Q. Solve the differential equation dy/dx = y/x. (2023)
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A.
y = Cx
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B.
y = Cx^2
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C.
y = C/x
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D.
y = C ln(x)
Solution
This is a separable equation. Integrating gives y = Cx.
Correct Answer: A — y = Cx
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Q. Solve the differential equation y' = 5 - 2y.
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A.
y = 5/2 + Ce^(-2x)
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B.
y = 5 + Ce^(-2x)
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C.
y = 2 + Ce^(2x)
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D.
y = 5/2 - Ce^(-2x)
Solution
This is a linear first-order equation. The solution is y = 5/2 + Ce^(-2x).
Correct Answer: A — y = 5/2 + Ce^(-2x)
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Q. Solve the differential equation y' = 5y + 3.
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A.
y = (3/5) + Ce^(5x)
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B.
y = (5/3) + Ce^(5x)
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C.
y = Ce^(5x) - 3
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D.
y = Ce^(3x) + 5
Solution
Using the integrating factor method, we find the solution y = (3/5) + Ce^(5x).
Correct Answer: A — y = (3/5) + Ce^(5x)
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Q. Solve the differential equation y'' - 3y' + 2y = 0.
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A.
y = C1e^(2x) + C2e^(x)
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B.
y = C1e^(x) + C2e^(2x)
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C.
y = C1e^(-x) + C2e^(-2x)
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D.
y = C1e^(3x) + C2e^(x)
Solution
The characteristic equation is r^2 - 3r + 2 = 0, which factors to (r - 1)(r - 2) = 0. The general solution is y = C1e^(x) + C2e^(2x).
Correct Answer: B — y = C1e^(x) + C2e^(2x)
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Q. Solve the equation y' = 6y + 12.
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A.
y = 2 - Ce^(-6x)
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B.
y = Ce^(6x) - 2
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C.
y = 2 + Ce^(6x)
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D.
y = 6Ce^(-x)
Solution
This is a first-order linear equation. The integrating factor method gives the solution y = 2 - Ce^(-6x).
Correct Answer: A — y = 2 - Ce^(-6x)
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Q. Solve the first-order differential equation dy/dx = y/x.
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A.
y = Cx
-
B.
y = Cx^2
-
C.
y = C/x
-
D.
y = C ln(x)
Solution
This is a separable equation. Separating variables and integrating gives y = Cx.
Correct Answer: A — y = Cx
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Q. Solve the first-order linear differential equation dy/dx + 2y = 6.
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A.
y = 3 - Ce^(-2x)
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B.
y = 3 + Ce^(-2x)
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C.
y = 6 - Ce^(-2x)
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D.
y = 6 + Ce^(-2x)
Solution
Using an integrating factor e^(2x), we solve to get y = 3 - Ce^(-2x).
Correct Answer: A — y = 3 - Ce^(-2x)
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Q. Solve the first-order linear differential equation dy/dx + y/x = 1.
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A.
y = x + C/x
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B.
y = Cx - x
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C.
y = Cx + x
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D.
y = C/x + x
Solution
Using the integrating factor e^(∫(1/x)dx) = x, we solve to get y = x + C/x.
Correct Answer: A — y = x + C/x
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Q. Solve the first-order linear differential equation dy/dx = y/x.
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A.
y = Cx
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B.
y = Cx^2
-
C.
y = C/x
-
D.
y = C ln(x)
Solution
This is separable: dy/y = dx/x. Integrating gives ln|y| = ln|x| + C, thus y = Cx.
Correct Answer: A — y = Cx
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Q. The quadratic equation x² + 4x + 4 = 0 has how many distinct roots? (2021)
Solution
The discriminant is 4² - 4*1*4 = 0, indicating one distinct root.
Correct Answer: B — 1
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Q. The roots of the equation x² + 2x + k = 0 are real and distinct if k is: (2020)
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A.
< 1
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B.
≥ 1
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C.
≤ 1
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D.
> 1
Solution
For real and distinct roots, the discriminant must be positive: 2² - 4*1*k > 0, which simplifies to k < 1.
Correct Answer: A — < 1
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Q. The roots of the equation x² + 4x + 4 = 0 are: (2020)
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A.
-2 and -2
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B.
2 and 2
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C.
0 and 4
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D.
1 and 3
Solution
The equation can be factored as (x + 2)² = 0, giving the double root x = -2.
Correct Answer: A — -2 and -2
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Q. The roots of the equation x² + 4x + k = 0 are 2 and -6. What is the value of k? (2021)
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A.
-12
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B.
-8
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C.
-10
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D.
-14
Solution
Using the product of roots: k = 2 * (-6) = -12. The sum is 2 + (-6) = -4, which matches.
Correct Answer: B — -8
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Q. The roots of the equation x² - 8x + k = 0 are 4 and 4. Find k. (2021)
Solution
Using the product of roots: k = 4 * 4 = 16.
