Q. If the revenue function is R(x) = 100x - 2x^2, find the number of units that maximizes revenue. (2021)
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Solution
Max revenue occurs at x = -b/(2a) = 100/(2*2) = 25.
Correct Answer: B — 50
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Q. If the revenue function is R(x) = 20x - 0.5x^2, find the quantity that maximizes revenue. (2021)
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Solution
R'(x) = 20 - x = 0 gives x = 20. This maximizes revenue.
Correct Answer: B — 20
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Q. If the roots of the equation x² + 2x + k = 0 are 1 and -3, what is the value of k? (2020)
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Solution
Using the sum and product of roots: 1 + (-3) = -2 and 1 * (-3) = -3, thus k = 3.
Correct Answer: C — 3
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Q. If the roots of the equation x² + 2x + k = 0 are real and distinct, what is the condition for k? (2020)
A.
k > 1
B.
k < 1
C.
k > 4
D.
k < 4
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Solution
The discriminant must be greater than zero: 2² - 4*1*k > 0, which simplifies to k < 1.
Correct Answer: C — k > 4
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Q. If the roots of the equation x² + 5x + 6 = 0 are a and b, what is the value of a + b? (2019)
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Solution
The sum of the roots is given by -b/a = -5/1 = -5.
Correct Answer: A — 5
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Q. If the roots of the equation x² + 5x + k = 0 are -2 and -3, find k. (2020)
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Solution
Using the product of roots: k = (-2)(-3) = 6.
Correct Answer: A — 6
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Q. If the roots of the equation x² + 5x + k = 0 are 1 and 4, find k. (2020)
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Solution
Using the sum and product of roots: k = 1*4 = 4, and sum = 1 + 4 = 5, thus k = 7.
Correct Answer: D — 7
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Q. If the roots of the equation x² + 5x + q = 0 are 1 and 4, find q. (2019)
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Solution
Using the product of roots: q = 1 * 4 = 4.
Correct Answer: A — 5
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Q. If the roots of the equation x² + 7x + k = 0 are -3 and -4, find k. (2022)
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Solution
Using the sum of roots (-3 + -4 = -7) and product of roots (-3*-4 = 12), we find k = 12.
Correct Answer: A — 12
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Q. If the roots of the equation x² + 7x + k = 0 are 1 and 6, what is the value of k? (2020)
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Solution
Using the product of roots, k = 1*6 = 6.
Correct Answer: C — 8
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Q. If the roots of the equation x² + 7x + p = 0 are -3 and -4, find p. (2019)
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Solution
Using the sum of roots (-3 + -4 = -7) and product of roots (-3*-4 = 12), we find p = 12.
Correct Answer: A — 12
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Q. If the roots of the equation x² + 7x + p = 0 are -3 and -4, find the value of p. (2019)
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Solution
Using the product of roots: p = (-3)(-4) = 12.
Correct Answer: A — 12
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Q. If the roots of the equation x² + mx + n = 0 are 1 and -1, find m and n. (2020)
A.
0, 1
B.
2, 1
C.
0, 0
D.
1, 1
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Solution
The sum of the roots is 0 (m = 0) and the product is -1 (n = 1).
Correct Answer: A — 0, 1
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Q. If the roots of the equation x² + px + 12 = 0 are 3 and 4, find p. (2020)
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Solution
Using the sum of the roots: p = -(3 + 4) = -7.
Correct Answer: A — -7
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Q. If the roots of the equation x² + px + 12 = 0 are 3 and 4, what is the value of p? (2020)
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Solution
The sum of the roots is -p = 3 + 4 = 7, hence p = -7.
Correct Answer: A — -7
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Q. If the roots of the equation x² + px + q = 0 are 3 and -2, what is the value of p? (2019)
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Solution
Using the sum of roots formula, p = -(3 + (-2)) = -1, hence p = -1.
Correct Answer: C — 5
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Q. If the speed of light in a vacuum is 3 x 10^8 m/s, what is its speed in water (n = 1.33)?
A.
2.26 x 10^8 m/s
B.
2.25 x 10^8 m/s
C.
2.27 x 10^8 m/s
D.
2.28 x 10^8 m/s
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Solution
The speed of light in a medium is given by v = c/n. Thus, v = (3 x 10^8 m/s) / 1.33 ≈ 2.26 x 10^8 m/s.
Correct Answer: A — 2.26 x 10^8 m/s
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Q. If the speed of light in a vacuum is approximately 3 x 10^8 m/s, what is its value in km/s?
A.
300 km/s
B.
3000 km/s
C.
30000 km/s
D.
300000 km/s
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Solution
To convert m/s to km/s, divide by 1000. Therefore, 3 x 10^8 m/s = 300000 km/s.
