Algebra
Q. Evaluate the determinant \( \begin{vmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 6 \end{vmatrix} \)
Solution
The determinant is 0 because the first column is repeated.
Correct Answer: A — 0
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Q. Evaluate the determinant \( \begin{vmatrix} 1 & 2 & 1 \\ 2 & 3 & 1 \\ 3 & 4 & 1 \end{vmatrix} \)
Solution
The determinant is 0 because the rows are linearly dependent.
Correct Answer: A — 0
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Q. Evaluate the determinant \( \begin{vmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{vmatrix} \).
Solution
The determinant evaluates to -12.
Correct Answer: A — -12
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Q. Evaluate the determinant \( \begin{vmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{vmatrix} \)
Solution
The determinant is 0 because the rows are linearly dependent.
Correct Answer: A — 0
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Q. Evaluate the determinant \( \begin{vmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{vmatrix} \).
Solution
The determinant is 0 because the rows are linearly dependent.
Correct Answer: A — 0
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Q. Evaluate the determinant \( \begin{vmatrix} 3 & 1 & 2 \\ 1 & 2 & 3 \\ 2 & 3 & 1 \end{vmatrix} \).
Solution
The determinant is 0 because the rows are linearly dependent.
Correct Answer: A — 0
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Q. Evaluate the determinant \( \det \begin{pmatrix} 2 & 1 & 3 \\ 1 & 0 & 2 \\ 3 & 4 & 1 \end{pmatrix} \).
Solution
The determinant is calculated as \( 2(0*1 - 2*4) - 1(1*1 - 2*3) + 3(1*4 - 0*3) = -10 \).
Correct Answer: A — -10
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Q. Evaluate the determinant \( |C| \) where \( C = \begin{pmatrix} 2 & 1 & 3 \\ 1 & 0 & 2 \\ 3 & 2 & 1 \end{pmatrix} \).
-
A.
-12
-
B.
-10
-
C.
-8
-
D.
-6
Solution
The determinant is calculated as 2(0*1 - 2*2) - 1(1*1 - 2*3) + 3(1*2 - 0*3) = 2(0 - 4) - 1(1 - 6) + 3(2) = -8 + 5 + 6 = 3.
Correct Answer: A — -12
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Q. Evaluate the determinant | 1 1 1 | | 1 2 3 | | 1 3 6 |.
Solution
The rows are linearly dependent, hence the determinant is 0.
Correct Answer: A — 0
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Q. Evaluate the determinant | 1 1 1 | | 2 2 2 | | 3 3 3 |.
Solution
The determinant is 0 because the rows are linearly dependent.
Correct Answer: A — 0
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Q. Evaluate the determinant: | 1 0 0 | | 0 1 0 | | 0 0 1 |
Solution
The determinant of the identity matrix is 1.
Correct Answer: A — 1
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Q. Evaluate the determinant: | 1 2 3 | | 4 5 6 | | 7 8 9 |
Solution
The determinant of a matrix with linearly dependent rows is 0. Here, the rows are linearly dependent.
Correct Answer: A — 0
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Q. Evaluate the expression sin^(-1)(1) + cos^(-1)(0).
Solution
sin^(-1)(1) = π/2 and cos^(-1)(0) = π/2. Therefore, π/2 + π/2 = π.
Correct Answer: A — π/2
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Q. Evaluate the expression: tan^(-1)(1) + tan^(-1)(1) + tan^(-1)(0).
Solution
tan^(-1)(1) = π/4, so the expression becomes π/4 + π/4 + 0 = π/2.
Correct Answer: A — π/2
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Q. Evaluate the expression: tan^(-1)(1) + tan^(-1)(1) = ?
-
A.
π/2
-
B.
π/4
-
C.
π/3
-
D.
π/6
Solution
tan^(-1)(1) = π/4, thus tan^(-1)(1) + tan^(-1)(1) = π/4 + π/4 = π/2.
