POST UTME ACHIEVERS UNIVERSITY 2023 Mathematics | Objective

Practice these randomly selected questions to test your readiness.

Question 1
Solve the system of equations: \begin{align*} x+y+z&=3,\ x+2y+3z&=6,\ 2x+3y+4z&=9.\end{align*}
A. \\begin{pmatrix}1\\ 2\\ 0\\end{pmatrix}
B. \\begin{pmatrix}0\\ 1\\ 2\\end{pmatrix}
C. \\begin{pmatrix}1\\ 1\\ 1\\end{pmatrix}
D. \\begin{pmatrix}2\\ 1\\ 0\\end{pmatrix}
Question 2
Find the area under the curve \( y = x^2 - 2x + 1 \) from \( x = 0 \) to \( x = 2 \) u\sing the definite integral.
A. \frac{1}{3}
B. \frac{2}{3}
C. \frac{3}{2}
D. \frac{4}{3}
Question 3
Solve the inequality \( 2x^2 + 5x - 3 > 0 \) u\sing the quadratic formula.
A. \left\( -\frac{5}{4}, \frac{3}{2} \right \)
B. \left\( -\frac{3}{2}, \frac{5}{4} \right \)
C. \left\( -\infty, -\frac{3}{2} \right \) \cup \left\( \frac{5}{4}, \infty \right \)
D. \left\( -\infty, \frac{5}{4} \right \) \cup \left\( -\frac{3}{2}, \infty \right \)
Question 4
Find the area under the curve y = x^2 from x = 0 to x = 1 u\sing integration.
A. 1/3
B. 1/2
C. 2/3
D. 1
Question 5
A matrix A has the following form: \begin{pmatrix}a&b\ c&d\end{pmatrix}. If the determinant of A is 6, what is the value of ad-bc?
A. 6
B. 12
C. 18
D. 24
Question 6
Solve for ( x ) in the equation \( \frac{1}{2}x + 3 = \frac{3}{4}x - 2 \).
A. \( x = -\frac{14}{5} \)
B. \( x = \frac{14}{5} \)
C. \( x = -\frac{7}{5} \)
D. \( x = \frac{7}{5} \)
Question 7
A fair six-sided die is rolled. What is the probability that the number rolled is greater than 4?
A. \frac{1}{3}
B. \frac{1}{2}
C. \frac{2}{3}
D. \frac{1}{6}
Question 8
Find the derivative of ( f(x) = \frac{1}{x^2 + 1} ) u\sing the chain rule.
A. -2x/\( x^2 + 1 \)^2
B. 2x/\( x^2 + 1 \)^2
C. -x/\( x^2 + 1 \)^2
D. x/\( x^2 + 1 \)^2
Question 9
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 \( mathrm{proj}_{mathbf{a}}\( mathbf{b} \ \) = \frac{mathbf{a} cdot mathbf{b}}{|mathbf{a}|^2} mathbf{a} ).
A. \begin{pmatrix} \frac{7}{13} \\ \frac{12}{13} \end{pmatrix}
B. \begin{pmatrix} \frac{1}{13} \\ -\frac{2}{13} \end{pmatrix}
C. \begin{pmatrix} \frac{2}{13} \\ -\frac{3}{13} \end{pmatrix}
D. \begin{pmatrix} \frac{3}{13} \\ -\frac{4}{13} \end{pmatrix}
Question 10
Let A = \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix} and B = \begin{pmatrix} 5 & 6 \ 7 & 8 \end{pmatrix}. Find the product AB.
A. \begin{pmatrix} 19 & 22 \ 43 & 50 \end{pmatrix}
B. \begin{pmatrix} 23 & 26 \ 47 & 54 \end{pmatrix}
C. \begin{pmatrix} 17 & 20 \ 39 & 46 \end{pmatrix}
D. \begin{pmatrix} 21 & 24 \ 41 & 48 \end{pmatrix}
Question 11
Solve the inequality \( 2x^2 + 5x - 3 > 0 \).
A. \( -∞, -1 \) ∪ (3, ∞)
B. \( -∞, -3 \) ∪ (1, ∞)
C. \( -∞, 1 \) ∪ (3, ∞)
D. \( -∞, -3 \) ∪ \( -1, ∞ \)
Question 12
A right triangle has a hypotenuse of length 10 cm and one leg of length 6 cm. What is the length of the other leg?
A. 8
B. 8.66
C. 8.94
D. 9.22
Question 13
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. \( \frac{5}{12} \)
B. \( \frac{1}{3} \)
C. \( \frac{2}{3} \)
D. \( \frac{7}{12} \)
Question 14
Let ( f(x) = egin{cases} x^2 & x geq 0 \ 0 & x < 0 \end{cases} ). Find ( f'(x) ) u\sing the chain rule.
A. ( f'(x) = egin{cases} 2x & x > 0 \ 0 & x < 0 \end{cases} )
B. ( f'(x) = egin{cases} 2x & x < 0 \ 0 & x > 0 \end{cases} )
C. ( f'(x) = egin{cases} 2x & x geq 0 \ 0 & x < 0 \end{cases} )
D. ( f'(x) = egin{cases} 0 & x geq 0 \ 2x & x < 0 \end{cases} )
Question 15
Find the volume of the solid formed by rotating the region bounded by the curves \( y = x^2 \) and \( y = 2x \) about the x-axis.
A. \( \frac{1}{3} \)
B. \( \frac{2}{3} \)
C. \( \frac{4}{3} \)
D. \( \frac{8}{3} \)

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