Correct Answer: D — 300000 km/s
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Q. If the speed of light in a vacuum is approximately 3 x 10^8 m/s, what is the wavelength of light with a frequency of 6 x 10^14 Hz?
A.
500 nm
B.
600 nm
C.
700 nm
D.
800 nm
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Solution
Using the formula c = fλ, where c = 3 x 10^8 m/s and f = 6 x 10^14 Hz. Thus, λ = c/f = (3 x 10^8)/(6 x 10^14) = 500 nm.
Correct Answer: A — 500 nm
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Q. If the speed of light in vacuum is 3 x 10^8 m/s, what is its speed in a medium with refractive index 1.5?
A.
2 x 10^8 m/s
B.
2.5 x 10^8 m/s
C.
3 x 10^8 m/s
D.
1.5 x 10^8 m/s
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Solution
The speed of light in a medium is given by v = c/n, where c is the speed of light in vacuum and n is the refractive index. Thus, v = 3 x 10^8 m/s / 1.5 = 2 x 10^8 m/s.
Correct Answer: A — 2 x 10^8 m/s
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Q. If the speed of light in vacuum is 3 x 10^8 m/s, what is its speed in a medium with a refractive index of 1.5?
A.
2 x 10^8 m/s
B.
2.5 x 10^8 m/s
C.
3 x 10^8 m/s
D.
1.5 x 10^8 m/s
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Solution
Speed of light in a medium = speed of light in vacuum / refractive index = (3 x 10^8 m/s) / 1.5 = 2 x 10^8 m/s.
Correct Answer: A — 2 x 10^8 m/s
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Q. If the speed of light in vacuum is 3 x 10^8 m/s, what is its speed in glass (n = 1.5)?
A.
2 x 10^8 m/s
B.
2.5 x 10^8 m/s
C.
3 x 10^8 m/s
D.
1.5 x 10^8 m/s
Show solution
Solution
The speed of light in a medium is given by v = c/n. Therefore, v = (3 x 10^8 m/s) / 1.5 = 2 x 10^8 m/s.
Correct Answer: A — 2 x 10^8 m/s
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Q. If the speed of light is 3 x 10^8 m/s, what is its value in km/s? (2021)
A.
3 x 10^5 km/s
B.
3 x 10^6 km/s
C.
3 x 10^7 km/s
D.
3 x 10^8 km/s
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Solution
To convert m/s to km/s, divide by 1000. Therefore, 3 x 10^8 m/s = 3 x 10^5 km/s.
Correct Answer: A — 3 x 10^5 km/s
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Q. If the total current in an AC circuit is 5A and the voltage is 120V, what is the impedance? (2023)
A.
24Ω
B.
20Ω
C.
30Ω
D.
15Ω
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Solution
Impedance (Z) = Voltage (V) / Current (I) = 120V / 5A = 24Ω.
Correct Answer: B — 20Ω
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Q. If the voltage across a capacitor in an AC circuit is 100 V and the capacitive reactance is 50 Ω, what is the current through the capacitor? (2021)
A.
2 A
B.
0.5 A
C.
5 A
D.
10 A
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Solution
Using Ohm's law, I = V/Xc = 100 V / 50 Ω = 2 A.
Correct Answer: B — 0.5 A
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Q. If the voltage across a capacitor in an AC circuit is 100 V and the capacitive reactance is 50 ohms, what is the current through the capacitor? (2021)
A.
2 A
B.
1 A
C.
0.5 A
D.
0.25 A
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Solution
Current I = V/Xc = 100 V / 50 ohms = 2 A.
Correct Answer: B — 1 A
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Q. If the voltage across a capacitor in an AC circuit is 100 V and the capacitive reactance is 50 ohms, what is the current? (2020)
A.
2 A
B.
0.5 A
C.
5 A
D.
10 A
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Solution
Using Ohm's law, I = V/Xc = 100 V / 50 ohms = 2 A.
Correct Answer: B — 0.5 A
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Q. If the voltage across a capacitor in an AC circuit is 100V and the capacitive reactance is 50Ω, what is the current through the capacitor? (2023)
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Solution
Current (I) = Voltage (V) / Capacitive Reactance (XC) = 100V / 50Ω = 2A.
Correct Answer: B — 2A
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Q. If the voltage across a capacitor in an AC circuit is 100V and the capacitive reactance is 50Ω, what is the current? (2021)
A.
2A
B.
0.5A
C.
5A
D.
100A
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Solution
Using Ohm's law, I = V/Xc = 100V/50Ω = 2A.
Correct Answer: B — 0.5A
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Q. If the voltage across a capacitor in an AC circuit is 100V and the capacitive reactance is 50 ohms, what is the current? (2023)
A.
2A
B.
0.5A
C.
5A
D.
10A
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Solution
Using Ohm's law, I = V/Xc = 100V/50Ω = 2A.
Correct Answer: A — 2A
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