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. Find the coordinates of the midpoint of the line segment joining A(2, -1, 3) and B(4, 3, 5). (2022)

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

Solution

Midpoint M = ((2+4)/2, (-1+3)/2, (3+5)/2) = (3, 1, 4).

Correct Answer: A — (3, 1, 4)

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Q. Find the coordinates of the midpoint of the line segment joining A(2, 3, 4) and B(4, 5, 6). (2023)

A. (3, 4, 5)
B. (2, 3, 4)
C. (4, 5, 6)
D. (5, 6, 7)

Solution

Midpoint M = ((2+4)/2, (3+5)/2, (4+6)/2) = (3, 4, 5).

Correct Answer: A — (3, 4, 5)

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Q. Find the critical points of f(x) = x^4 - 8x^2 + 16. (2021)

A. (0, 16)
B. (2, 0)
C. (4, 0)
D. (1, 15)

Solution

f'(x) = 4x^3 - 16x. Setting f'(x) = 0 gives x = 0, ±2. f(2) = 0 is a critical point.

Correct Answer: B — (2, 0)

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Q. Find the critical points of the function f(x) = x^4 - 8x^2 + 16. (2019)

A. (0, 16)
B. (2, 0)
C. (4, 0)
D. (1, 9)

Solution

Setting f'(x) = 4x^3 - 16x = 0 gives x = 0, ±2. Evaluating f(2) = 0 shows (2, 0) is a critical point.

Correct Answer: B — (2, 0)

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Q. Find the derivative of f(x) = sin(x) + cos(x).

A. cos(x) - sin(x)
B. -sin(x) - cos(x)
C. sin(x) + cos(x)
D. -cos(x) + sin(x)

Solution

The derivative f'(x) = d/dx(sin(x) + cos(x)) = cos(x) - sin(x).

Correct Answer: A — cos(x) - sin(x)

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Q. Find the derivative of f(x) = tan(x). (2022) 2022

A. sec^2(x)
B. csc^2(x)
C. sec(x)
D. tan^2(x)

Solution

The derivative f'(x) = d/dx(tan(x)) = sec^2(x).

Correct Answer: A — sec^2(x)

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Q. Find the derivative of f(x) = x^5 + 3x^3 - 2x.

A. 5x^4 + 9x^2 - 2
B. 5x^4 + 6x^2 - 2
C. 3x^2 + 5x^4 - 2
D. 5x^4 + 3x^2 - 2

Solution

The derivative f'(x) = d/dx(x^5 + 3x^3 - 2x) = 5x^4 + 9x^2 - 2.

Correct Answer: A — 5x^4 + 9x^2 - 2

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Q. Find the derivative of f(x) = x^5 - 3x + 2.

A. 5x^4 - 3
B. 5x^4 + 3
C. 4x^3 - 3
D. 5x^4 - 2

Solution

The derivative f'(x) = d/dx(x^5) - d/dx(3x) + d/dx(2) = 5x^4 - 3.

Correct Answer: A — 5x^4 - 3

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Q. Find the derivative of f(x) = x^5 - 3x^3 + 2. (2022)

A. 5x^4 - 9x^2
B. 5x^4 + 9x^2
C. 3x^2 - 9x
D. 5x^4 - 3x^2

Solution

The derivative f'(x) = d/dx(x^5 - 3x^3 + 2) = 5x^4 - 9x^2.

Correct Answer: A — 5x^4 - 9x^2

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Q. Find the determinant of E = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]. (2019)

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

Solution

Determinant of E = 0 (rows are linearly dependent).

Correct Answer: A — 0

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Q. Find the determinant of E = [[3, 2], [1, 4]]. (2022)

A. 10
B. 14
C. 6
D. 8

Solution

Det(E) = (3*4) - (2*1) = 12 - 2 = 10.

Correct Answer: A — 10

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Q. Find the determinant of E = [[4, 2], [1, 3]]. (2023)

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

Solution

Det(E) = (4*3) - (2*1) = 12 - 2 = 10.

Correct Answer: A — 10

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Q. Find the determinant of F = [[4, 3], [2, 1]]. (2018)

A. -2
B. 2
C. 6
D. 8

Solution

Det(F) = (4*1) - (3*2) = 4 - 6 = -2.

Correct Answer: A — -2

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Q. Find the determinant of F = [[4, 5], [6, 7]]. (2020)

A. -2
B. 2
C. 10
D. 12

Solution

Det(F) = (4*7) - (5*6) = 28 - 30 = -2.

