Q. Find the derivative of f(x) = x^5 - 3x + 2.
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A.
5x^4 - 3
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B.
5x^4 + 3
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C.
4x^3 - 3
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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)
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A.
5x^4 - 9x^2
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B.
5x^4 + 9x^2
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C.
3x^2 - 9x
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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 derivative of g(x) = sin(x) + cos(x). (2020)
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A.
cos(x) - sin(x)
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B.
-sin(x) - cos(x)
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C.
sin(x) + cos(x)
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D.
-cos(x) + sin(x)
Solution
Using the derivatives of sine and cosine, g'(x) = cos(x) - sin(x).
Correct Answer: A — cos(x) - sin(x)
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Q. Find the determinant of E = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]. (2019)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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 \( D = \begin{pmatrix} 1 & 0 & 2 \\ 0 & 1 & 3 \\ 0 & 0 & 1 \end{pmatrix} \).
Solution
The determinant of an upper triangular matrix is the product of its diagonal elements, which is 1.
Correct Answer: B — 1
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Q. Find the determinant of the matrix \( E = \begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix} \). (2021)
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 \( I = \begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix} \).
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)
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 \( \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \).
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 \( \begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix} \).
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 \( \begin{pmatrix} 2 & 3 & 1 \\ 1 & 0 & 2 \\ 4 & 1 & 0 \end{pmatrix} \).
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 \( \begin{pmatrix} 2 & 3 & 1 \\ 1 & 0 & 4 \\ 5 & 2 & 1 \end{pmatrix} \).
Solution
The determinant evaluates to 0.
Correct Answer: A — -1
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Q. Find the determinant of the matrix \( \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix} \).
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 \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \).
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A.
ad - bc
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B.
bc - ad
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C.
a + b + c + d
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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 |.
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 |.
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 |.
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)
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)
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)
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A.
10, 10
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B.
15, 15
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C.
12, 12
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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)
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A.
5, 10
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B.
7, 7.14
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C.
8, 6.25
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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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