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 distance from the point (1, 2) to the line x + y = 5? (2020)

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

Solution

Distance = |1 + 2 - 5| / √(1² + 1²) = | -2 | / √2 = 2/√2 = √2.

Correct Answer: A — 3

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Q. What is the distance from the point (2, 3) to the line x + y - 5 = 0? (2022)

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

Solution

Distance = |(1*2 + 1*3 - 5)| / √(1² + 1²) = |5 - 5| / √2 = 0

Correct Answer: A — 1

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Q. What is the distance from the point (2, 3) to the line x + y = 5? (2020)

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

Solution

Distance = |(1*2 + 1*3 - 5)| / √(1² + 1²) = |5 - 5| / √2 = 0/√2 = 0

Correct Answer: B — 2

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Q. What is the distance from the point (3, 4) to the line x + y = 5? (2022)

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

Solution

Distance = |3 + 4 - 5| / √(1² + 1²) = |2| / √2 = 2/√2 = √2.

Correct Answer: B — 2

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Q. What is the dot product of vectors A = 2i + 3j and B = 4i - j? (2022)

A. 5
B. 10
C. 11
D. 14

Solution

A · B = (2)(4) + (3)(-1) = 8 - 3 = 5.

Correct Answer: B — 10

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Q. What is the equation of a plane passing through the point (1, 2, 3) with normal vector (1, 1, 1)? (2022)

A. x + y + z = 6
B. x + y + z = 3
C. x + y + z = 1
D. x + y + z = 0

Solution

Equation of the plane: 1(x-1) + 1(y-2) + 1(z-3) = 0 => x + y + z = 6.

Correct Answer: A — x + y + z = 6

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Q. What is the equation of a plane passing through the point (1, 2, 3) with normal vector (2, -1, 3)? (2021)

A. 2x - y + 3z = 12
B. 2x + y - 3z = 0
C. 2x - y + 3z = 0
D. 2x + y + 3z = 12

Solution

Equation of the plane: 2(x-1) - 1(y-2) + 3(z-3) = 0 simplifies to 2x - y + 3z = 12.

Correct Answer: C — 2x - y + 3z = 0

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Q. What is the equation of the line parallel to y = 3x + 4 that passes through the point (2, 5)? (2022)

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

Solution

Since parallel lines have the same slope, the equation is y - 5 = 3(x - 2) which simplifies to y = 3x + 1.

Correct Answer: B — y = 3x + 1

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Q. What is the equation of the line parallel to y = 3x - 5 and passing through (2, 1)?

A. y = 3x - 8
B. y = 3x + 5
C. y = 3x - 1
D. y = 3x + 1

Solution

Parallel lines have the same slope. The slope is 3. Using point-slope form: y - 1 = 3(x - 2) gives y = 3x - 8.

Correct Answer: A — y = 3x - 8

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

A. y = 3x - 8
B. y = 3x + 5
C. y = 3x - 1
D. y = 3x + 1

Solution

Since the lines are parallel, they have the same slope. Using point-slope form: y - 1 = 3(x - 2) gives y = 3x - 8.

Correct Answer: A — y = 3x - 8

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Q. What is the equation of the line passing through (0, 0) and (2, 4)? (2019)

A. y = 2x
B. y = x
C. y = 4x
D. y = 3x

Solution

Slope = (4-0)/(2-0) = 2, so the equation is y = 2x.

Correct Answer: A — y = 2x

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Q. What is the equation of the line passing through (0, 0) with a slope of 3? (2021)

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

Solution

Equation of line: y = mx + c; here m = 3, c = 0 => y = 3x

Correct Answer: A — y = 3x

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Q. What is the equation of the line passing through (2, 3) with a slope of 2? (2021)

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

Solution

Using point-slope form: y - 3 = 2(x - 2) => y = 2x - 4 + 3 => y = 2x - 1.

Correct Answer: B — y = 2x + 1

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Q. What is the equation of the line passing through the points (1, 2) and (3, 4)?

