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Q. Find the angle between the vectors A = (3, -2, 1) and B = (1, 1, 1) if A · B = |A||B|cos(θ).

A. 60°
B. 45°
C. 90°
D. 30°

Solution

A · B = 3*1 + (-2)*1 + 1*1 = 3 - 2 + 1 = 2. |A| = √(3^2 + (-2)^2 + 1^2) = √14, |B| = √3. cos(θ) = 2/(√14 * √3). θ = 60°.

Correct Answer: A — 60°

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Q. Find the angle between the vectors A = (3, -2, 1) and B = (1, 1, 1).

A. 60°
B. 45°
C. 90°
D. 30°

Solution

cos(θ) = (A · B) / (|A| |B|). A · B = 3*1 + (-2)*1 + 1*1 = 2. |A| = √(3^2 + (-2)^2 + 1^2) = √14, |B| = √3. θ = cos^(-1)(2/(√14 * √3)).

Correct Answer: A — 60°

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Q. Find the angle between the vectors A = 2i + 2j and B = 2i - 2j. (2022)

A. 0 degrees
B. 45 degrees
C. 90 degrees
D. 180 degrees

Solution

cos(θ) = (A · B) / (|A||B|). A · B = 0, hence θ = 90 degrees.

Correct Answer: C — 90 degrees

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Q. Find the angle between the vectors A = i + j and B = 2i + 2j.

A. 0 degrees
B. 45 degrees
C. 90 degrees
D. 60 degrees

Solution

cos(θ) = (A · B) / (|A| |B|). A · B = 1*2 + 1*2 = 4; |A| = √2, |B| = 2√2. Thus, cos(θ) = 4 / (√2 * 2√2) = 1, θ = 0 degrees.

Correct Answer: A — 0 degrees

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Q. Find the angle between the vectors A = i + j and B = i - j.

A. 0 degrees
B. 45 degrees
C. 90 degrees
D. 135 degrees

Solution

cos(θ) = (A · B) / (|A||B|) = (1 - 1) / (√2 * √2) = 0, θ = 90 degrees.

Correct Answer: C — 90 degrees

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Q. Find the angle between the vectors A = i + j and B = j - i. (2022)

A. 90 degrees
B. 45 degrees
C. 60 degrees
D. 30 degrees

Solution

cos(θ) = (A · B) / (|A| |B|). A · B = 0, hence θ = 90 degrees.

Correct Answer: A — 90 degrees

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Q. Find the angle θ between the vectors A = i + 2j and B = 2i + 3j if A · B = |A||B|cos(θ).

A. 60°
B. 45°
C. 30°
D. 90°

Solution

A · B = 1*2 + 2*3 = 8. |A| = √(1^2 + 2^2) = √5, |B| = √(2^2 + 3^2) = √13. cos(θ) = 8/(√5*√13).

Correct Answer: B — 45°

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Q. Find the angle θ between vectors A = 4i + 3j and B = 1i + 2j if A · B = |A||B|cos(θ).

A. 60°
B. 45°
C. 30°
D. 90°

Solution

A · B = 4*1 + 3*2 = 10; |A| = √(4^2 + 3^2) = 5; |B| = √(1^2 + 2^2) = √5; cos(θ) = 10/(5√5) = 2/√5; θ = 45°.

Correct Answer: B — 45°

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Q. Find the area between the curves y = x and y = x^2 from x = 0 to x = 1.

A. 0.5
B. 1
C. 0.25
D. 0.75

Solution

The area between the curves is given by ∫(from 0 to 1) (x - x^2) dx = [x^2/2 - x^3/3] from 0 to 1 = (1/2 - 1/3) = 1/6 = 0.5.

Correct Answer: A — 0.5

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Q. Find the area between the curves y = x^2 and y = 4 from x = -2 to x = 2.

A. 8/3
B. 16/3
C. 8
D. 4

Solution

The area between the curves is given by ∫(from -2 to 2) (4 - x^2) dx = [4x - x^3/3] from -2 to 2 = 16/3.

Correct Answer: B — 16/3

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Q. Find the area between the curves y = x^2 and y = 4 from x = 0 to x = 2.

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

Solution

The area between the curves y = x^2 and y = 4 is given by ∫(from 0 to 2) (4 - x^2) dx = [4x - x^3/3] from 0 to 2 = (8 - 8/3) = 4/3.

Correct Answer: A — 4

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Q. Find the area between the curves y = x^3 and y = x from x = 0 to x = 1.

A. 1/4
B. 1/3
C. 1/2
D. 1/6

Solution

The area between the curves is given by ∫(from 0 to 1) (x - x^3) dx = [x^2/2 - x^4/4] from 0 to 1 = (1/2 - 1/4) = 1/4.

Correct Answer: B — 1/3

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Q. Find the area of a triangle with vertices at A(0, 0, 0), B(1, 0, 0), and C(0, 1, 0). (2023)

A. 0.5
B. 1
C. 2
D. 0

Solution

Area = 0.5 * base * height = 0.5 * 1 * 1 = 0.5 square units.

