Mathematics for Competitive Exam Success

Mathematics

3D Geometry Applications of Derivatives Binomial Theorem Circles Complex Numbers Conic Sections Coordinate Geometry Differential Equations Differentiation Integration Limits & Continuity Matrices & Determinants Permutations & Combinations Probability Quadratic Equations Sequences & Series Sets & Relations Straight Lines Trigonometry Vector Algebra

Q. What is the term containing x^2 in the expansion of (x + 4)^6?

A. 240
B. 360
C. 480
D. 600

Solution

The term containing x^2 is given by C(6,2) * (4)^4 * (x)^2 = 15 * 256 * x^2 = 3840.

Correct Answer: C — 480

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Q. What is the value of (1 + i)(1 - i)? (2019)

A. 2
B. 0
C. 1
D. 1 + 0i

Solution

(1 + i)(1 - i) = 1^2 - i^2 = 1 - (-1) = 1 + 1 = 2.

Correct Answer: A — 2

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Q. What is the value of (2 + 3) × (4 - 1)? (2023)

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

Solution

Using BODMAS: (2 + 3) = 5 and (4 - 1) = 3, so 5 × 3 = 15.

Correct Answer: C — 10

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Q. What is the value of (2 + 3)^3?

A. 25
B. 30
C. 35
D. 40

Solution

Using the binomial theorem, (2 + 3)^3 = C(3,0)(2)^3 + C(3,1)(2)^2(3) + C(3,2)(2)(3)^2 + C(3,3)(3)^3 = 8 + 36 + 54 + 27 = 125.

Correct Answer: B — 30

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Q. What is the value of (2 + 3i) / (1 + i)? (2015)

A. 1 + 2i
B. 2 - i
C. 3 + 2i
D. 1 - 2i

Solution

Multiply numerator and denominator by the conjugate of the denominator: (2 + 3i)(1 - i) / (1 + i)(1 - i) = (2 - 2i + 3i + 3) / (1 + 1) = (5 + i) / 2 = 2.5 + 0.5i.

Correct Answer: B — 2 - i

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Q. What is the value of (2 + 3i)(2 - 3i)? (2019)

A. 13
B. 1
C. 12
D. 7

Solution

Using the difference of squares: (a + b)(a - b) = a² - b². Here, (2)² - (3i)² = 4 - (-9) = 4 + 9 = 13.

Correct Answer: A — 13

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Q. What is the value of (2 - 3)^5?

A. -1
B. -32
C. -243
D. -125

Solution

Using the binomial theorem, (2 - 3)^5 = (-1)^5 = -1.

Correct Answer: D — -125

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Q. What is the value of (3 + 4i) / (1 + i)? (2021)

A. 2 + i
B. 1 + 2i
C. 3 - 4i
D. 1 - i

Solution

Multiplying numerator and denominator by the conjugate of the denominator: (3 + 4i)(1 - i) / (1 + i)(1 - i) = (3 - 3i + 4i + 4) / 2 = (7 + i) / 2 = 2 + i.

Correct Answer: A — 2 + i

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Q. What is the value of (3 + 5) × (2 + 4)? (2020)

A. 48
B. 56
C. 42
D. 36

Solution

(3 + 5) = 8 and (2 + 4) = 6, so 8 × 6 = 48.

Correct Answer: B — 56

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Q. What is the value of (3 + 5) × (6 - 2)? (2020)

A. 32
B. 24
C. 40
D. 28

Solution

(3 + 5) = 8 and (6 - 2) = 4, so 8 × 4 = 32.

Correct Answer: B — 24

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Q. What is the value of (3^3) - (2^3)? (2016)

A. 19
B. 27
C. 21
D. 18

Solution

3^3 = 27 and 2^3 = 8, so 27 - 8 = 19.

Correct Answer: A — 19

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Q. What is the value of (x + 1)^5 when x = 2?

