Q. Find the constant term in the expansion of (x - 2/x)^6. (2022)
Solution
The constant term occurs when the power of x is zero. Setting 6 - 2k = 0 gives k = 3. The term is C(6,3)(-2)^3 = -64.
Correct Answer: A — -64
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Q. Find the coordinates of the centroid of the triangle with vertices (0, 0), (6, 0), and (0, 8). (2022)
-
A.
(2, 2)
-
B.
(2, 3)
-
C.
(3, 2)
-
D.
(4, 4)
Solution
Centroid = ((0+6+0)/3, (0+0+8)/3) = (2, 2).
Correct Answer: A — (2, 2)
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Q. Find the coordinates of the centroid of the triangle with vertices A(0, 0, 0), B(4, 0, 0), C(0, 3, 0). (2021)
-
A.
(4/3, 1, 0)
-
B.
(2, 1, 0)
-
C.
(1, 1, 0)
-
D.
(0, 0, 0)
Solution
Centroid G = ((0+4+0)/3, (0+0+3)/3, (0+0+0)/3) = (4/3, 1, 0).
Correct Answer: B — (2, 1, 0)
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Q. Find the coordinates of the centroid of the triangle with vertices at (0, 0), (6, 0), and (0, 8).
-
A.
(2, 2)
-
B.
(2, 3)
-
C.
(3, 2)
-
D.
(4, 4)
Solution
Centroid = ((0+6+0)/3, (0+0+8)/3) = (2, 2).
Correct Answer: A — (2, 2)
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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 = [[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, 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 \( E = \begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix} \). (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 \( G = \begin{pmatrix} 2 & 3 \\ 5 & 7 \end{pmatrix} \). (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)
-
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)
Solution
Distance = |d1 - d2| / √(A² + B² + C²) = |5 - 10| / √(2² + 3² + 1²) = 5 / √14.
Correct Answer: A — 5
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