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 discriminant of the equation x² + 6x + 9 = 0? (2020)

A. 0
B. 6
C. 9
D. 36

Solution

The discriminant is b² - 4ac = 6² - 4*1*9 = 0.

Correct Answer: A — 0

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Q. What is the distance between the center of a circle and a point on its circumference if the radius is 12 cm? (2022)

A. 6 cm
B. 12 cm
C. 18 cm
D. 24 cm

Solution

The distance from the center to a point on the circumference is the radius, which is 12 cm.

Correct Answer: B — 12 cm

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Q. What is the distance between the center of a circle and a point on its circumference if the radius is 10 cm? (2022)

A. 5 cm
B. 10 cm
C. 15 cm
D. 20 cm

Solution

The distance from the center to a point on the circumference is the radius, which is 10 cm.

Correct Answer: B — 10 cm

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Q. What is the distance between the center of a circle and a point on its circumference if the radius is 8 cm? (2023)

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

Solution

The distance from the center to a point on the circumference is the radius, which is 8 cm.

Correct Answer: A — 8 cm

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Q. What is the distance between the parallel planes 2x + 3y - z = 5 and 2x + 3y - z = 10? (2021)

A. 5/√14
B. 10/√14
C. 15/√14
D. 20/√14

Solution

Distance = |d1 - d2| / √(A² + B² + C²) = |5 - 10| / √(2² + 3² + (-1)²) = 5/√14.

Correct Answer: B — 10/√14

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Q. What is the distance between the point (2, 3, 4) and the plane x + y + z = 10? (2022)

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

Solution

Distance = |(2 + 3 + 4 - 10) / √(1² + 1² + 1²)| = |(-1)/√3| = 1/√3.

Correct Answer: A — 1

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Q. What is the distance between the points (-1, -1) and (2, 3)? (2022) 2022

A. 5
B. 4
C. 6
D. 7

Solution

Distance = √[(2 - (-1))² + (3 - (-1))²] = √[3² + 4²] = √25 = 5.

Correct Answer: A — 5

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Q. What is the distance between the points (-1, -1) and (2, 3)? (2023)

A. 5
B. 4
C. 3
D. 6

Solution

Distance = √[(2 - (-1))² + (3 - (-1))²] = √[3² + 4²] = √25 = 5.

Correct Answer: A — 5

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Q. What is the distance between the points (2, 3) and (5, 7)?

A. 5
B. 4
C. 3
D. 6

Solution

Using the distance formula, d = √((5 - 2)² + (7 - 3)²) = √(9 + 16) = √25 = 5.

Correct Answer: A — 5

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Q. What is the distance between the points (3, 4) and (7, 1)? (2021) 2021

A. 5
B. 4
C. 3
D. 6

Solution

Distance = √[(7-3)² + (1-4)²] = √[4 + 9] = √13 ≈ 5.

Correct Answer: A — 5

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Q. What is the distance between the points A(1, 2, 3) and B(4, 5, 6)? (2023)

A. 3√2
B. 3√3
C. 3
D. √27

Solution

Distance = √[(4-1)² + (5-2)² + (6-3)²] = √[3² + 3² + 3²] = √27 = 3√3.

Correct Answer: A — 3√2

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Q. What is the distance from the point (1, 1) to the line x + y = 2? (2023)

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

Solution

Distance = |1 + 1 - 2| / √(1² + 1²) = 0 / √2 = 0.

Correct Answer: A — 1

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Q. What is the distance from the point (1, 2) to the line x + y - 3 = 0? (2023)

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

Solution

Distance = |1 + 2 - 3| / √(1² + 1²) = |0| / √2 = 0, closest option is 1.

Correct Answer: B — 2

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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 (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 (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 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 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 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 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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Mathematics

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