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. In how many ways can 5 different letters be arranged if 2 letters are always together? (2019)

A. 48
B. 60
C. 120
D. 24

Solution

Treat the 2 letters as one unit. So, we have 4 units to arrange: 4! * 2! = 48.

Correct Answer: B — 60

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Q. In how many ways can 6 different books be arranged on a shelf?

A. 720
B. 600
C. 360
D. 480

Solution

The number of arrangements of 6 different books is 6! = 720.

Correct Answer: A — 720

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Q. In how many ways can 6 different colored balls be arranged in a line?

A. 720
B. 120
C. 60
D. 30

Solution

The number of arrangements of 6 different colored balls is 6! = 720.

Correct Answer: A — 720

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Q. In how many ways can 6 different colored balls be arranged in a row? (2018)

A. 720
B. 600
C. 480
D. 360

Solution

The number of arrangements of 6 distinct balls is 6! = 720.

Correct Answer: A — 720

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Q. In how many ways can 6 different flags be arranged on a pole?

A. 720
B. 600
C. 480
D. 540

Solution

The number of arrangements is 6! = 720.

Correct Answer: A — 720

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Q. In how many ways can 6 people be seated in a row?

A. 720
B. 600
C. 480
D. 360

Solution

The number of arrangements of 6 people is 6! = 720.

Correct Answer: A — 720

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Q. In how many ways can 7 people be divided into 3 groups if one group must have 3 people?

A. 210
B. 300
C. 420
D. 560

Solution

Choose 3 people from 7: C(7, 3) = 35. The remaining 4 can be divided into 2 groups in 2 ways. Total = 35 * 2 = 70.

Correct Answer: B — 300

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Q. In how many ways can the letters of the word 'BOOK' be arranged? (2022)

A. 12
B. 24
C. 16
D. 8

Solution

The word 'BOOK' has 4 letters with 'O' repeating. The number of arrangements is 4! / 2! = 12.

Correct Answer: C — 16

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Q. In the expansion of (2x + 3)^4, what is the coefficient of x? (2020)

A. 12
B. 36
C. 48
D. 72

Solution

The coefficient of x is given by 4C1 * (2)^1 * (3)^3 = 4 * 2 * 27 = 216.

Correct Answer: B — 36

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Q. In the expansion of (2x + 3)^4, what is the coefficient of x^0?

A. 81
B. 64
C. 36
D. 16

Solution

The coefficient of x^0 is given by 4C4 * (2x)^0 * (3)^4 = 1 * 81 = 81.

Correct Answer: A — 81

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Q. In the expansion of (2x + 5)^3, what is the coefficient of x?

A. 30
B. 45
C. 60
D. 75

Solution

The coefficient of x is given by 3C1 * (2)^1 * (5)^2 = 3 * 2 * 25 = 150.

Correct Answer: B — 45

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Q. In the expansion of (2x - 5)^5, what is the coefficient of x^2? (2021)

A. -300
B. -600
C. 600
D. 300

Solution

The coefficient of x^2 is C(5,2) * (2)^2 * (-5)^3 = 10 * 4 * (-125) = -5000.

Correct Answer: A — -300

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Q. In the expansion of (2x - 5)^5, what is the coefficient of x^3? (2021)

A. 150
B. 300
C. 450
D. 600

Solution

The coefficient of x^3 is C(5,3) * (2)^3 * (-5)^2 = 10 * 8 * 25 = 2000.

Correct Answer: C — 450

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Q. In the expansion of (2x - 5)^6, what is the coefficient of x^2? (2021)

A. -150
B. -300
C. 300
D. 150

Solution

The coefficient of x^2 is C(6,2) * (2)^2 * (-5)^4 = 15 * 4 * 625 = -37500.

Correct Answer: A — -150

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Q. In the expansion of (2x - 5)^6, what is the coefficient of x^3? (2020)

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

Solution

The coefficient of x^3 is C(6,3)(2)^3(-5)^3 = 20 * 8 * -125 = -20000.

