Class 11th

The internal energy of one mole of an ideal gas is given by U = (f/2)RT, where f is the number of degrees of freedom.
For a non-linear triatomic molecule (assumed to be a rigid rotator), there are:

3 translational degrees of freedom.

3 rotational degrees of freedom.
Vibrational modes are generally not considered at temperature T unless specified.
Total degrees of freedom f = 3 + 3 = 6.
U = (6/2)RT = 3RT.

The name of the element is derived from its atomic number using the IUPAC nomenclature for elements with Z > 100.
un = 1
nil = 0
enn = 9
So, unnilennium corresponds to the atomic number 109.

p →~ (p ∧ ~ q)
=~ p ∨ ~ (p ∧ ~ q)
=~ p ∨ ~ p ∨ q
=~ p ∨ q

The least count (LC) of the screw gauge is the pitch divided by the number of divisions on the circular scale.
LC = Pitch / N = 0.1 cm / 50 = 0.002 cm.
A measurement taken with this instrument must be a multiple of its least count. We check the options:
(A) 2.121 / 0.002 = 1060.5 (Not a multiple)
(B) 2.123 / 0.002 = 1061.5 (Not a multiple)
(C) 2.124 / 0.002 = 1062 (Is a multiple)
(D) 2.125 / 0.002 = 1062.5 (Not a multiple)
The correctly recorded measurement is 2.124 cm.

∴ σ² ≤ 1/4 (M - m)²
Where M and m are upper and lower bounds of values of any random variable.
∴ σ² < 1/4 (10 - 0)
⇒ 0 < < 5
∴ σ ≠ 6

Let P = (3t², 6t); N = (3t²,0)
M = (3t², 3t)
Equation of MQ: y = 3t
∴ Q = (3/4 t², 3t)
Equation of NQ
y = ( 3t / (3/4 t² - 3t²) ) (x - 3t²)
y - intercept of NQ = 4t = 4/3 ⇒ t = 1/3
∴ MQ = 9/4 t² = 1/4
PN = 6t = 2

A: D ≥ 0
⇒ (m + 1)² - 4 (m + 4) ≥ 0
⇒ m² + 2m + 1 - 4m - 16 ≥ 0
⇒ m² - 2m - 15 ≥ 0
⇒ (m - 5) (m + 3) ≥ 0
⇒ m ∈ (-∞, -3] U [5, ∞)
∴ A = (-∞, -3] U [5, ∞)
B = [-3,5)
A − B = (-∞, −3) U [5, ∞)
A ∩ B = {-3}
B - A = (-3,5)
A U B = R

Ellipse: x²/4 + y²/3 = 1
eccentricity = √ (1 - 3/4) = 1/2
∴ foci = (±1,0)
For hyperbola, given 2a = √2 ⇒ a = 1/√2
∴ hyperbola will be x²/ (1/2) - y²/b² = 1
eccentricity = √ (1 + 2b²)
∴ foci = (±√ ( (1+2b²)/2 ), 0)
∴ Ellipse and hyperbola have same foci
√ ( (1+2b²)/2 ) = 1
⇒ b² = 1/2
∴ Equation of hyperbola: x²/ (1/2) - y²/ (1/2) = 1
⇒ x² - y² = 1/2
Clearly, (√3/2, 1/2) does not lie on it.

Burning of fossil fuels (which contain sulphur and nitrogenous matter) such as coal and oil in power stations and furnaces produce sulphur dioxide and nitrogen oxides which causes acid rain.