Q. What is the equation of the plane passing through the point (1, 2, 3) with normal vector (1, -1, 1)? (2023)
-
A.
x - y + z = 0
-
B.
x + y + z = 6
-
C.
x - y + z = 1
-
D.
x + y - z = 0
Solution
Equation of the plane: 1(x-1) - 1(y-2) + 1(z-3) = 0 => x - y + z = 1.
Correct Answer: C — x - y + z = 1
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Q. What is the equation of the plane passing through the points (1, 2, 3), (4, 5, 6), and (7, 8, 9)? (2021)
-
A.
0 = 0
-
B.
x + y + z = 12
-
C.
x + y + z = 10
-
D.
x + y + z = 9
Solution
The points are collinear, hence the equation of the plane is 0 = 0.
Correct Answer: A — 0 = 0
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Q. What is the general solution of the equation y' + 4y = 0?
-
A.
y = Ce^(-4x)
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B.
y = Ce^(4x)
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C.
y = 4x + C
-
D.
y = Cx^4
Solution
This is a separable equation, and integrating gives y = Ce^(-4x).
Correct Answer: A — y = Ce^(-4x)
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Q. What is the general solution of the equation y' = 4y + 3?
-
A.
y = Ce^(4x) - 3/4
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B.
y = Ce^(4x) + 3/4
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C.
y = 3e^(4x)
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D.
y = Ce^(3x) + 4
Solution
The integrating factor is e^(-4x). The solution is y = Ce^(4x) + 3/4.
Correct Answer: B — y = Ce^(4x) + 3/4
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Q. What is the general solution of the equation y'' - 3y' + 2y = 0?
-
A.
y = C1 e^(x) + C2 e^(2x)
-
B.
y = C1 e^(2x) + C2 e^(x)
-
C.
y = C1 e^(3x) + C2 e^(0)
-
D.
y = C1 e^(0) + C2 e^(3x)
Solution
The characteristic equation is r^2 - 3r + 2 = 0, which factors to (r - 1)(r - 2) = 0. Thus, the general solution is y = C1 e^(2x) + C2 e^(x).
Correct Answer: B — y = C1 e^(2x) + C2 e^(x)
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Q. What is the general solution of the equation y'' - 4y' + 4y = 0?
-
A.
y = (C1 + C2x)e^(2x)
-
B.
y = C1 e^(2x) + C2 e^(-2x)
-
C.
y = C1 e^(4x) + C2 e^(-4x)
-
D.
y = C1 cos(2x) + C2 sin(2x)
Solution
The characteristic equation has a repeated root r = 2. The general solution is y = (C1 + C2x)e^(2x).
Correct Answer: A — y = (C1 + C2x)e^(2x)
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Q. What is the integral of 1/x with respect to x? (2022)
-
A.
ln
-
B.
x
-
C.
+ C
-
D.
1/x + C
-
.
x + C
-
.
e^x + C
Solution
The integral of 1/x is ln|x| + C.
Correct Answer: A — ln
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Q. What is the integral of 1/x? (2022)
-
A.
ln
-
B.
x
-
C.
+ C
-
D.
1/x + C
-
.
x + C
-
.
x^2 + C
Solution
The integral of 1/x is ln|x| + C.
Correct Answer: A — ln
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Q. What is the integral of cos(2x) dx? (2021)
-
A.
(1/2)sin(2x) + C
-
B.
sin(2x) + C
-
C.
(1/2)cos(2x) + C
-
D.
2sin(2x) + C
Solution
The integral of cos(2x) is (1/2)sin(2x) + C.
Correct Answer: A — (1/2)sin(2x) + C
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Q. What is the integral of e^x with respect to x? (2023)
-
A.
e^x + C
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B.
xe^x + C
-
C.
e^x/x + C
-
D.
x^2e^x + C
Solution
The integral of e^x is e^x + C.
Correct Answer: A — e^x + C
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Q. What is the integral of e^x? (2023)
-
A.
e^x + C
-
B.
xe^x + C
-
C.
e^x/x + C
-
D.
x^2e^x + C
Solution
The integral of e^x is e^x + C.
Correct Answer: A — e^x + C
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Q. What is the integral of sin(x) dx? (2021)
-
A.
-cos(x) + C
-
B.
cos(x) + C
-
C.
sin(x) + C
-
D.
-sin(x) + C
Solution
The integral of sin(x) is -cos(x) + C.
Correct Answer: A — -cos(x) + C
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Q. What is the integral of tan(x) dx? (2023)
-
A.
-ln
-
B.
cos(x)
-
C.
+ C
-
D.
ln
-
.
sin(x)
-
.
+ C
-
.
ln
-
.
tan(x)
-
.
+ C
-
.
-ln
-
.
sin(x)
-
.
+ C
Solution
The integral of tan(x) is -ln|cos(x)| + C.
Correct Answer: A — -ln
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Q. What is the integral of tan(x) with respect to x? (2021)
-
A.
-ln
-
B.
cos(x)
-
C.
+ C
-
D.
ln
-
.
sin(x)
-
.
+ C
-
.
ln
-
.
tan(x)
-
.
+ C
-
.
-ln
-
.
sin(x)
-
.
+ C
Solution
The integral of tan(x) is -ln|cos(x)| + C.
Correct Answer: A — -ln
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Q. What is the integral of tan(x)? (2021)
-
A.
-ln
-
B.
cos(x)
-
C.
+ C
-
D.
ln
-
.
sin(x)
-
.
+ C
-
.
ln
-
.
tan(x)
-
.
+ C
-
.
-ln
-
.
sin(x)
-
.
+ C
Solution
The integral of tan(x) is -ln|cos(x)| + C.
Correct Answer: A — -ln
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Q. What is the integral of x^2 with respect to x? (2021)
-
A.
