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Q. Calculate the determinant of D = [[4, 5, 6], [7, 8, 9], [1, 2, 3]]. (2020)

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

Solution

Determinant of D = 4(8*3 - 9*2) - 5(7*3 - 9*1) + 6(7*2 - 8*1) = 0.

Correct Answer: A — 0

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Q. Calculate the determinant of F = [[1, 2, 3], [0, 1, 4], [5, 6, 0]]. (2023)

A. -14
B. 14
C. 0
D. 10

Solution

Using the determinant formula, det(F) = 1(1*0 - 4*6) - 2(0*0 - 4*5) + 3(0*6 - 1*5) = -24 + 40 - 15 = 1.

Correct Answer: A — -14

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Q. Calculate the determinant of G = [[1, 1, 1], [1, 2, 3], [1, 3, 6]]. (2022)

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

Solution

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

Correct Answer: A — 0

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Q. Calculate the determinant of G = [[1, 2, 1], [0, 1, 0], [2, 3, 1]]. (2023)

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

Solution

Using cofactor expansion, det(G) = 1(1*1 - 0*3) - 2(0*1 - 0*2) + 1(0*3 - 1*2) = 1 - 0 - 2 = -1.

Correct Answer: A — -1

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Q. Calculate the determinant of G = [[1, 2, 1], [0, 1, 4], [1, 0, 0]]. (2021)

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

Solution

Using the determinant formula for 3x3 matrices, det(G) = 1(1*0 - 4*0) - 2(0*0 - 4*1) + 1(0*0 - 1*1) = 0 + 8 - 1 = 7.

Correct Answer: A — -2

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Q. Calculate the determinant of H = [[1, 2, 1], [0, 1, 2], [1, 0, 1]]. (2020)

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

Solution

Det(H) = 1(1*1 - 2*0) - 2(0*1 - 2*1) + 1(0*0 - 1*1) = 1(1) - 2(-2) + 1(-1) = 1 + 4 - 1 = 4.

Correct Answer: A — 0

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Q. Calculate the determinant of H = [[1, 2, 1], [0, 1, 3], [1, 0, 1]]. (2023)

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

Solution

Det(H) = 1(1*1 - 3*0) - 2(0*1 - 3*1) + 1(0*0 - 1*1) = 1(1) - 2(-3) + 1(-1) = 1 + 6 - 1 = 6.

Correct Answer: A — -1

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Q. Calculate the determinant of H = [[1, 2, 1], [0, 1, 3], [2, 1, 0]]. (2020)

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

Solution

Det(H) = 1(1*0 - 3*1) - 2(0*0 - 3*2) + 1(0*1 - 1*2) = 1(0 - 3) - 2(0 - 6) + 1(0 - 2) = -3 + 12 - 2 = 7.

Correct Answer: A — -5

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Q. Calculate the determinant of H = [[1, 2, 3], [0, 1, 4], [5, 6, 0]]. (2021)

A. -14
B. 14
C. 0
D. 10

Solution

Det(H) = 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. Calculate the determinant of H = [[3, 2], [1, 4]]. (2021)

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

Solution

Determinant of H = (3*4) - (2*1) = 12 - 2 = 10.

Correct Answer: A — 10

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Q. Calculate the determinant of H = [[5, 4], [2, 3]]. (2021)

A. 7
B. 8
C. 9
D. 10

Solution

Det(H) = (5*3) - (4*2) = 15 - 8 = 7.

Correct Answer: B — 8

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Q. Calculate the determinant of I = [[1, 2, 1], [0, 1, 2], [1, 0, 1]]. (2021)

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

Solution

Det(I) = 1(1*1 - 2*0) - 2(0*1 - 2*1) + 1(0*0 - 1*1) = 1(1) - 2(-2) + 1(-1) = 1 + 4 - 1 = 4.

Correct Answer: B — 1

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Q. Calculate the determinant of I = [[2, 3, 1], [1, 0, 2], [0, 1, 3]]. (2015)

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

Solution

Det(I) = 2(0*3 - 2*1) - 3(1*3 - 2*0) + 1(1*1 - 0*0) = 2(0 - 2) - 3(3) + 1(1) = -4 - 9 + 1 = -12.

