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 solution of the equation dy/dx = 6 - 2y? (2021)

A. y = 3 - Ce^(-2x)
B. y = 3 + Ce^(-2x)
C. y = 2 - Ce^(2x)
D. y = 6 - Ce^(2x)

Solution

Rearranging gives dy/(6 - 2y) = dx. Integrating both sides leads to y = 3 - Ce^(-2x).

Correct Answer: A — y = 3 - Ce^(-2x)

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Q. What is the solution of the equation y' + 4y = 0?

A. y = Ce^(-4x)
B. y = Ce^(4x)
C. y = 4Ce^x
D. y = Ce^(x/4)

Solution

This is a separable equation. The solution is y = Ce^(-4x).

Correct Answer: A — y = Ce^(-4x)

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Q. What is the solution of the equation y' = -ky, where k is a constant?

A. y = Ce^(kt)
B. y = Ce^(-kt)
C. y = -Ce^(kt)
D. y = -Ce^(-kt)

Solution

This is a separable equation. Integrating gives y = Ce^(-kt).

Correct Answer: B — y = Ce^(-kt)

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Q. What is the solution to the differential equation dy/dx = -y/x?

A. y = Cx
B. y = C/x
C. y = Cx^2
D. y = Cx^(-1)

Solution

This is a separable equation. Separating variables and integrating gives y = C/x.

Correct Answer: B — y = C/x

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Q. What is the solution to the differential equation y' = 5y + 3?

A. y = (3/5) + Ce^(5x)
B. y = (5/3) + Ce^(5x)
C. y = Ce^(5x) - 3
D. y = Ce^(3x) + 5

Solution

Using the integrating factor method, we find the solution to be y = (3/5) + Ce^(5x).

Correct Answer: A — y = (3/5) + Ce^(5x)

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Q. What is the solution to the equation dy/dx = -5y?

A. y = Ce^(-5x)
B. y = -5Ce^x
C. y = Ce^(5x)
D. y = 5Ce^(-x)

Solution

This is a separable differential equation. The solution is y = Ce^(-5x), where C is a constant.

Correct Answer: A — y = Ce^(-5x)

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Q. What is the solution to the equation dy/dx = y^2? (2022)

A. y = 1/(C - x)
B. y = C/(x - 1)
C. y = Cx^2
D. y = ln(Cx)

Solution

This is a separable equation. Integrating gives y = 1/(C - x).

Correct Answer: A — y = 1/(C - x)

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Q. What is the solution to the equation y' + 2y = 0?

A. y = Ce^(-2x)
B. y = Ce^(2x)
C. y = 2Ce^x
D. y = Ce^x

Solution

This is a separable equation. The solution is y = Ce^(-2x).

Correct Answer: A — y = Ce^(-2x)

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Q. What is the solution to the equation y' + 3y = 0?

A. y = Ce^(-3x)
B. y = Ce^(3x)
C. y = 3Ce^(-x)
D. y = Ce^(-x/3)

Solution

This is a first-order linear differential equation. The solution is y = Ce^(-3x).

Correct Answer: A — y = Ce^(-3x)

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Q. What is the solution to the equation y' = 3y + 6?

A. y = Ce^(3x) - 2
B. y = Ce^(3x) + 2
C. y = 2e^(3x)
D. y = 3Ce^(x)

Solution

This is a first-order linear equation. The integrating factor is e^(3x), leading to the solution y = Ce^(3x) + 2.

Correct Answer: B — y = Ce^(3x) + 2

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Q. What is the solution to the equation y'' + 4y = 0?

A. y = C1 cos(2x) + C2 sin(2x)
B. y = C1 e^(2x) + C2 e^(-2x)
C. y = C1 e^(4x) + C2 e^(-4x)
D. y = C1 sin(4x) + C2 cos(4x)

Solution

The characteristic equation is r^2 + 4 = 0, giving complex roots. The general solution is y = C1 cos(2x) + C2 sin(2x).

Correct Answer: A — y = C1 cos(2x) + C2 sin(2x)

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Q. What is the solution to the equation y'' - 3y' + 2y = 0?

A. y = C1 e^(2x) + C2 e^(x)
B. y = C1 e^(x) + C2 e^(2x)
C. y = C1 e^(-x) + C2 e^(-2x)
D. y = C1 + C2x

Solution

The characteristic equation r^2 - 3r + 2 = 0 has roots 1 and 2, leading to y = C1 e^(x) + C2 e^(2x).

Correct Answer: B — y = C1 e^(x) + C2 e^(2x)

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Q. What is the square of the modulus of the complex number 1 + 2i? (2014)

A. 5
B. 4
C. 1
D. 2

Solution

The modulus is √(1^2 + 2^2) = √(1 + 4) = √5. The square of the modulus is 5.

Correct Answer: A — 5

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Q. What is the square of the modulus of the complex number 1 + i? (2020)

A. 2
B. 1
C. 0
D. 4

Solution

The modulus is √(1^2 + 1^2) = √2, and the square of the modulus is 2.

Correct Answer: A — 2

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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 (3x - 2)^4? (2023)

A. 1
B. 81
C. 64
D. 256

Solution

To find the sum of the coefficients, substitute x = 1. Thus, (3(1) - 2)^4 = (3 - 2)^4 = 1^4 = 1.

Correct Answer: B — 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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Mathematics

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