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Let the two numbers be x, 12 – x. 

Their product p = x (12 – x) = 12x – x² 

To find the maximum product. 

p'(x) = 12 – 2x 

p”(x) = -2 

p'(x) = 0 ⇒ 12 – 2x = 0 ⇒ 2x = 12 

⇒ x = 6 

at x = 6, p”(x) = -2 = -ve 

⇒ p is maximum at x = 6

when x = 6, 12 – x = 12 – 6 = 6 

So the two numbers are 6, 6

A) Density of ice is < Density of water

(d) 9

f(x) = |3 – x| + 9 

Minimum value of |3 – x| = 0 

Minimum value of |3 – x| + 9 = 0 + 9 = 9

1. As compared to coefficient of linear expansion (α) and coefficient of areal expansion (β), γ changes more with temperature.

2. It is constant only at high temperatures.

i. Thermal resistivity (ρ_T) is the reciprocal of thermal conductivity (k).

ii. It is characteristic of a material.

(i) f(x) = |1/x|

f(x) is not continuous at x = 0. So Rolle’s Theorem is not applicable.

(ii) f(x) = tan x 

f(x) is not continuous at x = π/2. So Rolle’s Theorem is not applicable.

(iii) f(x) = x – 2 log x 

f(x) = x – 2 log x 

f(2) = 2 – 2 log 2 = 2 – log 4 

f(7) = 7 – 2 log 7 = 7 – log 49 

f(2) ≠ f(7) 

So Rolle’s theorem is not applicable.

(a) The pressure drop is greater. If the dissipative forces are present, there will be a loss of energy along the sides of the tube and therefore, the pressure drop will be greater.

(b) With the increase in velocity the rate of loss of energy also increase. As a result, the dissipative forces become more significant.

(i) f(x) = (x + 1)/x

The function is not continuous at x = 0. So Lagrange’s mean value theorem is not applicable in the given interval.

(ii) f(x) = |3x + 1|, x ∈ [-1, 3]

The function is not differentiable at x = -1/3. So Lagrange’s mean value theorem is not applicable in the given interval.

They are electromagnetic waves.

Charles’ law:

At constant pressure, volume of a fixed mass of gas is directly proportional to its absolute temperature.

Mathematically,

V ∝ T … ( at constant pressure)

∴ V = kT

where k is constant of proportionality 

∴ \(\frac{V}{T}\) = k = constant

For two gases, \(\frac{V_1}{T_1}=\frac{V_2}{T_2}\) = constant.

Pressure (Gay Lussac’s) law:

At constant volume, pressure of a fixed mass of gas is directly proportional to its absolute temperature.

Mathematically,

P ∝ T .. .(at constant volume)

∴ P = kT

Where, k is constant of proportionality P

∴ \(\frac{P}{T}\) = k = constant

For two gases, \(\frac{P_1}{T_1}=\frac{P_2}{T_2}\) = constant

1. The relation between three variables of a gas i.e., pressure, volume and absolute temperature is called as ideal gas equation.

From Boyle’s law,

V ∝ \(\frac{T}{P}\), at constant temperature ….(1)

From Charles’ law,

V ∝ T, at constant pressure … .(2)

2. Combining equations (1) and (2) 

we get,

∴ V ∝ \(\frac{T}{P}\)

∴ \(\frac{PV}{T}\) = constant

3. For one mole of a gas,

\(\frac{PV}{T}\) = R or PV = RT … (3)

where R is the constant of proportionality.

4. Equation (3) is called ideal gas equation. The value of constant R is same for all gases. Therefore, R is called as universal gas constant. R = 8.31 JK^-1mol^-1.

5. For ‘n’ moles of gas, i.e. if the gas contains ‘n’ moles, equation (3) can be written as,

PV = nRT

(c) π/2

The angle between the parabolas is the angle between the axes = π/2

Boyle’s law:

At constant temperature, the volume of a fixed mass of a gas is inversely proportional to its pressure.

Mathematically,

V ∝ \(\frac{1}{P}\) .. ..(at constant temperature)

V = k × \(\frac{1}{P}\)

where, k is constant of proportionality.

PV = k = constant For two gases,

P₁V₁ = P₂V₂ = constant

At a given temperature, the ratio of the emissive power to the absorptive power for all bodies is a constant and is equal to the emissive power of a perfect black body.

If e is the emissive power and a is the absorptive power then, e/a = constant = 1.

∴ e = a (∵ emissive power of a perfectly black body = 1) 

Thus, the emissive power of a body is equal to its absorptive power, which means good emitter of thermal radiation is also a good absorber of heat.

(i) Gravitational force 

(ii) Electromagnetic force

(iii) nuclear force 

(iv) Weak force

The relative strengths of these forces are

Fg :Fw:Fe:Fs=1:10²⁵:10³⁶:10³⁸ .

(A) gravitational constant = [M^-1 L³ T^-2],

(B) Power = [ML²T^-3]

(C) Coefficient of viscosity = [ ML^-1 T^-1]

(D) Surface tension = [ ML⁰T^-2]

For x = a/b ,

Δx/x = ± ( Δa/a + Δb/b)

Unit of a is meter and unit of b is m/sec².

