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. Find the equation of the line parallel to y = 3x + 2 and passing through (4, 5).

A. y = 3x - 7
B. y = 3x + 5
C. y = 3x + 2
D. y = 3x - 2

Solution

Since the line is parallel, it has the same slope. Using point-slope form: y - 5 = 3(x - 4) gives y = 3x - 7.

Correct Answer: A — y = 3x - 7

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Q. Find the equation of the line parallel to y = 3x + 2 that passes through the point (4, 1).

A. y = 3x - 11
B. y = 3x + 1
C. y = 3x + 2
D. y = 3x - 2

Solution

Since the line is parallel, it has the same slope (3). Using point-slope form: y - 1 = 3(x - 4) gives y = 3x - 11.

Correct Answer: A — y = 3x - 11

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Q. Find the equation of the line parallel to y = 5x - 2 that passes through the point (2, 3).

A. y = 5x - 7
B. y = 5x + 2
C. y = 5x - 5
D. y = 5x + 1

Solution

Since the slope is the same (5), using point-slope form: y - 3 = 5(x - 2) gives y = 5x - 7.

Correct Answer: A — y = 5x - 7

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Q. Find the equation of the line parallel to y = 5x - 3 that passes through the point (2, 1).

A. y = 5x - 9
B. y = 5x + 1
C. y = 5x - 7
D. y = 5x + 3

Solution

Since the slope is the same (5), using point-slope form: y - 1 = 5(x - 2) gives y = 5x - 9.

Correct Answer: A — y = 5x - 9

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Q. Find the equation of the line that passes through (0, 0) and has a slope of 5.

A. y = 5x
B. y = x/5
C. y = 5/x
D. y = 1/5x

Solution

Using the slope-intercept form y = mx + b, with m = 5 and b = 0, we get y = 5x.

Correct Answer: A — y = 5x

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Q. Find the equation of the line that passes through (2, 3) and is perpendicular to the line y = 1/3x + 2.

A. y = -3x + 9
B. y = 3x - 3
C. y = -1/3x + 4
D. y = 1/3x + 1

Solution

The slope of the given line is 1/3, so the perpendicular slope is -3. Using point-slope form, y - 3 = -3(x - 2) gives y = -3x + 9.

Correct Answer: A — y = -3x + 9

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Q. Find the equation of the line that passes through (2, 3) and is perpendicular to the line y = 4x - 1.

A. y = -1/4x + 4
B. y = 4x - 5
C. y = -4x + 11
D. y = 1/4x + 2

Solution

The slope of the given line is 4, so the perpendicular slope is -1/4. Using point-slope form, we get y - 3 = -1/4(x - 2) which simplifies to y = -1/4x + 11/4.

Correct Answer: C — y = -4x + 11

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Q. Find the equation of the line that passes through the origin and has a slope of -3.

A. y = -3x
B. y = 3x
C. y = -x/3
D. y = 1/3x

Solution

Using the slope-intercept form, the equation is y = -3x.

Correct Answer: A — y = -3x

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Q. Find the equation of the line that passes through the point (2, -3) and has a slope of 4.

A. y = 4x - 11
B. y = 4x + 5
C. y = 4x - 3
D. y = 4x + 3

Solution

Using point-slope form, y + 3 = 4(x - 2) simplifies to y = 4x - 11.

Correct Answer: A — y = 4x - 11

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Q. Find the equation of the line that passes through the point (4, -1) and is perpendicular to the line y = 3x + 2.

A. y = -1/3x + 5/3
B. y = 3x - 13
C. y = -3x + 11
D. y = 1/3x - 5/3

Solution

The slope of the given line is 3, so the slope of the perpendicular line is -1/3. Using point-slope form, we get y + 1 = -1/3(x - 4), which simplifies to y = -1/3x + 11/3.

Correct Answer: C — y = -3x + 11

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Q. Find the equation of the line that passes through the point (4, 5) and is perpendicular to the line y = 1/3x + 2.

A. y = -3x + 17
B. y = 3x - 7
C. y = -3x + 5
D. y = 1/3x + 5

Solution

The slope of the given line is 1/3, so the slope of the perpendicular line is -3. Using point-slope form, we get y - 5 = -3(x - 4), which simplifies to y = -3x + 17.

Correct Answer: A — y = -3x + 17

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Q. Find the equation of the line that passes through the points (2, 3) and (4, 7).

A. y = 2x - 1
B. y = 2x + 1
C. y = 3x - 3
D. y = x + 1

Solution

The slope m = (7 - 3) / (4 - 2) = 2. Using point-slope form: y - 3 = 2(x - 2) gives y = 2x + 1.

Correct Answer: B — y = 2x + 1

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Q. Find the general solution of dy/dx = 3x^2. (2020)

A. y = x^3 + C
B. y = 3x^3 + C
C. y = x^2 + C
D. y = 3x + C

Solution

Integrating 3x^2 gives y = x^3 + C.

