Mathematics Study Resources for Competitive Exams

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Q. The function f(x) = 2x + 1 is continuous at which of the following intervals?

A. (-∞, ∞)
B. (0, 1)
C. (1, 2)
D. (2, 3)

Solution

f(x) = 2x + 1 is a linear function and is continuous over the entire real line (-∞, ∞).

Correct Answer: A — (-∞, ∞)

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Q. The function f(x) = 2x + 3 is continuous at which of the following intervals?

A. (-∞, ∞)
B. [0, 1]
C. [1, 2]
D. [2, 3]

Solution

f(x) = 2x + 3 is a linear function and is continuous over the entire real line (-∞, ∞).

Correct Answer: A — (-∞, ∞)

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Q. The function f(x) = 2x + 3 is continuous at which of the following? (2023)

A. x = -1
B. x = 0
C. x = 1
D. All of the above

Solution

f(x) = 2x + 3 is a linear function and is continuous for all x.

Correct Answer: D — All of the above

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Q. The function f(x) = sin(x) + cos(x) has a maximum value at which point? (2022)

A. π/4
B. π/2
C. 0
D. 3π/4

Solution

To find the maximum, set f'(x) = cos(x) - sin(x) = 0. This occurs at x = π/4.

Correct Answer: A — π/4

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Q. The function f(x) = x^2 + 3 is continuous for which of the following intervals? (2023)

A. (-∞, ∞)
B. (0, 1)
C. (1, 2)
D. (2, 3)

Solution

f(x) = x^2 + 3 is a polynomial function and is continuous for all x in (-∞, ∞).

Correct Answer: A — (-∞, ∞)

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Q. The function f(x) = x^2 is continuous at which of the following points? (2023)

A. x = -1
B. x = 0
C. x = 1
D. All of the above

Solution

The function f(x) = x^2 is a polynomial function and is continuous at all points, including -1, 0, and 1.

Correct Answer: D — All of the above

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Q. The function f(x) = x^3 - 3x is continuous at which of the following points? (2023)

A. x = -2
B. x = 0
C. x = 2
D. All of the above

Solution

The function f(x) = x^3 - 3x is a polynomial function and is continuous at all points, including -2, 0, and 2.

Correct Answer: D — All of the above

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Q. The function f(x) = { x + 1, x < 1; 2, x = 1; x^2, x > 1 } is continuous at x = 1?

A. Yes
B. No
C. Only left continuous
D. Only right continuous

Solution

Left limit as x approaches 1 is 2, right limit is 1, but f(1) = 2. Hence, it is discontinuous at x = 1.

Correct Answer: B — No

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Q. The function f(x) = { x^2, x < 0; 0, x = 0; x + 1, x > 0 } is:

A. Continuous
B. Not continuous
C. Continuous from the left
D. Continuous from the right

Solution

The left limit as x approaches 0 is 0, but the right limit is 1. Hence, it is not continuous at x = 0.

Correct Answer: B — Not continuous

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Q. The function f(x) = { x^2, x < 0; 2, x = 0; x + 1, x > 0 } is continuous at x = 0?

A. Yes
B. No
C. Only left continuous
D. Only right continuous

Solution

At x = 0, lim x→0- f(x) = 0 and lim x→0+ f(x) = 1, hence it is discontinuous at x = 0.

Correct Answer: B — No

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Q. The lengths of the sides of triangle ABC are 5 cm, 12 cm, and 13 cm. What is the area of the triangle? (2019)

A. 30 cm²
B. 60 cm²
C. 24 cm²
D. 40 cm²

Solution

Using Heron's formula, s = (5 + 12 + 13)/2 = 15. Area = √[s(s-a)(s-b)(s-c)] = √[15(15-5)(15-12)(15-13)] = √[15*10*3*2] = √[900] = 30 cm².

Correct Answer: A — 30 cm²

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Q. The lengths of the sides of triangle ABC are 7 cm, 24 cm, and 25 cm. Is this triangle a right triangle? (2020)

A. Yes
B. No
C. Cannot be determined
D. Only if angle A is 90 degrees

Solution

Using the Pythagorean theorem, 7^2 + 24^2 = 49 + 576 = 625 = 25^2. Therefore, triangle ABC is a right triangle.

Correct Answer: A — Yes

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Q. The lengths of the sides of triangle GHI are 5 cm, 12 cm, and 13 cm. What is the area of the triangle? (2019)

A. 30 cm²
B. 60 cm²
C. 65 cm²
D. 78 cm²

Solution

This is a right triangle. Area = 1/2 * base * height = 1/2 * 5 * 12 = 30 cm².

