Mathematics
Q. What is the radius of a circle if the diameter is 14 cm? (2021)
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A.
7 cm
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B.
14 cm
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C.
21 cm
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D.
28 cm
Solution
Radius is half of the diameter. Therefore, radius = 14 cm / 2 = 7 cm.
Correct Answer: A — 7 cm
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Q. What is the real part of the complex number 4 + 5i? (2023)
Solution
The real part of the complex number 4 + 5i is 4.
Correct Answer: A — 4
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Q. What is the real part of the complex number 7 - 4i? (2023)
Solution
The real part of the complex number 7 - 4i is 7.
Correct Answer: A — 7
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Q. What is the slope of the line passing through the points (1, 2) and (3, 6)? (2020) 2020
Solution
Slope = (6-2)/(3-1) = 4/2 = 2.
Correct Answer: A — 2
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Q. What is the slope of the line perpendicular to the line 3x + 4y = 12?
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A.
-3/4
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B.
4/3
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C.
3/4
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D.
-4/3
Solution
The slope of the line 3x + 4y = 12 is -3/4. The slope of the perpendicular line is the negative reciprocal, which is 4/3.
Correct Answer: B — 4/3
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Q. What is the slope of the line perpendicular to the line 4x + 3y = 12?
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A.
-3/4
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B.
4/3
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C.
3/4
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D.
-4/3
Solution
The slope of the line 4x + 3y = 12 is -4/3. The slope of the perpendicular line is the negative reciprocal, which is 3/4.
Correct Answer: D — -4/3
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Q. What is the slope of the line perpendicular to the line 5x + 2y = 10?
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A.
-2/5
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B.
5/2
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C.
2/5
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D.
-5/2
Solution
The slope of the line is -5/2. The slope of the perpendicular line is the negative reciprocal, which is 2/5.
Correct Answer: A — -2/5
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Q. What is the slope of the line perpendicular to the line 7x - 2y + 3 = 0?
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A.
1/2
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B.
-1/2
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C.
2
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D.
-2
Solution
The slope of the line is 7/2. The slope of the perpendicular line is the negative reciprocal, which is -2/7.
Correct Answer: B — -1/2
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Q. What is the slope of the line perpendicular to the line 7x - 2y = 14?
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A.
1/7
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B.
-1/7
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C.
2/7
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D.
-2/7
Solution
The slope of the line is 7/2. The slope of the perpendicular line is the negative reciprocal, which is -2/7.
Correct Answer: B — -1/7
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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
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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
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Q. What is the solution of the differential equation y' = 2y + 3?
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A.
y = Ce^(2x) - 3/2
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B.
y = Ce^(2x) + 3/2
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C.
y = 3e^(2x)
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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
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Q. What is the solution of the differential equation y' = 5y + 3?
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A.
y = (3/5) + Ce^(5x)
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B.
y = Ce^(5x) - (3/5)
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C.
y = (3/5)e^(5x)
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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)
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Q. What is the solution of the equation dy/dx = 3x^2?
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A.
y = x^3 + C
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B.
y = 3x^3 + C
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C.
y = x^2 + C
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D.
y = 3x^2 + C
Solution
Integrating both sides gives y = ∫3x^2 dx = x^3 + C.
Correct Answer: A — y = x^3 + C
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Q. What is the solution of the equation dy/dx = 4y + 2? (2021)
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A.
y = Ce^(4x) - 1/2
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B.
y = Ce^(-4x) + 1/2
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C.
y = 2e^(4x) + C
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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
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Q. What is the solution of the equation dy/dx = 6 - 2y? (2021)
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A.
y = 3 - Ce^(-2x)
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B.
y = 3 + Ce^(-2x)
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C.
y = 2 - Ce^(2x)
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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)
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Q. What is the solution of the equation y' = -ky, where k is a constant?
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A.
y = Ce^(kt)
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B.
y = Ce^(-kt)
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C.
y = -Ce^(kt)
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D.
y = -Ce^(-kt)
Solution
This is a separable equation. Integrating gives y = Ce^(-kt).
Correct Answer: B — y = Ce^(-kt)
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Q. What is the solution to the differential equation dy/dx = -y/x?
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A.
y = Cx
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B.
y = C/x
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C.
y = Cx^2
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D.
y = Cx^(-1)
Solution
This is a separable equation. Separating variables and integrating gives y = C/x.
Correct Answer: B — y = C/x
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Q. What is the solution to the equation dy/dx = -5y?
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A.
y = Ce^(-5x)
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B.
y = -5Ce^x
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C.
y = Ce^(5x)
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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)
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Q. What is the solution to the equation dy/dx = y^2? (2022)
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A.
y = 1/(C - x)
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B.
y = C/(x - 1)
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C.
y = Cx^2
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D.
y = ln(Cx)
Solution
This is a separable equation. Integrating gives y = 1/(C - x).
Correct Answer: A — y = 1/(C - x)
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Q. What is the solution to the equation y' + 2y = 0?
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A.
y = Ce^(-2x)
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B.
y = Ce^(2x)
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C.
y = 2Ce^x
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D.
y = Ce^x
Solution
This is a separable equation. The solution is y = Ce^(-2x).
Correct Answer: A — y = Ce^(-2x)
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Q. What is the solution to the equation y' + 3y = 0?
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A.
y = Ce^(-3x)
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B.
y = Ce^(3x)
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C.
y = 3Ce^(-x)
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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)
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Q. What is the solution to the equation y' = 3y + 6?
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A.
y = Ce^(3x) - 2
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B.
y = Ce^(3x) + 2
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C.
y = 2e^(3x)
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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
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Q. What is the solution to the equation y'' + 4y = 0?
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A.
y = C1 cos(2x) + C2 sin(2x)
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B.
y = C1 e^(2x) + C2 e^(-2x)
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C.
y = C1 e^(4x) + C2 e^(-4x)
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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)
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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
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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
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Q. What is the square root of 64? (2020)
Solution
The square root of 64 is 8.
Correct Answer: C — 8
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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
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Q. What is the sum of the complex numbers 3 + 2i and 1 - 4i? (2023)
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A.
4 - 2i
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B.
2 - 2i
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C.
4 + 2i
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D.
2 + 2i
Solution
(3 + 2i) + (1 - 4i) = (3 + 1) + (2 - 4)i = 4 - 2i.
Correct Answer: A — 4 - 2i
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Q. What is the sum of the roots of the equation 2x² - 4x + 1 = 0? (2023)
Solution
The sum of the roots is given by -b/a = 4/2 = 2.
Correct Answer: A — 2
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