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Q. Find the range of the data set: 12, 15, 22, 30, 5.

A. 25
B. 22
C. 30
D. 15

Solution

Range = max - min = 30 - 5 = 25.

Correct Answer: A — 25

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Q. Find the range of the data set: 8, 12, 15, 20, 25.

A. 12
B. 15
C. 17
D. 20

Solution

Range = Maximum - Minimum = 25 - 8 = 17.

Correct Answer: A — 12

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Q. Find the real part of the complex number 4 + 5i. (2023)

A. 4
B. 5
C. 9
D. 0

Solution

The real part of the complex number 4 + 5i is 4.

Correct Answer: A — 4

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Q. Find the real part of the complex number z = 2 + 3i.

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

Solution

The real part of z = 2 + 3i is 2.

Correct Answer: A — 2

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Q. Find the real part of the complex number z = 2e^(iπ/3).

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

Solution

The real part is 2 * cos(π/3) = 2 * 1/2 = 1.

Correct Answer: B — 2

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Q. Find the real part of the complex number z = 3 + 4i.

A. 3
B. 4
C. 7
D. 0

Solution

The real part of z is 3.

Correct Answer: A — 3

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Q. Find the real part of the complex number z = 4 + 3i.

A. 4
B. 3
C. 7
D. 0

Solution

The real part of z = 4 + 3i is 4.

Correct Answer: A — 4

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Q. Find the real part of the complex number z = 4(cos(π/3) + i sin(π/3)).

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

Solution

The real part is 4 * cos(π/3) = 4 * 1/2 = 2.

Correct Answer: A — 2

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Q. Find the real part of the complex number z = 5 - 2i.

A. 5
B. -2
C. 2
D. 0

Solution

The real part of z = 5 - 2i is 5.

Correct Answer: A — 5

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Q. Find the roots of the equation 3x² - 12x + 12 = 0. (2021)

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

Solution

Dividing by 3 gives x² - 4x + 4 = 0, which factors to (x - 2)² = 0, hence the root is 2.

Correct Answer: A — 2

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Q. Find the roots of the equation 4x² - 12x + 9 = 0. (2023)

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

Solution

This is a perfect square: (2x - 3)² = 0, hence the root is x = 1.5.

Correct Answer: B — 2

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Q. Find the roots of the equation x^2 + 4x + 4 = 0.

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

Solution

The equation factors to (x + 2)^2 = 0, giving a double root x = -2.

Correct Answer: A — -2

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Q. Find the roots of the equation x^2 + 5x + 6 = 0.

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

Solution

The roots are x = -2 and x = -3.

Correct Answer: C — -3, -2

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Q. Find the roots of the equation x² + 2x - 8 = 0. (2022)

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

Solution

Factoring gives (x + 4)(x - 2) = 0, hence the roots are 4 and -2.

Correct Answer: B — 4 and -2

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Q. Find the roots of the quadratic equation x^2 + 4x + 4 = 0.

A. {-2}
B. {2, -2}
C. {-4, 0}
D. {0, 4}

Solution

The equation factors to (x + 2)(x + 2) = 0, giving a double root x = -2.

Correct Answer: A — {-2}

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Q. Find the scalar product of A = (1, 2, 3) and B = (4, 5, 6).

A. 32
B. 30
C. 28
D. 26

Solution

A · B = 1*4 + 2*5 + 3*6 = 4 + 10 + 18 = 32.

Correct Answer: B — 30

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Q. Find the scalar product of A = 2i + 3j + k and B = i + 2j + 3k. (2020)

A. 14
B. 12
C. 10
D. 8

Solution

A · B = (2)(1) + (3)(2) + (1)(3) = 2 + 6 + 3 = 11

Correct Answer: A — 14

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Q. Find the scalar product of A = 6i + 8j and B = 2i + 3j.

A. 42
B. 54
C. 48
D. 36

Solution

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

Correct Answer: A — 42

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Q. Find the scalar product of the vectors (3, -2, 5) and (1, 4, -1).

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

Solution

Scalar product = 3*1 + (-2)*4 + 5*(-1) = 3 - 8 - 5 = -10.

Correct Answer: A — -1

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Q. Find the scalar product of the vectors (4, 5) and (1, 2).

A. 14
B. 13
C. 12
D. 11

Solution

Scalar product = 4*1 + 5*2 = 4 + 10 = 14.

Correct Answer: A — 14

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Q. Find the scalar product of the vectors (7, 8, 9) and (0, 1, 2).

A. 26
B. 27
C. 28
D. 29

Solution

Scalar product = 7*0 + 8*1 + 9*2 = 0 + 8 + 18 = 26.

Correct Answer: A — 26

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Q. Find the scalar product of the vectors A = (2, 3) and B = (4, -1).

A. -1
B. 5
C. 10
D. 11

Solution

A · B = 2*4 + 3*(-1) = 8 - 3 = 5.

Correct Answer: C — 10

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Q. Find the scalar product of the vectors A = 5i + 12j and B = 3i - 4j.

A. -33
B. 33
C. 39
D. 45

Solution

A · B = (5)(3) + (12)(-4) = 15 - 48 = -33.

Correct Answer: A — -33

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Q. Find the scalar product of the vectors A = 7i - 2j + k and B = 3i + 4j - 5k.

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

Solution

A · B = (7)(3) + (-2)(4) + (1)(-5) = 21 - 8 - 5 = 8.

Correct Answer: A — -1

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Q. Find the scalar product of the vectors G = (2, -3, 1) and H = (4, 0, -2).

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

Solution

G · H = 2*4 + (-3)*0 + 1*(-2) = 8 + 0 - 2 = 6.

Correct Answer: A — -2

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Q. Find the scalar product of the vectors G = (5, -3, 2) and H = (1, 1, 1).

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

Solution

G · H = 5*1 + (-3)*1 + 2*1 = 5 - 3 + 2 = 4.

Correct Answer: D — 3

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Q. Find the scalar product of vectors A = 7i + 1j + 2k and B = 3i + 4j + 5k.

A. 43
B. 37
C. 35
D. 41

Solution

A · B = (7)(3) + (1)(4) + (2)(5) = 21 + 4 + 10 = 35.

Correct Answer: A — 43

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Q. Find the scalar projection of vector A = (3, 4) onto vector B = (1, 0).

A. 3
B. 4
C. 1
D. 0

Solution

Scalar projection = (A · B) / |B| = (3*1 + 4*0) / 1 = 3.

Correct Answer: A — 3

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Q. Find the scalar triple product of vectors A = (1, 2, 3), B = (4, 5, 6), and C = (7, 8, 9).

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

Solution

Scalar triple product = A · (B × C) = 0, as vectors are coplanar.

Correct Answer: A — 0

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Q. Find the second derivative of f(x) = 4x^4 - 2x^3 + x. (2019)

A. 48x^2 - 12x + 1
B. 48x^3 - 6
C. 12x^2 - 6
D. 12x^3 - 6x

Solution

First derivative f'(x) = 16x^3 - 6x^2 + 1. Second derivative f''(x) = 48x^2 - 12x.

Correct Answer: A — 48x^2 - 12x + 1

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Major Competitive Exams

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