Correct Answer: A — 30 cm²

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Q. The magnitude of the vector A = 4i - 3j + 12k is:

A. 13
B. 14
C. 15
D. 16

Solution

Magnitude |A| = √(4^2 + (-3)^2 + 12^2) = √(16 + 9 + 144) = √169 = 13.

Correct Answer: B — 14

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Q. The mean of 10 numbers is 50. If one number is removed, the mean becomes 48. What is the removed number?

A. 52
B. 50
C. 48
D. 46

Solution

Total of 10 numbers = 10 * 50 = 500. Total of 9 numbers = 9 * 48 = 432. Removed number = 500 - 432 = 68.

Correct Answer: A — 52

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Q. The mean of 10, 20, 30, 40, and x is 25. What is the value of x? (2022)

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

Solution

Mean = (10 + 20 + 30 + 40 + x) / 5 = 25. Therefore, 100 + x = 125, so x = 25.

Correct Answer: A — 10

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Q. The mean of 4, 6, 8, and x is 7. What is the value of x?

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

Solution

Mean = (4 + 6 + 8 + x) / 4 = 7. Thus, 18 + x = 28, so x = 10.

Correct Answer: D — 10

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Q. The mean of 5, 10, 15, 20, and x is 12. What is the value of x?

A. 5
B. 10
C. 15
D. 20

Solution

Mean = (5 + 10 + 15 + 20 + x) / 5 = 12.\n\n50 + x = 60\n\nx = 60 - 50 = 10.

Correct Answer: A — 5

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Q. The mean of 5, 10, 15, and 20 is what?

A. 10
B. 12.5
C. 15
D. 17.5

Solution

Mean = (5 + 10 + 15 + 20) / 4 = 50 / 4 = 12.5.

Correct Answer: C — 15

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Q. The mean of 5, 10, 15, and x is 12. What is the value of x?

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

Solution

Mean = (5 + 10 + 15 + x) / 4 = 12. Therefore, 30 + x = 48, so x = 18.

Correct Answer: A — 8

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Q. The mean of 5, 15, 25, and x is 20. Find the value of x. (2023)

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

Solution

Mean = (5 + 15 + 25 + x) / 4 = 20. Solving gives x = 30.

Correct Answer: C — 30

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Q. The mean of 5, 15, and x is 10. What is the value of x? (2020)

A. 5
B. 10
C. 15
D. 20

Solution

Mean = (5 + 15 + x) / 3 = 10. Therefore, 20 + x = 30, so x = 10.

Correct Answer: D — 20

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Q. The mean of 6, 8, 10, and x is 9. What is the value of x? (2022)

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

Solution

Mean = (6 + 8 + 10 + x) / 4 = 9. Therefore, 24 + x = 36, so x = 12.

Correct Answer: D — 12

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Q. The mean of 7, 9, 11, 13, and x is 10. Find the value of x.

A. 5
B. 7
C. 9
D. 11

Solution

Mean = (7 + 9 + 11 + 13 + x) / 5 = 10. Solving gives x = 5.

Correct Answer: A — 5

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Q. The mean of 7, 9, 11, 13, and x is 10. What is the value of x?

A. 5
B. 7
C. 9
D. 11

Solution

Mean = (7 + 9 + 11 + 13 + x) / 5 = 10.\n\n40 + x = 50\n\nx = 50 - 40 = 10.

Correct Answer: A — 5

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Q. The mean of 7, 9, 11, and 13 is what? (2019)

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

Solution

Mean = (7 + 9 + 11 + 13) / 4 = 40 / 4 = 10.

Correct Answer: B — 10

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Q. The mean of 8, 12, 16, and 20 is what? (2019)

A. 14
B. 15
C. 16
D. 17

Solution

Mean = (8 + 12 + 16 + 20) / 4 = 56 / 4 = 14.

Correct Answer: B — 15

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Q. The mean of 8, 12, 16, and x is 14. Find the value of x.

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

Solution

Mean = (8 + 12 + 16 + x) / 4 = 14. Therefore, 36 + x = 56, so x = 20.

Correct Answer: A — 10

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Q. The mean of a data set is 10 and the variance is 4. What is the standard deviation? (2019)

A. 2
B. 4
C. 8
D. 10

Solution

Standard deviation is the square root of variance. Therefore, SD = √4 = 2.

Correct Answer: A — 2

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Q. The mean of a set of numbers is 15. If one number is removed, the mean becomes 12. What was the removed number?

A. 18
B. 20
C. 24
D. 30

Solution

Let the total number of items be n. Then, 15n - x = 12(n - 1). Solving gives x = 24.

Correct Answer: C — 24

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Mathematics

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