Mathematics Study Resources for Competitive Exams

Mathematics

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Q. What is the value of the derivative of f(x) = 3x^3 - 2x at x = 1?

A. 7
B. 5
C. 3
D. 1

Solution

f'(x) = 9x^2 - 2. At x = 1, f'(1) = 9*1^2 - 2 = 7.

Correct Answer: A — 7

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Q. What is the value of the determinant of the matrix G = [[1, 2, 3], [0, 1, 4], [5, 6, 0]]? (2022)

A. -34
B. 34
C. 0
D. 10

Solution

Using the determinant formula, det(G) = 1(1*0 - 4*6) - 2(0 - 4*5) + 3(0 - 1*5) = -24 + 40 - 15 = 1.

Correct Answer: A — -34

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Q. What is the value of the discriminant for the equation 3x^2 + 12x + 12 = 0?

A. 0
B. 12
C. 36
D. 24

Solution

The discriminant is b^2 - 4ac = 12^2 - 4(3)(12) = 144 - 144 = 0.

Correct Answer: A — 0

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Q. What is the value of the discriminant for the equation 3x^2 + 12x + 9 = 0?

A. 0
B. 36
C. 12
D. 24

Solution

The discriminant is b^2 - 4ac = 12^2 - 4*3*9 = 144 - 108 = 36.

Correct Answer: A — 0

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Q. What is the value of the discriminant for the quadratic equation 3x^2 + 12x + 9 = 0? (2019)

A. 0
B. 9
C. 36
D. 27

Solution

The discriminant D = b^2 - 4ac = 12^2 - 4*3*9 = 0, indicating equal roots.

Correct Answer: A — 0

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Q. What is the value of the discriminant for the quadratic equation 3x^2 + 6x + 2 = 0? (2023)

A. 0
B. 4
C. 12
D. 36

Solution

The discriminant is b^2 - 4ac = 6^2 - 4(3)(2) = 36 - 24 = 12.

Correct Answer: B — 4

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Q. What is the value of the term containing x^4 in the expansion of (3x + 4)^6? (1944)

A. 864
B. 1296
C. 1728
D. 2160

Solution

The term containing x^4 is given by C(6, 4)(3x)^4(4)^2 = 15 * 81 * 16 = 19440.

Correct Answer: B — 1296

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Q. What is the value of x if 3x - 7 = 2x + 5? (2021)

A. 12
B. 11
C. 10
D. 9

Solution

Rearranging gives 3x - 2x = 5 + 7, thus x = 12.

Correct Answer: A — 12

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Q. What is the value of x if the matrix A = [[x, 2], [3, 4]] is singular?

A. 1
B. 2
C. 3
D. 4

Solution

A matrix is singular if its determinant is zero. Det(A) = (x*4) - (2*3) = 4x - 6 = 0. Solving gives x = 1.5.

Correct Answer: C — 3

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Q. What is the value of x in the equation 3x - 7 = 2x + 5? (2023)

A. 5
B. 6
C. 7
D. 8

Solution

Rearranging gives 3x - 2x = 5 + 7, thus x = 12.

Correct Answer: B — 6

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Q. What is the value of x in the equation 4x + 3 = 19? (2023)

A. 4
B. 3
C. 5
D. 2

Solution

Solving for x gives 4x = 16, thus x = 4.

Correct Answer: A — 4

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Q. What is the value of x in the equation 4x - 5 = 11? (2022)

A. 2
B. 3
C. 4
D. 5

Solution

Add 5 to both sides: 4x = 16. Then divide by 4: x = 4.

Correct Answer: B — 3

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Q. What is the value of x in the equation 4x - 7 = 9? (2022)

A. 4
B. 5
C. 6
D. 3

Solution

Adding 7 to both sides gives 4x = 16. Dividing by 4 gives x = 4.

Correct Answer: A — 4

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Q. What is the value of |z| if z = -1 - i?

A. √2
B. 2
C. 1
D. √3

Solution

The modulus |z| = √((-1)² + (-1)²) = √(1 + 1) = √2.

Correct Answer: A — √2

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Q. What is the value of |−8| + |−3|? (2023)

A. 5
B. 8
C. 11
D. 14

Solution

|−8| = 8 and |−3| = 3, so 8 + 3 = 11.

Correct Answer: C — 11

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Q. What is the value of √(144) + √(64)?

A. 16
B. 20
C. 18
D. 22

Solution

√(144) = 12 and √(64) = 8, so 12 + 8 = 20.

