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Q. What is the value of tan^(-1)(1) + tan^(-1)(2)?

A. π/4
B. π/3
C. π/2
D. π/6

Solution

Using the formula tan^(-1)(a) + tan^(-1)(b) = tan^(-1)((a+b)/(1-ab)), we have tan^(-1)(1) + tan^(-1)(2) = tan^(-1)((1+2)/(1-1*2)) = tan^(-1)(3/-1) = π - tan^(-1)(3) = π/4.

Correct Answer: A — π/4

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Q. What is the value of tan^(-1)(1)?

A. π/4
B. π/3
C. π/2
D. 0

Solution

tan^(-1)(1) = π/4, since tan(π/4) = 1.

Correct Answer: A — π/4

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Q. What is the value of tan^(-1)(√3)?

A. π/3
B. π/4
C. π/6
D. π/2

Solution

tan^(-1)(√3) corresponds to the angle π/3.

Correct Answer: A — π/3

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Q. What is the value of the 3rd term in the expansion of (3x - 4)^5? (2020)

A. -240
B. 240
C. -180
D. 180

Solution

The 3rd term is C(5,2) * (3x)^3 * (-4)^2 = 10 * 27x^3 * 16 = -240.

Correct Answer: A — -240

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Q. What is the value of the 3rd term in the expansion of (a + b)^6?

A. 15ab^4
B. 20ab^4
C. 30ab^4
D. 35ab^4

Solution

The 3rd term in the expansion of (a + b)^6 is given by C(6, 2) * a^4 * b^2 = 15 * a^4 * b^2.

Correct Answer: B — 20ab^4

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Q. What is the value of the 3rd term in the expansion of (x + 2)^5?

A. 40
B. 80
C. 60
D. 100

Solution

The 3rd term in the expansion of (x + 2)^5 is C(5, 2)(x)^2(2)^3 = 10 * x^2 * 8 = 80.

Correct Answer: A — 40

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Q. What is the value of the 5th term in the expansion of (3x - 2)^6? (2023)

A. -540
B. 540
C. 720
D. 360

Solution

The 5th term is given by C(6,4) * (3x)^4 * (-2)^2 = 15 * 81 * 4 = -4860.

Correct Answer: A — -540

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Q. What is the value of the 5th term in the expansion of (x + 2)^6?

A. 80
B. 120
C. 160
D. 240

Solution

The 5th term is given by C(6,4) * (x)^4 * (2)^2 = 15 * x^4 * 4 = 240.

Correct Answer: B — 120

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Q. What is the value of the 5th term in the expansion of (x + 2)^7?

A. 672
B. 672x^4
C. 672x^3
D. 672x^2

Solution

The 5th term is C(7,4) * (2)^4 * x^3 = 35 * 16 * x^3 = 560x^3.

Correct Answer: C — 672x^3

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Q. What is the value of the azimuthal quantum number (l) for a d orbital? (2021) 2021

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

Solution

The azimuthal quantum number l for d orbitals is 2.

Correct Answer: C — 2

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Q. What is the value of the azimuthal quantum number (l) for a p-orbital? (2015)

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

Solution

The azimuthal quantum number (l) for a p-orbital is 1.

Correct Answer: B — 1

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Q. What is the value of the coefficient of x^0 in the expansion of (x + 5)^3?

A. 0
B. 5
C. 15
D. 125

Solution

The coefficient of x^0 in (x + 5)^3 is given by 3C0 * (5)^3 = 1 * 125 = 125.

Correct Answer: D — 125

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Q. What is the value of the coefficient of x^2 in the expansion of (3x - 2)^4?

A. -36
B. 36
C. -54
D. 54

Solution

The coefficient of x^2 is given by 4C2 * (3^2) * (-2)^2 = 6 * 9 * 4 = 216.

Correct Answer: A — -36

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Q. What is the value of the coefficient of x^3 in the expansion of (x - 1)^6?

A. -20
B. -30
C. -40
D. -10

Solution

The coefficient of x^3 is C(6,3) * (-1)^3 = 20 * (-1) = -20.

Correct Answer: B — -30

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Q. What is the value of the coefficient of x^4 in the expansion of (2x - 5)^6?

A. -600
B. -720
C. 720
D. 600

Solution

The coefficient of x^4 in (2x - 5)^6 is given by 6C2 * (2)^4 * (-5)^2 = 15 * 16 * 25 = 600. The coefficient is -720.

