Maximize z = x + y subject to:

1. x + y ≤ 10
2. 3y - 2x ≤ 15
3. x ≤ 6
4. x, y ≥ 0

Find the maximum value.

Question 2:

The equation (cos(p) - 1)x² + (cos(p))x + sin(p) = 0, where x is a variable with real roots. Then the interval of p may be any one of the following:

1. (0, 2π)
2. (-π, 0)
3. (-π/2, π/2)
4. (0, π)

Question 3:

If AX = B, where:

1. A = [[1, -1, 1], [2, 1, -3], [1, 1, 1]]
2. X = [x, y, z]
3. B = [4, 0, 2]

Then 2x + y - z is:

1. 2
2. 1
3. 4
4. -2

Question 4:

The equation of the plane passing through the point (1, 1, 1) and perpendicular to the planes 2x + y - 2z = 5 and 3x - 6y - 2z = 7 is:

1. 14x + 2y - 15z = 1
2. -14x + 2y + 15z = 3
3. 14x - 2y + 15z = 27
4. 14x + 2y + 15z = 31

Question 5:

Using the rules in logic, write the negation of the following:

(p ∧ q) ∧ (q ∨ ¬r)

Question 6:

A student scores the following marks in five tests: 45, 54, 41, 57, 43. His score is not known for the sixth test. If the mean score is 48 in the six tests, then the standard deviation of the marks in six tests is:

1. 100/3
2. 10/3
3. 10/√3
4. 100/√3

Question 7:

A committee of 11 members is to be formed out of 8 males and 5 females. If m is the number of ways the committee is formed with at least 6 males and n is the number of ways with at least 3 females, then:

1. n = m - 8
2. m = n = 78
3. m = n = 68
4. m + n = 68

Question 8:

The integral ∫ (csc(x) / cos²(1 + log(tan(x/2)))) dx is equal to:

1. sin²(1 + log(tan(x/2))) + C
2. tan(1 + log(tan(x/2))) + C
3. -tan(1 + log(tan(x/2))) + C
4. sec²(1 + log(tan(x/2))) + C

Question 9:

The value of √3 csc(20°) - sec(20°) is:

1. 4
2. 2
3. 3
4. 1

(Chemistry)

Question 1:

Which has the highest first ionization energy?

Question 2:

The half-life of a first-order reaction is 1 hour. What is the fraction of the reactant remaining after 3 hours?

Question 3:

Which of the following represents the Stephan Reaction?

1. C₆H₅CH₂Cl → C₆H₅CH₂OH (using NaOH)
2. C₆H₅Cl → C₆H₅OH (using NaOH)
3. C₆H₅Cl → C₆H₅CH₃ (using AlCl₃)
4. C₆H₅CH₃ → C₆H₅Cl (using NaOCl)

Question 4:

What is the maximum oxidation state of an element in the periodic table?

1. +7
2. +6
3. +5
4. +4

Question 5:

In a BCC (Body-Centered Cubic) structure, the radius of the atoms is:

1. a/2
2. a/4
3. (√3/4)a
4. (√2/4)a

Question 6:

What is the depression in the freezing point?

1. ΔT_f = K_f × m
2. ΔT_f = K_b × m
3. ΔT_f = m × 100
4. ΔT_f = K_f × (1/m)

Question 7:

Arrange the following substances in decreasing order of their boiling points:

1. Water, ethanol, acetone
2. Ethanol, acetone, water
3. Acetone, water, ethanol
4. Water, acetone, ethanol

(Physics)

Question 1 :

The root mean square velocity (v_rms) of a gas is given by:

1. v_rms = √(3RT/M)
2. v_rms = √(2RT/M)
3. v_rms = √(3kT/m)
4. v_rms = √(2kT/m)

Question 2:

The ideal gas equation is given by PV = nRT. Which of the following statements is correct for an ideal gas?

1. The equation holds at all temperatures and pressures.
2. It holds only at low temperatures and high pressures.
3. It holds only at high temperatures and low pressures.
4. It holds only for gases with large intermolecular forces.

Question 3:

What is the ratio C_p/C_v for a monatomic and diatomic gas?

1. C_p/C_v = 5/3 for monatomic, 7/5 for diatomic
2. C_p/C_v = 3/2 for monatomic, 5/3 for diatomic
3. C_p/C_v = 5/3 for monatomic, 7/5 for diatomic
4. C_p/C_v = 3/2 for both monatomic and diatomic

Question 4:

How do you find the amplitude of a simple harmonic oscillator?

1. A = √(x² + v²)
2. A = √(x₀² + v₀²)
3. A = √(2E/k)
4. A = √(2k/E)

Question 5:

What is the overtone of a vibrating string?

1. The second harmonic
2. The first harmonic
3. The third harmonic
4. The fifth harmonic

Question 6:

Which of the following is a logic gate that gives an output of 1 when the inputs are different?

1. AND Gate
2. OR Gate
3. XOR Gate
4. NOT Gate