POST UTME MOUNTAIN TOP UNIVERSITY 2023 Mathematics | Objective

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Question 1
Find the sum of the first 10 terms of the geometric series with first term 2 and common ratio 3.
Correct A. 2 \cdot \frac{3^{10} - 1}{3 - 1}
B. 2 \cdot \frac{3^{11} - 1}{3 - 1}
C. 2 \cdot \frac{3^{12} - 1}{3 - 1}
D. 2 \cdot \frac{3^{13} - 1}{3 - 1}

Correct Answer: A

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Question 2
A rec\tangular prism has a length of 5 cm, a width of 3 cm, and a height of 2 cm. Find its volume.
Correct A. 5 \cdot 3 \cdot 2
B. 5 \cdot 3 + 2
C. 5 \cdot 2 + 3
D. 3 \cdot 2 + 5

Correct Answer: A

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Question 3
Solve for x in the equation \frac{x}{2} + 3 = 7.
Correct A. x = 8
B. x = 10
C. x = 12
D. x = 14

Correct Answer: A

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Question 4
Find the determinant of the 3x3 matrix \begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9 \end{bmatrix}.
Correct A. 0
B. 1
C. 2
D. 3

Correct Answer: A

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Question 5
A polynomial f(x) has roots x = 1, x = 2, and x = 3. Write the polynomial in factored form.
Correct A. \( x - 1 \)\( x - 2 \)\( x - 3 \)
B. \( x - 1 \)\( x - 2 \)\( x + 3 \)
C. \( x - 1 \)\( x + 2 \)\( x - 3 \)
D. \( x + 1 \)\( x - 2 \)\( x - 3 \)

Correct Answer: A

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Question 6
Let \( mathbf{a} = egin{pmatrix} 2 \ 3 \end{pmatrix} \) and \( mathbf{b} = egin{pmatrix} 4 \ 5 \end{pmatrix} \). Find the unit vector in the direction of \( mathbf{a} + mathbf{b} \).
Correct A. \( egin{pmatrix} 0.6 \ 0.8 \end{pmatrix} \)
B. \( egin{pmatrix} 0.8 \ 0.6 \end{pmatrix} \)
C. \( egin{pmatrix} 0.4 \ 0.6 \end{pmatrix} \)
D. \( egin{pmatrix} 0.6 \ 0.4 \end{pmatrix} \)

Correct Answer: A

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Question 7
A random variable ( X ) has a probability distribution given by \( P\( X = x \ \) = \frac{1}{2} ) for \( x = 1, 2, 3 \). Find the probability that ( X ) is greater than 2.
A. \( \frac{1}{4} \)
B. \( \frac{1}{2} \)
Correct C. \( \frac{3}{4} \)
D. \( \frac{1}{3} \)

Correct Answer: C

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Question 8
Find the area under the curve \( y = x^2 \) from \( x = 0 \) to \( x = 2 \) u\sing integration by substitution.
Correct A. \( \frac{8}{3} \)
B. \( \frac{16}{3} \)
C. \( \frac{24}{3} \)
D. \( \frac{32}{3} \)

Correct Answer: A

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Question 9
Find the derivative of the function ( f(x) = \sin^2 x ) u\sing the chain rule.
Correct A. \( 2 \sin x \cos x \)
B. \( \cos^2 x \)
C. \( \sin^2 x \)
D. \( \tan x \)

Correct Answer: A

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Question 10
A set of data has a mean of 25 and a s\tandard deviation of 5. Find the probability that a randomly selected value from the set is greater than 30.
A. \( \frac{1}{4} \)
B. \( \frac{1}{2} \)
Correct C. \( \frac{3}{4} \)
D. \( \frac{1}{3} \)

Correct Answer: C

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Question 11
Find the area under the curve \( y = \frac{1}{2}x^2 + 3x - 2 \) from \( x = 0 \) to \( x = 4 \).
A. 64
B. 80
Correct C. 96
D. 112

Correct Answer: C

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Question 12
Solve for ( x ) in the matrix equation \( \begin{bmatrix} 2 & 1 \ 1 & 2 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} 3 \ 4 \end{bmatrix} \).
Correct A. x = 1, y = 2
B. x = 2, y = 1
C. x = 1, y = 1
D. x = 2, y = 2

Correct Answer: A

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Question 13
A fair six-sided die is rolled twice. What is the probability that the sum of the two rolls is 7?
A. 1/6
Correct B. 1/12
C. 1/36
D. 1/24

Correct Answer: B

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Question 14
The mean of a set of 5 numbers is 10. If one of the numbers is 15, what is the sum of the other 4 numbers?
Correct A. 40
B. 45
C. 50
D. 55

Correct Answer: A

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Question 15
The polynomial ( p(x) = x^3 - 6x^2 + 11x - 6 ) has a root at \( x = 1 \). What is the quotient when ( p(x) ) is divided by \( x - 1 \)?
Correct A. x^2 - 5x + 6
B. x^2 - 5x + 5
C. x^2 - 5x + 4
D. x^2 - 5x + 3

Correct Answer: A

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Question 16
In a certain number base $b$, the number $101_5$ is equivalent to $b^2 + 1$ in base $10$. Find the value of $b$.
A. 2
Correct B. 3
C. 4
D. 5

Correct Answer: B

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Question 17
The line $y = 2x + 3$ intersects the circle $x^2 + y^2 = 16$ at two points. Find the coordinates of the point where the line is \tangent to the circle.
Correct A. \( 4, -1 \)
B. \( -4, -1 \)
C. (4, 1)
D. \( -4, 1 \)

Correct Answer: A

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Question 18
The quadratic equation $x^2 + 4x + 4 = 0$ has two equal roots. Find the value of $x$.
Correct A. -2
B. -1
C. 0
D. 1

Correct Answer: A

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Question 19
The mean of a set of numbers is $\bar{x} = 10$. If the sum of the numbers is $S = 50$, find the median of the set.
A. 10
B. 15
Correct C. 20
D. 25

Correct Answer: C

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Question 20
The polynomial $P(x) = x^3 - 6x^2 + 11x - 6$ has a root at $x = 1$. Find the value of $P(2)$.
A. 0
B. 2
Correct C. 4
D. 6

Correct Answer: C

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Question 21
A set of 10 numbers has a mean of 20 and a s\tandard deviation of 5. If the set is multiplied by 2, what is the new s\tandard deviation?
A. 10
B. 20
Correct C. 40
D. 50

Correct Answer: C

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Question 22
Solve for 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 23
A histogram of exam scores has a mean of 70 and a s\tandard deviation of 10. If the scores are normally distributed, what is the probability that a randomly selected score is between 60 and 80?
A. 0.68
B. 0.80
Correct C. 0.95
D. 0.99

Correct Answer: C

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Question 24
A sequence is defined by the formula: \( a_n = 2a_{n-1} + 1 \). If \( a_1 = 3 \), find the value of \( a_5 \)
A. 31
B. 33
Correct C. 35
D. 37

Correct Answer: C

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Question 25
A polynomial function is defined by: \( f(x) = x^3 - 6x^2 + 11x - 6 \). Find the value of \( f\( -1 \)
A. -2
Correct B. 0
C. 2
D. 4

Correct Answer: B

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