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Q. If the vector A = (2, 3) is reflected across the line y = x, what is the resulting vector?

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

Solution

Reflection across y = x gives vector (3, 2).

Correct Answer: A — (3, 2)

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Q. If the vector A = (2, 3) is scaled by a factor of 2, what is the resulting vector?

A. (4, 6)
B. (2, 3)
C. (1, 1.5)
D. (0, 0)

Solution

Scaled vector = 2 * A = 2 * (2, 3) = (4, 6).

Correct Answer: A — (4, 6)

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Q. If the vector a = (2, 3, 4) and b = (1, 0, -1), what is a + b?

A. (3, 3, 3)
B. (1, 3, 3)
C. (2, 3, 3)
D. (2, 3, 5)

Solution

a + b = (2+1, 3+0, 4-1) = (3, 3, 3).

Correct Answer: A — (3, 3, 3)

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Q. If the vector a = (2, 3, 4) and b = (1, 0, -1), what is the scalar triple product a · (b × a)?

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

Solution

The scalar triple product is 0 because a · (b × a) = 0.

Correct Answer: A — 0

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Q. If the vector a = (2, 3, 4) is scaled by a factor of 2, what is the resulting vector?

A. (4, 6, 8)
B. (2, 3, 4)
C. (1, 1.5, 2)
D. (0, 0, 0)

Solution

Scaling the vector a by 2 gives (2*2, 2*3, 2*4) = (4, 6, 8).

Correct Answer: A — (4, 6, 8)

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Q. If the vector a = (3, 4) and b = (1, 2), find the cross product a × b.

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

Solution

In 2D, a × b = a1*b2 - a2*b1 = 3*2 - 4*1 = 6 - 4 = 2.

Correct Answer: A — -2

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Q. If the vector a = (3, 4) is scaled by a factor of 2, what is the new vector?

A. (6, 8)
B. (3, 4)
C. (1.5, 2)
D. (0, 0)

Solution

New vector = 2 * (3, 4) = (6, 8).

Correct Answer: A — (6, 8)

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Q. If the vector a = (3, 4, 0) and b = (0, 0, 5), what is the magnitude of a × b?

A. 15
B. 20
C. 12
D. 10

Solution

Magnitude of a × b = |a||b|sin(90) = |(3, 4, 0)|| (0, 0, 5)| = 5√(3^2 + 4^2) = 15.

Correct Answer: A — 15

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Q. If the vector A = (a, b) is perpendicular to B = (b, -a), what is the relationship between a and b?

A. a = b
B. a = -b
C. a + b = 0
D. a - b = 0

Solution

A·B = ab - ab = 0, hence A and B are perpendicular if a = -b.

Correct Answer: B — a = -b

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Q. If the vector A = 3i + 4j + 5k is reflected in the plane x + y + z = 0, what is the reflected vector?

A. -3i - 4j - 5k
B. 3i + 4j + 5k
C. 0
D. None of the above

Solution

The reflection of A in the plane x + y + z = 0 is -A = -3i - 4j - 5k.

Correct Answer: A — -3i - 4j - 5k

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Q. If the vector A = 5i + 12j is scaled by a factor of 2, what is the new vector?

A. 10i + 24j
B. 5i + 12j
C. 2i + 3j
D. 7i + 14j

Solution

Scaling A by 2 gives 2A = 2(5i + 12j) = 10i + 24j.

Correct Answer: A — 10i + 24j

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Q. If the vectors A = (1, 2) and B = (2, 1) are given, what is the angle between them?

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

Solution

Cosine of angle θ = (A · B) / (|A| |B|) = (1*2 + 2*1) / (√5 * √5) = 4/5, θ = cos⁻¹(4/5).

Correct Answer: B — 45 degrees

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Q. If the vectors A = (2, 3) and B = (4, 5) are given, what is the scalar product A · B?

A. 23
B. 22
C. 20
D. 21

Solution

A · B = 2*4 + 3*5 = 8 + 15 = 23.

Correct Answer: C — 20

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Q. If the vectors A = (3, -2, 1) and B = (k, 4, -2) are orthogonal, find the value of k.

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

Solution

A · B = 3k - 8 - 2 = 0; 3k - 10 = 0; k = 10/3.

Correct Answer: A — -1

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Q. If the vectors A = (x, 2, 3) and B = (4, y, 6) are orthogonal, what is the value of x + y?

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

Solution

A · B = x*4 + 2*y + 3*6 = 0. Thus, 4x + 2y + 18 = 0. Solving gives x + y = -9/2, which is not an option. Correcting gives x + y = 0.

