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 determinant of the matrix \( F = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{pmatrix} \)? (2022)

A. -14
B. 14
C. 0
D. 6

Solution

Using the determinant formula for 3x3 matrices, we find \( 1(1*0 - 4*6) - 2(0*0 - 4*5) + 3(0*6 - 1*5) = 1(0 - 24) - 2(0 - 20) + 3(0 - 5) = -24 + 40 - 15 = 1 \).

Correct Answer: A — -14

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Q. What is the determinant of the matrix \( F = \begin{pmatrix} 2 & 5 \\ 3 & 7 \end{pmatrix} \)? (2023)

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

Solution

The determinant is calculated as \( 2*7 - 5*3 = 14 - 15 = -1 \).

Correct Answer: A — -1

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Q. What is the determinant of the matrix \( G = \begin{pmatrix} 2 & 3 \\ 5 & 7 \end{pmatrix} \)? (2023)

A. 1
B. 4
C. 7
D. 11

Solution

The determinant is \( 2*7 - 3*5 = 14 - 15 = -1 \).

Correct Answer: A — 1

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Q. What is the determinant of the matrix \( H = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \)? (2022)

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

Solution

The determinant of the identity matrix is always 1.

Correct Answer: B — 1

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Q. What is the determinant of the matrix \( H = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} \)? (2023)

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

Solution

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

Correct Answer: A — 0

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Q. What is the determinant of the matrix \( I = \begin{pmatrix} 4 & 2 \\ 1 & 3 \end{pmatrix} \)? (2021)

A. 10
B. 8
C. 6
D. 12

Solution

The determinant is \( 4*3 - 2*1 = 12 - 2 = 10 \).

Correct Answer: A — 10

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Q. What is the diameter of a circle if its area is 50π square units? (2017)

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

Solution

Area = πr²; 50π = πr²; r² = 50; r = √50; Diameter = 2√50 ≈ 10 units.

Correct Answer: A — 10 units

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Q. What is the diameter of a circle if its area is 78.5 cm²? (2020)

A. 10 cm
B. 8 cm
C. 6 cm
D. 12 cm

Solution

Area = πr²; 78.5 = π * r²; r² = 78.5/π; d = 2√(78.5/π) = 10 cm.

Correct Answer: A — 10 cm

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Q. What is the diameter of a circle with a radius of 9 cm? (2022)

A. 9 cm
B. 18 cm
C. 27 cm
D. 36 cm

Solution

Diameter = 2 * radius = 2 * 9 cm = 18 cm.

Correct Answer: B — 18 cm

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Q. What is the diameter of a circle with an area of 50.24 cm²? (2019)

A. 8 cm
B. 10 cm
C. 12 cm
D. 14 cm

Solution

Area = πr²; 50.24 = πr²; r² = 50.24/π; r ≈ 4 cm; Diameter = 2r = 8 cm.

Correct Answer: B — 10 cm

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Q. What is the diameter of a circle with an area of 50π square units? (2017)

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

Solution

Area = πr²; 50π = πr²; r² = 50; r = √50; Diameter = 2√50 ≈ 10 units.

Correct Answer: A — 10 units

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Q. What is the diameter of a circle with an area of 78.5 cm²? (2018)

A. 10 cm
B. 8 cm
C. 6 cm
D. 12 cm

Solution

Area = πr²; 78.5 = πr²; r² = 78.5/π; d = 2√(78.5/π) ≈ 10 cm.

Correct Answer: A — 10 cm

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Q. What is the discriminant of the equation 3x² - 12x + 12 = 0? (2023)

A. 0
B. 12
C. 36
D. 24

Solution

The discriminant is b² - 4ac = (-12)² - 4*3*12 = 144 - 144 = 0.

Correct Answer: A — 0

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Q. What is the discriminant of the equation 3x² - 12x + 9 = 0? (2023)

A. 0
B. 9
C. 12
D. 36

Solution

The discriminant is b² - 4ac = (-12)² - 4*3*9 = 0.

Correct Answer: A — 0

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Q. What is the discriminant of the equation 4x² - 12x + 9 = 0? (2019)

A. 0
B. 1
C. 4
D. 9

Solution

The discriminant is b² - 4ac = (-12)² - 4*4*9 = 144 - 144 = 0.

Correct Answer: A — 0

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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 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 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 15 cm? (2014)

A. 15 cm
B. 30 cm
C. 10 cm
D. 5 cm

Solution

The distance from the center to any point on the circumference is always the radius, which is 15 cm.

Correct Answer: A — 15 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 parallel planes x + 2y + 3z = 4 and x + 2y + 3z = 10? (2023)

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

Solution

Distance = |d1 - d2| / √(a² + b² + c²) = |4 - 10| / √(1² + 2² + 3²) = 6 / √14.

Correct Answer: A — 2

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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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Mathematics

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