Mathematics Study Resources for Competitive Exams

Mathematics

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Q. If sin²A + cos²A = 1, what is the value of sin²A when cos A = 0?

A. 0
B. 1
C. 1/2
D. Undefined

Solution

If cos A = 0, then sin²A = 1 - cos²A = 1 - 0 = 1.

Correct Answer: B — 1

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Q. If tan θ = 1/√3, what is the value of sin θ?

A. 1/2
B. √3/2
C. 1/√2
D. √2/2

Solution

Since tan θ = sin θ / cos θ, and tan θ = 1/√3, we can use the 30-60-90 triangle where sin θ = 1/2.

Correct Answer: A — 1/2

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Q. If the angle between two vectors A and B is 60 degrees and |A| = 5, |B| = 10, what is A · B?

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

Solution

A · B = |A||B|cos(60°) = 5 * 10 * 0.5 = 25.

Correct Answer: B — 30

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Q. If the angle between two vectors A and B is 60 degrees and |A| = 5, |B| = 10, what is the scalar product A · B? (2020)

A. 25
B. 30
C. 35
D. 50

Solution

A · B = |A||B|cos(60°) = 5 * 10 * 0.5 = 25

Correct Answer: B — 30

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Q. If the angle between two vectors A and B is 90 degrees, what is A · B?

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

Solution

A · B = |A||B|cos(90°) = |A||B| * 0 = 0.

Correct Answer: A — 0

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Q. If the angle of elevation of a hill from a point on the ground is 30 degrees and the distance from the point to the base of the hill is 100 m, what is the height of the hill? (2021)

A. 50 m
B. 30 m
C. 20 m
D. 10 m

Solution

Height = distance * tan(30) = 100 * (1/√3) ≈ 57.74 m, which rounds to 50 m.

Correct Answer: A — 50 m

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Q. If the angle of elevation of the sun is 30 degrees, how tall is a 10 m pole casting a shadow? (2023)

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

Solution

Height = shadow * tan(angle) = 10 * tan(30) = 10 * (1/√3) ≈ 10 m.

Correct Answer: B — 10 m

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Q. If the angle of elevation of the sun is 30 degrees, how tall is a 10-meter pole casting a shadow of 10√3 meters? (2023)

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

Solution

Height = Shadow * tan(30) = 10√3 * (1/√3) = 10 m.

Correct Answer: B — 10 m

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Q. If the angles of a quadrilateral are in the ratio 1:2:3:4, what is the measure of the largest angle?

A. 90 degrees
B. 120 degrees
C. 150 degrees
D. 180 degrees

Solution

Let the angles be x, 2x, 3x, and 4x. The sum of angles in a quadrilateral is 360 degrees. Therefore, x + 2x + 3x + 4x = 360, which gives 10x = 360, so x = 36 degrees. The largest angle = 4x = 144 degrees.

Correct Answer: C — 150 degrees

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Q. If the angles of triangle JKL are in the ratio 2:3:4, what is the measure of the largest angle? (2023)

A. 40 degrees
B. 60 degrees
C. 80 degrees
D. 90 degrees

Solution

Let the angles be 2x, 3x, and 4x. Then, 2x + 3x + 4x = 180 degrees. Therefore, 9x = 180, x = 20. The largest angle is 4x = 80 degrees.

Correct Answer: C — 80 degrees

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Q. If the angles of triangle MNO are in the ratio 2:3:4, what is the measure of the largest angle? (2023)

A. 40 degrees
B. 60 degrees
C. 80 degrees
D. 90 degrees

Solution

Let the angles be 2x, 3x, and 4x. Then, 2x + 3x + 4x = 180 degrees. Thus, 9x = 180 degrees, x = 20 degrees. The largest angle = 4x = 80 degrees.

Correct Answer: C — 80 degrees

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Q. If the area of a circle is 50π cm², what is the diameter? (2022)

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

Solution

Area = πr² = 50π. Thus, r² = 50, r = √50 = 5√2 cm. Diameter = 2r = 10√2 cm ≈ 20 cm.

Correct Answer: B — 20 cm

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Q. If the area of a rectangle is 48 square units and the length is 8 units, what is the width? (2022)

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

Solution

Area = Length × Width. Therefore, Width = Area / Length = 48 / 8 = 6 units.

Correct Answer: A — 4 units

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Q. If the area of a triangle is 30 cm² and the base is 10 cm, what is the height? (2020)

A. 3 cm
B. 6 cm
C. 9 cm
D. 12 cm

Solution

Area = 1/2 * base * height. Thus, 30 = 1/2 * 10 * height. Solving gives height = 6 cm.

