Q. Calculate the coefficient of x^2 in the expansion of (x + 1/x)^6. (2019)
Solution
The coefficient of x^2 is given by 6C2 * (1)^4 = 15.
Correct Answer: A — 15
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Q. Calculate the coefficient of x^2 in the expansion of (x + 4)^6.
-
A.
96
-
B.
144
-
C.
192
-
D.
256
Solution
The coefficient of x^2 is given by C(6,2) * 4^4 = 15 * 256 = 3840.
Correct Answer: A — 96
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Q. Calculate the coefficient of x^3 in the expansion of (x + 1/2)^6.
Solution
The coefficient of x^3 is given by C(6,3) * (1/2)^3 = 20 * 1/8 = 2.5.
Correct Answer: A — 20
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Q. Calculate the coefficient of x^3 in the expansion of (x - 1)^5.
Solution
The coefficient of x^3 is C(5,3) * (-1)^2 = 10.
Correct Answer: B — 10
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Q. Calculate the coefficient of x^4 in the expansion of (3x - 2)^6.
-
A.
540
-
B.
720
-
C.
810
-
D.
960
Solution
The coefficient of x^4 is C(6,4) * (3)^4 * (-2)^2 = 15 * 81 * 4 = 4860.
Correct Answer: A — 540
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Q. Calculate the coefficient of x^4 in the expansion of (x + 1/2)^6. (2021)
-
A.
15/8
-
B.
45/8
-
C.
5/8
-
D.
1/8
Solution
The coefficient of x^4 is C(6,4)(1/2)^2 = 15 * 1/4 = 15/4.
Correct Answer: B — 45/8
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Q. Calculate the coefficient of x^4 in the expansion of (x + 2)^6.
Solution
The coefficient of x^4 is C(6,4) * 2^2 = 15 * 4 = 60.
Correct Answer: C — 90
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Q. Calculate the coefficient of x^4 in the expansion of (x + 3)^6. (2021)
-
A.
54
-
B.
81
-
C.
108
-
D.
729
Solution
The coefficient of x^4 is C(6,4)(3)^2 = 15 * 9 = 135.
Correct Answer: C — 108
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Q. Calculate the coefficient of x^4 in the expansion of (x + 5)^6.
-
A.
150
-
B.
600
-
C.
750
-
D.
1000
Solution
The coefficient of x^4 is C(6,4) * 5^2 = 15 * 25 = 375.
Correct Answer: C — 750
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Q. Calculate the coefficient of x^5 in the expansion of (x + 2)^7.
Solution
The coefficient of x^5 is C(7,5) * (2)^2 = 21 * 4 = 84.
Correct Answer: C — 63
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Q. Calculate the coefficient of x^5 in the expansion of (x - 3)^7. (2021)
-
A.
-189
-
B.
-243
-
C.
-126
-
D.
-21
Solution
The coefficient of x^5 is C(7,5) * (-3)^2 = 21 * 9 = -189.
Correct Answer: A — -189
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Q. Calculate the derivative of f(x) = 5x^5. (2016)
-
A.
25x^4
-
B.
5x^4
-
C.
20x^4
-
D.
10x^4
Solution
The derivative f'(x) = d/dx(5x^5) = 25x^4.
Correct Answer: A — 25x^4
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Q. Calculate the derivative of f(x) = e^(2x).
-
A.
2e^(2x)
-
B.
e^(2x)
-
C.
2xe^(2x)
-
D.
e^(x)
Solution
Using the chain rule, f'(x) = d/dx(e^(2x)) = 2e^(2x).
Correct Answer: A — 2e^(2x)
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Q. Calculate the derivative of f(x) = ln(x^2 + 1).
-
A.
2x/(x^2 + 1)
-
B.
1/(x^2 + 1)
-
C.
2/(x^2 + 1)
-
D.
x/(x^2 + 1)
Solution
Using the chain rule, f'(x) = d/dx(ln(x^2 + 1)) = (2x)/(x^2 + 1).
Correct Answer: A — 2x/(x^2 + 1)
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Q. Calculate the derivative of f(x) = x^2 * e^x.
-
A.
(2x + x^2)e^x
-
B.
2xe^x
-
C.
x^2e^x
-
D.
(x^2 + 2x)e^x
Solution
Using the product rule, f'(x) = d/dx(x^2 * e^x) = (x^2 + 2x)e^x.
Correct Answer: D — (x^2 + 2x)e^x
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Q. Calculate the derivative of f(x) = x^2 * e^x. (2023) 2023
-
A.
e^x(2x + 1)
-
B.
e^x(2x - 1)
-
C.
2xe^x
-
D.
x^2e^x
Solution
Using the product rule, f'(x) = d/dx(x^2 * e^x) = e^x(2x + 1).
Correct Answer: A — e^x(2x + 1)
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Q. Calculate the determinant of D = [[2, 3, 1], [1, 0, 2], [4, 1, 0]]. (2020)
Solution
Det(D) = 2(0*0 - 2*1) - 3(1*0 - 2*4) + 1(1*1 - 0*4) = 2(0 - 2) - 3(0 - 8) + 1(1) = -4 + 24 + 1 = 21.
Correct Answer: A — -10
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Q. Calculate the determinant of D = [[3, 2, 1], [1, 0, 2], [0, 1, 3]]. (2023)
Solution
Det(D) = 3(0*3 - 2*1) - 2(1*3 - 0*2) + 1(1*1 - 0*0) = 3(0 - 2) - 2(3) + 1(1) = -6 - 6 + 1 = -11.
Correct Answer: A — 1
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Q. Calculate the determinant of D = [[3, 2, 1], [1, 0, 2], [2, 1, 3]]. (2020)
Solution
Det(D) = 3(0*3 - 2*1) - 2(1*3 - 2*2) + 1(1*1 - 0*2) = 3(0 - 2) - 2(3 - 4) + 1(1) = -6 + 2 + 1 = -3.
Correct Answer: A — 1
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Q. Calculate the determinant of D = [[4, 2], [1, 3]]. (2020)
Solution
Determinant of D = (4*3) - (2*1) = 12 - 2 = 10.
Correct Answer: A — 10
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Q. Calculate the determinant of D = [[4, 2], [3, 1]]. (2020)
Solution
Determinant of D = (4*1) - (2*3) = 4 - 6 = -2.
Correct Answer: A — -2
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Q. Calculate the determinant of D = [[4, 5, 6], [7, 8, 9], [1, 2, 3]]. (2020)
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)
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)
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)
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)
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)
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)
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)
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)
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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