Longest range force is gravitational force and nuclear force is shortest range force.

One such laws is the Newton’s gravitation law, According to this law everybody in this nature are attracts with other body with a force of attraction which is directly proportional to the product of their masses and inversely proportionally To the square of the distance between them.

Electromagnetic force acts between on all electrically charged particles.

% error in heat dissipated is ±6 %.

When we add or subtract a length from length we get length, So L + L = L AND L – L =L, justify.

No, a quantity having dimension must have some units of its measurement.

Slap x speed = power

Or slap = power/speed = [MLT^-2]

Any group of physical quantities, like work , energy and torque and their dimensions [ ML² T^-2].

Energy = Work done = Force x length

So when the units are doubled, then the unit of energy will increase four times.

Yes, like strain.

Percentage error = 2 %

Let G.P.be a, ar, ar² ,………….. , then

T_n = ar^{n - 1}

Given: a = - 3 and T₄ = (T₂)²

since T₄ = (T₂)² then

ar³ = (ar)²

⇒ (-3)r₃ = (-3r)² = 9r²

⇒ r³/r² = 9/-3 ⇒ r = - 3

Hence, 7^th term = ar⁶ = (-3)(-3)⁶

= (-3)⁷ = - 2187

The first term, a of the G.P. is given as –3.

The nth term of the G.P. is given by a_n = ar^n-1

∴Second term, a₂ = ar = –3r

4th term, a₄ = ar³ = –3r³

It is given that, –3r³= (–3r)² .

∴ –3r³ = 9r²   ⇒ r = –3

Thus, 7th term, a₇ = ar6 = –3*(–3)⁶ = –2187

If - 2/7, x , - 7/2 are in G.P.

then, x² = (-2/7)(-7/2) (By definition)

⇒ x² = 1 ⇒ x = 1

∴ x = 1 or x = - 1

x²=(-2/7)(-7/2)

x=+1,-1.

for x is equal to -1 or 1, Given numbers are in G.P.

Work done against frictional force equals the kinetic energy of the body.
when a body of mass m, moves with velocity v, it has kinetic energy k= 1/2mv² , this energy is utilized in doing work against the frictional force between the tyres of the car and road.
∴ kinetic energy = work done against friction force 1/2m² =μ mgs
where s is the distance in which the car is stopped and μ is coefficient of kinetic friction.
Given v=v₀ 

s=v₀²/2μg

He was debarred from appearing at the examination.

You must tell me the truth.

When a ray of white light passes through, prism is, split up into its constituent colours. This is known as ''dispersion of light.".

The constituent colours i.e. seven colour may be easily remembered by word "VIBGYOR". The violet rays are deviated most and the red rays are least. The deviation of the remaining five colours lies between two colours.

(B) Good manners

Correct answer is (C) Opuntia

Correct answer is (B) Cheetah

Correct answer is (B) Dodo

Correct Answer is: (C) Lactobacillus 

When Lactococcus lactis is added to milk, the bacterium uses enzymes to produce energy (ATP) from lactose. 

The byproduct of ATP production is lactic acid. The lactic acid curdles the milk that then separates to form curds, which are used to produce cheese and whey.

Correct Answer is: (B) Nematodes 

First demonstrated in the model free living nematode, Caenorhabtidis elegans; the phenomenon of RNA i has been successfully used to suppress essential genes of plant parasitic nematodes involved in parasitism, nematode development and mRNA metabolism.

In nematodes, the body is round bilaterally symmetrical triploblastic and pseudocoelomates 

Eg: Wucheraria Bancroft, Ascaris.

Correct Answer is: (B) Dodo 

It's commonly believed that the dodo went extinct because Dutch sailors ate the beast to extinction after finding that the bird was incredibly easy to catch due to the fact it had no fear of humans, (why it didn't fear the creature many times its size is a mystery for another day).

The modulus of elasticity of a material is the ratio of stress to the corresponding strain.

1. Young’s modulus indicates the resistance of an elastic solid to elongation or compression.

2. Young’s modulus of a material is useful for characterization of an object subjected to compression or tension.

• When a person cannot see the distant objects clearly he is said to be myopic. 
• Myopia is a condition where image is formed in front of retina. 
• When a person cannot see the nearer objects clearly he is said to be hypermetropic. 
• Hypermetropia is a condition where image is formed behind the retina.

Correction of Myopia 

• Myopia can be corrected by using a concave lens of suitable power. 
• A concave lens of suitable power will bring the image back on to the retina and thus the defect is corrected.

 Correction of Hypermetropia

• Hypermetropia can be corrected by using a convex lens of suitable power.
• Eye-glasses with converging lenses provide the additional focussing power required for forming the image on the retina.

Correct option (B) Both statements 1 and 2 are true but statement 2 is not the correct explanation of statement 1.   

Explanation:

The prism Y recombines the seven colours to form white light.

Correct option: (C) 24 g

1 mole of ozone(O₃) = 48 g

∴ 0.5 mole of ozone(O₃) = \(\cfrac{0.5\, \times\,48}{1}\) = 24 g.