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. If H = [[2, 3], [4, 5]], find det(H). (2022)

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

Solution

Det(H) = (2*5) - (3*4) = 10 - 12 = -2.

Correct Answer: D — 7

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Q. If I = [[1, 0, 2], [0, 1, 3], [1, 1, 0]], find det(I). (2023)

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

Solution

Using the determinant formula for 3x3 matrices, det(I) = 1(1*0 - 3*1) - 0(0 - 3*1) + 2(0 - 1*1) = 0 - 0 - 2 = -2.

Correct Answer: A — -1

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Q. If I = [[1, 2], [2, 4]], what is det(I)? (2021)

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

Solution

The determinant of I is 0 because the rows are linearly dependent.

Correct Answer: A — 0

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Q. If J = [[1, 1], [1, 1]], what is det(J)? (2019)

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

Solution

Det(J) = (1*1) - (1*1) = 1 - 1 = 0.

Correct Answer: A — 0

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Q. If J = [[1, 2, 1], [0, 1, 3], [2, 1, 0]], calculate det(J). (2023)

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

Solution

Using the determinant formula, det(J) = 1*(1*0 - 3*1) - 2*(0*0 - 3*2) + 1*(0*1 - 1*2) = 1*(-3) - 2*(-6) + 1*(-2) = -3 + 12 - 2 = 7.

Correct Answer: A — -4

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Q. If J = [[1, 2], [2, 4]], what is det(J)? (2022)

A. 0
B. 1
C. 2
D. 4

Solution

Det(J) = (1*4) - (2*2) = 4 - 4 = 0.

Correct Answer: A — 0

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Q. If one root of the equation x² - 7x + k = 0 is 3, find k. (2023)

A. 10
B. 12
C. 15
D. 9

Solution

Using the root, substitute x = 3: 3² - 7*3 + k = 0, which gives k = 10.

Correct Answer: A — 10

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Q. If one root of the equation x² - 7x + k = 0 is 3, what is the value of k? (2020)

A. 10
B. 12
C. 15
D. 9

Solution

Using the root, substitute x = 3: 3² - 7*3 + k = 0, which gives k = 9.

Correct Answer: D — 9

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Q. If sin(θ) = 0.5, what is θ in degrees? (2014)

A. 30°
B. 45°
C. 60°
D. 90°

Solution

sin(30°) = 0.5, so θ = 30°.

Correct Answer: A — 30°

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Q. If sin(θ) = 0.8, what is cos(θ) using Pythagorean identity? (2020)

A. 0.6
B. 0.8
C. 0.4
D. 0.2

Solution

Using sin²(θ) + cos²(θ) = 1, cos²(θ) = 1 - 0.64 = 0.36, cos(θ) = 0.6

Correct Answer: A — 0.6

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Q. If sin(θ) = 0.8, what is cos(θ)? (2022)

A. 0.6
B. 0.8
C. 0.4
D. 0.2

Solution

Using sin²(θ) + cos²(θ) = 1, cos(θ) = √(1 - 0.8²) = √(0.36) = 0.6.

Correct Answer: A — 0.6

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Q. If sin(θ) = 0.866, what is θ in degrees? (2020)

A. 30°
B. 45°
C. 60°
D. 90°

Solution

sin(60°) = √3/2 ≈ 0.866, so θ = 60°.

Correct Answer: C — 60°

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Q. If sin(θ) = 1, what is the value of θ? (2023)

A. 0°
B. 90°
C. 180°
D. 270°

Solution

sin(90°) = 1, so θ = 90°.

Correct Answer: B — 90°

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Q. If sin(θ) = 1/2, what is the possible value of θ? (2022)

A. 30°
B. 60°
C. 90°
D. 45°

Solution

sin(30°) = 1/2, so θ = 30°.

Correct Answer: A — 30°

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Q. If sin(θ) = 1/√2, what is θ in degrees?

A. 30°
B. 45°
C. 60°
D. 90°

Solution

sin(45°) = 1/√2, so θ = 45°

Correct Answer: B — 45°

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Q. If tan(θ) = 1, what is the value of θ? (2020)

A. 0°
B. 30°
C. 45°
D. 60°

Solution

tan(45°) = 1, so θ = 45°.

