Mathematics Study Resources for Competitive Exams

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Q. The quadratic equation x^2 + 6x + 9 = 0 can be expressed in which of the following forms? (2020)

A. (x + 3)^2
B. (x - 3)^2
C. (x + 6)^2
D. (x - 6)^2

Solution

This is a perfect square trinomial: (x + 3)(x + 3) = 0.

Correct Answer: A — (x + 3)^2

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Q. The quadratic equation x^2 + 6x + k = 0 has equal roots. What is the value of k? (2020)

A. 9
B. 12
C. 16
D. 36

Solution

For equal roots, b^2 - 4ac = 0. Here, 6^2 - 4(1)(k) = 0, so k = 9.

Correct Answer: A — 9

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Q. The quadratic equation x^2 + 6x + k = 0 has no real roots. What is the condition on k? (2020)

A. k < 9
B. k > 9
C. k = 9
D. k ≤ 9

Solution

For no real roots, the discriminant must be less than zero: 6^2 - 4*1*k < 0, which gives k > 9.

Correct Answer: B — k > 9

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Q. The quadratic equation x^2 + 6x + k = 0 has roots that are both negative. What is the condition for k? (2020)

A. k > 9
B. k < 9
C. k = 9
D. k = 0

Solution

For both roots to be negative, k must be greater than the square of half the coefficient of x, hence k > 9.

Correct Answer: A — k > 9

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Q. The quadratic equation x^2 - 4x + 4 = 0 can be expressed in which of the following forms? (2022)

A. (x - 2)^2
B. (x + 2)^2
C. (x - 4)^2
D. (x + 4)^2

Solution

The equation can be factored as (x - 2)(x - 2) = 0, which is (x - 2)^2.

Correct Answer: A — (x - 2)^2

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Q. The quadratic equation x^2 - 6x + 9 = 0 can be expressed as which of the following? (2021)

A. (x - 3)^2 = 0
B. (x + 3)^2 = 0
C. (x - 2)(x - 4) = 0
D. (x + 2)(x + 4) = 0

Solution

The equation can be factored as (x - 3)(x - 3) = 0, or (x - 3)^2 = 0.

Correct Answer: A — (x - 3)^2 = 0

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Q. The quadratic equation x^2 - 6x + 9 = 0 can be expressed in the form (x - a)^2 = 0. What is the value of a? (2021)

A. 3
B. 6
C. 9
D. 0

Solution

The equation can be factored as (x - 3)^2 = 0, hence a = 3.

Correct Answer: A — 3

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Q. The radius of a circle is increased by 50%. What is the percentage increase in the area of the circle? (2020)

A. 50%
B. 75%
C. 100%
D. 125%

Solution

If r is the original radius, new radius = 1.5r. Area increases from πr² to π(1.5r)² = 2.25πr². Percentage increase = (2.25 - 1) × 100% = 125%.

Correct Answer: C — 100%

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Q. The roots of the equation 4x^2 - 12x + 9 = 0 are: (2019)

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

Solution

The equation can be factored as (2x - 3)(2x - 3) = 0, hence the roots are 3 and 3.

Correct Answer: B — 3 and 3

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Q. The roots of the equation x^2 + 2x + k = 0 are -1 and -3. What is the value of k?

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

Solution

The sum of the roots is -1 + (-3) = -4, so -2 = -4, which is correct. The product of the roots is (-1)(-3) = 3, so k = 3.

Correct Answer: B — 3

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Q. The roots of the equation x^2 + 3x + k = 0 are -1 and -2. What is the value of k? (2021)

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

Solution

The sum of the roots is -1 + (-2) = -3, and the product is (-1)(-2) = 2. Thus, k = 2.

Correct Answer: A — 2

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Q. The roots of the equation x^2 + 4x + k = 0 are equal. What is the value of k?

A. 4
B. 8
C. 16
D. 0

Solution

For the roots to be equal, the discriminant must be zero. Thus, 4^2 - 4*1*k = 0 => 16 - 4k = 0 => k = 4.

Correct Answer: B — 8

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Q. The roots of the equation x^2 - 10x + 21 = 0 are: (2020)

A. 3 and 7
B. 4 and 6
C. 5 and 5
D. 2 and 8

Solution

Factoring gives (x - 3)(x - 7) = 0, so the roots are 3 and 7.

Correct Answer: A — 3 and 7

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Q. The roots of the equation x^2 - 2x + k = 0 are real and distinct if k is:

A. < 1
B. ≥ 1
C. ≤ 1
D. > 1

Solution

The discriminant must be greater than zero: (-2)^2 - 4*1*k > 0, which simplifies to k < 1.

Correct Answer: A — < 1

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Q. The roots of the equation x^2 - 5x + k = 0 are equal. What is the value of k? (2020)

A. 6.25
B. 5
C. 4
D. 0

Solution

For the roots to be equal, the discriminant must be zero. Thus, (-5)^2 - 4*1*k = 0 leads to 25 - 4k = 0, giving k = 6.25.

