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 value of \( |D| \) for the matrix \( D = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \)? (2023)

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

Solution

The determinant is \( 0*0 - 1*1 = 0 - 1 = -1 \).

Correct Answer: A — 1

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Q. What is the vertex of the parabola represented by the equation y = x² - 4x + 3? (2022)

A. (2, -1)
B. (2, 1)
C. (1, 2)
D. (3, 0)

Solution

The vertex can be found using the formula x = -b/2a. Here, x = 2, and substituting back gives y = -1.

Correct Answer: A — (2, -1)

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Q. What is the vertex of the parabola represented by the equation y = x² - 6x + 8? (2023)

A. (3, -1)
B. (3, -5)
C. (2, -4)
D. (2, -2)

Solution

The vertex can be found using the formula x = -b/2a = 6/2 = 3. Substituting x = 3 into the equation gives y = -1.

Correct Answer: A — (3, -1)

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Q. What is the volume of a cube with side length 3 units? (2023)

A. 9
B. 27
C. 81
D. 6

Solution

Volume = side³ = 3³ = 27 cubic units.

Correct Answer: B — 27

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Q. What is the volume of a cylinder with radius 2 and height 5? (2023)

A. 20π
B. 10π
C. 15π
D. 25π

Solution

Volume = πr²h = π(2²)(5) = 20π cubic units.

Correct Answer: A — 20π

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Q. What is the volume of a sphere with radius 5 units? (2021)

A. 500/3π
B. 125/3π
C. 100/3π
D. 200/3π

Solution

Volume = (4/3)πr³ = (4/3)π(5)³ = (4/3)π(125) = 500/3π cubic units.

Correct Answer: A — 500/3π

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Q. What is the x-intercept of the line 5x - 2y + 10 = 0?

A. -2
B. 2
C. 0
D. 5

Solution

Setting y = 0 in the equation gives 5x + 10 = 0, thus x = -2.

Correct Answer: B — 2

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Q. What is the x-intercept of the line 6x - 2y = 12?

A. 2
B. 3
C. 4
D. 6

Solution

Setting y = 0 in the equation gives 6x = 12, thus x = 2. The x-intercept is 2.

Correct Answer: B — 3

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Q. What is the y-intercept of the line 3x - 2y = 6? (2020) 2020

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

Solution

Setting x = 0, we get -2y = 6, so y = -3.

Correct Answer: A — 3

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Q. What is the y-intercept of the line 3x - 4y = 12? (2023)

A. 3
B. 4
C. 0
D. 12

Solution

Setting x = 0, we get -4y = 12, so y = -3.

Correct Answer: A — 3

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Q. Which of the following is the conjugate of the complex number 2 - 3i? (2020)

A. 2 + 3i
B. 2 - 3i
C. 3 - 2i
D. 3 + 2i

Solution

The conjugate of a complex number a + bi is a - bi. Thus, the conjugate of 2 - 3i is 2 + 3i.

Correct Answer: A — 2 + 3i

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Q. Which of the following is the conjugate of the complex number 5 - 3i? (2020)

A. 5 + 3i
B. 5 - 3i
C. -5 + 3i
D. -5 - 3i

Solution

The conjugate of a complex number a + bi is a - bi. Thus, the conjugate of 5 - 3i is 5 + 3i.

Correct Answer: A — 5 + 3i

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Q. Which of the following points lies on the line 4x - 5y + 20 = 0?

A. (5, 0)
B. (0, 4)
C. (4, 0)
D. (0, -4)

Solution

Substituting (0, -4) into the equation gives 4(0) - 5(-4) + 20 = 0, which is true.

Correct Answer: D — (0, -4)

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Mathematics

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