POST UTME UI 2025 Mathematics | Objective

Practice these randomly selected questions to test your readiness.

Question 1
Solve the inequality \( \frac{x^2 - 4}{x^2 - 9} > 0 \).
A. \( x in \( -infty, -3 \ \) cup (3, infty) )
B. \( x in \( -infty, -3 \ \) cup (3, 4) cup (4, infty) )
C. \( x in \( -infty, -3 \ \) cup (3, 4) )
D. \( x in \( -infty, -3 \ \) cup (4, infty) )
Question 2
A histogram of exam scores has a mean of 75 and a s\tandard deviation of 10. If the scores are normally distributed, what is the probability that a randomly selected score is between 60 and 90?
A. 0.68
B. 0.69
C. 0.70
D. 0.71
Question 3
Find the derivative of the function f(x) = \[\frac{x^2 + 2x - 3}{x^2 - 4}\] u\sing the quotient rule.
A. \[\frac{2x + 2}{\( x^2 - 4 \)^2}\]
B. \[\frac{2x^2 + 4x - 6}{\( x^2 - 4 \)^2}\]
C. \[\frac{2x^2 + 4x - 6}{\( x^2 - 4 \)^2} + \frac{2x + 2}{x^2 - 4}\]
D. \[\frac{2x^2 + 4x - 6}{\( x^2 - 4 \)^2} - \frac{2x + 2}{x^2 - 4}\]
Question 4
A circle has an equation of the form x^2 + y^2 + 2gx + 2fy + c = 0. Find the center of the circle.
A. \( -g, -f \)
B. (g, f)
C. \( -f, -g \)
D. (f, g)
Question 5
Find the derivative of the function ( f(x) = \frac{1}{x^2 + 1} ) u\sing the chain rule.
A. ( f'(x) = -\frac{2x}{\( x^2 + 1 \)^2} )
B. ( f'(x) = \frac{2x}{\( x^2 + 1 \)^2} )
C. ( f'(x) = -\frac{2}{\( x^2 + 1 \)^2} )
D. ( f'(x) = \frac{2}{\( x^2 + 1 \)^2} )
Question 6
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}{2}
B. \frac{1}{3}
C. \frac{2}{3}
D. \frac{3}{4}
Question 7
In a right-angled triangle, the length of the hypotenuse is 10 units and one of the acute angles is 30°. Find the length of the side opposite the 30° angle.
A. 5.773
B. 6.928
C. 7.071
D. 8.062
Question 8
Find the derivative of the function ( 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 9
A vector ( mathbf{a} ) has a magnitude of 5 and makes an angle of 30° with the positive x-axis. Find the x and y components of ( mathbf{a} ).
A. 4.33, 2.5
B. 4.5, 2.5
C. 4.33, 2.33
D. 4.5, 2.33
Question 10
Find the value of $\lim_{x\to\infty} \left\( \frac{\ln x}{x}\right \)$.
A. 0
B. 1
C. ∞
D. -∞
Question 11
Find the equation of the line pas\sing through the points ( (2,3) ) and ( (4,5) ).
A. \( y = x + 1 \)
B. \( y = x - 1 \)
C. \( y = -x + 1 \)
D. \( y = x - 2 \)
Question 12
In a binary operation \( * \) on the set of vectors \( mathbb{R}^2 \), the operation is defined as ((mathbf{a}, mathbf{b}) mapsto mathbf{a} ast mathbf{b} = \( a_1b_1, a_2b_2 \)). Find the value of (mathbf{a} ast (mathbf{b} ast mathbf{c})) if \( mathbf{a} = \( 2, 3 \ \)), \( mathbf{b} = \( 4, 5 \ \)), and \( mathbf{c} = \( 6, 7 \ \)).
A. (48, 105)
B. (56, 105)
C. (48, 105)
D. (56, 105)
Question 13
Solve the equation \( x^3 - 6x^2 + 11x - 6 = 0 \) u\sing the Rational Root Theorem.
A. x = 1
B. x = 2
C. x = 3
D. x = 4
Question 14
Let ( X ) and ( Y ) be indep\endent events with ( P(X) = \frac{3}{5} ) and ( P(Y) = \frac{2}{3} ). Find the probability that both events occur.
A. \( \frac{6}{15} \)
B. \( \frac{9}{15} \)
C. \( \frac{12}{15} \)
D. \( \frac{15}{15} \)
Question 15
Find the area of the triangle with vertices ( (0, 0), (1, 0), ) and ( (0, 1) ) u\sing the formula for the area of a triangle.
A. \frac{1}{2}
B. \frac{1}{3}
C. \frac{1}{4}
D. \frac{1}{6}

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