the area under the curve \( y = x^2 \) from \( x = 0 \) to \( x = 2 \) u\sing integration by substitution.
A. \( \frac{8}{3} \)
B. \( \frac{16}{3} \)
C. \( \frac{24}{3} \)
D. \( \frac{32}{3} \)
Question 9
In a triangle $ABC$, the length of side $AB$ is 5 cm, the length of side $BC$ is 6 cm, and the length of side $AC$ is 7 cm. If the angle $B$ is $60^{circ}$, what is the area of the triangle?
A. 10 cm^2
B. 12 cm^2
C. 15 cm^2
D. 18 cm^2
Question 10
A set of 10 numbers has a mean of 20 and a s\tandard deviation of 5. If the set is multiplied by 2, what is the new s\tandard deviation?
A. 10
B. 20
C. 40
D. 50
Question 11
Find the value of x in the equation \( egin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} egin{bmatrix} x \ 1 \end{bmatrix} = egin{bmatrix} 7 \ 11 \end{bmatrix} \).
A. 3
B. 4
C. 5
D. 6
Question 12
Find the derivative of the function \( f(x) = \frac{x^2}{x^2 + 1} \)
A. \frac{2x\( x^2 + 1 \) - 2x^3}{\( x^2 + 1 \)^2}
B. \frac{2x\( x^2 + 1 \) + 2x^3}{\( x^2 + 1 \)^2}
C. \frac{2x\( x^2 + 1 \) - 2x^3}{\( x^2 + 1 \)^2}
D. \frac{2x\( x^2 + 1 \) + 2x^3}{\( x^2 + 1 \)^2}
Question 13
A binary operation $*$ on the set of real numbers is defined as $a*b = a^2 + b^2$. Find the value of $2*3$.
A. 13
B. 17
C. 19
D. 25
Question 14
In a certain number base $b$, the number $101_5$ is equivalent to $b^2 + 1$ in base $10$. Find the value of $b$.
A. 2
B. 3
C. 4
D. 5
Question 15
The polynomial ( p(x) = x^3 - 6x^2 + 11x - 6 ) has a root at \( x = 1 \). What is the quotient when ( p(x) ) is divided by \( x - 1 \)?
A. x^2 - 5x + 6
B. x^2 - 5x + 5
C. x^2 - 5x + 4
D. x^2 - 5x + 3

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