Correct Answer: A — -2

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Q. Find the determinant of G = [[1, 2], [2, 4]]. (2020)

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

Solution

Determinant of G = (1*4) - (2*2) = 4 - 4 = 0.

Correct Answer: A — 0

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Q. Find the determinant of H = [[3, 1], [2, 5]]. (2021)

A. 7
B. 8
C. 6
D. 5

Solution

Determinant of H = (3*5) - (1*2) = 15 - 2 = 13.

Correct Answer: A — 7

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Q. Find the determinant of J = [[5, 2], [1, 3]]. (2020)

A. 10
B. 11
C. 12
D. 13

Solution

The determinant of J is calculated as (5*3) - (2*1) = 15 - 2 = 13.

Correct Answer: A — 10

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Q. Find the determinant of the matrix D = [[3, 2, 1], [1, 0, 2], [2, 1, 3]]. (2020)

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

Solution

The determinant of D can be calculated using the rule of Sarrus or cofactor expansion, which results in 0.

Correct Answer: A — 0

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Q. Find the determinant of the matrix D = [[4, 2], [3, 1]]. (2023)

A. -2
B. 2
C. 10
D. 12

Solution

The determinant of D is calculated as (4*1) - (2*3) = 4 - 6 = -2.

Correct Answer: A — -2

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Q. Find the determinant of the matrix \( E = \begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix} \). (2021)

A. 10
B. 14
C. 5
D. 6

Solution

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

Correct Answer: A — 10

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Q. Find the determinant of the matrix \( J = \begin{pmatrix} 2 & 3 \\ 5 & 7 \end{pmatrix} \). (2022)

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

Solution

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

Correct Answer: A — 1

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Q. Find the determinant of \( G = \begin{pmatrix} 2 & 3 \\ 5 & 7 \end{pmatrix} \). (2021)

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

Solution

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

Correct Answer: A — 1

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Q. Find the determinant of \( G = \begin{pmatrix} 4 & 2 \\ 3 & 1 \end{pmatrix} \). (2020)

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

Solution

The determinant is \( 4*1 - 2*3 = 4 - 6 = -2 \).

Correct Answer: A — -2

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Q. Find the dimensions of a box with a square base that maximizes volume given a surface area of 600 sq. units. (2020)

A. 10, 10
B. 15, 15
C. 12, 12
D. 20, 20

Solution

Let x be the side of the base and h the height. The surface area constraint gives 2x^2 + 4xh = 600. Max volume occurs at x = 12.

Correct Answer: C — 12, 12

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Q. Find the dimensions of a rectangle with a fixed area of 50 m^2 that minimizes the perimeter. (2021)

A. 5, 10
B. 7, 7.14
C. 8, 6.25
D. 10, 5

Solution

For a fixed area, the minimum perimeter occurs when the rectangle is a square. Thus, dimensions are approximately 7 m by 7.14 m.

Correct Answer: B — 7, 7.14

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Q. Find the dimensions of a rectangle with a fixed area of 50 square units that minimizes the perimeter. (2022) 2022

A. 5, 10
B. 7, 7.14
C. 10, 5
D. 8, 6.25

Solution

For minimum perimeter, the rectangle should be a square. Thus, side = sqrt(50) ≈ 7.07.

Correct Answer: B — 7, 7.14

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Q. Find the dimensions of a rectangle with a fixed area of 50 square units that minimizes the perimeter. (2020)

A. 5, 10
B. 7, 7
C. 10, 5
D. 8, 6.25

Solution

For a fixed area, the perimeter is minimized when the rectangle is a square. Thus, side = √50.

Correct Answer: B — 7, 7

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Q. Find the distance between the parallel planes 2x + 3y + 4z = 5 and 2x + 3y + 4z = 10. (2023)

A. 5/√29
B. 10/√29
C. 15/√29
D. 20/√29

Solution

Distance = |d1 - d2| / √(a² + b² + c²) = |5 - 10| / √(2² + 3² + 4²) = 5 / √29.

Correct Answer: B — 10/√29

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Q. Find the distance between the parallel planes 2x + 3y + z = 5 and 2x + 3y + z = 10. (2022)

A. 5
B. 10
C. √14
D. √10

Solution

Distance = |d1 - d2| / √(A² + B² + C²) = |5 - 10| / √(2² + 3² + 1²) = 5 / √14.

Correct Answer: A — 5

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Q. Find 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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Mathematics

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