Correct Answer: B — 6 cm

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Q. If the area of triangle ABC is 60 square units and the base AC is 12 units, what is the height from point B to line AC? (2019)

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

Solution

Area = 1/2 * base * height. Therefore, 60 = 1/2 * 12 * height, which gives height = 10 units.

Correct Answer: A — 5 units

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Q. If the area of triangle ABC is 60 square units and the base is 10 units, what is the height? (2019)

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

Solution

Area = 1/2 * base * height. Therefore, 60 = 1/2 * 10 * height, which gives height = 12 units.

Correct Answer: A — 6 units

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Q. If the area of triangle ABC is 60 square units and the base is 10 units, what is the height of the triangle? (2019)

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

Solution

Area = 1/2 * base * height. Therefore, 60 = 1/2 * 10 * height, which gives height = 12 units.

Correct Answer: A — 6 units

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Q. If the area of triangle ABC is 60 square units and the base is 12 units, what is the height of the triangle? (2019)

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

Solution

Area = 1/2 * base * height. Therefore, 60 = 1/2 * 12 * height, which gives height = 10 units.

Correct Answer: A — 5 units

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Q. If the area of triangle ABC is 60 square units and the base is 12 units, what is the height? (2019)

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

Solution

Area = 1/2 * base * height. Therefore, 60 = 1/2 * 12 * height, which gives height = 10 units.

Correct Answer: A — 5 units

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Q. If the average of 10 numbers is 50, what is the sum of these numbers?

A. 500
B. 600
C. 550
D. 450

Solution

Sum = Average * Number of items = 50 * 10 = 500.

Correct Answer: A — 500

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Q. If the average of 6 numbers is 30, what is the average if one number is increased by 10?

A. 30
B. 32
C. 34
D. 36

Solution

Total of 6 numbers = 6 * 30 = 180.\n\nNew total = 180 + 10 = 190.\n\nNew average = 190 / 6 = 31.67.

Correct Answer: B — 32

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Q. If the average of 7, 9, 11, and 13 is x, what is the value of x? (2023)

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

Solution

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

Correct Answer: B — 10

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Q. If the average of a set of numbers is 12 and there are 6 numbers, what is the total sum of the numbers?

A. 60
B. 72
C. 84
D. 96

Solution

Total sum = Average * Number of items = 12 * 6 = 72.

Correct Answer: B — 72

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Q. If the average of five numbers is 60, what is the total of these numbers?

A. 300
B. 250
C. 350
D. 400

Solution

The total of the five numbers is calculated as average * number of items = 60 * 5 = 300.

Correct Answer: A — 300

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Q. If the base of a triangle is 10 cm and the height is 5 cm, what is the area of the triangle? (2022)

A. 25 cm²
B. 50 cm²
C. 15 cm²
D. 30 cm²

Solution

Area = (1/2) * base * height = (1/2) * 10 * 5 = 25 cm².

Correct Answer: A — 25 cm²

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Q. If the center of a circle is at (3, 4) and it passes through the point (7, 1), what is the radius? (2019)

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

Solution

Radius = distance from center to point = √[(7-3)² + (1-4)²] = √[16 + 9] = √25 = 5 units.

Correct Answer: A — 5 units

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Q. If the coordinates of a point are (x, y) and it lies on the line 2x + 3y = 12, what is the value of y when x = 2?

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

Solution

Substituting x = 2: 2(2) + 3y = 12 => 4 + 3y = 12 => 3y = 8 => y = 8/3 ≈ 2.67, closest is 3.

Correct Answer: B — 3

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Q. If the coordinates of a point are (x, y) and it lies on the line 2x + 3y = 6, what is the value of y when x = 0? (2023)

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

Solution

Substituting x = 0 in 2(0) + 3y = 6 gives 3y = 6, so y = 2.

Correct Answer: B — 3

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Q. If the coordinates of a point are (x, y) and it lies on the line 3x + 4y = 12, what is y when x = 0?

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

Solution

Substituting x = 0: 3(0) + 4y = 12 => 4y = 12 => y = 3.

Correct Answer: B — 4

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Q. If the coordinates of point A are (1, 2) and point B are (4, 6), what is the slope of line AB?

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

Solution

Slope = (6-2)/(4-1) = 4/3.

Correct Answer: A — 2

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Mathematics

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