Q. What is the induced EMF in a loop of wire if the magnetic flux through it changes at a rate of dΦ/dt?
A.
-dΦ/dt
B.
dΦ/dt
C.
Φ
D.
0
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Solution
According to Faraday's law of electromagnetic induction, the induced EMF is given by ε = -dΦ/dt.
Correct Answer: A — -dΦ/dt
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Q. What is the induced EMF in a loop of wire when the magnetic field changes at a rate of 5 T/s?
A.
0 V
B.
5 V
C.
10 V
D.
15 V
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Solution
The induced EMF (ε) is given by Faraday's law of electromagnetic induction: ε = -dΦ/dt. If the change in magnetic field is 5 T/s, then ε = 5 V.
Correct Answer: B — 5 V
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Q. What is the induced EMF in a loop of wire when the magnetic field through it changes at a rate of 5 T/s?
A.
0 V
B.
5 V
C.
10 V
D.
15 V
Show solution
Solution
The induced EMF (ε) is given by Faraday's law of electromagnetic induction: ε = -dΦ/dt. If the rate of change of magnetic field is 5 T/s, then ε = 5 V.
Correct Answer: B — 5 V
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Q. What is the induced EMF in a loop of wire when the magnetic field through it changes at a rate of dB/dt?
A.
-dB/dt
B.
dB/dt
C.
μ₀dB/dt
D.
0
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Solution
According to Faraday's law of electromagnetic induction, the induced EMF is given by ε = -dΦ/dt, where Φ is the magnetic flux.
Correct Answer: A — -dB/dt
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Q. What is the inradius of a triangle with sides 7 cm, 8 cm, and 9 cm?
A.
3 cm
B.
4 cm
C.
5 cm
D.
6 cm
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Solution
Using the formula r = A/s, where A is the area and s is the semi-perimeter. Area = 26 cm², s = 12 cm, so r = 26/12 = 4 cm.
Correct Answer: B — 4 cm
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Q. What is the integral form of Ampere's Law?
A.
∮B·dl = μ₀I_enclosed
B.
∮E·dl = -dΦ/dt
C.
∮F·dl = m*a
D.
∮V·dl = Q/C
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Solution
The integral form of Ampere's Law states that the line integral of the magnetic field B around a closed loop is equal to μ₀ times the enclosed current I.
Correct Answer: A — ∮B·dl = μ₀I_enclosed
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Q. What is the integral of 1/x dx?
A.
ln
B.
x
C.
+ C
D.
x + C
.
1/x + C
.
x^2/2 + C
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Solution
The integral of 1/x dx is ln|x| + C.
Correct Answer: A — ln
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Q. What is the integral of cos(x)dx?
A.
sin(x) + C
B.
cos(x) + C
C.
-sin(x) + C
D.
-cos(x) + C
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Solution
The integral of cos(x) is sin(x) + C.
Correct Answer: A — sin(x) + C
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Q. What is the integral of f(x) = 2x from 0 to 3?
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Solution
∫(2x)dx from 0 to 3 = [x^2] from 0 to 3 = 9 - 0 = 9.
Correct Answer: B — 6
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Q. What is the integral of f(x) = 2x with respect to x?
A.
x^2 + C
B.
2x^2 + C
C.
x^2 + 2C
D.
2x + C
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Solution
The integral ∫2x dx = x^2 + C.
Correct Answer: A — x^2 + C
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Q. What is the integral of f(x) = 2x?
A.
x^2 + C
B.
2x^2 + C
C.
x^2 + 2C
D.
2x + C
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Solution
∫2x dx = x^2 + C.
Correct Answer: A — x^2 + C
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Q. What is the integral of f(x) = 4x^3?
A.
x^4 + C
B.
x^4
C.
x^3 + C
D.
4x^4 + C
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Solution
The integral ∫4x^3 dx = x^4 + C.
Correct Answer: A — x^4 + C
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Q. What is the integral of f(x) = cos(x)?
A.
sin(x) + C
B.
-sin(x) + C
C.
tan(x) + C
D.
sec(x) + C
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Solution
The integral is sin(x) + C.
Correct Answer: A — sin(x) + C
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Q. What is the integrated rate law for a first-order reaction?
A.
ln[A] = -kt + ln[A]0
B.
[A] = [A]0 - kt
C.
[A] = [A]0e^(-kt)
D.
1/[A] = kt + 1/[A]0
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Solution
The integrated rate law for a first-order reaction is ln[A] = -kt + ln[A]0.
Correct Answer: A — ln[A] = -kt + ln[A]0
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Q. What is the integrating factor for the equation dy/dx + 2y = 3x?
A.
e^(2x)
B.
e^(-2x)
C.
e^(3x)
D.
e^(-3x)
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Solution
The integrating factor is e^(∫2dx) = e^(2x).
Correct Answer: A — e^(2x)
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Q. What is the integrating factor for the equation dy/dx + 3y = 6x?
