Mathematics Study Resources for Competitive Exams

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Q. Solve the differential equation dy/dx = 6x.

A. y = 3x^2 + C
B. y = 6x^2 + C
C. y = 2x^2 + C
D. y = 3x + C

Solution

Integrating gives y = 3x^2 + C, where C is the constant of integration.

Correct Answer: A — y = 3x^2 + C

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Q. Solve the differential equation dy/dx = y^2.

A. y = 1/(C - x)
B. y = Cx
C. y = C + x^2
D. y = C - x

Solution

This is a separable equation. Integrating gives y = 1/(C - x), where C is the constant.

Correct Answer: A — y = 1/(C - x)

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Q. The ages of a group of people are: 22, 25, 22, 30, 22, 25, 30. What is the mode of their ages?

A. 22
B. 25
C. 30
D. None

Solution

The age 22 appears 3 times, which is more than any other age, making it the mode.

Correct Answer: A — 22

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Q. The ages of a group of people are: 22, 25, 22, 30, 25, 22, 28. What is the mode of their ages?

A. 22
B. 25
C. 28
D. 30

Solution

The mode is 22, as it appears 3 times, more than any other age.

Correct Answer: A — 22

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Q. The angle between the vectors A = i + j and B = j + k is:

A. 45°
B. 60°
C. 90°
D. 30°

Solution

cos(θ) = (A · B) / (|A||B|) = (0) / (√2 * √2) = 0, thus θ = 90°.

Correct Answer: C — 90°

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

A. 60
B. 50
C. 40
D. 30

Solution

Total of 10 numbers = 10 * 50 = 500.\n\nTotal of 9 numbers = 9 * 48 = 432.\n\nRemoved number = 500 - 432 = 68.

Correct Answer: A — 60

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Q. The average of 10, 20, 30, and x is 25. What is the value of x?

A. 20
B. 30
C. 40
D. 50

Solution

Mean = (10 + 20 + 30 + x) / 4 = 25. Solving gives x = 50.

Correct Answer: C — 40

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Q. The average of 4 numbers is 15. If one number is increased by 5, what will be the new average? (2019)

A. 15
B. 16
C. 17
D. 18

Solution

Total of 4 numbers = 4 * 15 = 60. New total = 60 + 5 = 65. New average = 65 / 4 = 16.25.

Correct Answer: B — 16

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

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

Solution

Mean = (4 + 6 + 8 + x) / 4 = 7.\n\n18 + x = 28\n\nx = 28 - 18 = 10.

Correct Answer: C — 10

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Q. The average of 7 numbers is 14. If one number is added which is 10 more than the average, what will be the new average?

A. 15
B. 16
C. 17
D. 18

Solution

New total = 7 * 14 + (14 + 10) = 98 + 24 = 122. New average = 122 / 8 = 15.25.

Correct Answer: B — 16

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Q. The average of 8 numbers is 15. If one number is excluded, the average of the remaining numbers becomes 12. What is the excluded number?

A. 24
B. 20
C. 18
D. 16

Solution

The total sum of the 8 numbers is 8 * 15 = 120. Let the excluded number be x. The sum of the remaining 7 numbers is 120 - x, and their average is (120 - x)/7 = 12. Thus, 120 - x = 84, leading to x = 36.

Correct Answer: B — 20

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Q. The average of five consecutive integers is 12. What is the smallest of these integers?

A. 10
B. 11
C. 12
D. 13

Solution

Let the integers be x, x+1, x+2, x+3, x+4.\n\nMean = (5x + 10) / 5 = 12\n\n5x + 10 = 60\n\n5x = 50\n\nx = 10.

Correct Answer: B — 11

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Q. The average of five numbers is 20. If one number is excluded, the average of the remaining numbers becomes 18. What is the excluded number?

A. 22
B. 24
C. 20
D. 18

Solution

Total of five numbers = 5 * 20 = 100. Total of remaining four = 4 * 18 = 72. Excluded number = 100 - 72 = 28.

Correct Answer: A — 22

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Q. The average of five numbers is 20. If one number is removed, the average becomes 18. What is the number removed?

A. 22
B. 24
C. 20
D. 18

Solution

Total of five numbers = 5 * 20 = 100. Total of four numbers = 4 * 18 = 72. Number removed = 100 - 72 = 28.

Correct Answer: A — 22

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Q. The average of three consecutive integers is 20. What are the integers?

A. 19, 20, 21
B. 18, 19, 20
C. 20, 21, 22
D. 17, 18, 19

Solution

Let the integers be x, x+1, x+2. (x + (x + 1) + (x + 2)) / 3 = 20. Therefore, 3x + 3 = 60, so x = 19.

