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. Solve the first-order linear differential equation dy/dx + 2y = 6.

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

Solution

Using an integrating factor e^(2x), we solve to get y = 3 - Ce^(-2x).

Correct Answer: A — y = 3 - Ce^(-2x)

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Q. Solve the first-order linear differential equation dy/dx + y/x = 1.

A. y = x + C/x
B. y = Cx - x
C. y = Cx + x
D. y = C/x + x

Solution

Using the integrating factor e^(∫(1/x)dx) = x, we solve to get y = x + C/x.

Correct Answer: A — y = x + C/x

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Q. Solve the first-order linear 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 separable: dy/y = dx/x. Integrating gives ln|y| = ln|x| + C, thus y = Cx.

Correct Answer: A — y = Cx

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Q. The quadratic equation x² + 4x + 4 = 0 has how many distinct roots? (2021)

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

Solution

The discriminant is 4² - 4*1*4 = 0, indicating one distinct root.

Correct Answer: B — 1

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Q. The roots of the equation x² + 2x + k = 0 are real and distinct if k is: (2020)

A. < 1
B. ≥ 1
C. ≤ 1
D. > 1

Solution

For real and distinct roots, the discriminant must be positive: 2² - 4*1*k > 0, which simplifies to k < 1.

Correct Answer: A — < 1

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Q. The roots of the equation x² + 4x + 4 = 0 are: (2020)

A. -2 and -2
B. 2 and 2
C. 0 and 4
D. 1 and 3

Solution

The equation can be factored as (x + 2)² = 0, giving the double root x = -2.

Correct Answer: A — -2 and -2

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Q. The roots of the equation x² + 4x + k = 0 are 2 and -6. What is the value of k? (2021)

A. -12
B. -8
C. -10
D. -14

Solution

Using the product of roots: k = 2 * (-6) = -12. The sum is 2 + (-6) = -4, which matches.

Correct Answer: B — -8

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Q. The roots of the equation x² - 8x + k = 0 are 4 and 4. Find k. (2021)

A. 16
B. 8
C. 0
D. 4

Solution

Using the product of roots: k = 4 * 4 = 16.

Correct Answer: A — 16

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Q. What are the roots of the equation 3x² - 12x + 12 = 0? (2019)

A. 2
B. 4
C. 0
D. 3

Solution

Dividing the equation by 3 gives x² - 4x + 4 = 0, which factors to (x - 2)² = 0, hence the root is 2.

Correct Answer: B — 4

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Q. What are the roots of the equation x² - 2x - 8 = 0? (2022)

A. -2 and 4
B. 2 and -4
C. 4 and -2
D. 0 and 8

Solution

Factoring gives (x - 4)(x + 2) = 0, hence the roots are 4 and -2.

Correct Answer: C — 4 and -2

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Q. What are the roots of the equation x² - 5x + 6 = 0? (2021)

A. 1 and 6
B. 2 and 3
C. 3 and 2
D. 0 and 5

Solution

The roots can be found using the factorization method: (x - 2)(x - 3) = 0, hence the roots are 2 and 3.

Correct Answer: B — 2 and 3

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Q. What is the 10th term of the arithmetic sequence where the first term is 5 and the common difference is 3?

A. 32
B. 35
C. 30
D. 28

Solution

The nth term of an arithmetic sequence is given by a_n = a + (n-1)d. Here, a = 5, d = 3, n = 10. So, a_10 = 5 + (10-1) * 3 = 5 + 27 = 32.

Correct Answer: B — 35

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Q. What is the 12th term of the arithmetic sequence where the first term is 7 and the common difference is 5?

A. 62
B. 67
C. 72
D. 77

Solution

Using the formula a_n = a + (n-1)d, we have a_12 = 7 + (12-1) * 5 = 7 + 55 = 62.

Correct Answer: A — 62

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Q. What is the 3rd term in the expansion of (2x + 3)^4?

A. 108x^2
B. 216x^2
C. 324x^2
D. 432x^2

Solution

The 3rd term is given by C(4,2) * (2x)^2 * (3)^2 = 6 * 4x^2 * 9 = 216x^2.

Correct Answer: B — 216x^2

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Q. What is the 3rd term in the expansion of (2x + 5)^6? (2000)

A. 600x^4
B. 1500x^4
C. 1800x^4
D. 2000x^4

Solution

The 3rd term is given by C(6,2) * (2x)^2 * (5)^4 = 15 * 4 * 625 = 37500.