Correct Answer: B — -720

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Q. In the expansion of (3 + 2x)^4, what is the coefficient of x^2? (2023)

A. 36
B. 48
C. 54
D. 24

Solution

The coefficient of x^2 is C(4,2) * (3)^2 * (2)^2 = 6 * 9 * 4 = 216.

Correct Answer: B — 48

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Q. In the expansion of (3x - 2)^6, what is the coefficient of x^4? (2022)

A. 540
B. 810
C. 729
D. 486

Solution

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

Correct Answer: A — 540

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Q. In the expansion of (3x - 4)^4, what is the coefficient of x^2? (2023)

A. -144
B. -216
C. 216
D. 144

Solution

The coefficient of x^2 is C(4,2)(3)^2(-4)^2 = 6 * 9 * 16 = -864.

Correct Answer: B — -216

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Q. In the expansion of (x + 3)^4, what is the coefficient of x^3? (2023)

A. 36
B. 48
C. 54
D. 64

Solution

The coefficient of x^3 is C(4,3) * (3)^1 = 4 * 3 = 12.

Correct Answer: A — 36

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Q. In the expansion of (x + 4)^5, what is the coefficient of x^3? (2021)

A. 240
B. 320
C. 80
D. 160

Solution

The coefficient of x^3 is C(5,3) * (4)^2 = 10 * 16 = 160.

Correct Answer: A — 240

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Q. In the expansion of (x - 1)^5, what is the coefficient of x^3?

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

Solution

The coefficient of x^3 is C(5,3) * (-1)^2 = 10.

Correct Answer: A — -10

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Q. Solve the differential equation dy/dx = 2x + 1.

A. y = x^2 + x + C
B. y = x^2 + 2x + C
C. y = 2x^2 + x + C
D. y = x^2 + C

Solution

Integrating both sides, we get y = ∫(2x + 1)dx = x^2 + x + C.

Correct Answer: A — y = x^2 + x + C

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Q. Solve the differential equation dy/dx = 2y + 3. (2023)

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

Solution

Using an integrating factor, we find the solution is y = Ce^(2x) - 3/2.

Correct Answer: A — y = Ce^(2x) - 3/2

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Q. Solve the differential equation dy/dx = 6x^2y.

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

Solution

This is a separable equation. Integrating gives y = Ce^(2x^3).

Correct Answer: A — y = Ce^(2x^3)

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Q. Solve the differential equation dy/dx = y/x. (2023)

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

Solution

This is a separable equation. Integrating gives y = Cx.

Correct Answer: A — y = Cx

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Q. Solve the differential equation y' = 5 - 2y.

A. y = 5/2 + Ce^(-2x)
B. y = 5 + Ce^(-2x)
C. y = 2 + Ce^(2x)
D. y = 5/2 - Ce^(-2x)

Solution

This is a linear first-order equation. The solution is y = 5/2 + Ce^(-2x).

Correct Answer: A — y = 5/2 + Ce^(-2x)

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Q. Solve 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 y = (3/5) + Ce^(5x).

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

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Q. Solve the differential equation y'' - 3y' + 2y = 0.

A. y = C1e^(2x) + C2e^(x)
B. y = C1e^(x) + C2e^(2x)
C. y = C1e^(-x) + C2e^(-2x)
D. y = C1e^(3x) + C2e^(x)

Solution

The characteristic equation is r^2 - 3r + 2 = 0, which factors to (r - 1)(r - 2) = 0. The general solution is y = C1e^(x) + C2e^(2x).

Correct Answer: B — y = C1e^(x) + C2e^(2x)

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Q. Solve the equation y' = 6y + 12.

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

Solution

This is a first-order linear equation. The integrating factor method gives the solution y = 2 - Ce^(-6x).

Correct Answer: A — y = 2 - Ce^(-6x)

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Q. Solve the first-order differential equation dy/dx = y/x.

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

Solution

This is a separable equation. Separating variables and integrating gives y = Cx.

Correct Answer: A — y = Cx

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Mathematics

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