Correct Answer: A — π/2
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Q. Evaluate the expression: tan^(-1)(1) + tan^(-1)(√3).
-
A.
π/3
-
B.
π/2
-
C.
2π/3
-
D.
π
Solution
tan^(-1)(1) = π/4 and tan^(-1)(√3) = π/3. Therefore, π/4 + π/3 = 7π/12.
Correct Answer: A — π/3
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Q. Evaluate \( \begin{vmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{vmatrix} \)
Solution
The determinant of the identity matrix is 1.
Correct Answer: B — 1
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Q. Evaluate \( \begin{vmatrix} 1 & 2 & 1 \\ 0 & 1 & 0 \\ 2 & 3 & 1 \end{vmatrix} \)
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 — 2
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Q. Evaluate \( \begin{vmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{vmatrix} \)
Solution
The determinant is calculated as \( 1(1*0 - 4*6) - 2(0 - 4*5) + 3(0 - 1*5) = -24 + 40 - 15 = 1 \).
Correct Answer: A — -12
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Q. Evaluate \( \begin{vmatrix} x & 1 \\ 1 & y \end{vmatrix} \) when \( x = 2 \) and \( y = 3 \).
Solution
The determinant is \( 2*3 - 1*1 = 6 \).
Correct Answer: B — 6
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Q. Evaluate: sin^(-1)(0) + cos^(-1)(0).
Solution
sin^(-1)(0) = 0 and cos^(-1)(0) = π/2, thus the sum is 0 + π/2 = π/2.
Correct Answer: B — π/2
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Q. Evaluate: sin^(-1)(1) + cos^(-1)(0).
Solution
sin^(-1)(1) = π/2 and cos^(-1)(0) = π/2. Therefore, π/2 + π/2 = π.
Correct Answer: A — π/2
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Q. Find the 10th term of the sequence defined by a_n = 3n + 2.
Solution
a_10 = 3(10) + 2 = 30 + 2 = 32.
Correct Answer: A — 32
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Q. Find the 10th term of the sequence defined by a_n = 3n^2 + 2n.
-
A.
320
-
B.
302
-
C.
290
-
D.
310
Solution
a_10 = 3(10^2) + 2(10) = 300 + 20 = 320.
Correct Answer: B — 302
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Q. Find the argument of the complex number z = -1 - i.
-
A.
-3π/4
-
B.
3π/4
-
C.
π/4
-
D.
-π/4
Solution
The argument of z = -1 - i is θ = tan^(-1)(-1/-1) = 3π/4.
Correct Answer: A — -3π/4
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Q. Find the coefficient of x^0 in the expansion of (2x + 3)^4.
Solution
The coefficient of x^0 is C(4, 0) * (2x)^0 * 3^4 = 1 * 81 = 81.
Correct Answer: A — 81
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Q. Find the coefficient of x^1 in the expansion of (x + 2)^5.
Solution
The coefficient of x^1 is C(5,1) * 2^4 = 5 * 16 = 80.
Correct Answer: B — 20
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Q. Find the coefficient of x^2 in the expansion of (3x - 4)^6.
-
A.
540
-
B.
720
-
C.
480
-
D.
360
Solution
The coefficient of x^2 is C(6,2) * (3)^2 * (-4)^4 = 15 * 9 * 256 = 34560.
Correct Answer: B — 720
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Q. Find the coefficient of x^3 in the expansion of (2x - 3)^6.
-
A.
-540
-
B.
-720
-
C.
540
-
D.
720
Solution
The coefficient of x^3 is C(6,3)(2)^3(-3)^3 = 20 * 8 * (-27) = -4320.
Correct Answer: A — -540
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Q. Find the coefficient of x^3 in the expansion of (x + 1/2)^6.
Solution
The coefficient of x^3 is C(6,3) * (1/2)^3 = 20 * 1/8 = 2.5.
Correct Answer: B — 15
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