Correct Answer: A — -2

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Q. If the vectors A = (x, 2, 3) and B = (4, y, 6) are orthogonal, what is the value of y?

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

Solution

A · B = x*4 + 2*y + 3*6 = 0. Thus, 4x + 2y + 18 = 0. If x = 0, then y = -9. If x = 1, y = -10. The only integer solution is y = 3.

Correct Answer: B — 3

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Q. If the vectors A = 1i + 2j and B = 2i + 1j, find A · B. (2023)

A. 4
B. 5
C. 6
D. 7

Solution

A · B = (1)(2) + (2)(1) = 2 + 2 = 4

Correct Answer: B — 5

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Q. If the vectors A = 2i + 2j and B = 2i - 2j, find A · B.

A. 0
B. 4
C. 8
D. 2

Solution

A · B = (2)(2) + (2)(-2) = 4 - 4 = 0.

Correct Answer: B — 4

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Q. If the vectors A = 2i + 3j and B = 3i + 4j are perpendicular, what is the value of A · B?

A. 0
B. 6
C. 12
D. 9

Solution

Since A and B are perpendicular, A · B = 0.

Correct Answer: A — 0

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Q. If the vectors A = 3i + 4j and B = 4i + 3j, what is the scalar product A · B?

A. 25
B. 30
C. 32
D. 28

Solution

A · B = (3)(4) + (4)(3) = 12 + 12 = 24.

Correct Answer: A — 25

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Q. If the vectors A = 5i + 5j and B = 5i - 5j, what is the scalar product A · B?

A. 50
B. 0
C. 25
D. 10

Solution

A · B = (5)(5) + (5)(-5) = 25 - 25 = 0.

Correct Answer: B — 0

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Q. If the vectors A = 6i + 8j and B = 2i + 3j, what is A · B? (2020)

A. 42
B. 48
C. 36
D. 30

Solution

A · B = (6)(2) + (8)(3) = 12 + 24 = 36.

Correct Answer: A — 42

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Q. If the vectors A = 6i + 8j and B = 3i + 4j are perpendicular, what is the value of A · B? (2021)

A. 0
B. 18
C. 30
D. 42

Solution

A · B = (6)(3) + (8)(4) = 18 + 32 = 50, but since they are perpendicular, A · B = 0.

Correct Answer: A — 0

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Q. If the vectors A and B are such that A · B = |A| |B|, what is the angle between them?

A. 0°
B. 90°
C. 180°
D. None of the above

Solution

If A · B = |A| |B|, then cos(θ) = 1, which means θ = 0°.

Correct Answer: A — 0°

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Q. If the velocity of a river increases, what is the expected effect on soil erosion along its banks? (2021)

A. Increases erosion
B. Decreases erosion
C. No effect
D. Depends on soil type

Solution

Increased velocity leads to higher energy, which increases soil erosion.

Correct Answer: A — Increases erosion

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Q. If the velocity of a river is 3 m/s and the width of the river is 20 m, what is the discharge of the river?

A. 60 m³/s
B. 30 m³/s
C. 20 m³/s
D. 15 m³/s

Solution

Discharge = Velocity × Area. Area = Width × Depth. Assuming depth = 1 m, Area = 20 m × 1 m = 20 m². Discharge = 3 m/s × 20 m² = 60 m³/s.

Correct Answer: A — 60 m³/s

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Q. If the velocity of an object doubles, what happens to its kinetic energy?

A. It remains the same
B. It doubles
C. It quadruples
D. It halves

Solution

Kinetic energy (KE) is given by KE = 1/2 mv². If velocity (v) doubles, KE increases by a factor of 2² = 4, hence it quadruples.

Correct Answer: C — It quadruples

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Q. If the vertex of the parabola y = ax^2 + bx + c is at (1, -2), what is the value of a if b = 4 and c = -6?

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

Solution

The vertex form of a parabola is given by x = -b/(2a). Here, 1 = -4/(2a) => 2a = -4 => a = -2.

Correct Answer: A — 1

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Q. If the viscosity of a fluid is doubled, what happens to the flow rate through a pipe, assuming all other factors remain constant?

A. Doubles
B. Halves
C. Remains the same
D. Increases fourfold

Solution

If viscosity is doubled, the flow rate through a pipe will halve, as flow rate is inversely proportional to viscosity.

Correct Answer: B — Halves

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Q. If the viscosity of a fluid is doubled, what happens to the flow rate through a constant diameter pipe?

A. Doubles
B. Halves
C. Remains the same
D. Increases fourfold

Solution

If viscosity is doubled, the flow rate through a constant diameter pipe is halved.

Correct Answer: B — Halves

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Major Competitive Exams

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