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POST UTME LEAD CITY UNIVERSITY 2024 Mathematics | Objective

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Question 1

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

Correct A. $ \frac{1}{2} \( 1 - x^2 \ $^{-\frac{3}{2}} )

B. $ \frac{1}{2} \( 1 - x^2 \ $^{-\frac{1}{2}} )

C. $ \frac{x}{\( 1 - x^2 \ $^{\frac{3}{2}}} )

D. $ \frac{x}{\( 1 - x^2 \ $^{\frac{1}{2}}} )

Correct Answer: A

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Question 2

Solve the system of equations $ egin{cases} x + y + z = 6 \ 2x + 3y - z = 2 \ x - 2y + 3z = -3 \end{cases} $ u\sing matrices.

Correct A. $ x = 1, y = 2, z = 3 $

B. $ x = 2, y = 1, z = 3 $

C. $ x = 3, y = 2, z = 1 $

D. $ x = 1, y = 3, z = 2 $

Correct Answer: A

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Question 3

Find the volume of the solid formed by revolving the region bounded by the curves $ y = x^2 $ and $ y = 2x $ about the x-axis.

A. $ \frac{1}{3} pi $

B. $ \frac{2}{3} pi $

Correct C. $ \frac{4}{3} pi $

D. $ \frac{8}{3} pi $

Correct Answer: C

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Question 4

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

Correct A. $ x = \frac{pi}{4}, \frac{3pi}{4}, \frac{5pi}{4}, \frac{7pi}{4} $

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

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

D. $ x = \frac{pi}{2}, \frac{3pi}{2}, \frac{5pi}{2} $

Correct Answer: A

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Question 5

Find the area of the triangle with vertices ( A(1, 2), B(3, 4), C(5, 6) ).

A. ( 10 )

B. ( 15 )

Correct C. ( 20 )

D. ( 25 )

Correct Answer: C

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Question 6

In a probability experiment, two events A and B are indep\endent. If P(A) = 0.4 and P(B) = 0.6, what is the probability that both events occur?

Correct A. 0.24

B. 0.48

C. 0.64

D. 0.76

Correct Answer: A

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Question 7

Find the area under the curve y = 3x^2 + 2x - 5 from x = 0 to x = 2.

A. 22

Correct B. 24

C. 26

D. 28

Correct Answer: B

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Question 8

Solve the trigonometric equation \sin^2(x) + \cos^2(x) = 1.

Correct A. x = \frac{\pi}{4}

B. x = \frac{\pi}{2}

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

D. x = \frac{5\pi}{4}

Correct Answer: A

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Question 9

A quadratic equation has roots 2 and 3. Write the equation in the form ax^2 + bx + c = 0.

Correct A. x^2 - 5x + 6 = 0

B. x^2 - 4x + 3 = 0

C. x^2 - 6x + 4 = 0

D. x^2 - 3x + 2 = 0

Correct Answer: A

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Question 10

A number is chosen at random from the set {1, 2, 3, 4, 5, 6}. What is the probability that the number is even?

Correct A. 0.5

B. 0.6

C. 0.7

D. 0.8

Correct Answer: A

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Question 11

Find the volume of the solid formed by revolving the region bounded by the curve $y = \frac{1}{2}x^2 + 1$, the $x$-axis, and the line $x = 2$ about the $x$-axis.

Correct A. \frac{8\pi}{3}

B. \frac{16\pi}{3}

C. \frac{32\pi}{3}

D. \frac{64\pi}{3}

Correct Answer: A

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Question 12

Solve the equation $\log_{10}$ x^2 $ = 4$ for $x$.

A. \pm 10

Correct B. \pm 20

C. \pm 40

D. \pm 80

Correct Answer: B

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Question 13

Find the determinant of the matrix $\begin{bmatrix} 2 & 1 & 1 \ 1 & 2 & 1 \ 1 & 1 & 2 \end{bmatrix}$.

A. 4

B. 6

Correct C. 8

D. 10

Correct Answer: C

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Question 14

A histogram of exam scores has a mean of 70 and a s\tandard deviation of 10. If 80% of the scores fall below 90, what is the value of the score that corresponds to the 80th percentile?

A. 80

Correct B. 85

C. 90

D. 95

Correct Answer: B

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Question 15

Find the area of the triangle formed by the points $(0, 0)$, $(3, 0)$, and $(0, 4)$.

A. 6

B. 12

Correct C. 18

D. 24

Correct Answer: C

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Question 16

Find the derivative of the function ( f(x) = \frac{x^2 + 2x - 3}{x + 1} ) u\sing the quotient rule.

A. \frac{2x + 2}{$ x + 1 $^2}

Correct B. \frac{2x + 2}{$ x + 1 $^2} + \frac{1}{x + 1}

C. \frac{2x + 2}{$ x + 1 $^2} - \frac{1}{x + 1}

D. \frac{2x + 2}{$ x + 1 $^2} - \frac{2}{x + 1}

Correct Answer: B

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Question 17

Solve the system of linear equations u\sing matrices: \begin{align*} 2x + 3y &= 7 \ 4x - 2y &= -2 \end{align*}

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

B. \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} 2 \ 3 \end{pmatrix}

C. \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} 3 \ 2 \end{pmatrix}

D. \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} 4 \ 1 \end{pmatrix}

Correct Answer: A

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Question 18

Find the equation of the circle pas\sing through the points (1, 2), (3, 4), and (5, 6).

Correct A. $ x - 3 $^2 + $ y - 4 $^2 = 16

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

C. $ x - 1 $^2 + $ y - 2 $^2 = 4

D. $ x - 4 $^2 + $ y - 5 $^2 = 25

Correct Answer: A

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Question 19

Solve the inequality \frac{x^2 - 4}{x + 2} > 0.

Correct A. x > -2

B. x < -2

C. x > 2

D. x < 2

Correct Answer: A

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Question 20

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

Correct A. \cos^2 x

B. -\cos^2 x

C. \sin^2 x

D. -\sin^2 x

Correct Answer: A

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Question 21

Solve the system of linear equations u\sing matrices: \begin{align*} 2x + 3y &= 7 \ 4x - 2y &= -2 \end{align*}

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

B. \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} 2 \ 3 \end{pmatrix}

C. \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} 3 \ 2 \end{pmatrix}

D. \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} 4 \ 1 \end{pmatrix}

Correct Answer: A

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Question 22

Find the equation of the circle pas\sing through the points (1, 2), (3, 4), and (5, 6).

Question 23

Find the value of $\int_0^1 \frac{1}{1+x^2} dx$.

Correct A. \frac{\pi}{4}

B. \frac{\pi}{2}

C. \frac{\pi}{3}

D. \frac{\pi}{6}

Correct Answer: A

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Question 24

Solve the inequality $|x-2| > 3$.

Correct A. $ -\infty, -1 $ \cup $ 5, \infty $

B. $ -\infty, -1 $ \cup $ 1, \infty $

C. $ -\infty, 1 $ \cup $ 5, \infty $

D. $ -\infty, 5 $ \cup $ 1, \infty $

Correct Answer: A

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Question 25

A histogram of exam scores has a mean of 75 and a s\tandard deviation of 10. If 75% of the scores fall below 85, what is the z-score corresponding to 85?

A. 0.5

Correct B. 1

C. 1.5

D. 2

Correct Answer: B

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