Correct Answer: B — -720

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Q. What is the value of the coefficient of x^4 in the expansion of (3x - 2)^6?

A. -540
B. 540
C. -720
D. 720

Solution

The coefficient of x^4 in (3x - 2)^6 is given by 6C4 * (3^4) * (-2)^2 = 15 * 81 * 4 = 4860.

Correct Answer: A — -540

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Q. What is the value of the coefficient of x^4 in the expansion of (x + 5)^6?

A. 150
B. 300
C. 600
D. 750

Solution

The coefficient of x^4 in (x + 5)^6 is given by 6C4 * 5^2 = 15 * 25 = 375.

Correct Answer: B — 300

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Q. What is the value of the coefficient of x^4 in the expansion of (x - 2)^6? (2023)

A. -15
B. -60
C. -90
D. -120

Solution

The coefficient of x^4 is C(6,4) * (-2)^2 = 15 * 4 = -60.

Correct Answer: B — -60

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Q. What is the value of the coefficient of x^5 in the expansion of (3x - 2)^8?

A. -6720
B. 6720
C. 13440
D. -13440

Solution

The coefficient is C(8,5) * (3^5) * (-2)^3 = 56 * 243 * (-8) = -6720.

Correct Answer: A — -6720

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Q. What is the value of the coefficient of x^5 in the expansion of (x + 1)^8?

A. 56
B. 70
C. 80
D. 90

Solution

The coefficient of x^5 is given by C(8, 5) = 56.

Correct Answer: B — 70

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Q. What is the value of the coefficient of x^5 in the expansion of (x + 1/2)^8?

A. 56
B. 112
C. 128
D. 64

Solution

The coefficient of x^5 is given by 8C5 * (1/2)^3 = 56 * 1/8 = 7.

Correct Answer: A — 56

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Q. What is the value of the coefficient of x^5 in the expansion of (x + 2)^7?

A. 21
B. 42
C. 56
D. 70

Solution

The coefficient of x^5 is given by 7C5 * (2)^2 = 21 * 4 = 84.

Correct Answer: C — 56

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Q. What is the value of the coefficient of x^5 in the expansion of (x + 3)^7?

A. 21
B. 63
C. 126
D. 189

Solution

The coefficient of x^5 is C(7,5) * (3)^2 = 21 * 9 = 189.

Correct Answer: C — 126

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Q. What is the value of the coefficient of x^5 in the expansion of (x + 3)^8?

A. 1680
B. 168
C. 840
D. 280

Solution

The coefficient of x^5 is C(8, 5) * (3)^3 = 56 * 27 = 1512.

Correct Answer: A — 1680

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Q. What is the value of the derivative of f(x) = 3x^3 - 2x at x = 1?

A. 7
B. 5
C. 3
D. 1

Solution

f'(x) = 9x^2 - 2. At x = 1, f'(1) = 9*1^2 - 2 = 7.

Correct Answer: A — 7

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Q. What is the value of the derivative of f(x) = ln(x^2 + 1) at x = 1?

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

Solution

f'(x) = (2x)/(x^2 + 1). At x = 1, f'(1) = 2/(1 + 1) = 1.

Correct Answer: B — 1/2

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Q. What is the value of the determinant of the matrix B = [[2, 3], [5, 7]]? (2020)

A. 1
B. 4
C. 11
D. 14

Solution

Determinant of B = (2*7) - (3*5) = 14 - 15 = -1.

Correct Answer: C — 11

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Q. What is the value of the determinant of the matrix B = [[5, 1], [2, 3]]? (2022)

A. 10
B. 15
C. 7
D. 5

Solution

Determinant of B = (5*3) - (1*2) = 15 - 2 = 13.

Correct Answer: A — 10

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Q. What is the value of the determinant of the matrix G = [[1, 2, 3], [0, 1, 4], [5, 6, 0]]? (2022)

A. -34
B. 34
C. 0
D. 10

Solution

Using the determinant formula, det(G) = 1(1*0 - 4*6) - 2(0 - 4*5) + 3(0 - 1*5) = -24 + 40 - 15 = 1.

Correct Answer: A — -34

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Q. What is the value of the determinant of the matrix [[1, 2], [3, 4]]?

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

Solution

The determinant is calculated as (1*4) - (2*3) = 4 - 6 = -2.

Correct Answer: A — -2

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Major Competitive Exams

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