A. 32
B. 64
C. 128
D. 256

Solution

Using the binomial theorem, (x + 1)^5 = C(5,0)(2)^5 + C(5,1)(2)^4 + C(5,2)(2)^3 + C(5,3)(2)^2 + C(5,4)(2)^1 + C(5,5)(2)^0 = 32 + 80 + 80 + 40 + 10 + 1 = 243.

Correct Answer: C — 128

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Q. What is the value of 100 - 37? (2014)

A. 63
B. 67
C. 73
D. 60

Solution

100 - 37 = 63.

Correct Answer: A — 63

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Q. What is the value of 12 × 12? (2020)

A. 144
B. 124
C. 156
D. 132

Solution

12 × 12 = 144.

Correct Answer: A — 144

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Q. What is the value of 12% of 250? (2020)

A. 30
B. 25
C. 35
D. 20

Solution

12% of 250 = (12/100) * 250 = 30.

Correct Answer: A — 30

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Q. What is the value of 15% of 200? (2014)

A. 30
B. 25
C. 35
D. 20

Solution

15% of 200 = (15/100) * 200 = 30.

Correct Answer: A — 30

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Q. What is the value of 5! (5 factorial)? (2023) 2023

A. 120
B. 100
C. 150
D. 110

Solution

5! = 5 × 4 × 3 × 2 × 1 = 120.

Correct Answer: A — 120

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Q. What is the value of cot(45°)?

A. 0
B. 1
C. √3
D. ∞

Solution

cot(45°) = 1/tan(45°) = 1/1 = 1

Correct Answer: B — 1

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Q. What is the value of cot(90°)? (2018)

A. 0
B. 1
C. undefined
D. ∞

Solution

cot(90°) = 1/tan(90°) = 1/0 = undefined.

Correct Answer: C — undefined

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Q. What is the value of k for which the equation x² - 6x + k = 0 has roots 2 and 4? (2022)

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

Solution

The product of the roots is k = 2 * 4 = 8.

Correct Answer: B — 12

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Q. What is the value of k if the equation x² + kx + 16 = 0 has no real roots? (2022)

A. -8
B. -4
C. 4
D. 8

Solution

For no real roots, the discriminant must be less than zero: k² - 4*1*16 < 0, thus k < -8 or k > 8.

Correct Answer: A — -8

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Q. What is the value of k if the equation x² - 4x + k = 0 has no real roots? (2021)

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

Solution

For no real roots, the discriminant must be less than zero: (-4)² - 4*1*k < 0, hence k > 4.

Correct Answer: B — 6

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Q. What is the value of k if the equation x² - kx + 16 = 0 has roots 4 and 4? (2021)

A. 8
B. 16
C. 4
D. 12

Solution

Since the roots are equal, k = 4 + 4 = 8.

Correct Answer: A — 8

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Q. What is the value of sec(0°)?

A. 0
B. 1
C. √2
D. ∞

Solution

sec(0°) = 1/cos(0°) = 1/1 = 1

Correct Answer: B — 1

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Q. What is the value of sec(30°)? (2015)

A. √3/2
B. 2/√3
C. 2
D. √3

Solution

sec(30°) = 1/cos(30°) = 1/(√3/2) = 2/√3 = √3

Correct Answer: D — √3

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Q. What is the value of sec(45°)? (2019)

A. 1
B. √2
C. 2
D. √3

Solution

sec(45°) = 1/cos(45°) = 1/(√2/2) = √2.

Correct Answer: B — √2

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Q. What is the value of sin(45°)? (2016)

A. 1/√2
B. 1/2
C. √3/2
D. 1

Solution

sin(45°) = 1/√2

Correct Answer: A — 1/√2

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Q. What is the value of tan(0°)? (2018)

A. 0
B. 1
C. undefined
D. ∞

Solution

tan(0°) = 0

Correct Answer: A — 0

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Q. What is the value of tan(90°)? (2022)

A. 0
B. 1
C. undefined
D. ∞

Solution

tan(90°) is undefined.

Correct Answer: C — undefined

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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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Mathematics

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