Defence Exams Preparation Resources

Defence Exams

NDA

Q. If vector A = 4i + 3j and vector B = -i + 2j, what is the resultant vector A + B? (2019)

A. 3i + 5j
B. 5i + j
C. 3i + j
D. 5i + 5j

Solution

A + B = (4 - 1)i + (3 + 2)j = 3i + 5j.

Correct Answer: A — 3i + 5j

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Q. If vector A = 4i + 3j and vector B = -i + 2j, what is the resultant vector R = A + B? (2019)

A. 3i + 5j
B. 5i + j
C. 3i + j
D. 5i + 5j

Solution

R = A + B = (4 - 1)i + (3 + 2)j = 3i + 5j.

Correct Answer: A — 3i + 5j

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Q. If vector A = 4i + 3j and vector B = 3i + 4j, what is the angle between A and B?

A. 45 degrees
B. 60 degrees
C. 90 degrees
D. 135 degrees

Solution

cos(θ) = (A · B) / (|A||B|) = (12 + 12) / (5 * 5) = 24/25, θ = cos^(-1)(24/25) ≈ 60 degrees.

Correct Answer: B — 60 degrees

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Q. If vector A = 4i + 3j and vector B = 4i - 3j, what is the angle between A and B? (2019)

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

Solution

A · B = 16 - 9 = 7. |A| = 5, |B| = 5. cos(θ) = 7/(5*5) = 0.28, θ = 180 degrees.

Correct Answer: C — 180 degrees

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Q. If vector A = 5i + 12j and vector B = -5i + 12j, what is the angle between them?

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

Solution

A · B = (5)(-5) + (12)(12) = -25 + 144 = 119. Since A and B are in opposite directions, the angle is 180 degrees.

Correct Answer: C — 180 degrees

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Q. If vector A = 5i + 12j and vector B = -5i + 12j, what is the resultant vector A + B? (2021)

A. 0i + 24j
B. 10i + 0j
C. 0i + 12j
D. 5i + 12j

Solution

A + B = (5 - 5)i + (12 + 12)j = 0i + 24j.

Correct Answer: A — 0i + 24j

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Q. If vector A = 5i + 12j and vector B = 12i - 5j, what is the angle between A and B?

A. 30 degrees
B. 60 degrees
C. 90 degrees
D. 120 degrees

Solution

cos(θ) = (A · B) / (|A||B|) = (5*12 + 12*(-5)) / (√(169) * √(169)) = 0, θ = 90 degrees.

Correct Answer: C — 90 degrees

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Q. If vector A = 5i + 12j and vector B = 12i - 5j, what is the value of A × B?

A. -85
B. 85
C. 0
D. 60

Solution

A × B = |i  j  k| |5  12  0| |12 -5  0| = (0 - 0)i - (0 - 0)j + (5*-5 - 12*12)k = -85k.

Correct Answer: A — -85

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Q. If vector A = 5i + 12j and vector B = 5i - 12j, what is the angle between A and B?

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

Solution

A · B = 5*5 + 12*(-12) = 25 - 144 = -119. Since A · B < 0, angle is 180 degrees.

Correct Answer: C — 180 degrees

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Q. If vector A = 5i + 5j and vector B = 5i - 5j, what is the angle between A and B?

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

Solution

cos(θ) = (A · B) / (|A||B|) = (25 - 25) / (√(50) * √(50)) = 0, θ = 90 degrees.

Correct Answer: C — 90 degrees

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Q. If vector A = 5i + 5j and vector B = 5i - 5j, what is the angle between them?

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

Solution

cos(θ) = (A · B) / (|A||B|) = (25 - 25) / (5√2 * 5√2) = 0, θ = 90 degrees.

Correct Answer: C — 90 degrees

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Q. If vector A = 6i + 8j, what is the unit vector in the direction of A? (2023)

A. 3/5 i + 4/5 j
B. 6/10 i + 8/10 j
C. 1/5 i + 2/5 j
D. 2/5 i + 3/5 j

Solution

Magnitude |A| = √(6^2 + 8^2) = 10. Unit vector = (1/10)(6i + 8j) = (3/5)i + (4/5)j.

Correct Answer: A — 3/5 i + 4/5 j

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Q. If vector G = 4i - 3j + 2k, what is the y-component of G?

