Q. What is the characteristic polynomial of the matrix A = [[1, 2], [3, 4]]?
A.
λ^2 - 5λ - 2
B.
λ^2 - 5λ + 2
C.
λ^2 + 5λ + 2
D.
λ^2 + 5λ - 2
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Solution
The characteristic polynomial is det(A - λI) = det([[1-λ, 2], [3, 4-λ]]) = (1-λ)(4-λ) - 6 = λ^2 - 5λ + 2.
Correct Answer: B — λ^2 - 5λ + 2
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Q. What is the characteristic polynomial of the matrix A = [[2, 1], [1, 2]]?
A.
λ^2 - 3λ + 3
B.
λ^2 - 3λ + 1
C.
λ^2 - 5λ + 2
D.
λ^2 - 2λ + 1
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Solution
The characteristic polynomial is given by det(A - λI) = det([[2-λ, 1], [1, 2-λ]]) = (2-λ)(2-λ) - 1 = λ^2 - 3λ + 3.
Correct Answer: B — λ^2 - 3λ + 1
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Q. What is the coefficient of x^0 in the expansion of (3x - 4)^5?
A.
-1024
B.
243
C.
-243
D.
1024
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Solution
The coefficient of x^0 is (-4)^5 = -1024.
Correct Answer: A — -1024
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Q. What is the coefficient of x^0 in the expansion of (x + 5)^3?
A.
125
B.
100
C.
75
D.
50
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Solution
The coefficient of x^0 is C(3,0) * 5^3 = 1 * 125 = 125.
Correct Answer: A — 125
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Q. What is the coefficient of x^0 in the expansion of (x + 5)^5?
A.
3125
B.
625
C.
125
D.
25
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Solution
The coefficient of x^0 is C(5,0) * 5^5 = 1 * 3125 = 3125.
Correct Answer: A — 3125
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Q. What is the coefficient of x^2 in the expansion of (3x - 4)^4?
A.
144
B.
216
C.
432
D.
576
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Solution
The coefficient of x^2 is C(4,2) * (3)^2 * (-4)^2 = 6 * 9 * 16 = 864.
Correct Answer: C — 432
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Q. What is the coefficient of x^2 in the expansion of (x - 4)^6?
A.
240
B.
360
C.
480
D.
720
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Solution
Using the binomial theorem, the coefficient of x^2 in (a + b)^n is given by nCk * a^(n-k) * b^k. Here, n=6, a=x, b=-4, and k=2. Thus, the coefficient is 6C2 * (1)^4 * (-4)^2 = 15 * 16 = 240.
Correct Answer: B — 360
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Q. What is the coefficient of x^3 in the expansion of (2x + 3)^5?
A.
90
B.
100
C.
120
D.
150
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Solution
Using the binomial theorem, the coefficient of x^3 is given by C(5,3) * (2)^3 * (3)^(5-3) = 10 * 8 * 9 = 720.
Correct Answer: A — 90
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Q. What is the coefficient of x^4 in the expansion of (2x + 3)^6?
A.
540
B.
720
C.
810
D.
960
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Solution
The coefficient of x^4 is C(6,4) * (2)^4 * (3)^(6-4) = 15 * 16 * 9 = 2160.
Correct Answer: C — 810
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Q. What is the coefficient of x^4 in the expansion of (2x - 3)^5?
A.
240
B.
300
C.
360
D.
420
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Solution
The coefficient of x^4 is C(5, 4) * (2)^4 * (-3)^1 = 5 * 16 * (-3) = -240.
Correct Answer: C — 360
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Q. What is the coefficient of x^4 in the expansion of (x + 1)^6?
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Solution
The coefficient of x^4 is C(6,4) = 15.
Correct Answer: D — 35
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Q. What is the coefficient of x^5 in the expansion of (x + 1/2)^10?
A.
252
B.
210
C.
120
D.
300
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Solution
The coefficient of x^5 is C(10,5) * (1/2)^5 = 252 * 1/32 = 7.875.
Correct Answer: A — 252
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Q. What is the coefficient of x^5 in the expansion of (x + 2)^7?
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Solution
The coefficient of x^5 is C(7,5)(2)^2 = 21 * 4 = 84.
Correct Answer: C — 35
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Q. What is the coefficient of x^5 in the expansion of (x + 2)^8?