Correct Answer: A — y = x^3 + C

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Q. Find the general solution of the equation y' = 3x^2y.

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

Solution

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

Correct Answer: A — y = Ce^(x^3)

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Q. Find the general solution of the equation y' = 5y + 3.

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

Solution

The integrating factor method gives the general solution y = Ce^(5x) - 3/5.

Correct Answer: A — y = Ce^(5x) - 3/5

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Q. Find the general solution of 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^(3x) + C2 e^(0)
D. y = C1 e^(0) + C2 e^(3x)

Solution

The characteristic equation is r^2 - 3r + 2 = 0, which factors to (r-1)(r-2)=0. Thus, the general solution is y = C1 e^(x) + C2 e^(2x).

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

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Q. Find the integral of (2x + 1)^3 dx. (2019)

A. (1/4)(2x + 1)^4 + C
B. (1/3)(2x + 1)^4 + C
C. (1/5)(2x + 1)^4 + C
D. (1/2)(2x + 1)^4 + C

Solution

Using substitution, the integral is (1/4)(2x + 1)^4 + C.

Correct Answer: A — (1/4)(2x + 1)^4 + C

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Q. Find the integral of (2x + 3)dx. (2022)

A. x^2 + 3x + C
B. x^2 + 3x + 1
C. x^2 + 3 + C
D. 2x^2 + 3x + C

Solution

Integrating term by term: ∫2xdx = x^2 and ∫3dx = 3x. Thus, ∫(2x + 3)dx = x^2 + 3x + C.

Correct Answer: A — x^2 + 3x + C

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Q. Find the integral of cos(2x)dx. (2023)

A. (1/2)sin(2x) + C
B. sin(2x) + C
C. (1/2)cos(2x) + C
D. 2sin(2x) + C

Solution

The integral of cos(kx) is (1/k)sin(kx) + C. Here, k=2, so the integral is (1/2)sin(2x) + C.

Correct Answer: A — (1/2)sin(2x) + C

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Q. Find the integral of cos(x) with respect to x. (2023)

A. sin(x) + C
B. -sin(x) + C
C. cos(x) + C
D. -cos(x) + C

Solution

The integral of cos(x) is sin(x) + C.

Correct Answer: A — sin(x) + C

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Q. Find the integral of cos(x). (2023)

A. sin(x) + C
B. -sin(x) + C
C. cos(x) + C
D. -cos(x) + C

Solution

The integral of cos(x) is sin(x) + C.

Correct Answer: A — sin(x) + C

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Q. Find the integral of cos(x)dx. (2023)

A. sin(x) + C
B. -sin(x) + C
C. cos(x) + C
D. -cos(x) + C

Solution

The integral of cos(x) is sin(x) + C.

Correct Answer: A — sin(x) + C

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Q. Find the integral of e^x dx. (2022)

A. e^x + C
B. e^x
C. x e^x + C
D. ln(e^x) + C

Solution

The integral of e^x is e^x + C.

Correct Answer: A — e^x + C

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Q. Find the integral of sin(x) with respect to x. (2020)

A. -cos(x) + C
B. cos(x) + C
C. sin(x) + C
D. -sin(x) + C

Solution

The integral of sin(x) is -cos(x) + C.

Correct Answer: A — -cos(x) + C

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Q. Find the integral of sin(x). (2020)

A. -cos(x) + C
B. cos(x) + C
C. sin(x) + C
D. -sin(x) + C

Solution

The integral of sin(x) is -cos(x) + C.

Correct Answer: A — -cos(x) + C

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Q. Find the integral of sin(x)dx. (2020)

A. -cos(x) + C
B. cos(x) + C
C. sin(x) + C
D. -sin(x) + C

Solution

The integral of sin(x) is -cos(x) + C.

Correct Answer: A — -cos(x) + C

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Q. Find the integral of x^5 dx. (2020)

A. (1/6)x^6 + C
B. (1/5)x^6 + C
C. (1/4)x^6 + C
D. (1/7)x^6 + C

Solution

The integral is (1/6)x^6 + C.

Correct Answer: B — (1/5)x^6 + C

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Q. Find the length of the diagonal of a cuboid with dimensions 3, 4, and 12 units. (2020)

A. √169
B. √145
C. √153
D. √157

Solution

Diagonal = √(3² + 4² + 12²) = √(9 + 16 + 144) = √169 = 13.

Correct Answer: C — √153

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Q. Find the length of the diagonal of a rectangular box with dimensions 2, 3, and 6. (2023)

A. √49
B. √36
C. √45
D. √50

Solution

Diagonal = √(2² + 3² + 6²) = √(4 + 9 + 36) = √49 = 7.

Correct Answer: A — √49

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Q. Find the length of the diagonal of a rectangular box with dimensions 2, 3, and 6 units. (2022)

A. √49
B. √45
C. √36
D. √50

Solution

Diagonal = √(2² + 3² + 6²) = √(4 + 9 + 36) = √49 = 7 units.

Correct Answer: A — √49

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Mathematics

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