Exam Body

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POST UTME ACHIEVERS UNIVERSITY 2023 Mathematics | Objective

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Question 1

In the coordinate plane, the equation of a circle is given by \( x - h \ \)^2 + \( y - k \)^2 = r^2 ). If the center of the circle is at ( (h, k) = (3, 4) ) and the radius is 5 units, what is the equation of the circle?

Correct A. \( x - 3 \ \)^2 + \( y - 4 \)^2 = 25 )

B. \( x - 4 \ \)^2 + \( y - 3 \)^2 = 25 )

C. \( x - 3 \ \)^2 + \( y - 3 \)^2 = 25 )

D. \( x - 4 \ \)^2 + \( y - 4 \)^2 = 25 )

Correct Answer: A

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Question 2

Solve for ( x ) in the equation \( \frac{1}{2}x + 3 = \frac{3}{4}x - 2 \).

Correct A. \( x = -\frac{14}{5} \)

B. \( x = \frac{14}{5} \)

C. \( x = -\frac{7}{5} \)

D. \( x = \frac{7}{5} \)

Correct Answer: A

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Question 3

Find the determinant of the matrix \( egin{bmatrix} 2 & 3 & 4 \ 5 & 6 & 7 \ 8 & 9 & 10 \end{bmatrix} \).

Correct A. ( 0 )

B. ( 1 )

C. \( -1 \)

D. ( 2 )

Correct Answer: A

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Question 4

A bakery sells a total of 250 loaves of bread per day. They sell a combination of whole wheat and white bread. If the ratio of whole wheat to white bread is 5:3, how many loaves of white bread are sold per day?

A. ( 90 )

Correct B. ( 100 )

C. ( 110 )

D. ( 120 )

Correct Answer: B

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Question 5

In a right-angled triangle, the length of the hypotenuse is 10 cm and one of the other sides is 6 cm. What is the length of the third side?

A. ( 8 ) cm

Correct B. \( 6 \sqrt{5} \) cm

C. \( 8 \sqrt{5} \) cm

D. ( 10 ) cm

Correct Answer: B

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Question 6

Let \( mathbf{a} = egin{pmatrix} 2 \ 3 \end{pmatrix} \) and \( mathbf{b} = egin{pmatrix} -1 \ 4 \end{pmatrix} \). Find the projection of ( mathbf{b} ) onto ( mathbf{a} ).

Correct A. 0.8

B. -0.6

C. 0.6

D. 1.2

Correct Answer: A

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Question 7

Find the derivative of ( f(x) = \frac{1}{x^2 + 1} ) u\sing the chain rule.

Correct A. -2x/\( x^2 + 1 \)^2

B. 2x/\( x^2 + 1 \)^2

C. -x/\( x^2 + 1 \)^2

D. x/\( x^2 + 1 \)^2

Correct Answer: A

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Question 8

Solve the inequality \( 2x^2 + 5x - 3 > 0 \).

Correct A. \( -∞, -1 \) ∪ (3, ∞)

B. \( -∞, -3 \) ∪ (1, ∞)

C. \( -∞, 1 \) ∪ (3, ∞)

D. \( -∞, -3 \) ∪ \( -1, ∞ \)

Correct Answer: A

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Question 9

Find the mean of the following data set: 2, 4, 6, 8, 10.

Correct A. 6

B. 5

C. 7

D. 8

Correct Answer: A

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Question 10

A fair six-sided die is rolled. What is the probability that the number rolled is greater than 4?

Correct A. \frac{1}{3}

B. \frac{1}{2}

C. \frac{2}{3}

D. \frac{1}{6}

Correct Answer: A

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Question 11

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} ).

Correct 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}

Correct Answer: A

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Question 12

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}

Correct C. \frac{3}{2}

D. \frac{4}{3}

Correct Answer: C

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Question 13

Solve the inequality \( 2x^2 + 5x - 3 > 0 \) u\sing the quadratic formula.

