Mathematics Study Resources for Competitive Exams

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Q. If the numbers are 8, 8, 6, 5, 5, 5, 7, 9, what is the mode?

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

Solution

The mode is 5, as it appears 3 times, more than any other number.

Correct Answer: A — 5

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Q. If the perimeter of a rectangle is 50 cm and the length is 15 cm, what is the width? (2020)

A. 10 cm
B. 12.5 cm
C. 15 cm
D. 20 cm

Solution

Perimeter = 2(length + width). So, 50 = 2(15 + width). Solving gives width = 10 cm.

Correct Answer: B — 12.5 cm

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Q. If the perimeter of triangle DEF is 36 cm and the lengths of sides DE and EF are 10 cm and 12 cm respectively, what is the length of side DF? (2020)

A. 10 cm
B. 12 cm
C. 14 cm
D. 16 cm

Solution

Perimeter = DE + EF + DF. Therefore, DF = 36 - (10 + 12) = 14 cm.

Correct Answer: C — 14 cm

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Q. If the point (x, y) lies on the line 3x - 4y = 12, what is the value of y when x = 4? (2023)

A. 0
B. 3
C. 4
D. 2

Solution

Substituting x = 4 gives 3(4) - 4y = 12 => 12 - 4y = 12 => y = 0.

Correct Answer: D — 2

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Q. If the point (x, y) lies on the line 4x - 5y = 20, what is the value of y when x = 0?

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

Solution

Substituting x = 0: 4(0) - 5y = 20 => -5y = 20 => y = -4.

Correct Answer: B — 5

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Q. If the polynomial equation x^3 - 6x^2 + 11x - 6 = 0 has roots a, b, and c, what is the value of a + b + c? (2021)

A. 6
B. 11
C. 3
D. 0

Solution

By Vieta's formulas, the sum of the roots (a + b + c) of the polynomial x^3 - 6x^2 + 11x - 6 is equal to the coefficient of x^2 with the opposite sign, which is 6.

Correct Answer: A — 6

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Q. If the position vector of point P is given by r = 2i + 3j + 4k, what is the distance from the origin to point P?

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

Solution

Distance = |r| = √(2^2 + 3^2 + 4^2) = √(4 + 9 + 16) = √29.

Correct Answer: B — 6

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Q. If the position vector of point P is given by r = 2i + 3j + 4k, what is the x-coordinate of point P? (2020)

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

Solution

The x-coordinate of point P is the coefficient of i in the position vector, which is 2.

Correct Answer: A — 2

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Q. If the position vector of point P is given by r = 2i + 3j + 4k, what is the x-component of r?

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

Solution

The x-component of r is 2.

Correct Answer: A — 2

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Q. If the position vector of point P is given by r = 3i + 4j, what is the distance of point P from the origin?

A. 5
B. 7
C. 4
D. 3

Solution

Distance = |r| = √(3^2 + 4^2) = √(9 + 16) = √25 = 5.

Correct Answer: A — 5

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Q. If the probability of an event A is 0.4, what is the probability of the event not occurring? (2021)

A. 0.4
B. 0.6
C. 0.8
D. 0.2

Solution

Probability of not A = 1 - P(A) = 1 - 0.4 = 0.6.

Correct Answer: B — 0.6

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Q. If the probability of an event occurring is 0.3, what is the probability of it not occurring? (2023)

A. 0.3
B. 0.7
C. 0.5
D. 0.1

Solution

Probability of not occurring = 1 - P(event) = 1 - 0.3 = 0.7.

Correct Answer: B — 0.7

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Q. If the quadratic equation ax^2 + bx + c = 0 has roots p and q, what is the value of p + q? (2020)

A. -b/a
B. b/a
C. c/a
D. -c/a

Solution

By Vieta's formulas, the sum of the roots p + q = -b/a.

Correct Answer: A — -b/a

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Q. If the quadratic equation x^2 + 2x + 1 = 0 is solved, what are the roots? (2022)

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

Solution

The equation can be factored as (x + 1)(x + 1) = 0, giving the root -1 with multiplicity 2.

Correct Answer: A — -1

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Q. If the quadratic equation x^2 + 2x + 1 = 0 is solved, what is the nature of its roots? (2019)

A. Real and distinct
B. Real and equal
C. Complex
D. None of the above

Solution

The discriminant is 2^2 - 4*1*1 = 0, indicating that the roots are real and equal.