Correct Answer: C — 18

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Q. What is the value of √(16) + √(25)? (2021)

A. 5
B. 9
C. 11
D. 15

Solution

√(16) = 4 and √(25) = 5, so 4 + 5 = 9.

Correct Answer: C — 11

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Q. What is the variance of the following numbers: 1, 1, 1, 1, 1?

A. 0
B. 1
C. 2
D. 3

Solution

All values are the same, so variance = 0.

Correct Answer: A — 0

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Q. What is the variance of the following numbers: 1, 2, 3, 4, 5, 6? (2022)

A. 2.5
B. 3.5
C. 4.5
D. 5.5

Solution

Mean = (1 + 2 + 3 + 4 + 5 + 6) / 6 = 3.5. Variance = [(1-3.5)² + (2-3.5)² + (3-3.5)² + (4-3.5)² + (5-3.5)² + (6-3.5)²] / 6 = 2.92 (approx 2.5).

Correct Answer: A — 2.5

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Q. What is the variance of the following numbers: 2, 4, 4, 4, 5, 5, 7, 9?

A. 4
B. 6
C. 8
D. 10

Solution

Mean = 5. Variance = [(2-5)² + (4-5)² + (4-5)² + (4-5)² + (5-5)² + (5-5)² + (7-5)² + (9-5)²] / 8 = 6.

Correct Answer: B — 6

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Q. What is the variance of the following numbers: 5, 5, 5, 5? (2019)

A. 0
B. 1
C. 2
D. 3

Solution

All values are the same, so variance = 0.

Correct Answer: A — 0

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Q. What is the variance of the following numbers: 5, 7, 9, 11? (2020)

A. 4
B. 6
C. 8
D. 10

Solution

Mean = (5 + 7 + 9 + 11) / 4 = 8. Variance = [(5-8)² + (7-8)² + (9-8)² + (11-8)²] / 4 = 4.

Correct Answer: A — 4

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Q. What is the variance of the numbers: 1, 1, 1, 1, 1? (2023)

A. 0
B. 1
C. 2
D. 3

Solution

All values are the same, so variance = 0.

Correct Answer: A — 0

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Q. What is the variance of the numbers: 5, 5, 5, 5? (2023)

A. 0
B. 1
C. 2
D. 3

Solution

All values are the same, so variance = 0.

Correct Answer: A — 0

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Q. What is the vertex of the parabola given by the equation y = 2x^2 - 4x + 1?

A. (1, -1)
B. (1, 0)
C. (2, 1)
D. (0, 1)

Solution

To find the vertex, use the formula x = -b/(2a). Here, a = 2, b = -4, so x = 1. Substitute x = 1 into the equation to find y = -1.

Correct Answer: A — (1, -1)

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Q. What is the vertex of the parabola represented by the equation y = 3x^2 - 12x + 7? (2023)

A. (2, -5)
B. (2, -1)
C. (2, 1)
D. (2, 5)

Solution

The vertex can be found using x = -b/(2a). Here, x = 12/(2*3) = 2. Substituting x = 2 into the equation gives y = 3(2)^2 - 12(2) + 7 = -5.

Correct Answer: A — (2, -5)

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Q. What is the vertex of the quadratic function f(x) = 2x^2 - 8x + 5? (2021)

A. (2, -3)
B. (2, -7)
C. (4, -3)
D. (4, -7)

Solution

The vertex can be found using x = -b/(2a). Here, x = 8/(2*2) = 2. Substituting x = 2 into f(x) gives f(2) = -3, so the vertex is (2, -3).

Correct Answer: A — (2, -3)

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Q. What is the volume of a cube with a side length of 5 cm? (2022)

A. 25 cm³
B. 50 cm³
C. 125 cm³
D. 150 cm³

Solution

The volume of a cube is given by V = side³. For a side length of 5 cm, V = 5³ = 125 cm³.

Correct Answer: C — 125 cm³

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Q. What is the x-intercept of the line 2x + 3y = 6? (2019)

A. 2
B. 3
C. 0
D. 6

Solution

To find the x-intercept, set y = 0. Thus, 2x = 6, giving x = 3.

Correct Answer: A — 2

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Q. What is the x-intercept of the line given by the equation 4x + 5y - 20 = 0?

A. 4
B. 5
C. 20
D. 0

Solution

To find the x-intercept, set y = 0. Thus, 4x = 20, giving x = 5.

Correct Answer: A — 4

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Mathematics

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