A.
e^(3x)
B.
e^(-3x)
C.
e^(6x)
D.
e^(-6x)
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Solution
The integrating factor is e^(∫3dx) = e^(3x).
Correct Answer: A — e^(3x)
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Q. What is the interquartile range (IQR) of the data set {1, 3, 5, 7, 9, 11, 13, 15}?
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Solution
Q1 = 4, Q3 = 10. IQR = Q3 - Q1 = 10 - 4 = 6.
Correct Answer: B — 6
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Q. What is the interquartile range (IQR) of the data set {1, 3, 7, 8, 9, 10}?
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Solution
IQR = Q3 - Q1; Q1 = 3, Q3 = 8; IQR = 8 - 3 = 5.
Correct Answer: A — 5
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Q. What is the interquartile range (IQR) of the data set {1, 3, 7, 8, 9}?
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Solution
Q1 = 3, Q3 = 8. IQR = Q3 - Q1 = 8 - 3 = 5.
Correct Answer: A — 6
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Q. What is the interquartile range (IQR) of the data set: 1, 2, 3, 4, 5, 6, 7, 8, 9?
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Solution
Q1 = 3, Q3 = 7; IQR = Q3 - Q1 = 7 - 3 = 4.
Correct Answer: A — 4
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Q. What is the interquartile range (IQR) of the data set: 1, 3, 5, 7, 9, 11, 13?
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Solution
Q1 = 3, Q3 = 9. IQR = Q3 - Q1 = 9 - 3 = 6.
Correct Answer: B — 6
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Q. What is the intersection of the sets A = {1, 2, 3} and B = {2, 3, 4}?
A.
{1, 2, 3}
B.
{2, 3}
C.
{4}
D.
{1, 4}
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Solution
The intersection of two sets includes only the elements that are present in both sets. Here, the intersection is {2, 3}.
Correct Answer: B — {2, 3}
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Q. What is the inverse of the function f(x) = 2x + 3?
A.
(x - 3)/2
B.
(x + 3)/2
C.
2x - 3
D.
2(x - 3)
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Solution
To find the inverse, set y = 2x + 3, solve for x: x = (y - 3)/2.
Correct Answer: A — (x - 3)/2
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Q. What is the inverse of the function f(x) = 2x + 5?
A.
f^-1(x) = (x - 5)/2
B.
f^-1(x) = 2x - 5
C.
f^-1(x) = (x + 5)/2
D.
f^-1(x) = 5 - 2x
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Solution
To find the inverse, set y = 2x + 5, solve for x: x = (y - 5)/2, thus f^-1(x) = (x - 5)/2.
Correct Answer: A — f^-1(x) = (x - 5)/2
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Q. What is the inverse of the function f(x) = 3x + 1?
A.
(x-1)/3
B.
(x-1)/3
C.
(x-3)/1
D.
3(x-1)
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Solution
To find the inverse, set y = 3x + 1, solve for x: x = (y - 1)/3.
Correct Answer: A — (x-1)/3
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Q. What is the inverse of the function f(x) = 3x + 4?
A.
(x - 4)/3
B.
(x + 4)/3
C.
3/x - 4
D.
3/x + 4
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Solution
To find the inverse, set y = 3x + 4, solve for x: x = (y - 4)/3, hence the inverse is (x - 4)/3.
Correct Answer: A — (x - 4)/3
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Q. What is the inverse of the function f(x) = 3x - 5?
A.
(x + 5)/3
B.
(x - 5)/3
C.
3(x + 5)
D.
3(x - 5)
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Solution
To find the inverse, set y = 3x - 5, solve for x: x = (y + 5)/3, hence f^(-1)(x) = (x + 5)/3.
Correct Answer: A — (x + 5)/3
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Q. What is the inverse of the matrix A = [[0, 1], [1, 0]]?
A.
[[0, 1], [1, 0]]
B.
[[1, 0], [0, 1]]
C.
[[0, 0], [0, 0]]
D.
[[1, 1], [1, 1]]
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Solution
The inverse of A is A itself since A is an involutory matrix.
Correct Answer: A — [[0, 1], [1, 0]]
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Q. What is the inverse of the matrix A = [[1, 2], [3, 4]]?
A.
[[4, -2], [-3, 1]]
B.
[[-2, 1], [1.5, -0.5]]
C.
[[-2, 1], [1.5, -0.5]]
D.
[[2, -1], [-1.5, 0.5]]
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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. What is the isoelectric point (pI) of a protein?
A.
The pH at which the protein is positively charged
B.
The pH at which the protein is negatively charged
C.
The pH at which the protein has no net charge
D.
The pH at which the protein is denatured
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Solution
The isoelectric point (pI) of a protein is the pH at which the protein has no net charge, leading to minimal solubility.
Correct Answer: C — The pH at which the protein has no net charge
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