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

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

Solution

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

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 the matrix \( \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \).

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

A. 5
B. 10
C. 7
D. 6

Solution

The determinant is calculated as \( 2*4 - 1*3 = 8 - 3 = 5 \).

Correct Answer: A — 5

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Q. Find the determinant of the matrix \( \begin{pmatrix} 2 & 3 & 1 \\ 1 & 0 & 2 \\ 4 & 1 & 0 \end{pmatrix} \).

A. -10
B. 10
C. 0
D. 5

Solution

Using the determinant formula, we find it equals 10.

Correct Answer: A — -10

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Q. Find the determinant of the matrix \( \begin{pmatrix} 2 & 3 & 1 \\ 1 & 0 & 4 \\ 5 & 2 & 1 \end{pmatrix} \).

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

Solution

The determinant evaluates to 0.

Correct Answer: A — -1

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

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

Solution

The determinant is \( 2*4 - 3*1 = 8 - 3 = 5 \).

Correct Answer: A — 5

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Q. Find the determinant of the matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \).

A. ad - bc
B. bc - ad
C. a + b + c + d
D. a^2 + b^2

Solution

The determinant is given by the formula \( ad - bc \).

Correct Answer: A — ad - bc

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Q. Find the determinant of the matrix | 1 0 0 | | 0 1 0 | | 0 0 1 |.

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

Solution

This is the identity matrix, and its determinant is 1.

Correct Answer: B — 1

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Q. Find the determinant of the matrix | 1 2 3 | | 0 1 4 | | 5 6 0 |.

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

Solution

The determinant evaluates to 0 as the third row can be expressed as a linear combination of the first two.

Correct Answer: A — -12

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Q. Find the determinant of the matrix: | 1 2 | | 3 5 |.

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

Solution

det = (1*5) - (2*3) = 5 - 6 = -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 directrix of the parabola y^2 = -8x.

A. x = 2
B. x = -2
C. x = 4
D. x = -4

Solution

For the parabola y^2 = 4px, here 4p = -8, so p = -2. The directrix is given by x = -p, which is x = 2.

Correct Answer: B — x = -2

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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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Q. Find the distance between the points (-1, -1) and (2, 2).

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

Solution

Using the distance formula: d = √[(2 - (-1))² + (2 - (-1))²] = √[9 + 9] = √18 = 3√2.

Correct Answer: C — 5

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Q. Find the distance between the points (-2, -3) and (4, 5).

A. 8
B. 7
C. 6
D. 9

Solution

Using the distance formula: d = √[(4 - (-2))² + (5 - (-3))²] = √[(4 + 2)² + (5 + 3)²] = √[36 + 64] = √100 = 10.

Correct Answer: B — 7

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Q. Find the distance between the points (0, 0) and (x, y) where x = 6 and y = 8.

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

Solution

Using the distance formula: d = √[(6 - 0)² + (8 - 0)²] = √[36 + 64] = √100 = 10.

Correct Answer: A — 10

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Q. Find the distance between the points (1, 1) and (4, 5). (2023)

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

Solution

Using the distance formula: d = √[(4 - 1)² + (5 - 1)²] = √[9 + 16] = √25 = 5.

Correct Answer: A — 5

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Q. Find the distance between the points (1, 2) and (4, 6).

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

Solution

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

Correct Answer: A — 5

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Q. Find the distance between the points (3, 3) and (3, 7).

A. 4
B. 5
C. 6
D. 7

Solution

Using the distance formula: d = √[(3 - 3)² + (7 - 3)²] = √[0 + 16] = √16 = 4.

Correct Answer: A — 4

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Q. Find the distance between the points (3, 4) and (7, 1).

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

Solution

Distance = √[(7-3)² + (1-4)²] = √[4 + 9] = √13 ≈ 3.6, closest option is 4.

Correct Answer: A — 5

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Q. Find the distance between the points (3, 7) and (3, 1).

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

Solution

Using the distance formula: d = √((3 - 3)² + (1 - 7)²) = √(0 + 36) = √36 = 6.

Correct Answer: A — 6

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Q. Find the distance between the points (5, 5) and (5, 1).

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

Solution

Using the distance formula: d = √[(5 - 5)² + (1 - 5)²] = √[0 + 16] = √16 = 4.

Correct Answer: A — 4

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Major Competitive Exams

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