Q. Solve the equation y' = 6y + 12.
-
A.
y = 2 - Ce^(-6x)
-
B.
y = Ce^(6x) - 2
-
C.
y = 2 + Ce^(6x)
-
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)
Learn More →
Q. Solve the first-order differential equation dy/dx = y/x.
-
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
Learn More →
Q. Solve the first-order linear differential equation dy/dx + 2y = 6.
-
A.
y = 3 - Ce^(-2x)
-
B.
y = 3 + Ce^(-2x)
-
C.
y = 6 - Ce^(-2x)
-
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)
Learn More →
Q. Solve the first-order linear differential equation dy/dx + y/x = 1.
-
A.
y = x + C/x
-
B.
y = Cx - x
-
C.
y = Cx + x
-
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
Learn More →
Q. Solve the first-order linear differential equation dy/dx = y/x.
-
A.
y = Cx
-
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
Learn More →
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
Learn More →
Q. The roots of the equation x² + 2x + k = 0 are real and distinct if k is: (2020)
-
A.
< 1
-
B.
≥ 1
-
C.
≤ 1
-
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
Learn More →
Q. The roots of the equation x² + 4x + 4 = 0 are: (2020)
-
A.
-2 and -2
-
B.
2 and 2
-
C.
0 and 4
-
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
Learn More →
Q. The roots of the equation x² + 4x + k = 0 are 2 and -6. What is the value of k? (2021)
-
A.
-12
-
B.
-8
-
C.
-10
-
D.
-14
Solution
Using the product of roots: k = 2 * (-6) = -12. The sum is 2 + (-6) = -4, which matches.
Correct Answer: B — -8
Learn More →
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
Learn More →
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
Learn More →
Q. What are the roots of the equation x² - 2x - 8 = 0? (2022)
-
A.
-2 and 4
-
B.
2 and -4
-
C.
4 and -2
-
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
Learn More →
Q. What are the roots of the equation x² - 5x + 6 = 0? (2021)
-
A.
1 and 6
-
B.
2 and 3
-
C.
3 and 2
-
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
Learn More →
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
Learn More →
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
Learn More →
Q. What is the 3rd term in the expansion of (2x + 3)^4?
-
A.
108x^2
-
B.
216x^2
-
C.
324x^2
-
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
Learn More →
Q. What is the 3rd term in the expansion of (2x + 5)^6? (2000)
-
A.
600x^4
-
B.
1500x^4
-
C.
1800x^4
-
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
Learn More →
Q. What is the 3rd term in the expansion of (2x - 3)^5? (2022)
-
A.
-90x^3
-
B.
90x^3
-
C.
-60x^3
-
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
Learn More →
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
Learn More →
Q. What is the 3rd term in the expansion of (x + 2)^8? (2022)
-
A.
112
-
B.
128
-
C.
256
-
D.
64
Solution
The 3rd term is given by 8C2 * (2)^2 * (x)^6 = 28 * 4 * x^6 = 112x^6.
Correct Answer: A — 112
Learn More →
Q. What is the 3rd term in the expansion of (x + 3)^4? (2023)
-
A.
36x^2
-
B.
54x^2
-
C.
72x^2
-
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
Learn More →
Q. What is the 3rd term in the expansion of (x + 3)^5? (2023)
-
A.
45
-
B.
90
-
C.
135
-
D.
180
Solution
The 3rd term is C(5,2) * (3)^2 * (x)^3 = 10 * 9 * x^3 = 90.
Correct Answer: B — 90
Learn More →
Q. What is the 3rd term in the expansion of (x + 3)^7? (2023)
-
A.
189
-
B.
441
-
C.
729
-
D.
1024
Solution
The 3rd term is C(7,2) * (3)^2 * (x)^5 = 21 * 9 * x^5 = 189.
Correct Answer: B — 441
Learn More →
Q. What is the 3rd term in the expansion of (x + 4)^5? (2023)
-
A.
80x^3
-
B.
160x^3
-
C.
240x^3
-
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
Learn More →
Q. What is the 3rd term in the expansion of (x + 4)^6? (2020)
-
A.
240x^4
-
B.
360x^4
-
C.
480x^4
-
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
Learn More →
Q. What is the 4th term in the expansion of (2x - 3)^6? (2020)
-
A.
-540
-
B.
540
-
C.
-720
-
D.
720
Solution
The 4th term is C(6,3) * (2x)^3 * (-3)^3 = 20 * 8x^3 * -27 = -540.
Correct Answer: A — -540
Learn More →
Q. What is the 5th term in the expansion of (2 + 3x)^6? (2023)
-
A.
486
-
B.
540
-
C.
729
-
D.
810
Solution
The 5th term is given by 6C4 * (2)^(6-4) * (3x)^4 = 15 * 4 * 81x^4 = 4860x^4.
Correct Answer: B — 540
Learn More →
Q. What is the 5th term in the expansion of (3x - 2)^7? (2023)
-
A.
-1680x^3
-
B.
1680x^3
-
C.
-2520x^3
-
D.
2520x^3
Solution
The 5th term is given by C(7,4) * (3x)^4 * (-2)^3 = 35 * 81 * (-8) = -1680x^3.
Correct Answer: A — -1680x^3
Learn More →
Q. What is the 5th term of the sequence defined by a_n = 2n^2 + 3n - 1?
Solution
To find the 5th term, substitute n = 5 into the formula: a_5 = 2(5^2) + 3(5) - 1 = 2(25) + 15 - 1 = 50 + 15 - 1 = 64.
Correct Answer: B — 47
Learn More →
Q. What is the 8th term of the sequence defined by a_n = 3n - 2?
Solution
Substituting n = 8 into the formula: a_8 = 3(8) - 2 = 24 - 2 = 22.
Correct Answer: A — 22
Learn More →
Showing 541 to 570 of 893 (30 Pages)
