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 product of the roots of the equation 3x² - 6x + 2 = 0? (2023)

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

Solution

The product of the roots is given by c/a = 2/3.

Correct Answer: B — 2/3

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Q. What is the product of the roots of the equation x² + 5x + 6 = 0? (2022)

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

Solution

The product of the roots is given by c/a = 6/1 = 6.

Correct Answer: A — 6

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Q. What is the product of the roots of the equation x² - 10x + 24 = 0? (2021)

A. 24
B. 10
C. 16
D. 14

Solution

The product of the roots is given by c/a = 24/1 = 24.

Correct Answer: A — 24

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Q. What is the product of the roots of the equation x² - 8x + 15 = 0? (2022)

A. 15
B. 8
C. 7
D. 6

Solution

The product of the roots is given by c/a = 15/1 = 15.

Correct Answer: A — 15

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Q. What is the projection of vector A = 3i + 4j onto vector B = i + j? (2020)

A. 3
B. 4
C. 5
D. 7

Solution

proj_B(A) = (A · B / |B|^2)B = ((3*1 + 4*1) / (1^2 + 1^2))(i + j) = (7/2)(i + j) = 3.5.

Correct Answer: C — 5

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Q. What is the radius of a circle if its circumference is 62.8 cm? (2022)

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

Solution

Circumference = 2πr; 62.8 = 2πr; r = 10 cm.

Correct Answer: A — 10 cm

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Q. What is the radius of a circle if the area is 154 square units? (2023)

A. 7 units
B. 14 units
C. 11 units
D. 10 units

Solution

Area = πr²; 154 = 22/7 * r²; r² = 154 * 7/22 = 49; r = 7 units.

Correct Answer: A — 7 units

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Q. What is the radius of a circle if the circumference is 31.4 m? (2020)

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

Solution

Circumference = 2πr; 31.4 = 2πr; r = 31.4/(2π) ≈ 5 m.

Correct Answer: A — 5 m

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Q. What is the radius of a circle if the circumference is 62.8 cm? (2016)

A. 10 cm
B. 15 cm
C. 20 cm
D. 25 cm

Solution

Circumference = 2πr; 62.8 = 2πr; r = 62.8/(2π) ≈ 10 cm.

Correct Answer: A — 10 cm

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Q. What is the radius of a circle if the diameter is 14 cm? (2021)

A. 7 cm
B. 14 cm
C. 21 cm
D. 28 cm

Solution

Radius is half of the diameter. Therefore, radius = 14 cm / 2 = 7 cm.

Correct Answer: A — 7 cm

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Q. What is the real part of the complex number 4 + 5i? (2023)

A. 4
B. 5
C. 9
D. 0

Solution

The real part of the complex number 4 + 5i is 4.

Correct Answer: A — 4

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Q. What is the real part of the complex number 7 - 4i? (2023)

A. 7
B. -4
C. 4
D. 0

Solution

The real part of the complex number 7 - 4i is 7.

Correct Answer: A — 7

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Q. What is the slope of the line passing through the points (1, 2) and (3, 6)? (2020) 2020

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

Solution

Slope = (6-2)/(3-1) = 4/2 = 2.

Correct Answer: A — 2

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Q. What is the slope of the line perpendicular to the line 3x + 4y = 12?

A. -3/4
B. 4/3
C. 3/4
D. -4/3

Solution

The slope of the line 3x + 4y = 12 is -3/4. The slope of the perpendicular line is the negative reciprocal, which is 4/3.

Correct Answer: B — 4/3

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Q. What is the slope of the line perpendicular to the line 4x + 2y = 8?

A. -1/2
B. 1/2
C. 2
D. -2

Solution

The slope of the line is -2 (from y = -2x + 4). The slope of the perpendicular line is the negative reciprocal, which is 1/2.

Correct Answer: A — -1/2

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Q. What is the slope of the line perpendicular to the line 4x + 3y = 12?