A.
672
B.
1280
C.
960
D.
720
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Solution
The coefficient of x^5 is C(8,5) * (2)^3 = 56 * 8 = 448.
Correct Answer: A — 672
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Q. What is the coefficient of x^5 in the expansion of (x + 3)^8?
A.
1680
B.
1440
C.
1260
D.
1080
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Solution
Using the binomial theorem, the coefficient of x^5 in (x + 3)^8 is given by 8C5 * (3)^3 = 56 * 27 = 1512.
Correct Answer: A — 1680
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Q. What is the coefficient of x^6 in the expansion of (x + 5)^8?
A.
6720
B.
5600
C.
4480
D.
3360
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Solution
The coefficient of x^6 is C(8,6) * (5)^2 = 28 * 25 = 700.
Correct Answer: A — 6720
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Q. What is the common ratio of the geometric series 4, 12, 36, ...?
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Solution
The common ratio r = 12/4 = 3.
Correct Answer: A — 3
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Q. What is the conjugate of the complex number z = 2 + 5i?
A.
2 - 5i
B.
2 + 5i
C.
-2 + 5i
D.
-2 - 5i
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Solution
The conjugate of z = 2 + 5i is z̅ = 2 - 5i.
Correct Answer: A — 2 - 5i
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Q. What is the conjugate of the complex number z = 5 - 2i?
A.
5 + 2i
B.
5 - 2i
C.
2 - 5i
D.
2 + 5i
Show solution
Solution
The conjugate of z = 5 - 2i is 5 + 2i.
Correct Answer: A — 5 + 2i
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Q. What is the conjugate of the complex number z = 5 - 6i?
A.
5 + 6i
B.
5 - 6i
C.
6 - 5i
D.
6 + 5i
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Solution
The conjugate of z = 5 - 6i is 5 + 6i.
Correct Answer: A — 5 + 6i
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Q. What is the derivative of sin^(-1)(x)?
A.
1/√(1-x^2)
B.
-1/√(1-x^2)
C.
1/x
D.
0
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Solution
The derivative of sin^(-1)(x) is 1/√(1-x^2)
Correct Answer: A — 1/√(1-x^2)
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Q. What is the derivative of y = sin^(-1)(x)?
A.
1/√(1-x^2)
B.
1/(1+x^2)
C.
1/(1-x^2)
D.
√(1-x^2)
Show solution
Solution
The derivative of y = sin^(-1)(x) is 1/√(1-x^2)
Correct Answer: A — 1/√(1-x^2)
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Q. What is the determinant of the matrix [[0, 1], [1, 0]]?
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Solution
The determinant is calculated as (0*0) - (1*1) = 0 - 1 = -1.
Correct Answer: C — -1
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Q. What is the determinant of the matrix \( E = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \)?
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Solution
The determinant is calculated as \( 1*4 - 2*3 = 4 - 6 = -2 \).
Correct Answer: A — -2
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Q. What is the determinant of the matrix \( H = \begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix} \)?
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Solution
The determinant is 0 because the rows are linearly dependent.
Correct Answer: A — 0
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Q. What is the determinant of the matrix \( \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \)?
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Solution
The determinant of the identity matrix is 1.
Correct Answer: B — 1
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Q. What is the determinant of the matrix \( \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} \)?
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Solution
The determinant is calculated as (1*1) - (1*1) = 1 - 1 = 0.
Correct Answer: A — 0
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Q. What is the determinant of the matrix \( \begin{pmatrix} 1 & 2 & 1 \\ 0 & 1 & 0 \\ 2 & 3 & 1 \end{pmatrix} \)?
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Solution
The determinant is calculated as \( 1(1*1 - 0*3) - 2(0*1 - 0*2) + 1(0*3 - 1*2) = 1 - 0 - 2 = -1 \).
Correct Answer: B — 1
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Q. What is the determinant of the matrix \( \begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix} \)?
Show solution
Solution
The determinant is 0 because the rows are linearly dependent.
Correct Answer: A — 0
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Q. What is the determinant of the matrix \( \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \)?
Show solution
Solution
The determinant is calculated as \( 1*4 - 2*3 = 4 - 6 = -2 \).
Correct Answer: A — -2
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