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

Solution

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

Correct Answer: A — y = x + 1

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Q. What is the equation of the line passing through the points (1, 2) and (3, 6)?

A. y = 2x
B. y = 3x - 1
C. y = x + 1
D. y = 4x - 2

Solution

The slope m = (6 - 2) / (3 - 1) = 2. Using point-slope form, y - 2 = 2(x - 1) gives y = 2x.

Correct Answer: A — y = 2x

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

A. y = -1/3x + 4
B. y = 3x - 3
C. y = -3x + 9
D. y = 1/3x + 2

Solution

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

Correct Answer: A — y = -1/3x + 4

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Q. What is the equation of the line that passes through the origin and has a slope of -5?

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

Solution

The equation of a line through the origin with slope m is y = mx. Thus, y = -5x.

Correct Answer: A — y = -5x

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Q. What is the equation of the line that passes through the origin and has a slope of -1?

A. y = -x
B. y = x
C. y = -2x
D. y = 2x

Solution

The equation of a line through the origin with slope m is y = mx. Thus, y = -1x or y = -x.

Correct Answer: A — y = -x

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Q. What is the equation of the plane passing through the point (1, 2, 3) with normal vector (1, -1, 1)? (2023)

A. x - y + z = 0
B. x + y + z = 6
C. x - y + z = 1
D. x + y - z = 0

Solution

Equation of the plane: 1(x-1) - 1(y-2) + 1(z-3) = 0 => x - y + z = 1.

Correct Answer: C — x - y + z = 1

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Q. What is the equation of the plane passing through the points (1, 2, 3), (4, 5, 6), and (7, 8, 9)? (2021)

A. 0 = 0
B. x + y + z = 12
C. x + y + z = 10
D. x + y + z = 9

Solution

The points are collinear, hence the equation of the plane is 0 = 0.

Correct Answer: A — 0 = 0

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

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

Solution

This is a separable equation, and integrating gives y = Ce^(-4x).

Correct Answer: A — y = Ce^(-4x)

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Q. What is the general solution of the equation y' = 4y + 3?

A. y = Ce^(4x) - 3/4
B. y = Ce^(4x) + 3/4
C. y = 3e^(4x)
D. y = Ce^(3x) + 4

Solution

The integrating factor is e^(-4x). The solution is y = Ce^(4x) + 3/4.

Correct Answer: B — y = Ce^(4x) + 3/4

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

A. y = C1 e^(x) + C2 e^(2x)
B. y = C1 e^(2x) + C2 e^(x)
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^(2x) + C2 e^(x).

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

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

A. y = (C1 + C2x)e^(2x)
B. y = C1 e^(2x) + C2 e^(-2x)
C. y = C1 e^(4x) + C2 e^(-4x)
D. y = C1 cos(2x) + C2 sin(2x)

Solution

The characteristic equation has a repeated root r = 2. The general solution is y = (C1 + C2x)e^(2x).

Correct Answer: A — y = (C1 + C2x)e^(2x)

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Q. What is the integral of 1/x with respect to x? (2022)

A. ln
B. x
C. + C
D. 1/x + C
. x + C
. e^x + C

Solution

The integral of 1/x is ln|x| + C.

Correct Answer: A — ln

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Q. What is the integral of 1/x? (2022)

A. ln
B. x
C. + C
D. 1/x + C
. x + C
. x^2 + C

Solution

The integral of 1/x is ln|x| + C.

Correct Answer: A — ln

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Q. What is the integral of cos(2x) dx? (2021)

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(2x) is (1/2)sin(2x) + C.

Correct Answer: A — (1/2)sin(2x) + C

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Q. What is the integral of e^x with respect to x? (2023)

A. e^x + C
B. xe^x + C
C. e^x/x + C
D. x^2e^x + C

Solution

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

Correct Answer: A — e^x + C

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Q. What is the integral of e^x? (2023)

A. e^x + C
B. xe^x + C
C. e^x/x + C
D. x^2e^x + C

Solution

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

Correct Answer: A — e^x + C

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Q. What is the integral of sin(x) dx? (2021)

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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Mathematics

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