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

Solution

The y-component of G is -3.

Correct Answer: B — -3

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Q. If vector J = 5i + 12j, what is the angle between J and the positive x-axis?

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

Solution

tan(θ) = 12/5; θ = tan^(-1)(12/5) which is approximately 67.38 degrees, closest to 60 degrees.

Correct Answer: C — 60 degrees

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Q. If vector K = 2i + 2j and vector L = -i + 3j, what is the resultant vector K + L?

A. i + 5j
B. i + j
C. 3i + 5j
D. 3i + j

Solution

K + L = (2i + 2j) + (-i + 3j) = (2 - 1)i + (2 + 3)j = 1i + 5j = i + 5j.

Correct Answer: A — i + 5j

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Q. If vectors A = 3i + 4j and B = 2i - j, what is the dot product A · B?

A. -1
B. 2
C. 10
D. 11

Solution

A · B = (3)(2) + (4)(-1) = 6 - 4 = 2.

Correct Answer: C — 10

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Q. If vectors A and B are perpendicular, then A · B equals:

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

Solution

If A and B are perpendicular, then by definition A · B = |A||B|cos(90°) = 0.

Correct Answer: A — 0

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Q. If x + y = 10 and xy = 21, what is the value of x^2 + y^2? (2021)

A. 49
B. 59
C. 61
D. 41

Solution

We know that x^2 + y^2 = (x + y)^2 - 2xy. Substituting the values, we get (10)^2 - 2(21) = 100 - 42 = 58.

Correct Answer: A — 49

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Q. If x + y = 10 and xy = 21, what is the value of x² + y²?

A. 49
B. 59
C. 61
D. 41

Solution

x² + y² = (x + y)² - 2xy = 10² - 2*21 = 100 - 42 = 58.

Correct Answer: A — 49

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Q. If x = -3, what is the value of |x| + x?

A. 0
B. -3
C. -6
D. -9

Solution

|-3| = 3, so |x| + x = 3 - 3 = 0.

Correct Answer: B — -3

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Q. If x = -5, what is the value of |x|?

A. -5
B. 0
C. 5
D. 10

Solution

The absolute value |x| of -5 is 5.

Correct Answer: C — 5

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Q. If x = 1/2, what is the value of 1/x? (2022)

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

Solution

1/x = 1/(1/2) = 2.

Correct Answer: A — 2

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Q. If x = 3 and y = 4, what is the value of x² + y²?

A. 25
B. 20
C. 15
D. 12

Solution

x² + y² = 3² + 4² = 9 + 16 = 25.

Correct Answer: A — 25

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Q. If x = 3, what is the value of 2x^2 + 3x - 5?

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

Solution

Substituting x = 3 into the expression: 2(3^2) + 3(3) - 5 = 2(9) + 9 - 5 = 18 + 9 - 5 = 22.

Correct Answer: B — 12

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Q. If x = 4 and y = 5, what is the value of x² - y²?

A. -9
B. 9
C. 20
D. 1

Solution

x² - y² = 4² - 5² = 16 - 25 = -9.

Correct Answer: B — 9

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Q. If x = 4, what is the value of 2x - 3? (2022)

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

Solution

2x - 3 = 2(4) - 3 = 8 - 3 = 5.

Correct Answer: C — 7

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Q. If x = 4, what is the value of 3x - 2? (2019)

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

Solution

3x - 2 = 3(4) - 2 = 12 - 2 = 10.

Correct Answer: B — 12

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Q. If x = 5, what is the value of (x - 1)²?

A. 16
B. 20
C. 25
D. 36

Solution

(x - 1)² = (5 - 1)² = 4² = 16.

Correct Answer: A — 16

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Q. If x^2 - 5x + 6 = 0, what are the values of x? (2019)

A. 2, 3
B. 1, 6
C. 3, 2
D. 0, 6

Solution

Factoring gives (x - 2)(x - 3) = 0, so x = 2 or x = 3.

Correct Answer: A — 2, 3

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Q. If z = -1 + i, what is the square of the modulus of z?

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

Solution

The modulus of z = -1 + i is |z| = √((-1)² + 1²) = √(1 + 1) = √2. The square of the modulus is |z|² = 2.

Correct Answer: D — 5

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Defence Exams

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