Q. What is the slope of the line that passes through the points (4, 5) and (6, 9)?
Solution
The slope m = (9 - 5) / (6 - 4) = 4/2 = 2.
Correct Answer: A — 2
Learn More →
Q. What is the slope of the tangent line to f(x) = x^2 + 2x at x = 1? (2023)
Solution
f'(x) = 2x + 2. At x = 1, f'(1) = 2(1) + 2 = 4.
Correct Answer: B — 3
Learn More →
Q. What is the slope of the tangent line to the curve y = x^2 - 4x + 5 at x = 3? (2023)
Solution
The slope is given by f'(x) = 2x - 4. At x = 3, f'(3) = 2(3) - 4 = 2.
Correct Answer: C — 2
Learn More →
Q. What is the slope of the tangent to the curve y = x^2 + 2x at x = 1? (2023)
Solution
f'(x) = 2x + 2. At x = 1, f'(1) = 2(1) + 2 = 4.
Correct Answer: B — 3
Learn More →
Q. What is the solution of the differential equation dy/dx = y^2?
-
A.
y = 1/(C - x)
-
B.
y = C/(x + 1)
-
C.
y = Cx
-
D.
y = e^(x + C)
Solution
Separating variables and integrating gives 1/y = x + C, leading to y = 1/(C - x).
Correct Answer: A — y = 1/(C - x)
Learn More →
Q. What is the solution of the differential equation y' = 2y + 3?
-
A.
y = Ce^(2x) - 3/2
-
B.
y = Ce^(2x) + 3/2
-
C.
y = 3e^(2x)
-
D.
y = 2e^(x) + C
Solution
The integrating factor is e^(-2x). Solving gives y = Ce^(2x) + 3/2.
Correct Answer: B — y = Ce^(2x) + 3/2
Learn More →
Q. What is the solution of the differential equation y' = 5y + 3?
-
A.
y = (3/5) + Ce^(5x)
-
B.
y = Ce^(5x) - (3/5)
-
C.
y = (3/5)e^(5x)
-
D.
y = Ce^(3x) + 5
Solution
Using the integrating factor method, we find the general solution to be y = Ce^(5x) - (3/5).
Correct Answer: B — y = Ce^(5x) - (3/5)
Learn More →
Q. What is the solution of the equation dy/dx = 3x^2?
-
A.
y = x^3 + C
-
B.
y = 3x^3 + C
-
C.
y = x^2 + C
-
D.
y = 3x^2 + C
Solution
Integrating both sides gives y = ∫3x^2 dx = x^3 + C.
Correct Answer: A — y = x^3 + C
Learn More →
Q. What is the solution of the equation dy/dx = 4y + 2? (2021)
-
A.
y = Ce^(4x) - 1/2
-
B.
y = Ce^(-4x) + 1/2
-
C.
y = 2e^(4x) + C
-
D.
y = 4e^(4x) + C
Solution
Using an integrating factor, the solution is y = Ce^(4x) - 1/2.
Correct Answer: A — y = Ce^(4x) - 1/2
Learn More →
Q. What is the solution of the equation dy/dx = 6 - 2y? (2021)
-
A.
y = 3 - Ce^(-2x)
-
B.
y = 3 + Ce^(-2x)
-
C.
y = 2 - Ce^(2x)
-
D.
y = 6 - Ce^(2x)
Solution
Rearranging gives dy/(6 - 2y) = dx. Integrating both sides leads to y = 3 - Ce^(-2x).
Correct Answer: A — y = 3 - Ce^(-2x)
Learn More →
Q. What is the solution of the equation y' + 4y = 0?
-
A.
y = Ce^(-4x)
-
B.
y = Ce^(4x)
-
C.
y = 4Ce^x
-
D.
y = Ce^(x/4)
Solution
This is a separable equation. The solution is y = Ce^(-4x).
Correct Answer: A — y = Ce^(-4x)
Learn More →
Q. What is the solution of the equation y' = -ky, where k is a constant?
-
A.
y = Ce^(kt)
-
B.
y = Ce^(-kt)
-
C.
y = -Ce^(kt)
-
D.
y = -Ce^(-kt)
Solution
This is a separable equation. Integrating gives y = Ce^(-kt).
Correct Answer: B — y = Ce^(-kt)
Learn More →
Q. What is the solution to the differential equation dy/dx = -y/x?
-
A.
y = Cx
-
B.
y = C/x
-
C.
y = Cx^2
-
D.
y = Cx^(-1)
Solution
This is a separable equation. Separating variables and integrating gives y = C/x.
Correct Answer: B — y = C/x
Learn More →
Q. What is the solution to the differential equation y' = 5y + 3?
-
A.
y = (3/5) + Ce^(5x)
-
B.
y = (5/3) + Ce^(5x)
-
C.
y = Ce^(5x) - 3
-
D.
y = Ce^(3x) + 5
Solution
Using the integrating factor method, we find the solution to be y = (3/5) + Ce^(5x).
Correct Answer: A — y = (3/5) + Ce^(5x)
Learn More →
Q. What is the solution to the equation dy/dx = -5y?
-
A.
y = Ce^(-5x)
-
B.
y = -5Ce^x
-
C.
y = Ce^(5x)
-
D.
y = 5Ce^(-x)
Solution
This is a separable differential equation. The solution is y = Ce^(-5x), where C is a constant.
