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 square root of 64? (2020)

A. 6
B. 7
C. 8
D. 9

Solution

The square root of 64 is 8.

Correct Answer: C — 8

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Q. What is the square root of the complex number -1? (2021)

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

Solution

The square root of -1 is defined as i.

Correct Answer: A — i

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Q. What is the square root of the complex number -4? (2020)

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

Solution

The square root of -4 is √(-1) * √4 = 2i.

Correct Answer: A — 2i

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Q. What is the sum of the coefficients in the expansion of (2 + 3)^4? (2022)

A. 81
B. 64
C. 100
D. 125

Solution

The sum of the coefficients is (2 + 3)^4 = 5^4 = 625.

Correct Answer: A — 81

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Q. What is the sum of the coefficients in the expansion of (x + 2)^5? (2021)

A. 32
B. 64
C. 96
D. 128

Solution

The sum of the coefficients is found by substituting x=1. So, (1 + 2)^5 = 3^5 = 243.

Correct Answer: B — 64

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Q. What is the sum of the complex numbers 1 + 2i and 3 - 4i? (2023)

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

Solution

(1 + 2i) + (3 - 4i) = (1 + 3) + (2 - 4)i = 4 - 2i.

Correct Answer: A — 4 - 2i

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Q. What is the sum of the complex numbers 3 + 2i and 1 - 4i? (2023)

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

Solution

(3 + 2i) + (1 - 4i) = (3 + 1) + (2 - 4)i = 4 - 2i.

Correct Answer: A — 4 - 2i

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Q. What is the sum of the roots of the equation 2x² - 4x + 1 = 0? (2023)

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

Solution

The sum of the roots is given by -b/a = 4/2 = 2.

Correct Answer: A — 2

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Q. What is the sum of the roots of the equation 3x² + 12x + 9 = 0? (2021)

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

Solution

The sum of the roots is given by -b/a = -12/3 = -4.

Correct Answer: A — -4

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

A. -1440
B. 720
C. -720
D. 1440

Solution

The term containing x^2 is given by C(6,2) * (3x)^2 * (-4)^(6-2) = 15 * 9 * 256 = -1440.

Correct Answer: A — -1440

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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 term containing x^5 in the expansion of (2x + 3)^6?

A. 540
B. 720
C. 810
D. 900

Solution

The term containing x^5 occurs when k = 5. Using the binomial theorem, the term is C(6,5) * (2x)^5 * 3^1 = 6 * 32 * 3 = 576.

Correct Answer: B — 720

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

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

Solution

The term containing x^5 occurs when k = 5. Using the binomial theorem, the term is C(8,5) * (2x)^5 * (-3)^3 = 56 * 32 * (-27) = -48384.

Correct Answer: A — -6720

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Q. What is the term independent of x in the expansion of (2x - 3)^8?

A. -256
B. 256
C. -512
D. 512

Solution

The term independent of x is given by C(8,4) * (2x)^4 * (-3)^4 = 70 * 16 * 81 = -256.

Correct Answer: A — -256

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Q. What is the term independent of x in the expansion of (3x - 2)^6?

A. 20
B. 30
C. 40
D. 50

Solution

The term independent of x occurs when k = 3. Using the binomial theorem, the term is C(6,3) * (3x)^3 * (-2)^(6-3) = 20 * 27 * -8 = -4320.

Correct Answer: C — 40

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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) × 4? (2022)

A. 20
B. 25
C. 15
D. 10

Solution

(2 + 3) × 4 = 5 × 4 = 20.

Correct Answer: A — 20

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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 - 2)^5 using the binomial theorem?

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

Solution

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

Correct Answer: A — 1

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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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Mathematics

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