Correct Answer: C — 45°

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Q. If the circumference of a circle is 31.4 cm, what is its radius? (2022)

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

Solution

Circumference = 2πr; 31.4 = 2 * π * r; r = 5 cm.

Correct Answer: A — 5 cm

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Q. If the circumference of a circle is 31.4 cm, what is its radius? (Use π = 3.14) (2015)

A. 5 cm
B. 7 cm
C. 10 cm
D. 8 cm

Solution

Circumference = 2πr, so r = circumference / (2π) = 31.4 / (2 * 3.14) = 5 cm.

Correct Answer: A — 5 cm

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Q. If the circumference of a circle is 31.4 cm, what is the radius? (2020) 2020

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

Solution

Circumference = 2πr. Therefore, r = Circumference / (2π) = 31.4 cm / (2 * 3.14) = 5 cm.

Correct Answer: A — 5 cm

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Q. If the coordinates of point A are (1, 2) and point B are (4, 6), what is the length of AB? (2023)

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

Solution

Length AB = √[(4-1)² + (6-2)²] = √[9 + 16] = √25 = 5.

Correct Answer: A — 5

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Q. If the coordinates of point C are (5, 5) and point D are (5, 10), what is the length of CD? (2021)

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

Solution

Length CD = |10 - 5| = 5.

Correct Answer: A — 5

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Q. If the coordinates of point C are (5, 5) and the slope of the line is 1, what is the equation of the line? (2021) 2021

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

Solution

Using point-slope form: y - 5 = 1(x - 5) => y = x.

Correct Answer: A — y = x

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Q. If the cost function is C(x) = 3x^2 + 12x + 5, find the minimum cost. (2020)

A. 5
B. 8
C. 12
D. 10

Solution

The minimum cost occurs at x = -b/(2a) = -12/(2*3) = -2. C(-2) = 3(-2)^2 + 12(-2) + 5 = 8.

Correct Answer: B — 8

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Q. If the cost function is C(x) = 3x^2 + 12x + 5, find the minimum cost. (2020) 2020

A. 5
B. 8
C. 12
D. 10

Solution

The minimum cost occurs at x = -b/(2a) = -12/(2*3) = -2. C(-2) = 3(-2)^2 + 12(-2) + 5 = 8.

Correct Answer: B — 8

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Q. If the cost function is C(x) = 5x^2 + 20x + 100, find the minimum cost. (2020)

A. 100
B. 120
C. 140
D. 160

Solution

The minimum cost occurs at x = -b/(2a) = -20/(2*5) = -2. C(-2) = 5(-2)^2 + 20(-2) + 100 = 120.

Correct Answer: B — 120

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Q. If the diameter of a circle is 20 cm, what is its circumference? (2022)

A. 62.83 cm
B. 31.42 cm
C. 20 cm
D. 40 cm

Solution

Circumference = πd; C = π * 20 ≈ 62.83 cm.

Correct Answer: A — 62.83 cm

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Q. If the line 2x + 3y = 6 intersects the x-axis, what is the x-coordinate of the intersection point?

A. 0
B. 2
C. 3
D. 6

Solution

Setting y = 0 gives 2x = 6, thus x = 3. The x-coordinate of the intersection point is 3.

Correct Answer: B — 2

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Q. If the line 3x - 4y + 12 = 0 is parallel to another line, what is the slope of the parallel line?

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

Solution

Rearranging gives y = (3/4)x + 3. The slope of the line is 3/4, so the slope of the parallel line is -3/4.

Correct Answer: B — -3/4

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Q. If the line 3x - 4y + 12 = 0 is parallel to which of the following lines?

A. 6x - 8y + 24 = 0
B. 3x + 4y - 12 = 0
C. x + 2y - 5 = 0
D. 2x - 3y + 6 = 0

Solution

Parallel lines have the same slope. The slope of the given line is 3/4, which is the same as the slope of 6x - 8y + 24 = 0.

Correct Answer: A — 6x - 8y + 24 = 0

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Q. If the radius of a circle is doubled, how does the area change? (2021)

A. It doubles
B. It triples
C. It quadruples
D. It remains the same

Solution

Area = πr²; if r is doubled, area = π(2r)² = 4πr², so it quadruples.

Correct Answer: C — It quadruples

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Mathematics

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