Correct Answer: B — 1500x^4

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Q. What is the 3rd term in the expansion of (2x - 3)^5? (2022)

A. -90x^3
B. 90x^3
C. -60x^3
D. 60x^3

Solution

The 3rd term is C(5,2) * (2x)^3 * (-3)^2 = 10 * 8x^3 * 9 = -720x^3.

Correct Answer: A — -90x^3

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Q. What is the 3rd term in the expansion of (x + 2)^5? (2021)

A. 40
B. 80
C. 60
D. 20

Solution

The 3rd term is given by C(5,2) * (x)^3 * (2)^2 = 10 * x^3 * 4 = 40.

Correct Answer: C — 60

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Q. What is the 3rd term in the expansion of (x + 2)^8? (2022)

A. 112
B. 128
C. 256
D. 64

Solution

The 3rd term is given by 8C2 * (2)^2 * (x)^6 = 28 * 4 * x^6 = 112x^6.

Correct Answer: A — 112

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Q. What is the 3rd term in the expansion of (x + 3)^4? (2023)

A. 36x^2
B. 54x^2
C. 72x^2
D. 108x^2

Solution

The 3rd term is C(4,2) * (3)^2 * (x)^2 = 6 * 9 * x^2 = 54x^2.

Correct Answer: B — 54x^2

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Q. What is the 3rd term in the expansion of (x + 3)^5? (2023)

A. 45
B. 90
C. 135
D. 180

Solution

The 3rd term is C(5,2) * (3)^2 * (x)^3 = 10 * 9 * x^3 = 90.

Correct Answer: B — 90

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Q. What is the 3rd term in the expansion of (x + 3)^7? (2023)

A. 189
B. 441
C. 729
D. 1024

Solution

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

Correct Answer: B — 441

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Q. What is the 3rd term in the expansion of (x + 4)^5? (2023)

A. 80x^3
B. 160x^3
C. 240x^3
D. 320x^3

Solution

The 3rd term is given by C(5,2) * (4)^2 * (x)^3 = 10 * 16 * x^3 = 160x^3.

Correct Answer: C — 240x^3

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Q. What is the 3rd term in the expansion of (x + 4)^6? (2020)

A. 240x^4
B. 360x^4
C. 480x^4
D. 600x^4

Solution

The 3rd term is C(6,2) * (4)^2 * (x)^4 = 15 * 16 * x^4 = 240x^4.

Correct Answer: B — 360x^4

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Q. What is the 4th term in the expansion of (2x - 3)^6? (2020)

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

Solution

The 4th term is C(6,3) * (2x)^3 * (-3)^3 = 20 * 8x^3 * -27 = -540.

Correct Answer: A — -540

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

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

Solution

The 5th term is given by 6C4 * (2)^(6-4) * (3x)^4 = 15 * 4 * 81x^4 = 4860x^4.

Correct Answer: B — 540

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Q. What is the 5th term in the expansion of (3x - 2)^7? (2023)

A. -1680x^3
B. 1680x^3
C. -2520x^3
D. 2520x^3

Solution

The 5th term is given by C(7,4) * (3x)^4 * (-2)^3 = 35 * 81 * (-8) = -1680x^3.

Correct Answer: A — -1680x^3

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Q. What is the 5th term in the expansion of (x - 1)^8? (2022)

A. -56x^4
B. 56x^4
C. -70x^4
D. 70x^4

Solution

The 5th term corresponds to k=4. Using the binomial theorem, it is given by 8C4 * (x)^4 * (-1)^4 = 70x^4.

Correct Answer: A — -56x^4

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Q. What is the 5th term of the sequence defined by a_n = 2n^2 + 3n - 1?

A. 45
B. 47
C. 49
D. 51

Solution

To find the 5th term, substitute n = 5 into the formula: a_5 = 2(5^2) + 3(5) - 1 = 2(25) + 15 - 1 = 50 + 15 - 1 = 64.

Correct Answer: B — 47

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Q. What is the 8th term of the sequence defined by a_n = 3n - 2?

A. 22
B. 24
C. 26
D. 28

Solution

Substituting n = 8 into the formula: a_8 = 3(8) - 2 = 24 - 2 = 22.

Correct Answer: A — 22

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Q. What is the angle between the lines y = 2x + 1 and y = -1/2x + 3?

A. 90 degrees
B. 60 degrees
C. 45 degrees
D. 30 degrees

Solution

The slopes are m1 = 2 and m2 = -1/2. The angle θ = tan^(-1) |(m1 - m2) / (1 + m1*m2)| = tan^(-1)(5/4) which is approximately 60 degrees.

Correct Answer: B — 60 degrees

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Mathematics

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