POST UTME OSUSTECH 2021 Mathematics | Objective

Practice these randomly selected questions to test your readiness.

Question 1
Let ( f(x) = x^2 + 2x - 3 \). Find the equation of the \tangent line to the graph of \( f \ \) at the point where \( x = 1 \ \).
A. y = 2x - 1
B. y = 2x + 1
C. y = x + 1
D. y = x - 1
Question 2
Solve the quadratic equation \( x^2 + 5x + 6 = 0 \) u\sing the quadratic formula. What is the value of ( x )?
A. 2
B. -3
C. -2
D. 1
Question 3
Solve the system of linear equations \( egin{cases} x + y = 4 \ 2x - 3y = 5 \end{cases} \).
A. \begin{cases} x = 2 \ y = 2 \end{cases}
B. \begin{cases} x = 3 \ y = 1 \end{cases}
C. \begin{cases} x = 1 \ y = 3 \end{cases}
D. \begin{cases} x = 4 \ y = 0 \end{cases}
Question 4
Solve the inequality \( |x - 2| > 3 \).
A. \( -\infty, -1 \) \cup \( 5, \infty \)
B. \( -\infty, 1 \) \cup \( 5, \infty \)
C. \( -\infty, -1 \) \cup \( 1, \infty \)
D. \( -\infty, 1 \) \cup \( 3, \infty \)
Question 5
Find the area of the triangle with vertices $A(2,3), B(4,5),$ and $C(6,7)$.
A. 10
B. 20
C. 30
D. 40
Question 6
Find the derivative of the function ( f(x) = \frac{x^2 + 3x - 2}{x^2 - 4} ) u\sing the quotient rule.
A. ( f'(x) = \frac{\( x^2 - 4 \)\( 2x + 3 \) - \( x^2 + 3x - 2 \)(2x)}{\( x^2 - 4 \)^2} )
B. ( f'(x) = \frac{\( x^2 - 4 \)\( 2x + 3 \) - \( x^2 + 3x - 2 \)(2x)}{\( x^2 - 4 \)^2} )
C. ( f'(x) = \frac{\( x^2 - 4 \)\( 2x + 3 \) + \( x^2 + 3x - 2 \)(2x)}{\( x^2 - 4 \)^2} )
D. ( f'(x) = \frac{\( x^2 - 4 \)\( 2x + 3 \) - \( x^2 + 3x - 2 \)(2x)}{\( x^2 - 4 \)^2} )
Question 7
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
Question 8
Solve the inequality \frac{x^2 - 4}{x^2 - 9} > 0.
A. \left\( -\infty, -3\right \) \cup \left\( 2, \infty\right \)
B. \left\( -\infty, -3\right \) \cup \left\( -3, 2\right \) \cup \left\( 2, \infty\right \)
C. \left\( -\infty, -3\right \) \cup \left\( 2, \infty\right \)
D. \left\( -\infty, -3\right \) \cup \left\( -3, 2\right \)
Question 9
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 10
A fair six-sided die is rolled. What is the probability that the number rolled is a multiple of 3?
A. \frac{1}{2}
B. \frac{1}{3}
C. \frac{2}{3}
D. \frac{1}{6}
Question 11
A histogram of exam scores has a mean of 60 and a s\tandard deviation of 10. If the scores are normally distributed, what is the probability that a randomly selected score is between 50 and 70?
A. 0.68
B. 0.84
C. 0.95
D. 0.99
Question 12
Find the volume of the frustum of a cone with height 8 cm, lower base radius 4 cm, and upper base radius 2 cm.
A. 256\pi cm^3
B. 512\pi cm^3
C. 768\pi cm^3
D. 1024\pi cm^3
Question 13
Solve the inequality \( 2x^2 + 5x - 3 > 0 \).
A. \( -∞, -1 \) ∪ (3, ∞)
B. \( -∞, -3 \) ∪ (1, ∞)
C. \( -∞, 1 \) ∪ (3, ∞)
D. \( -∞, -3 \) ∪ (1, ∞)
Question 14
Solve the system of equations: \begin{align*} x+y+z&=3 \ 2x+3y+4z&=7 \ 3x+2y+z&=5 \end{align*}
A. x=1, y=1, z=1
B. x=2, y=1, z=0
C. x=1, y=2, z=0
D. x=0, y=1, z=2
Question 15
Let \( mathbf{a} = egin{pmatrix} 2 \ 3 \end{pmatrix} \) and \( mathbf{b} = egin{pmatrix} 1 \ -2 \end{pmatrix} \). Find the projection of ( mathbf{b} ) onto ( mathbf{a} ) u\sing the formula \( \text{proj}_{mathbf{a}}mathbf{b} = \frac{mathbf{a} cdot mathbf{b}}{|mathbf{a}|^2} mathbf{a} \).
A. \begin{pmatrix} \frac{1}{5} \frac{3}{5} \end{pmatrix}
B. \begin{pmatrix} \frac{2}{5} \frac{6}{5} \end{pmatrix}
C. \begin{pmatrix} \frac{1}{5} \frac{6}{5} \end{pmatrix}
D. \begin{pmatrix} \frac{2}{5} \frac{3}{5} \end{pmatrix}

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