Q. Evaluate tan(sin^(-1)(3/5)).
-
A.
3/4
-
B.
4/3
-
C.
5/3
-
D.
3/5
Solution
Let θ = sin^(-1)(3/5). Then sin(θ) = 3/5 and using the Pythagorean theorem, cos(θ) = 4/5. Therefore, tan(θ) = sin(θ)/cos(θ) = (3/5)/(4/5) = 3/4.
Correct Answer: B — 4/3
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Q. Evaluate tan^(-1)(1) + tan^(-1)(1).
-
A.
π/2
-
B.
π/4
-
C.
π/3
-
D.
0
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 tan^(-1)(1) + tan^(-1)(√3).
-
A.
π/3
-
B.
π/4
-
C.
π/2
-
D.
π/6
Solution
tan^(-1)(1) = π/4 and tan^(-1)(√3) = π/3. Therefore, π/4 + π/3 = π/2.
Correct Answer: C — π/2
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Q. Evaluate tan^(-1)(√3).
-
A.
π/3
-
B.
π/4
-
C.
π/6
-
D.
π/2
Solution
tan^(-1)(√3) = π/3, since tan(π/3) = √3.
Correct Answer: A — π/3
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Q. Evaluate the definite integral ∫(0 to 1) (3x^2)dx.
-
A.
1
-
B.
0.5
-
C.
0.33
-
D.
0.25
Solution
The integral evaluates to 1.
Correct Answer: A — 1
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Q. Evaluate the definite integral ∫(1 to 2) (3x^2)dx.
Solution
The integral evaluates to 6.
Correct Answer: B — 6
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Q. Evaluate the derivative of f(x) = e^x + ln(x) at x = 1.
Solution
f'(x) = e^x + 1/x. At x = 1, f'(1) = e + 1.
Correct Answer: A — 1
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Q. Evaluate the determinant \( \begin{pmatrix} 2 & 1 & 3 \\ 1 & 0 & 2 \\ 3 & 2 & 1 \end{pmatrix} \).
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 \( \begin{vmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{vmatrix} \)
Solution
The determinant of the identity matrix is 1.
Correct Answer: B — 1
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Q. Evaluate 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 sin^(-1)(x) + cos^(-1)(x).
-
A.
0
-
B.
π/2
-
C.
π
-
D.
undefined
Solution
sin^(-1)(x) + cos^(-1)(x) = π/2 for all x in the domain [-1, 1].
Correct Answer: B — π/2
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Q. Evaluate the expression sin^(-1)(x) + sin^(-1)(√(1-x^2)).
-
A.
π/2
-
B.
π/4
-
C.
π/3
-
D.
0
Solution
Using the identity sin^(-1)(x) + sin^(-1)(√(1-x^2)) = π/2 for x in [0, 1], the value is π/2.
Correct Answer: A — π/2
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Q. Evaluate the expression: 2sin^(-1)(1/2) + 2cos^(-1)(1/2).
Solution
2sin^(-1)(1/2) = 2(π/6) = π/3 and 2cos^(-1)(1/2) = 2(π/3) = 2π/3. Therefore, the total is π/3 + 2π/3 = π.
Correct Answer: A — π
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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 the integral ∫ (1/x) dx.
-
A.
ln
-
B.
x
-
C.
+ C
-
D.
ln(x) + C
-
.
1/x + C
-
.
x + C
Solution
The integral of 1/x is ln|x|. Therefore, ∫ (1/x) dx = ln|x| + C.
Correct Answer: A — ln
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Q. Evaluate the integral ∫ (2x + 1)/(x^2 + x) dx.
-
A.
ln
-
B.
x^2 + x
-
C.
+ C
-
D.
ln
-
.
x
-
.
+ C
-
.
ln
-
.
x^2 + x
-
.
+ 1 + C
-
.
ln
-
.
x^2 + x
-
.
+ 1
Solution
Using partial fraction decomposition, we can integrate to find that ∫ (2x + 1)/(x^2 + x) dx = ln|x^2 + x| + C.
Correct Answer: A — ln
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