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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Q. Evaluate the integral ∫ (3x^2 + 2x + 1) dx.
-
A.
x^3 + x^2 + x + C
-
B.
x^3 + x^2 + C
-
C.
x^3 + x^2 + x
-
D.
3x^3 + 2x^2 + x + C
Solution
The integral of 3x^2 is x^3, the integral of 2x is x^2, and the integral of 1 is x. Therefore, ∫ (3x^2 + 2x + 1) dx = x^3 + x^2 + x + C.
Correct Answer: A — x^3 + x^2 + x + C
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Q. Evaluate the integral ∫ (sec^2(x)) dx.
-
A.
tan(x) + C
-
B.
sec(x) + C
-
C.
sin(x) + C
-
D.
cos(x) + C
Solution
The integral of sec^2(x) is tan(x) + C.
Correct Answer: A — tan(x) + C
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Q. Evaluate the integral ∫ (x^2 + 2x + 1)/(x + 1) dx.
-
A.
(1/3)x^3 + x^2 + C
-
B.
x^2 + x + C
-
C.
(1/3)x^3 + (1/2)x^2 + C
-
D.
x^2 + 2x + C
Solution
By simplifying the integrand, we can integrate to find that ∫ (x^2 + 2x + 1)/(x + 1) dx = (1/3)x^3 + x^2 + C.
Correct Answer: A — (1/3)x^3 + x^2 + C
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Q. Evaluate the integral ∫ cos(3x) dx.
-
A.
(1/3)sin(3x) + C
-
B.
sin(3x) + C
-
C.
(1/3)cos(3x) + C
-
D.
-(1/3)sin(3x) + C
Solution
The integral of cos(kx) is (1/k)sin(kx). Here, k = 3, so ∫ cos(3x) dx = (1/3)sin(3x) + C.
Correct Answer: A — (1/3)sin(3x) + C
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Q. Evaluate the integral ∫ cos(5x) dx.
-
A.
1/5 sin(5x) + C
-
B.
-1/5 sin(5x) + C
-
C.
5 sin(5x) + C
-
D.
sin(5x) + C
Solution
The integral of cos(kx) is (1/k)sin(kx). Here, k = 5, so ∫ cos(5x) dx = (1/5)sin(5x) + C.
Correct Answer: A — 1/5 sin(5x) + C
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Q. Evaluate the integral ∫ e^(3x) dx.
-
A.
(1/3)e^(3x) + C
-
B.
(1/3)e^(3x)
-
C.
3e^(3x) + C
-
D.
e^(3x) + C
Solution
The integral of e^(kx) is (1/k)e^(kx). Here, k = 3, so ∫ e^(3x) dx = (1/3)e^(3x) + C.
Correct Answer: A — (1/3)e^(3x) + C
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Q. Evaluate the integral ∫ from 0 to 1 of (x^2 + 2x) dx.
Solution
The integral evaluates to [x^3/3 + x^2] from 0 to 1 = (1/3 + 1) - (0) = 4/3.
Correct Answer: B — 2
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Q. Evaluate the integral ∫ from 0 to 1 of e^x dx.
Solution
The integral evaluates to [e^x] from 0 to 1 = e - 1.
Correct Answer: A — e - 1
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Q. Evaluate the integral ∫ from 1 to 3 of (2x + 1) dx.
Solution
The integral evaluates to [x^2 + x] from 1 to 3 = (9 + 3) - (1 + 1) = 10.
Correct Answer: B — 8
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Q. Evaluate the integral ∫(0 to 1) (x^3 + 2x^2)dx.
-
A.
1/4
-
B.
1/3
-
C.
1/2
-
D.
1
Solution
The integral evaluates to [x^4/4 + 2x^3/3] from 0 to 1 = 1/4 + 2/3 = 11/12.
Correct Answer: B — 1/3
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Q. Evaluate the integral ∫(1 to 2) (2x + 3)dx.
Solution
∫(2x + 3)dx = [x^2 + 3x] from 1 to 2 = (4 + 6) - (1 + 3) = 10 - 4 = 6.
Correct Answer: B — 8
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Q. Evaluate the integral ∫(1 to 2) (3x^2 - 2)dx.
Solution
The integral evaluates to [(x^3 - 2x)] from 1 to 2 = (8 - 4) - (1 - 2) = 5.
Correct Answer: A — 3
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Q. Evaluate the integral ∫(1 to 2) (x^2 + 2x)dx.
Solution
The integral ∫(x^2 + 2x)dx = [(1/3)x^3 + x^2] from 1 to 2 = 8.
Correct Answer: B — 8
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Q. Evaluate the integral ∫(2x^3 - 4x)dx.
-
A.
(1/2)x^4 - 2x^2 + C
-
B.
(1/4)x^4 - 2x^2 + C
-
C.
(1/2)x^4 - 4x^2 + C
-
D.
(1/3)x^4 - 2x^2 + C
Solution
The integral ∫(2x^3 - 4x)dx = (1/2)x^4 - 2x^2 + C.
Correct Answer: A — (1/2)x^4 - 2x^2 + C
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