A. -3/4
B. 4/3
C. 3/4
D. -4/3

Solution

The slope of the line 4x + 3y = 12 is -4/3. The slope of the perpendicular line is the negative reciprocal, which is 3/4.

Correct Answer: D — -4/3

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Q. What is the slope of the line perpendicular to the line 5x + 2y = 10?

A. -2/5
B. 5/2
C. 2/5
D. -5/2

Solution

The slope of the line is -5/2. The slope of the perpendicular line is the negative reciprocal, which is 2/5.

Correct Answer: A — -2/5

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Q. What is the slope of the line perpendicular to the line 7x - 2y + 3 = 0?

A. 1/2
B. -1/2
C. 2
D. -2

Solution

The slope of the line is 7/2. The slope of the perpendicular line is the negative reciprocal, which is -2/7.

Correct Answer: B — -1/2

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Q. What is the slope of the line perpendicular to the line 7x - 2y = 14?

A. 1/7
B. -1/7
C. 2/7
D. -2/7

Solution

The slope of the line is 7/2. The slope of the perpendicular line is the negative reciprocal, which is -2/7.

Correct Answer: B — -1/7

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Q. What is the slope of the line perpendicular to the line with slope 3?

A. -1/3
B. 1/3
C. 3
D. -3

Solution

Slope of perpendicular line = -1/m = -1/3.

Correct Answer: A — -1/3

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Q. What is the slope of the line perpendicular to the line y = 3x + 1? (2023)

A. -1/3
B. 1/3
C. 3
D. -3

Solution

Slope of perpendicular line = -1/(slope of original line) = -1/3.

Correct Answer: A — -1/3

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Q. What is the slope of the line that passes through the points (4, 5) and (6, 9)?

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

Solution

The slope m = (9 - 5) / (6 - 4) = 4/2 = 2.

Correct Answer: A — 2

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Q. What is the slope of the tangent line to f(x) = x^2 + 2x at x = 1? (2023)

A. 2
B. 3
C. 4
D. 5

Solution

f'(x) = 2x + 2. At x = 1, f'(1) = 2(1) + 2 = 4.

Correct Answer: B — 3

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Q. What is the slope of the tangent line to the curve y = x^2 - 4x + 5 at x = 3? (2023)

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

Solution

The slope is given by f'(x) = 2x - 4. At x = 3, f'(3) = 2(3) - 4 = 2.

Correct Answer: C — 2

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Q. What is the slope of the tangent to the curve y = x^2 + 2x at x = 1? (2023)

A. 2
B. 3
C. 4
D. 5

Solution

f'(x) = 2x + 2. At x = 1, f'(1) = 2(1) + 2 = 4.

Correct Answer: B — 3

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Q. What is the solution of the differential equation dy/dx = y^2?

A. y = 1/(C - x)
B. y = C/(x + 1)
C. y = Cx
D. y = e^(x + C)

Solution

Separating variables and integrating gives 1/y = x + C, leading to y = 1/(C - x).

Correct Answer: A — y = 1/(C - x)

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Q. What is the solution of the differential equation y' = 2y + 3?

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

Solution

The integrating factor is e^(-2x). Solving gives y = Ce^(2x) + 3/2.

Correct Answer: B — y = Ce^(2x) + 3/2

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Q. What is the solution of the differential equation y' = 5y + 3?

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

Solution

Using the integrating factor method, we find the general solution to be y = Ce^(5x) - (3/5).

Correct Answer: B — y = Ce^(5x) - (3/5)

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Q. What is the solution of the equation dy/dx = 3x^2?

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

Solution

Integrating both sides gives y = ∫3x^2 dx = x^3 + C.

Correct Answer: A — y = x^3 + C

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Q. What is the solution of the equation dy/dx = 4y + 2? (2021)

A. y = Ce^(4x) - 1/2
B. y = Ce^(-4x) + 1/2
C. y = 2e^(4x) + C
D. y = 4e^(4x) + C

Solution

Using an integrating factor, the solution is y = Ce^(4x) - 1/2.

Correct Answer: A — y = Ce^(4x) - 1/2

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Mathematics

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