Correct Answer: B — Real and equal

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Q. If the quadratic equation x^2 + 2x + 1 = 0 is solved, what is the nature of the roots? (2022)

A. Real and distinct
B. Real and equal
C. Complex
D. None of the above

Solution

The discriminant is 0, indicating that the roots are real and equal.

Correct Answer: B — Real and equal

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Q. If the quadratic equation x^2 + 2x + 1 = 0 is solved, what is the value of x? (2023)

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

Solution

The equation can be factored as (x + 1)^2 = 0, giving the root x = -1.

Correct Answer: A — -1

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Q. If the quadratic equation x^2 + 2x + k = 0 has one root equal to -1, what is the value of k? (2022)

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

Solution

Substituting x = -1 into the equation gives (-1)^2 + 2(-1) + k = 0, leading to 1 - 2 + k = 0, thus k = 1.

Correct Answer: B — 1

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Q. If the quadratic equation x^2 + 2x + k = 0 has roots 1 and -3, what is the value of k? (2022)

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

Solution

The product of the roots is 1 * (-3) = -3, hence k = -3.

Correct Answer: A — -3

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Q. If the quadratic equation x^2 + 2x + k = 0 has roots that are both negative, what is the condition for k? (2023)

A. k > 1
B. k < 1
C. k > 0
D. k < 0

Solution

For both roots to be negative, k must be greater than 0.

Correct Answer: C — k > 0

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Q. If the quadratic equation x^2 + 2x + k = 0 has roots that are both negative, what is the condition on k? (2023)

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

Solution

For both roots to be negative, k must be greater than 0.

Correct Answer: A — k > 0

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Q. If the quadratic equation x^2 + 2x + k = 0 has roots that are both positive, what is the condition on k? (2019)

A. k < 0
B. k > 0
C. k < 4
D. k > 4

Solution

For both roots to be positive, k must be less than 4 (from the condition of the sum and product of roots).

Correct Answer: C — k < 4

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Q. If the quadratic equation x^2 + 4x + 4 = 0 is solved, what is the nature of its roots? (2019)

A. Two distinct real roots
B. One real root
C. Two complex roots
D. No roots

Solution

The discriminant is 0, indicating one real root (a repeated root).

Correct Answer: B — One real root

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Q. If the quadratic equation x^2 + 5x + 6 = 0 is solved, what is the product of the roots? (2022)

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

Solution

The product of the roots is given by c/a = 6/1 = 6.

Correct Answer: A — 6

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Q. If the quadratic equation x^2 + 5x + k = 0 has one root as 2, what is the value of k? (2019)

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

Solution

Substituting x = 2 into the equation gives 2^2 + 5(2) + k = 0, leading to 4 + 10 + k = 0, thus k = -14.

Correct Answer: A — 6

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Q. If the quadratic equation x^2 + 6x + 9 = 0 is solved, what is the nature of its roots? (2019)

A. Two distinct real roots
B. One real root
C. Two complex roots
D. No roots

Solution

The discriminant is 0, indicating one real root (a repeated root).

Correct Answer: B — One real root

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Q. If the quadratic equation x^2 + 6x + k = 0 has roots that are both positive, what is the minimum value of k? (2021)

A. 0
B. 1
C. 4
D. 9

Solution

For both roots to be positive, k must be greater than 9 (since the sum of roots is -b/a and must be positive).

Correct Answer: C — 4

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Q. If the quadratic equation x^2 + 7x + k = 0 has roots that are both positive, what is the minimum value of k? (2021)

A. 6
B. 7
C. 8
D. 9

Solution

For both roots to be positive, k must be greater than 12 (from Vieta's formulas). The minimum integer k is 13.

Correct Answer: C — 8

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Q. If the quadratic equation x^2 + kx + 16 = 0 has roots that are both real and distinct, what is the condition for k? (2022)

A. k > 8
B. k < -8
C. k > -8
D. k < 8

Solution

The discriminant must be positive: k^2 - 4*1*16 > 0, which simplifies to k^2 > 64, hence k < -8 or k > 8.

Correct Answer: B — k < -8

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Q. If the quadratic equation x^2 + px + q = 0 has roots 3 and -2, what is the value of p? (2020)

A. -1
B. 1
C. 5
D. -5

Solution

Using the sum of roots formula, p = -(3 + (-2)) = -1. Therefore, p = 1.

Correct Answer: C — 5

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