Correct Answer: A — 19, 20, 21

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Q. The average of three numbers is 15. If one number is increased by 5, what will be the new average?

A. 15
B. 16
C. 17
D. 18

Solution

Total of three numbers = 3 * 15 = 45.\n\nNew total = 45 + 5 = 50.\n\nNew average = 50 / 3 = 16.67.

Correct Answer: B — 16

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Q. The average of three numbers is 40. If the first number is increased by 10, what will be the new average?

A. 42
B. 43
C. 44
D. 45

Solution

The sum of the three numbers is 3 * 40 = 120. Increasing the first number by 10 gives a new sum of 120 + 10 = 130. The new average is 130 / 3 = 43.33, which rounds to 42.

Correct Answer: A — 42

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Q. The average of two numbers is 30. If one number is 20, what is the other number?

A. 40
B. 50
C. 60
D. 70

Solution

Mean = (20 + x) / 2 = 30.\n\n20 + x = 60\n\nx = 60 - 20 = 40.

Correct Answer: A — 40

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Q. The circumference of a circle is 31.4 cm. What is the radius of the circle? (Use π = 3.14) (2019)

A. 5 cm
B. 10 cm
C. 7 cm
D. 15 cm

Solution

Circumference = 2πr => r = circumference / (2π) = 31.4 / (2 * 3.14) = 5 cm

Correct Answer: A — 5 cm

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Q. The coordinates of the centroid of a triangle with vertices at (1, 2), (3, 4), and (5, 6) are:

A. (3, 4)
B. (2, 3)
C. (4, 5)
D. (3, 5)

Solution

Centroid = ((1+3+5)/3, (2+4+6)/3) = (3, 4).

Correct Answer: A — (3, 4)

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Q. The coordinates of the centroid of a triangle with vertices at (2, 3), (4, 5), and (6, 7) are:

A. (4, 5)
B. (3, 4)
C. (5, 6)
D. (6, 5)

Solution

Centroid = ((2+4+6)/3, (3+5+7)/3) = (4, 5).

Correct Answer: B — (3, 4)

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Q. The coordinates of the foot of the perpendicular from the point (1, 2) to the line 2x + 3y = 6 are:

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

Solution

Using the formula for foot of perpendicular, we find the coordinates to be (2, 0).

Correct Answer: A — (2, 0)

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Q. The coordinates of the foot of the perpendicular from the point (1, 2) to the line 3x + 4y = 12 are:

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

Solution

Using the formula for foot of perpendicular, we find the coordinates to be (2, 1).

Correct Answer: A — (2, 1)

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Q. The coordinates of the foot of the perpendicular from the point (3, 4) to the line 2x + 3y = 6 are:

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

Solution

Using the formula for foot of perpendicular, we find the coordinates to be (2, 0).

Correct Answer: A — (2, 0)

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Q. The coordinates of the foot of the perpendicular from the point (3, 4) to the line 2x + 3y - 6 = 0 are:

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

Solution

Using the formula for foot of perpendicular, we find the coordinates to be (2, 0).

Correct Answer: A — (2, 0)

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Q. The equation of a line with slope 2 passing through the point (1, 3) is?

A. y = 2x + 1
B. y = 2x + 2
C. y = 2x + 3
D. y = 2x - 1

Solution

Using point-slope form: y - 3 = 2(x - 1) => y = 2x + 1.

Correct Answer: C — y = 2x + 3

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Q. The equation of a parabola with vertex at (0, 0) and directrix y = -3 is?

A. x^2 = -12y
B. y^2 = -12x
C. x^2 = 12y
D. y^2 = 12x

Solution

The distance from the vertex to the directrix is 3, so the equation is x^2 = -12y.

Correct Answer: A — x^2 = -12y

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Q. The equation of a parabola with vertex at (0, 0) and focus at (0, 3) is?

A. x^2 = 12y
B. y^2 = 12x
C. x^2 = 6y
D. y^2 = 6x

Solution

The distance from the vertex to the focus is 3, so the equation is x^2 = 12y.

Correct Answer: A — x^2 = 12y

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Q. The equation x^2 - 2x + 1 = 0 has how many distinct roots?

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

Solution

The discriminant is 0, indicating that there is exactly one distinct root.

Correct Answer: B — 1

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Q. The equation x^2 - 2x + k = 0 has roots that are both positive. What is the range of k?

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

Solution

For both roots to be positive, k must be greater than 1 (from Vieta's formulas).

Correct Answer: C — k > 1

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Mathematics

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