Correct Answer: A — 0.5

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Q. Find the area of the triangle formed by the points (0, 0), (4, 0), and (0, 3).

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

Solution

Area = 1/2 * base * height = 1/2 * 4 * 3 = 6.

Correct Answer: A — 6

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Q. Find the area of the triangle formed by the points (0, 0), (4, 0), and (0, 3). (2022) 2022

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

Solution

Area = 1/2 * base * height = 1/2 * 4 * 3 = 6.

Correct Answer: A — 6

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Q. Find the area of the triangle formed by the points A(0, 0, 0), B(1, 0, 0), and C(0, 1, 0). (2023)

A. 0.5
B. 1
C. 2
D. 0

Solution

Area = 0.5 * base * height = 0.5 * 1 * 1 = 0.5 square units.

Correct Answer: A — 0.5

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Q. Find the area of the triangle formed by the points A(1, 2, 3), B(4, 5, 6), and C(7, 8, 9) using the vector product.

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

Solution

Area = 0.5 * |AB × AC| = 0, as points are collinear.

Correct Answer: A — 0

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Q. Find the area of the triangle formed by the points A(1, 2, 3), B(4, 5, 6), and C(7, 8, 9). (2022)

A. 0
B. 6
C. 12
D. 3

Solution

The points are collinear, hence the area = 0.

Correct Answer: A — 0

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Q. Find the area of the triangle with vertices (0,0), (4,0), (0,3).

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

Solution

Area = 0.5 * base * height = 0.5 * 4 * 3 = 6.

Correct Answer: A — 6

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Q. Find the area of the triangle with vertices (0,0), (4,0), and (4,3).

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

Solution

Area = 1/2 * base * height = 1/2 * 4 * 3 = 6.

Correct Answer: B — 12

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Q. Find the area of the triangle with vertices at (0,0), (4,0), and (0,3).

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

Solution

Area = 1/2 * base * height = 1/2 * 4 * 3 = 6.

Correct Answer: A — 6

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Q. Find the area under the curve y = 3x^2 from x = 1 to x = 2.

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

Solution

The area under the curve is given by ∫(from 1 to 2) 3x^2 dx = [x^3] from 1 to 2 = (8 - 1) = 7.

Correct Answer: B — 6

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Q. Find the area under the curve y = e^x from x = 0 to x = 1.

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

Solution

The area is given by the integral from 0 to 1 of e^x dx. This evaluates to [e^x] from 0 to 1 = e - 1.

Correct Answer: A — e - 1

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Q. Find the area under the curve y = x^2 + 2x from x = 0 to x = 3.

A. 9
B. 12
C. 15
D. 18

Solution

The area under the curve is given by ∫(from 0 to 3) (x^2 + 2x) dx = [x^3/3 + x^2] from 0 to 3 = (27/3 + 9) = 18.

Correct Answer: C — 15

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Q. Find the area under the curve y = x^2 from x = 0 to x = 2.

A. 2
B. 4
C. 8/3
D. 3

Solution

The area under the curve y = x^2 from 0 to 2 is given by the integral ∫(from 0 to 2) x^2 dx = [x^3/3] from 0 to 2 = (2^3/3) - (0^3/3) = 8/3.

Correct Answer: C — 8/3

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Q. Find the area under the curve y = x^2 from x = 0 to x = 3.

A. 9
B. 18
C. 27
D. 36

Solution

Area = ∫ from 0 to 3 of x^2 dx = [1/3 * x^3] from 0 to 3 = 9.

Correct Answer: A — 9

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Q. Find the area under the curve y = x^2 from x = 1 to x = 3.

A. 8/3
B. 10/3
C. 9/3
D. 7/3

Solution

The area is given by the integral ∫ (x^2) dx from 1 to 3. This evaluates to [x^3/3] from 1 to 3 = (27/3 - 1/3) = 26/3.

Correct Answer: B — 10/3

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Q. Find the area under the curve y = x^4 from x = 0 to x = 1.

A. 1/5
B. 1/4
C. 1/3
D. 1/2

Solution

The area under the curve y = x^4 from 0 to 1 is given by ∫(from 0 to 1) x^4 dx = [x^5/5] from 0 to 1 = 1/5.

Correct Answer: A — 1/5

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Q. Find the area under the curve y = x^4 from x = 0 to x = 2.

A. 4
B. 8
C. 16
D. 32

Solution

The area is given by the integral from 0 to 2 of x^4 dx. This evaluates to [x^5/5] from 0 to 2 = (32/5) = 16.

Correct Answer: C — 16

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Q. Find the argument of the complex number z = -1 - i.

A. -3π/4
B. 3π/4
C. π/4
D. -π/4

Solution

The argument of z = -1 - i is θ = tan^(-1)(-1/-1) = 3π/4.

Correct Answer: A — -3π/4

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Major Competitive Exams

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