Algebra
Q. Find the coefficient of x^3 in the expansion of (x + 2)^6.
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A.
80
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B.
120
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C.
160
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D.
240
Solution
The coefficient of x^3 is C(6,3) * (2)^3 = 20 * 8 = 160.
Correct Answer: B — 120
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Q. Find 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: A — -10
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Q. Find the coefficient of x^3 in the expansion of (x - 1)^6.
-
A.
-20
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B.
-15
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C.
-10
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D.
-6
Solution
The coefficient of x^3 is C(6,3) * (-1)^3 = 20 * (-1) = -20.
Correct Answer: A — -20
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Q. Find the coefficient of x^3 in the expansion of (x - 3)^5.
-
A.
-135
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B.
-90
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C.
-60
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D.
-45
Solution
The coefficient of x^3 is C(5,3) * (-3)^2 = 10 * 9 = -90.
Correct Answer: A — -135
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Q. Find the coefficient of x^4 in the expansion of (3x - 2)^6.
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A.
540
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B.
720
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C.
810
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D.
960
Solution
Using the binomial theorem, the coefficient of x^4 in (3x - 2)^6 is given by 6C4 * (3)^4 * (-2)^2 = 15 * 81 * 4 = 4860.
Correct Answer: C — 810
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Q. Find the coefficient of x^5 in the expansion of (x + 1)^8.
Solution
The coefficient of x^5 is C(8,5) = 56.
Correct Answer: B — 70
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Q. Find the coefficient of x^5 in the expansion of (x + 3)^8.
-
A.
56
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B.
168
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C.
336
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D.
672
Solution
The coefficient of x^5 is C(8,5) * (3)^3 = 56 * 27 = 1512.
Correct Answer: B — 168
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Q. Find the coefficient of x^5 in the expansion of (x - 3)^7.
-
A.
-1890
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B.
-2187
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C.
-2401
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D.
-2430
Solution
The coefficient of x^5 is C(7,5) * (-3)^2 = 21 * 9 = -1890.
Correct Answer: A — -1890
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Q. Find the conjugate of the complex number z = 5 - 6i.
-
A.
5 + 6i
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B.
5 - 6i
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C.
-5 + 6i
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D.
-5 - 6i
Solution
The conjugate of z = 5 - 6i is z̅ = 5 + 6i.
Correct Answer: A — 5 + 6i
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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 \( I = \begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix} \).
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 \( \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \).
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 \( \begin{pmatrix} 2 & 3 & 1 \\ 1 & 0 & 4 \\ 5 & 2 & 1 \end{pmatrix} \).
Solution
The determinant evaluates to 0.
Correct Answer: A — -1
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Q. Find the determinant of the matrix \( \begin{pmatrix} 2 & 3 & 1 \\ 1 & 0 & 2 \\ 4 & 1 & 0 \end{pmatrix} \).
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 \( \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix} \).
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 eigenvalues of the matrix A = [[2, 1], [1, 2]].
-
A.
1, 3
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B.
2, 2
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C.
3, 1
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D.
0, 4
Solution
The characteristic polynomial is det(A - λI) = (2-λ)(2-λ) - 1 = λ^2 - 4λ + 3 = 0, giving eigenvalues 1 and 3.
Correct Answer: A — 1, 3
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Q. Find the inverse of the matrix A = [[1, 2], [3, 4]].
-
A.
[[4, -2]; [-3, 1]]
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B.
[[1, -2]; [-3, 4]]
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C.
[[-2, 1]; [3, 4]]
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D.
[[2, -1]; [-1.5, 0.5]]
Solution
The inverse of A is (1/det(A)) * adj(A) = (1/-2) * [[4, -2], [-3, 1]] = [[-2, 1]; [1.5, -0.5]].
Correct Answer: A — [[4, -2]; [-3, 1]]
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Q. Find the real part of the complex number z = 2 + 3i.
Solution
The real part of z = 2 + 3i is 2.
Correct Answer: A — 2
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Q. Find the real part of the complex number z = 2e^(iπ/3).
Solution
The real part is 2 * cos(π/3) = 2 * 1/2 = 1.
Correct Answer: B — 2
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Q. Find the real part of the complex number z = 3 + 4i.
Solution
The real part of z is 3.
Correct Answer: A — 3
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Q. Find the real part of the complex number z = 4 + 3i.
Solution
The real part of z = 4 + 3i is 4.
Correct Answer: A — 4
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Q. Find the real part of the complex number z = 4(cos(π/3) + i sin(π/3)).
Solution
The real part is 4 * cos(π/3) = 4 * 1/2 = 2.
Correct Answer: A — 2
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Q. Find the real part of the complex number z = 5 - 2i.
Solution
The real part of z = 5 - 2i is 5.
Correct Answer: A — 5
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Q. Find the solution for the inequality 2(x - 1) ≥ 3.
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A.
x ≥ 2.5
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B.
x ≤ 2.5
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C.
x ≥ 1.5
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D.
x ≤ 1.5
Solution
2(x - 1) ≥ 3 => x - 1 ≥ 1.5 => x ≥ 2.5.
Correct Answer: C — x ≥ 1.5
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Q. Find the solution for the inequality 2x + 3 ≤ 5.
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A.
x ≤ 1
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B.
x ≥ 1
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C.
x ≤ 2
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D.
x ≥ 2
Solution
2x + 3 ≤ 5 => 2x ≤ 2 => x ≤ 1.
Correct Answer: A — x ≤ 1
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