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Subject

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POST UTME LAUTECH 2023 Mathematics | Objective

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Question 1

Determine the volume of the frustum of a cone with height 6 cm, lower base radius 4 cm, and upper base radius 2 cm.

A. 24π cm³

Correct B. 48π cm³

C. 96π cm³

D. 192π cm³

Correct Answer: B

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Question 2

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 3

Solve for x in the equation $ egin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} egin{bmatrix} x \ 1 \end{bmatrix} = egin{bmatrix} 7 \ 10 \end{bmatrix} $.

A. x = 3

B. x = 4

Correct C. x = 5

D. x = 6

Correct Answer: C

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Question 4

Find the surface area of the sphere with radius 5 cm.

A. 50π cm²

B. 100π cm²

Correct C. 150π cm²

D. 200π cm²

Correct Answer: C

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Question 5

Determine the number of terms in the sequence 1, 2, 4, 8, ...

A. 5

B. 6

C. 7

Correct D. 8

Correct Answer: D

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Question 6

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

A. 10

B. 100

C. 1000

Correct D. 10000

Correct Answer: D

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Question 7

Find the value of y in the equation $ y = \frac{1}{2} \( 3x + 5 \ $ ) when x = 4.

A. 8

Correct B. 9

C. 10

D. 11

Correct Answer: B

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Question 8

A circle has a radius of 5 cm. Find the area of the circle.

Correct A. 25π

B. 50π

C. 75π

D. 100π

Correct Answer: A

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Question 9

Solve the inequality $ 2x - 5 > 3 $.

Correct A. x > 4

B. x < 4

C. x > 2

D. x < 2

Correct Answer: A

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Question 10

A sequence is defined by the formula $ a_n = 2n + 1 $. Find the 5th term of the sequence.

A. 9

B. 11

Correct C. 13

D. 15

Correct Answer: C

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Question 11

In the diagram below, the equation of the circle is given by $ x - 2 \ $^2 + $ y - 3 $^2 = 4 ). Find the equation of the \tangent line to the circle at the point ( (5, 7) ).

Correct A. y = -x + 12

B. y = x - 1

C. y = -x + 11

D. y = x + 1

Correct Answer: A

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Question 12

Let $ mathbf{a} = egin{pmatrix} 2 \ 3 \end{pmatrix} $ and $ mathbf{b} = egin{pmatrix} 4 \ 5 \end{pmatrix} $. Find the projection of ( mathbf{b} ) onto ( mathbf{a} ).

Correct A. $ egin{pmatrix} 1.6 \ 2.4 \end{pmatrix} $

B. $ egin{pmatrix} 2.4 \ 3.6 \end{pmatrix} $

C. $ egin{pmatrix} 3.2 \ 4.8 \end{pmatrix} $

D. $ egin{pmatrix} 4.8 \ 7.2 \end{pmatrix} $

Correct Answer: A

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Question 13

A random variable ( X ) has a probability distribution given by $ P\( X = x \ $ = egin{cases} 0.2 & \text{if } x = 1 \ 0.3 & \text{if } x = 2 \ 0.5 & \text{if } x = 3 \end{cases} ). Find the probability that ( X ) takes on a value greater than 2.

A. 0.4

Correct B. 0.5

C. 0.6

D. 0.7

Correct Answer: B

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Question 14

Solve the inequality $ 2x^2 - 5x - 3 > 0 $.

Correct A. $ x < -1 $ or $ x > \frac{3}{2} $

B. $ x < -1 $ or $ x < \frac{3}{2} $

C. $ x > -1 $ or $ x < \frac{3}{2} $

D. $ x > -1 $ or $ x > \frac{3}{2} $

Correct Answer: A

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Question 15

Find the value of $ \sin 2\theta $ given that $ \sin \theta = \frac{3}{5} $ and $ \cos \theta = \frac{4}{5} $.

Correct A. $ \frac{24}{25} $

B. $ \frac{16}{25} $

C. $ \frac{20}{25} $

D. $ \frac{12}{25} $

Correct Answer: A

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Question 16

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{8pi}{3} $

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

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

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

Correct Answer: A

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Question 17

Solve the inequality $|x^2 - 4x + 3| geq 2$.

Correct A. $ x leq -1 $ or ( x geq 3 )

B. ( x leq 1 ) or ( x geq 3 )

C. $ x leq -1 $ or ( x geq 2 )

D. ( x leq 1 ) or ( x geq 2 )

Correct Answer: A

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Question 18

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

Correct A. $ x^2 + y^2 - 6x + 10y + 11 = 0 $

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

C. $ x^2 + y^2 + 2x - 4y - 5 = 0 $

D. $ x^2 + y^2 - 2x + 4y - 7 = 0 $

Correct Answer: A

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Question 19

A right circular cone has a height of 8 cm and a base radius of 5 cm. Find the volume of the cone.

A. ( 100pi ) cm$^3$

B. ( 200pi ) cm$^3$

Correct C. ( 300pi ) cm$^3$

D. ( 400pi ) cm$^3$

Correct Answer: C

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Question 20

Solve the equation $x^3 - 6x^2 + 11x - 6 = 0$.

A. $ x = 1 $

B. $ x = 2 $

Correct C. $ x = 3 $

D. $ x = 6 $

Correct Answer: C

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Question 21

Solve the inequality $ \frac{x^2 - 4}{x + 2} > 0 $ for ( x in mathbb{R} ).

Correct A. $ x < -2 $ or $ x > 2 $

B. $ x > -2 $ or $ x < 2 $

C. $ x = -2 $ or $ x = 2 $

D. $ x = -2 $ and $ x = 2 $

Correct Answer: A

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Question 22

Find the value of ( k ) such that the function ( f(x) = \frac{kx^2 - 2x + 1}{x^2 - 4} ) is continuous at $ x = 1 $.

Correct A. $ k = 3 $

B. $ k = 2 $

C. $ k = 1 $

D. $ k = 0 $

Correct Answer: A

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Question 23

Solve the system of equations $ egin{cases} x^2 + y^2 = 4 \ x - 2y = -1 \end{cases} $.

Correct A. $ x = 1, y = 1 $

B. $ x = 2, y = 1 $

C. $ x = 1, y = 2 $

D. $ x = 2, y = 2 $

Correct Answer: A

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Question 24

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

Correct A. ( f'(x) = -\frac{2x}{$ x^2 + 1 $^2} )

B. ( f'(x) = \frac{2x}{$ x^2 + 1 $^2} )

C. ( f'(x) = -\frac{1}{$ x^2 + 1 $^2} )

D. ( f'(x) = \frac{1}{$ x^2 + 1 $^2} )

Correct Answer: A

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Question 25

Find the sum of the first 5 terms of the geometric series $ 2 + 6 + 18 + ldots $.

Correct A. ( 62 )

B. ( 64 )

C. ( 66 )

D. ( 68 )

Correct Answer: A

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