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POST UTME OSUSTECH 2017 Mathematics | Objective

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Question 1 View Details

Let X be a random variable with probability density function ( f(x) = egin{cases} 2x & 0 leq x leq 1 \ 0 & \text{otherwise} \end{cases} ). Find the probability that X is greater than 0.5.

Correct A. \(int_{0.5}^{1} 2x , dx\)

B. \(int_{0}^{0.5} 2x , dx\)

C. \(int_{0}^{1} 2x , dx\)

D. \(int_{0}^{1} 2x^2 , dx\)

Correct Answer: A

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Question 2 View Details

Find the derivative of the function ( f(x) = \(\frac{1}{x^2 + 1}\)\)\) ) using 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'\((x) =\)\(\frac{-2}{x^2 + 1\) ^2}\)\) )

Correct Answer: A

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Question 3 View Details

A histogram of exam scores has a mean of 80 and a standard deviation of 10. If the scores are normally distributed, find the probability that a randomly selected score is greater than 90.

Correct A. \(P X > 90 \\)\(= 0.1587 )\)

B. \(P X > 90 \\)\(= 0.8413 )\)

C. \(P X > 90 \\)\(= 0.5 )\)

D. \(P X > 90 \\)\(= 0.25 )\)

Correct Answer: A

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Question 4 View Details

Solve the system of equations using matrices: \(egin{cases} 2x + 3y = 7 \ x - 2y = -3 \end{cases}\).

Correct A. \(egin{pmatrix} x \ y \end{pmatrix} = egin{pmatrix} 19/7 \ 2/7 \end{pmatrix}\)

B. \(egin{pmatrix} x \ y \end{pmatrix} = egin{pmatrix} 2/7 \ 19/7 \end{pmatrix}\)

C. \(egin{pmatrix} x \ y \end{pmatrix} = egin{pmatrix} 7/19 \ 2/19 \end{pmatrix}\)

D. \(egin{pmatrix} x \ y \end{pmatrix} = egin{pmatrix} 2/19 \ 7/19 \end{pmatrix}\)

Correct Answer: A

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Question 5 View Details

Find the volume of the solid formed by revolving the region bounded by the parabola (y = \(x^2\)\) and the line \(y = 2x\\((y = x^2) and the line\) about the x-axis.

Correct A. \(V = \frac{8}{15} \pi\)

B. \(V = \frac{8}{3} \pi\)

C. \(V = \frac{4}{3} \pi\)

D. \(V = \frac{16}{5} \pi\)

Correct Answer: A

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Question 6 View Details

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

Correct A. \(x-2 \\)^2 + \(y-3\)^\(2 = 16\) )

B. \(x-2 \\)^2 + \(y-3\)^\(2 = 32\) )

C. \(x-2 \\)^2 + \(y-3\)^\(2 = 64\) )

D. \(x-2 \\)^2 + \(y-3\)^\(2 = 256\) )

Correct Answer: A

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Question 7 View Details

Solve for x in the equation \(2^x + 2^{x+1} = 3 \cdot 2^x\).

A. \(x = -1\)

B. \(x = 0\)

Correct C. \(x = 1\)

D. \(x = 2\)

Correct Answer: C

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Question 8 View Details

A fair six-sided die is rolled. What is the probability that the number rolled is even?

Correct A. \(\frac{1}{2}\)

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

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

D. \(\frac{5}{6}\)

Correct Answer: A

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Question 9 View Details

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

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

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

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

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

Correct Answer: A

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Question 10 View Details

A vector (mathbf{a}) has magnitude 5 and direction \((30^circ\)\) counterclockwise from the positive x-axis. Find the vector (mathbf{a} cdot mathbf{b}) if \(\\((30^circ) counterclockwise from the positive x-axis. Find the vector (mathbf{a} cdot mathbf{b}) if \mathbf{b} = egin{pmatrix} 2 \ 3 \end{pmatrix}\).

Correct A. ( 11 )

B. ( 13 )

C. ( 15 )

D. ( 17 )

Correct Answer: A

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Question 11 View Details

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

Correct A. 4

B. 2

C. 1

D. 0.5

Correct Answer: A

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Question 12 View Details

A sequence is defined as \(a_n = 2n + 1\). Find the sum of the first 5 terms of the sequence.

A. 15

B. 20

Correct C. 25

D. 30

Correct Answer: C

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Question 13 View Details

A curve is defined by the equation \(y = x^2 - 4x + 3\). Find the x-coordinate of the vertex of the curve.

Correct A. 2

B. 3

C. 4

D. 5

Correct Answer: A

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Question 14 View Details

A solid is formed by rotating the region bounded by the curve \(y = x^2\) and the x-axis about the x-axis. Find the volume of the solid.

A. π/2

Correct B. π

C. 2π

D. 3π

Correct Answer: B

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Question 15 View Details

A triangle has sides of length 3, 4, and 5. Find the area of the triangle.

A. 6

B. 8

Correct C. 10

D. 12

Correct Answer: C

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Question 16 View Details

Let ( f(x) = \(\frac{x^2 - 4}{x - 2}\)\)) ). Find the value of lim_{x \to 2} f\( x \\() ). Find the value of lim_{x to 2} f\ ).

A. 1

B. 2

C. 3

Correct D. 4

Correct Answer: D

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Question 17 View Details

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

A. 5

B. 10

Correct C. 15

D. 20

Correct Answer: C

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Question 18 View Details

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

A. 0

Correct B. \(\frac{\pi}{4}\)\)

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

D. \(\frac{3\pi}{4}\)\)

Correct Answer: B

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Question 19 View Details

Find the equation of the circle with center ( (3, 4) ) and radius ( 5 ).

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

B. \(x - 3\)^2 + \(y - 4\)^\(2 = 30\)

C. \(x - 3\)^2 + \(y - 4\)^\(2 = 35\)

D. \(x - 3\)^2 + \(y - 4\)^\(2 = 40\)

Correct Answer: A

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Question 20 View Details

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

Correct A. \(-\infty, -1\) \cup \(2, \infty\)

B. \(-\infty, -1\) \cup (1, 2)

C. \(-\infty, -1\) \cup \(2, \infty\)

D. \(-\infty, -1\) \cup (1, 2)

Correct Answer: A

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Question 21 View Details

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

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

B. \(\frac{x}{x^2 + 1 \\)^{\(3/2\)}} )

C. \(\frac{1}{x^2 + 1 \\)^{\(3/2\)}} )

D. \(\frac{-1}{x^2 + 1 \\)^{\(3/2\)}} )

Correct Answer: A

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Question 22 View Details

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

A. \(\frac{1}{6}\)

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

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

Correct D. \(\frac{5}{6}\)

Correct Answer: D

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Question 23 View Details

Solve the quadratic equation \(x^2 + 5x + 6 = 0\) using the quadratic formula.

Correct A. \(x = -2\)

B. \(x = -3\)

C. \(x = 2\)

D. \(x = 3\)

Correct Answer: A

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Question 24 View Details

Find the determinant of the matrix \(egin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9 \end{bmatrix}\).

Correct A. ( 0 )

B. ( 1 )

C. \(-1\)

D. ( 2 )

Correct Answer: A

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Question 25 View Details

A box contains 5 red balls and 3 blue balls. What is the probability of drawing a red ball first and then a blue ball?

A. \(\frac{1}{4}\)

B. \(\frac{1}{2}\)

Correct C. \(\frac{3}{4}\)

D. \(\frac{5}{6}\)

Correct Answer: C

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