(1/3)x^3 + C
-
B.
(1/2)x^2 + C
-
C.
(1/4)x^4 + C
-
D.
(1/5)x^5 + C
Solution
The integral of x^n is (1/(n+1))x^(n+1) + C. Here, n=2, so the integral is (1/3)x^3 + C.
Correct Answer: A — (1/3)x^3 + C
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Q. What is the integral of x^3 + 2x? (2021)
-
A.
(1/4)x^4 + x^2 + C
-
B.
(1/3)x^3 + x^2 + C
-
C.
(1/4)x^4 + (1/2)x^2 + C
-
D.
(1/5)x^5 + (1/2)x^2 + C
Solution
Integrating term by term, ∫x^3dx = (1/4)x^4 and ∫2xdx = x^2. Thus, ∫(x^3 + 2x)dx = (1/4)x^4 + (1/2)x^2 + C.
Correct Answer: C — (1/4)x^4 + (1/2)x^2 + C
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Q. What is the integral of x^n where n ≠ -1? (2021)
-
A.
(1/n)x^(n+1) + C
-
B.
(1/(n+1))x^n + C
-
C.
(1/(n+1))x^(n+1) + C
-
D.
nx^(n-1) + C
Solution
The integral of x^n is (1/(n+1))x^(n+1) + C, provided n ≠ -1.
Correct Answer: C — (1/(n+1))x^(n+1) + C
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Q. What is the integrating factor for the equation dy/dx + (1/x)y = 2?
-
A.
x
-
B.
e^(ln(x))
-
C.
e^(ln(x^2))
-
D.
1/x
Solution
The integrating factor is e^(∫(1/x)dx) = e^(ln(x)) = x.
Correct Answer: C — e^(ln(x^2))
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Q. What is the integrating factor for the equation dy/dx + 2y = 3?
-
A.
e^(2x)
-
B.
e^(-2x)
-
C.
e^(3x)
-
D.
e^(-3x)
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 + 2y = 6?
-
A.
e^(2x)
-
B.
e^(-2x)
-
C.
e^(6x)
-
D.
e^(-6x)
Solution
The integrating factor is e^(∫2dx) = e^(2x).
Correct Answer: A — e^(2x)
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Q. What is the length of a chord that is 6 cm away from the center of a circle with a radius of 10 cm? (2015)
-
A.
8 cm
-
B.
12 cm
-
C.
10 cm
-
D.
6 cm
Solution
Using Pythagoras: (radius)² = (distance from center)² + (half chord)²; 10² = 6² + (half chord)²; half chord = 8 cm, so full chord = 16 cm.
Correct Answer: A — 8 cm
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Q. What is the length of a chord that is 6 cm from the center of a circle with a radius of 10 cm? (2019)
-
A.
8 cm
-
B.
12 cm
-
C.
10 cm
-
D.
6 cm
Solution
Using Pythagoras: chord length = 2√(r² - d²) = 2√(10² - 6²) = 2√(100 - 36) = 2√64 = 16 cm.
Correct Answer: A — 8 cm
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Q. What is the length of a diameter of a circle with a radius of 7 cm? (2022)
-
A.
14 cm
-
B.
21 cm
-
C.
7 cm
-
D.
28 cm
Solution
Diameter = 2 × radius; Diameter = 2 × 7 cm = 14 cm.
Correct Answer: A — 14 cm
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Q. What is the length of an arc of a circle with a radius of 10 cm and a central angle of 60 degrees? (2021)
-
A.
10.47 cm
-
B.
12.57 cm
-
C.
15.71 cm
-
D.
20.94 cm
Solution
Arc length = (θ/360) * 2πr = (60/360) * 2 * 3.14 * 10 = 10.47 cm.
Correct Answer: B — 12.57 cm
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Q. What is the length of an arc of a circle with a radius of 10 cm and a central angle of 60 degrees? (2021) 2021
-
A.
10.47 cm
-
B.
15.71 cm
-
C.
20.94 cm
-
D.
25.13 cm
Solution
Arc length = (θ/360) * 2πr = (60/360) * 2 * 3.14 * 10 = 10.47 cm.
Correct Answer: A — 10.47 cm
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Q. What is the length of an arc of a circle with a radius of 10 cm and a central angle of 60 degrees? (Use π = 3.14) (2023)
-
A.
10.47 cm
-
B.
15.71 cm
-
C.
20.94 cm
-
D.
25.13 cm
Solution
Arc length = (θ/360) * 2πr = (60/360) * 2 * 3.14 * 10 = 10.47 cm.
Correct Answer: A — 10.47 cm
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Q. What is the length of an arc of a circle with a radius of 4 cm and a central angle of 90 degrees? (2021)
-
A.
2π cm
-
B.
4π cm
-
C.
π cm
-
D.
8 cm
Solution
Arc length = (θ/360) * 2πr = (90/360) * 2π * 4 = 2π cm.
Correct Answer: A — 2π cm
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Q. What is the length of an arc of a circle with a radius of 5 cm and a central angle of 60 degrees? (2020)
-
A.
5.24 cm
-
B.
3.14 cm
-
C.
5.00 cm
-
D.
10.47 cm
Solution
Arc length = (θ/360) * 2πr = (60/360) * 2π(5) = (1/6) * 10π ≈ 5.24 cm.
Correct Answer: A — 5.24 cm
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Q. What is the length of an arc of a circle with a radius of 6 cm and a central angle of 60 degrees? (Use π = 3.14) (2020)
-
A.
6.28 cm
-
B.
3.14 cm
-
C.
12.56 cm
-
D.
9.42 cm
Solution
Arc length = (θ/360) * 2πr = (60/360) * 2 * 3.14 * 6 cm = 6.28 cm.
Correct Answer: A — 6.28 cm
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