Correct Answer: A — y = Ce^(-5x)
Learn More →
Q. What is the solution to the equation dy/dx = y^2? (2022)
-
A.
y = 1/(C - x)
-
B.
y = C/(x - 1)
-
C.
y = Cx^2
-
D.
y = ln(Cx)
Solution
This is a separable equation. Integrating gives y = 1/(C - x).
Correct Answer: A — y = 1/(C - x)
Learn More →
Q. What is the solution to the equation y' + 2y = 0?
-
A.
y = Ce^(-2x)
-
B.
y = Ce^(2x)
-
C.
y = 2Ce^x
-
D.
y = Ce^x
Solution
This is a separable equation. The solution is y = Ce^(-2x).
Correct Answer: A — y = Ce^(-2x)
Learn More →
Q. What is the solution to the equation y' + 3y = 0?
-
A.
y = Ce^(-3x)
-
B.
y = Ce^(3x)
-
C.
y = 3Ce^(-x)
-
D.
y = Ce^(-x/3)
Solution
This is a first-order linear differential equation. The solution is y = Ce^(-3x).
Correct Answer: A — y = Ce^(-3x)
Learn More →
Q. What is the solution to the equation y' = 3y + 6?
-
A.
y = Ce^(3x) - 2
-
B.
y = Ce^(3x) + 2
-
C.
y = 2e^(3x)
-
D.
y = 3Ce^(x)
Solution
This is a first-order linear equation. The integrating factor is e^(3x), leading to the solution y = Ce^(3x) + 2.
Correct Answer: B — y = Ce^(3x) + 2
Learn More →
Q. What is the solution to the equation y'' + 4y = 0?
-
A.
y = C1 cos(2x) + C2 sin(2x)
-
B.
y = C1 e^(2x) + C2 e^(-2x)
-
C.
y = C1 e^(4x) + C2 e^(-4x)
-
D.
y = C1 sin(4x) + C2 cos(4x)
Solution
The characteristic equation is r^2 + 4 = 0, giving complex roots. The general solution is y = C1 cos(2x) + C2 sin(2x).
Correct Answer: A — y = C1 cos(2x) + C2 sin(2x)
Learn More →
Q. What is the solution to the equation y'' - 3y' + 2y = 0?
-
A.
y = C1 e^(2x) + C2 e^(x)
-
B.
y = C1 e^(x) + C2 e^(2x)
-
C.
y = C1 e^(-x) + C2 e^(-2x)
-
D.
y = C1 + C2x
Solution
The characteristic equation r^2 - 3r + 2 = 0 has roots 1 and 2, leading to y = C1 e^(x) + C2 e^(2x).
Correct Answer: B — y = C1 e^(x) + C2 e^(2x)
Learn More →
Q. What is the square of the modulus of the complex number 1 + 2i? (2014)
Solution
The modulus is √(1^2 + 2^2) = √(1 + 4) = √5. The square of the modulus is 5.
Correct Answer: A — 5
Learn More →
Q. What is the square of the modulus of the complex number 1 + i? (2020)
Solution
The modulus is √(1^2 + 1^2) = √2, and the square of the modulus is 2.
Correct Answer: A — 2
Learn More →
Q. What is the square root of 64? (2020)
Solution
The square root of 64 is 8.
Correct Answer: C — 8
Learn More →
Q. What is the square root of the complex number -1? (2021)
Solution
The square root of -1 is defined as i.
Correct Answer: A — i
Learn More →
Q. What is the square root of the complex number -4? (2020)
-
A.
2i
-
B.
-2i
-
C.
4i
-
D.
-4i
Solution
The square root of -4 is √(-1) * √4 = 2i.
Correct Answer: A — 2i
Learn More →
Q. What is the sum of the coefficients in the expansion of (2 + 3)^4? (2022)
-
A.
81
-
B.
64
-
C.
100
-
D.
125
Solution
The sum of the coefficients is (2 + 3)^4 = 5^4 = 625.
Correct Answer: A — 81
Learn More →
Q. What is the sum of the coefficients in the expansion of (x + 2)^5? (2021)
Solution
The sum of the coefficients is found by substituting x=1. So, (1 + 2)^5 = 3^5 = 243.
Correct Answer: B — 64
Learn More →
Q. What is the sum of the complex numbers 1 + 2i and 3 - 4i? (2023)
-
A.
4 - 2i
-
B.
4 + 2i
-
C.
2 - 2i
-
D.
2 + 2i
Solution
(1 + 2i) + (3 - 4i) = (1 + 3) + (2 - 4)i = 4 - 2i.
Correct Answer: A — 4 - 2i
Learn More →
Q. What is the sum of the complex numbers 3 + 2i and 1 - 4i? (2023)
-
A.
4 - 2i
-
B.
2 - 2i
-
C.
4 + 2i
-
D.
2 + 2i
Solution
(3 + 2i) + (1 - 4i) = (3 + 1) + (2 - 4)i = 4 - 2i.
Correct Answer: A — 4 - 2i
Learn More →
Showing 751 to 780 of 855 (29 Pages)
