Engineering Entrance
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 + 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 + 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 + 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 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
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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 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 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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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 is 100V and the capacitive reactance is 50 ohms, what is the current through the capacitor? (2022)
A.
2 A
B.
1 A
C.
0.5 A
D.
0.25 A
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Solution
I = V/Xc = 100/50 = 2 A.
Correct Answer: B — 1 A
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Q. If the voltage across a resistor is 240V and the resistance is 60 ohms, what is the current flowing through it? (2020)
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Solution
I = V / R = 240V / 60Ω = 4A.
Correct Answer: C — 6A
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Q. If the voltage across a resistor is halved, what happens to the current through it? (2020)
A.
It doubles
B.
It halves
C.
It remains the same
D.
It quadruples
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Solution
According to Ohm's law, if voltage is halved and resistance remains constant, the current also halves.
Correct Answer: B — It halves
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Q. If the volume of a gas is doubled at constant temperature, what happens to its pressure?
A.
It doubles
B.
It halves
C.
It remains the same
D.
It quadruples
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Solution
According to Boyle's Law, at constant temperature, pressure is inversely proportional to volume. Therefore, if the volume is doubled, the pressure is halved.
Correct Answer: B — It halves
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Q. If the volume of a gas is halved while keeping the temperature constant, what happens to the pressure of the gas?
A.
Halves
B.
Doubles
C.
Remains the same
D.
Increases by 50%
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Solution
According to Boyle's Law, if the volume is halved, the pressure doubles.
Correct Answer: B — Doubles
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Q. If the wavelength of light used in a single-slit diffraction experiment is increased, what will happen to the position of the first minimum? (2022)
A.
It moves closer to the center
B.
It moves away from the center
C.
It remains unchanged
D.
It disappears
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Solution
The position of the first minimum in single-slit diffraction is given by a sin(θ) = λ. Increasing λ will cause θ to increase, moving the minimum away from the center.
Correct Answer: B — It moves away from the center
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Q. If two charges +Q and -Q are placed at a distance 'd' apart, what is the electric field at the midpoint? (2020)
A.
0
B.
kQ/d^2
C.
kQ/(2d^2)
D.
kQ/d
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Solution
At the midpoint, the electric fields due to +Q and -Q cancel each other out, resulting in a net electric field of 0.
Correct Answer: A — 0
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Q. If two dice are rolled, what is the probability that the sum is 7?
A.
1/6
B.
1/12
C.
1/36
D.
5/36
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Solution
Possible combinations for sum 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6 combinations. Total outcomes = 36. Probability = 6/36 = 1/6.
Correct Answer: D — 5/36
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Q. If two identical capacitors of 4 µF each are connected in series, what is the equivalent capacitance?
A.
2 µF
B.
4 µF
C.
6 µF
D.
8 µF
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Solution
C_eq = 1 / (1/C1 + 1/C2) = 1 / (1/4 + 1/4) = 2 µF.
Correct Answer: A — 2 µF
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Q. If two resistors of 4Ω and 6Ω are connected in series, what is the total resistance? (2023)
A.
10Ω
B.
24Ω
C.
2.4Ω
D.
12Ω
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Solution
In series, R_eq = R1 + R2 = 4Ω + 6Ω = 10Ω.
Correct Answer: A — 10Ω
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Q. If x = 3, what is the value of 2x^2 - 5x + 1? (2022)
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Solution
2(3^2) - 5(3) + 1 = 18 - 15 + 1 = 4.
Correct Answer: D — 6
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Q. If x = 5, what is the value of 2x + 3? (2022)
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Solution
Substituting x = 5, we get 2(5) + 3 = 10 + 3 = 13.
Correct Answer: B — 13
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Q. If x = 5, what is the value of 3x + 2? (2022)
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Solution
3(5) + 2 = 15 + 2 = 17.
Correct Answer: A — 17
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Q. If you have 0.25 moles of sulfuric acid (H2SO4), what is its mass? (2023)
A.
49 g
B.
98 g
C.
25 g
D.
12.5 g
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Solution
Molar mass of H2SO4 = 2*1 + 32 + 4*16 = 98 g/mol. Mass = 0.25 moles x 98 g/mol = 24.5 g.
Correct Answer: B — 98 g
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Q. If z = -2 + 2i, what is the value of |z|? (2019)
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Solution
|z| = √((-2)^2 + (2)^2) = √(4 + 4) = √8 = 2√2.
Correct Answer: A — 2√2
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Q. If z = 1 + i, what is the value of z^2? (2022)
A.
2i
B.
1 + 2i
C.
2
D.
0
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Solution
z^2 = (1 + i)^2 = 1^2 + 2(1)(i) + i^2 = 1 + 2i - 1 = 2i.
Correct Answer: B — 1 + 2i
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Q. If z = 1 + i, what is the value of z²? (2022)
A.
0
B.
2i
C.
2
D.
1 + 2i
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Solution
z² = (1 + i)² = 1² + 2(1)(i) + i² = 1 + 2i - 1 = 2i.
Correct Answer: D — 1 + 2i
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Q. If z = 2 + 2i, what is |z|^2? (2021)
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Solution
|z|^2 = (2^2 + 2^2) = 4 + 4 = 8.
Correct Answer: A — 8
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Q. If z = 2 - 3i, what is the conjugate of z? (2020)
A.
2 + 3i
B.
2 - 3i
C.
-2 + 3i
D.
-2 - 3i
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Solution
The conjugate of z = 2 - 3i is 2 + 3i.
Correct Answer: A — 2 + 3i
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