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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)

A. 5
B. 6
C. 10
D. 4

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)

A. 6
B. 5
C. 7
D. 8

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)

A. 4
B. 5
C. 6
D. 7

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 + k = 0 are real and distinct, what is the condition for k? (2020)

A. k > 25
B. k < 25
C. k = 25
D. k ≤ 25

Solution

The discriminant must be greater than zero: 5² - 4*1*k > 0, thus k < 25.

Correct Answer: A — k > 25

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Q. If the roots of the equation x² + 5x + q = 0 are 1 and 4, find q. (2019)

A. 5
B. 4
C. 6
D. 7

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)

A. 12
B. 7
C. 15
D. 20

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)

A. 6
B. 7
C. 8
D. 9

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)

A. 12
B. 7
C. 15
D. 10

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)

A. 12
B. 7
C. 10
D. 15

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 + 7x + p = 0 are -3 and -4, what is the value of p? (2019)

A. 12
B. 7
C. 15
D. 10

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

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)

A. -7
B. -5
C. -6
D. -8

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)

A. -7
B. -5
C. 7
D. 5

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)

A. -1
B. 1
C. 5
D. -5

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 roots of the equation x² - 6x + p = 0 are 2 and 4, what is the value of p? (2023)

A. 8
B. 12
C. 6
D. 10

Solution

The product of the roots gives p = 2 * 4 = 8.

Correct Answer: B — 12

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Q. If the shear modulus of a material is 80 GPa, what is the ratio of shear stress to shear strain? (2021)

A. 80 GPa
B. 160 GPa
C. 40 GPa
D. 20 GPa

Solution

Shear modulus (G) = Shear stress / Shear strain, thus it is 80 GPa.

Correct Answer: A — 80 GPa

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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

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

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

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

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

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

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

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 temperature of a gas is doubled at constant volume, what happens to its pressure? (2021)

A. It doubles
B. It triples
C. It halves
D. It remains the same

Solution

According to Gay-Lussac's law, pressure is directly proportional to temperature at constant volume.

Correct Answer: A — It doubles

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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

Solution

Impedance (Z) = Voltage (V) / Current (I) = 120V / 5A = 24Ω.

Correct Answer: B — 20Ω

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Q. If the total impedance in an AC circuit is 20Ω and the current is 2A, what is the voltage across the circuit? (2023)

A. 10V
B. 20V
C. 30V
D. 40V

Solution

Using Ohm's law, V = IZ = 2A * 20Ω = 40V.

Correct Answer: B — 20V

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Q. If the vapor pressure of pure solvent is 100 mmHg, and the vapor pressure of the solution is 80 mmHg, what is the relative lowering of vapor pressure? (2022)

A. 0.2
B. 0.1
C. 0.5
D. 0.3

Solution

Relative lowering of vapor pressure = (P° - P) / P° = (100 - 80) / 100 = 0.2.

Correct Answer: A — 0.2

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Q. If the vapor pressure of pure solvent is 100 mmHg, and the vapor pressure of the solution is 80 mmHg, what is the mole fraction of the solute? (2022)

A. 0.2
B. 0.4
C. 0.5
D. 0.8

Solution

Using Raoult's Law: P_solution = X_solvent * P_solvent. X_solvent = P_solution / P_solvent = 80/100 = 0.8, thus X_solute = 1 - X_solvent = 0.2.

Correct Answer: B — 0.4

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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

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 Ω, what is the current through the capacitor? (2021)

A. 2 A
B. 0.5 A
C. 5 A
D. 10 A

Solution

Using Ohm's law, I = V/Xc = 100 V / 50 Ω = 2 A.

Correct Answer: B — 0.5 A

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