Correct Answer: C — 3

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Q. Calculate the determinant of J = [[1, 2, 1], [0, 1, 2], [1, 0, 1]]. (2023)

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

Solution

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

Correct Answer: C — 2

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Q. Calculate the determinant of J = [[5, 6], [7, 8]]. (2014)

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

Solution

Det(J) = (5*8) - (6*7) = 40 - 42 = -2.

Correct Answer: A — -2

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Q. Calculate the determinant of the matrix F = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]. (2022)

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

Solution

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

Correct Answer: A — 0

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Q. Calculate the determinant of the matrix \( C = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} \). (2020)

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

Solution

The determinant is \( 5*8 - 6*7 = 40 - 42 = -2 \).

Correct Answer: A — -2

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

A. 5
B. 10
C. 1
D. 8

Solution

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

Correct Answer: A — 5

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Q. Calculate the determinant of the matrix \( H = \begin{pmatrix} 1 & 0 & 2 \\ 0 & 1 & 3 \\ 0 & 0 & 1 \end{pmatrix} \). (2020)

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

Solution

The determinant of an upper triangular matrix is the product of its diagonal elements: \( 1*1*1 = 1 \).

Correct Answer: A — 1

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Q. Calculate the determinant of \( D = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \). (2021)

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. Calculate the determinant of \( D = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 6 \end{pmatrix} \). (2021)

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. Calculate the determinant of \( I = \begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix} \). (2022)

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

Solution

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

Correct Answer: B — 8

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Q. Calculate the distance from the point (1, 2, 3) to the origin (0, 0, 0). (2021)

A. √14
B. √6
C. √9
D. √12

Solution

Distance = √[(1-0)² + (2-0)² + (3-0)²] = √[1 + 4 + 9] = √14.

Correct Answer: A — √14

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Q. Calculate the distance from the point P(1, 2, 3) to the origin O(0, 0, 0). (2023)

A. 3
B. √14
C. √6
D. √9

Solution

Distance = √[(1-0)² + (2-0)² + (3-0)²] = √[1 + 4 + 9] = √14.

Correct Answer: B — √14

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Q. Calculate the gravitational potential energy of a 2 kg mass at a height of 5 m. (g = 9.8 m/s²)

A. 98 J
B. 19.6 J
C. 39.2 J
D. 49 J

Solution

Potential Energy (PE) = m * g * h = 2 kg * 9.8 m/s² * 5 m = 98 J

Correct Answer: C — 39.2 J

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Q. Calculate the limit: lim (x -> 0) (tan(5x)/x) (2022)

A. 0
B. 1
C. 5
D. Undefined

Solution

Using the limit property lim (x -> 0) (tan(kx)/x) = k, we have lim (x -> 0) (tan(5x)/x) = 5.

Correct Answer: C — 5

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Q. Calculate the limit: lim (x -> 1) (x^4 - 1)/(x - 1) (2021)

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

Solution

Factoring gives (x - 1)(x^3 + x^2 + x + 1)/(x - 1). Canceling gives lim (x -> 1) (x^3 + x^2 + x + 1) = 4.

Correct Answer: D — 4

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Q. Calculate the limit: lim (x -> ∞) (5x^2 + 3)/(2x^2 + 1) (2023)

A. 0
B. 5/2
C. 2/5
D. ∞

Solution

Dividing the numerator and denominator by x^2, we get lim (x -> ∞) (5 + 3/x^2)/(2 + 1/x^2) = 5/2.

Correct Answer: B — 5/2

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Q. Calculate the perimeter of a square with side length 4 cm. (2015)

A. 16 cm
B. 12 cm
C. 8 cm
D. 20 cm

Solution

Perimeter = 4 × side = 4 × 4 cm = 16 cm.

Correct Answer: A — 16 cm

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Q. Calculate the perimeter of a square with side length 6 cm. (2015)

A. 24 cm
B. 20 cm
C. 18 cm
D. 30 cm

Solution

Perimeter = 4 × side = 4 × 6 = 24 cm.

Correct Answer: A — 24 cm

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