Correct Answer: A — 6.25

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Q. The roots of the quadratic equation x^2 - 4x + k = 0 are 2 and 2. What is the value of k? (2022)

A. 0
B. 2
C. 4
D. 8

Solution

If the roots are both 2, then k = 2^2 - 4*2 = 4 - 8 = -4. Thus, k = 4.

Correct Answer: C — 4

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Q. The scalar product of two unit vectors is 0. What can be inferred about these vectors?

A. They are parallel
B. They are orthogonal
C. They are collinear
D. They are equal

Solution

If the scalar product is 0, the vectors are orthogonal.

Correct Answer: B — They are orthogonal

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Q. The scalar product of two unit vectors is 0.5. What is the angle between them?

A. 60°
B. 30°
C. 90°
D. 120°

Solution

cos(θ) = 0.5, θ = cos⁻¹(0.5) = 60°.

Correct Answer: A — 60°

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Q. The scalar product of two vectors A and B is 12, and the angle between them is 60°. If |A| = 4, find |B|.

A. 6
B. 8
C. 10
D. 12

Solution

A · B = |A||B|cos(θ) => 12 = 4|B|(0.5) => |B| = 6.

Correct Answer: B — 8

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Q. The scores of a class in a test are: 10, 12, 14, 16, 18. What is the variance? (2022)

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

Solution

Mean = 14. Variance = [(10-14)² + (12-14)² + (14-14)² + (16-14)² + (18-14)²] / 5 = 8.

Correct Answer: A — 4

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Q. The scores of a test are: 20, 30, 40, 50. Calculate the variance. (2020)

A. 100
B. 200
C. 300
D. 400

Solution

Mean = (20 + 30 + 40 + 50) / 4 = 35. Variance = [(20-35)² + (30-35)² + (40-35)² + (50-35)²] / 4 = 250.

Correct Answer: B — 200

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Q. The scores of a test are: 20, 30, 40, 50. What is the standard deviation? (2020)

A. 10
B. 15
C. 20
D. 25

Solution

Mean = (20 + 30 + 40 + 50) / 4 = 35. Variance = [(20-35)² + (30-35)² + (40-35)² + (50-35)²] / 4 = 125. SD = √125 = 11.18 (approx 10).

Correct Answer: B — 15

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Q. The scores of five students are: 20, 22, 24, 26, 28. Calculate the variance. (2020)

A. 8
B. 10
C. 12
D. 14

Solution

Mean = (20 + 22 + 24 + 26 + 28) / 5 = 24. Variance = [(20-24)² + (22-24)² + (24-24)² + (26-24)² + (28-24)²] / 5 = 8.

Correct Answer: A — 8

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Q. The scores of five students are: 20, 22, 24, 26, 28. What is the standard deviation? (2020)

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

Solution

Mean = (20 + 22 + 24 + 26 + 28) / 5 = 24. Variance = [(20-24)² + (22-24)² + (24-24)² + (26-24)² + (28-24)²] / 5 = 8. SD = √8 = 2.83 (approximately 3).

Correct Answer: A — 2

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Q. The scores of five students are: 20, 22, 24, 26, 28. What is the variance? (2020)

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

Solution

Mean = (20 + 22 + 24 + 26 + 28) / 5 = 24. Variance = [(20-24)² + (22-24)² + (24-24)² + (26-24)² + (28-24)²] / 5 = 8.

Correct Answer: A — 4

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Q. The slope of the tangent line to the curve y = x^2 at the point (2, 4) is: (2022)

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

Solution

The derivative y' = 2x. At x = 2, y' = 2(2) = 4.

Correct Answer: A — 2

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Q. The standard deviation of a data set is 5. If all values are increased by 2, what will be the new standard deviation? (2020)

A. 3
B. 5
C. 7
D. 10

Solution

Standard deviation is unaffected by adding a constant. Therefore, new SD = 5.

Correct Answer: B — 5

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Q. The standard deviation of a data set is 5. What is the variance? (2023)

A. 20
B. 25
C. 30
D. 35

Solution

Variance is the square of the standard deviation. Therefore, variance = 5² = 25.

Correct Answer: B — 25

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Q. The sum of the angles in a triangle is equal to how many degrees? (2021)

A. 90
B. 180
C. 270
D. 360

Solution

The sum of the angles in any triangle is always 180 degrees.

Correct Answer: B — 180

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Q. The sum of the roots of the equation x^2 - 7x + k = 0 is 7. What is the value of k if the product of the roots is 10? (2023)

A. 10
B. 15
C. 20
D. 25

Solution

Using Vieta's formulas, the sum of the roots is 7 and the product is k = 10.

Correct Answer: A — 10

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