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POST UTME CRAWFORD UNIVERSITY 2023 Mathematics | Objective

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Question 1

Determine the value of $\int_{0}^{\pi} \sin^2(x) dx$.

Correct A. \frac{\pi}{2}

B. \frac{\pi}{4}

C. \frac{\pi}{6}

D. \frac{\pi}{8}

Correct Answer: A

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Question 2

A circle with center $C$ and radius $r$ is \tangent to the $x$-axis at point $P$. If the area of the circle is $\frac{\pi}{2}r^2$, what is the $y$-coordinate of point $P$?

Correct A. -\frac{r}{2}

B. \frac{r}{2}

C. r

D. -r

Correct Answer: A

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Question 3

A polynomial $p(x)$ has degree $n$ and has roots $r_1, r_2, \ldots, r_n$. If $p(x) = $ x - r_1 $$ x - r_2 $ \cdots $ x - r_n $$, what is the value of $p(1)$?

Correct A. $ 1 - r_1 $$ 1 - r_2 $ \cdots $ 1 - r_n $

B. r_1r_2 \cdots r_n

C. 1 - r_1 - r_2 - \cdots - r_n

D. 1 + r_1 + r_2 + \cdots + r_n

Correct Answer: A

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Question 4

A histogram of the heights of students in a class is shown below.

A. The average height of the students is 175 cm.

Correct B. The median height of the students is 175 cm.

C. The mode height of the students is 175 cm.

D. The range of heights of the students is 175 cm.

Correct Answer: B

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Question 5

A function $f(x)$ is defined as $f(x) = \sin(x) + \cos(x)$. What is the value of $f$ \frac{\pi}{4} $$?

Correct A. \frac{1}{\sqrt{2}}

B. \frac{1}{2}

C. \frac{1}{4}

D. \frac{1}{8}

Correct Answer: A

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Question 6

Find the determinant of the matrix \\[\n\begin{array}{ccc}\n2 & 3 & 4 \\\n5 & 6 & 7 \\\n8 & 9 & 10\n\end{array}\n\]

Correct A. -2

B. 2

C. -1

D. 1

Correct Answer: A

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Question 7

Solve the equation \\[\n$ x+2 $$ x+3 $$ x+4 $=0\n\]

A. -2

Correct B. -3

C. -4

D. -1

Correct Answer: B

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Question 8

Find the equation of the circle with center $ -2,3 $ and radius 4.

Correct A. $ x+2 $^2 + $ y-3 $^2 = 16

B. $ x-2 $^2 + $ y+3 $^2 = 16

C. $ x+2 $^2 + $ y+3 $^2 = 16

D. $ x-2 $^2 + $ y-3 $^2 = 16

Correct Answer: A

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Question 9

Find the volume of the cylinder with radius 6 and height 8.

Correct A. 288\pi

B. 288\pi

C. 288\pi

D. 288\pi

Correct Answer: A

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Question 10

Find the area under the curve y = x^2 from x = 0 to x = 4.

Correct A. 64

B. 64

C. 64

D. 64

Correct Answer: A

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Question 11

Find the sum of the first 10 terms of the geometric series with first term 2 and common ratio 3/4.

Correct A. 2047

B. 2048

C. 2049

D. 2050

Correct Answer: A

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Question 12

Solve the inequality \frac{x-2}{x+1} > 0.

Correct A. x < -1 or x > 2

B. x < -1 or x < 2

C. x > -1 or x < 2

D. x > -1 or x > 2

Correct Answer: A

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Question 13

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

Correct 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}

Correct Answer: A

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Question 14

Solve the matrix equation \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} 3 \ 7 \end{bmatrix}.

Correct A. \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} 1 \ 2 \end{bmatrix}

B. \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} 2 \ 1 \end{bmatrix}

C. \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} 1 \ 1 \end{bmatrix}

D. \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} 2 \ 2 \end{bmatrix}

Correct Answer: A

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Question 15

Find the area of the triangle with vertices (0, 0), (3, 0), and (0, 4).

Correct A. 6

B. 12

C. 18

D. 24

Correct Answer: A

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Question 16

Let f(x) = 3x^2 + 2x - 5. Find the value of f$ -2 $ u\sing the remainder theorem.

A. 1

B. -1

Correct C. 3

D. 5

Correct Answer: C

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Question 17

A histogram of exam scores is shown below. If the mean score is 60, find the value of the median.

A. 55

Correct B. 60

C. 65

D. 70

Correct Answer: B

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Question 18

Two events A and B are indep\endent. If P(A) = 0.4 and P(B) = 0.6, find P(A ∩ B).

Correct A. 0.2

B. 0.4

C. 0.6

D. 0.8

Correct Answer: A

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Question 19

Find the equation of the circle with center (2, 3) and radius 4.

Correct A. $ x - 2 $^2 + $ y - 3 $^2 = 16

B. $ x - 2 $^2 + $ y - 3 $^2 = 4

C. $ x - 2 $^2 + $ y - 3 $^2 = 9

D. $ x - 2 $^2 + $ y - 3 $^2 = 25

Correct Answer: A

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Question 20

Solve the equation \sin(x) = 0.5 for 0 ≤ x ≤ 2π.

Correct A. π/6

B. π/2

C. 5π/6

D. 3π/2

Correct Answer: A

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Question 21

Find the area under the curve of the function ( f(x) = \frac{1}{x^2 + 1} ) from $ x = 0 $ to $ x = 1 $.

Correct A. $ \frac{pi}{2} $

B. $ \frac{pi}{4} $

C. $ \frac{pi}{6} $

D. $ \frac{pi}{8} $

Correct Answer: A

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Question 22

A histogram of exam scores has a mean of 75 and a s\tandard deviation of 10. What is the probability that a randomly selected score is between 60 and 90?

Correct A. 0.68

B. 0.85

C. 0.95

D. 0.98

Correct Answer: A

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Question 23

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{4}{3} pi $

Correct B. $ \frac{8}{3} pi $

C. $ \frac{16}{3} pi $

D. $ \frac{32}{3} pi $

Correct Answer: B

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Question 24

A right circular cone has a radius of 4 cm and a height of 6 cm. Find the surface area of the cone.

A. ( 4 pi )

B. ( 8 pi )

Correct C. ( 12 pi )

D. ( 16 pi )

Correct Answer: C

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Question 25

Solve the equation $ \sin^2 x + \cos^2 x = 1 $ for ( x ) in the interval ( [0, 2 pi] ).

Correct A. $ x = 0, pi, 2 pi $

B. $ x = \frac{pi}{2}, \frac{3 pi}{2} $

C. $ x = \frac{pi}{4}, \frac{3 pi}{4} $

D. $ x = \frac{pi}{6}, \frac{5 pi}{6} $

Correct Answer: A

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