Correct Answer: A — 16
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Q. What are the roots of the equation 3x² - 12x + 12 = 0? (2019)
Solution
Dividing the equation by 3 gives x² - 4x + 4 = 0, which factors to (x - 2)² = 0, hence the root is 2.
Correct Answer: B — 4
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Q. What are the roots of the equation x² - 2x - 8 = 0? (2022)
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A.
-2 and 4
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B.
2 and -4
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C.
4 and -2
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D.
0 and 8
Solution
Factoring gives (x - 4)(x + 2) = 0, hence the roots are 4 and -2.
Correct Answer: C — 4 and -2
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Q. What are the roots of the equation x² - 5x + 6 = 0? (2021)
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A.
1 and 6
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B.
2 and 3
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C.
3 and 2
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D.
0 and 5
Solution
The roots can be found using the factorization method: (x - 2)(x - 3) = 0, hence the roots are 2 and 3.
Correct Answer: B — 2 and 3
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Q. What is the 10th term of the arithmetic sequence where the first term is 5 and the common difference is 3?
Solution
The nth term of an arithmetic sequence is given by a_n = a + (n-1)d. Here, a = 5, d = 3, n = 10. So, a_10 = 5 + (10-1) * 3 = 5 + 27 = 32.
Correct Answer: B — 35
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Q. What is the 12th term of the arithmetic sequence where the first term is 7 and the common difference is 5?
Solution
Using the formula a_n = a + (n-1)d, we have a_12 = 7 + (12-1) * 5 = 7 + 55 = 62.
Correct Answer: A — 62
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Q. What is the 3rd term in the expansion of (2x + 3)^4?
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A.
108x^2
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B.
216x^2
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C.
324x^2
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D.
432x^2
Solution
The 3rd term is given by C(4,2) * (2x)^2 * (3)^2 = 6 * 4x^2 * 9 = 216x^2.
Correct Answer: B — 216x^2
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Q. What is the 3rd term in the expansion of (2x + 5)^6? (2000)
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A.
600x^4
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B.
1500x^4
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C.
1800x^4
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D.
2000x^4
Solution
The 3rd term is given by C(6,2) * (2x)^2 * (5)^4 = 15 * 4 * 625 = 37500.
Correct Answer: B — 1500x^4
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Q. What is the 3rd term in the expansion of (2x - 3)^5? (2022)
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A.
-90x^3
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B.
90x^3
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C.
-60x^3
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D.
60x^3
Solution
The 3rd term is C(5,2) * (2x)^3 * (-3)^2 = 10 * 8x^3 * 9 = -720x^3.
Correct Answer: A — -90x^3
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Q. What is the 3rd term in the expansion of (x + 2)^5? (2021)
Solution
The 3rd term is given by C(5,2) * (x)^3 * (2)^2 = 10 * x^3 * 4 = 40.
Correct Answer: C — 60
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Q. What is the 3rd term in the expansion of (x + 2)^8? (2022)
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A.
112
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B.
128
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C.
256
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D.
64
Solution
The 3rd term is given by 8C2 * (2)^2 * (x)^6 = 28 * 4 * x^6 = 112x^6.
Correct Answer: A — 112
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Q. What is the 3rd term in the expansion of (x + 3)^4? (2023)
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A.
36x^2
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B.
54x^2
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C.
72x^2
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D.
108x^2
Solution
The 3rd term is C(4,2) * (3)^2 * (x)^2 = 6 * 9 * x^2 = 54x^2.
Correct Answer: B — 54x^2
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Q. What is the 3rd term in the expansion of (x + 3)^5? (2023)
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A.
45
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B.
90
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C.
135
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D.
180
Solution
The 3rd term is C(5,2) * (3)^2 * (x)^3 = 10 * 9 * x^3 = 90.
Correct Answer: B — 90
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Q. What is the 3rd term in the expansion of (x + 3)^7? (2023)
-
A.
189
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B.
441
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C.
729
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D.
1024
Solution
The 3rd term is C(7,2) * (3)^2 * (x)^5 = 21 * 9 * x^5 = 189.
Correct Answer: B — 441
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Q. What is the 3rd term in the expansion of (x + 4)^5? (2023)
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A.
80x^3
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B.
160x^3
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C.
240x^3
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D.
320x^3
Solution
The 3rd term is given by C(5,2) * (4)^2 * (x)^3 = 10 * 16 * x^3 = 160x^3.
Correct Answer: C — 240x^3
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Q. What is the 3rd term in the expansion of (x + 4)^6? (2020)
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A.
240x^4
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B.
360x^4
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C.
480x^4
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D.
600x^4
Solution
The 3rd term is C(6,2) * (4)^2 * (x)^4 = 15 * 16 * x^4 = 240x^4.
Correct Answer: B — 360x^4
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Q. What is the 4th term in the expansion of (2x - 3)^6? (2020)
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A.
-540
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B.
540
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C.
-720
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D.
720
Solution
The 4th term is C(6,3) * (2x)^3 * (-3)^3 = 20 * 8x^3 * -27 = -540.
Correct Answer: A — -540
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