A. \left\( -\frac{5}{4}, \frac{3}{2} \right \)

B. \left\( -\frac{3}{2}, \frac{5}{4} \right \)

Correct C. \left\( -\infty, -\frac{3}{2} \right \) \cup \left\( \frac{5}{4}, \infty \right \)

D. \left\( -\infty, \frac{5}{4} \right \) \cup \left\( -\frac{3}{2}, \infty \right \)

Correct Answer: C

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Question 14

Find the value of ( x ) in the equation \( 2^x + 3^x = 5^x \).

A. 1

Correct B. 2

C. 3

D. 4

Correct Answer: B

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Question 15

Solve the system of linear equations \( egin{cases} 2x + 3y = 7 \ 4x - 2y = -3 \end{cases} \) u\sing substitution or elimination.

Correct A. \left\( x, y \right \) = \left\( 1, 2 \right \)

B. \left\( x, y \right \) = \left\( 2, 1 \right \)

C. \left\( x, y \right \) = \left\( 3, -1 \right \)

D. \left\( x, y \right \) = \left\( -1, 3 \right \)

Correct Answer: A

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Question 16

Let X and Y be indep\endent random variables with probability density functions f_X(x) = 2x, 0 < x < 1, and f_Y(y) = 3y^2, 0 < y < 1. Find the probability that X + Y < 1.

A. 1/3

B. 1/4

Correct C. 1/2

D. 2/3

Correct Answer: C

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Question 17

Find the derivative of the function f(x) = \frac{\sin(x)}{x^2} u\sing the chain rule.

Correct A. \frac{\cos(x)}{x^2} - \frac{2\sin(x)}{x^3}

B. \frac{\cos(x)}{x^2} + \frac{2\sin(x)}{x^3}

C. \frac{\cos(x)}{x^2} - \frac{\sin(x)}{x^3}

D. \frac{\cos(x)}{x^2} + \frac{\sin(x)}{x^3}

Correct Answer: A

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Question 18

Solve the system of linear equations u\sing the method of substitution: x + y = 2 and x - y = 1.

Correct A. x = 1, y = 1

B. x = 1, y = -1

C. x = -1, y = 1

D. x = -1, y = -1

Correct Answer: A

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Question 19

Find the area under the curve y = x^2 from x = 0 to x = 1 u\sing integration.

A. 1/3

Correct B. 1/2

C. 2/3

D. 1

Correct Answer: B

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Question 20

Let A = \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix} and B = \begin{pmatrix} 5 & 6 \ 7 & 8 \end{pmatrix}. Find the product AB.

Correct 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}

Correct Answer: A

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Question 21

Solve the quadratic equation x^2 + 4x + 4 = 0 u\sing the quadratic formula.

Correct A. x = -2

B. x = -1

C. x = 1

D. x = 2

Correct Answer: A

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Question 22

Find the derivative of the function f(x) = \sin\( x^2 \) u\sing the chain rule.

Correct A. 2x \cos\( x^2 \)

B. x^2 \cos\( x^2 \)

C. x \cos\( x^2 \)

D. \cos\( x^2 \)

Correct Answer: A

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Question 23

Let X and Y be indep\endent random variables with probability density functions f_X(x) = 2x, 0 < x < 1, and f_Y(y) = 3y^2, 0 < y < 1. Find the probability that X + Y < 1.

A. 1/3

B. 1/4

Correct C. 1/2

D. 2/3

Correct Answer: C

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Question 24

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 \( X + Y leq 1 \).

Correct A. \( \frac{5}{12} \)

B. \( \frac{1}{3} \)

C. \( \frac{2}{3} \)

D. \( \frac{7}{12} \)

Correct Answer: A

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Question 25

Find the equation of the line pas\sing through the points ( (2,3) ) and ( (4,5) ).

Correct A. \( y = \frac{1}{2}x + \frac{1}{2} \)

B. \( y = \frac{1}{2}x - \frac{1}{2} \)

C. \( y = 2x - 1 \)

D. \( y = 2x + 1 \)

Correct Answer: A

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