The number of moles of solute present in 1 dm 3 volume of solution

The number of moles of solute dissolved in 1 kg of solvent

The number of moles of solute dissolved in 1 kg of solution

The number of moles of solute dissolved in 100 dm 3 volume of solution

If y = sin - 1 x 2 , prove that 1 - x 2 d 2 y d x 2 - x d y d x - 2 = 0 .

Given y = sin - 1 x 2 Differentiating w.r.t. x , we get ⇒ d y d x = d sin - 1 x 2 d x ⇒ d y d x = 2 sin - 1 x × 1 1 - x 2 ⇒ d y d x = 2 sin - 1 x 1 - x 2 → 1 Again differentiating w.r.t x , we get ⇒ d 2 y d x 2 = 2 d d x sin - 1 x 1 - x 2 = 2 1 - x 2 × d sin - 1 x d x - sin - 1 x × d 1 - x 2 d x 1 - x 2 (Use Quotient Rule in Differentiation) = 2 1 - x 2 × 1 1 - x 2 - sin - 1 x × 1 2 1 - x 2 × ( - 2 x ) 1 - x 2 ⇒ d 2 y d x 2 = 2 1 + x sin - 1 x 1 - x 2 1 - x 2 ⇒ 1 - x 2 d 2 y d x 2 = 2 + 2 x sin - 1 x 1 - x 2 = 2 + x × 2 sin - 1 x 1 - x 2 ⇒ 1 - x 2 d 2 y d x 2 = 2 + x d y d x from 1 ⇒ 1 - x 2 d 2 y d x 2 - x d y d x - 2 = 0 Hence proved.

If y = sin ( sin x ) , prove that d 2 y d x 2 + tan x d y d x + y cos 2 x = 0 .

Given that, we have: y = sin ( sin x ) and to prove: d 2 y d x 2 + tan x d y d x + y cos 2 x = 0 Differentiating with respect to x , we get: d y d x = cos ( sin x ) d d x sin x ⇒ d y d x = cos ( sin x ) cos x Again differentiating with respect to x , we get d 2 y d x 2 = - sin ( sin x ) · cos x · cos x - sin x · cos ( sin x ) [Using Product Rule in Differentiation] ⇒ d 2 y d x 2 = - y cos 2 x - sin x d y d x cos x ⇒ d 2 y d x 2 = - y cos 2 x - tan x d y d x ⇒ d 2 y d x 2 + tan x d y d x + y cos 2 x = 0 Hence, proved.

MEDIUM

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Which measure of dispersion is calculated as the square root of the average of the squared deviations from the mean?

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Important Questions on Continuity and Differentiability

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Second Order Derivative

Molarity is:

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Second Order Derivative

A prism has a refracting angle of

60^{o} .

When placed in the position of minimum deviation, it produces a deviation of

30^{o} .

The angle of incidence is

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Second Order Derivative

5.2

molal aqueous solution of methyl alcohol,

{CH}_{3} OH

, is supplied. What is the mole fraction of methyl alcohol in the solution?

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Second Order Derivative

y = {(\sin^{- 1} x)}^{2}

, prove that

(1 - x^{2}) \frac{d^{2} y}{d x^{2}} - x \frac{d y}{d x} - 2 = 0

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Second Order Derivative

The graph between angle of deviation

(δ)

and angle of incidence

(i)

for a triangular prism is represented by :

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Second Order Derivative

The molarity of urea (molar mass

60 g {mol}^{- 1}

) solution obtained by dissolving

15 g

of urea in

500 {cm}^{3}

of water is

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Second Order Derivative

An antifreeze solution is prepared from

222.6 g

of ethylene glycol

(C_{2} H_{6} O_{2})

and

200 g

of water. Calculate the molality of the solution. If the density of the solution is

1.072 g {mL}^{- 1}

, then what shall be the molarity of the solution?

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Second Order Derivative

Let

f (x)

be a polynomial in

x

. Then the second derivative of

f (e^{x})

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Second Order Derivative

An equilateral glass prism has a refractive index

1.6

in the air. Calculate the angle of minimum deviation of the prism, when kept in a medium of refractive index

\frac{4 \sqrt{2}}{5}

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Second Order Derivative

A ray of light on passing through an equilateral glass prism, suffers a minimum deviation equal to the angle of the prism. The value of refractive index of the material of the prism is _____.

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Second Order Derivative

y = \sin (\sin x)

, prove that

\frac{d^{2} y}{d x^{2}} + \tan x \frac{d y}{d x} + y \cos^{2} x = 0

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Second Order Derivative

A small angled prism of

4 °

deviates a ray through

2.48 °

. Find the refractive Index of the prism.

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Second Order Derivative

For a glass prism, the angle of minimum deviation will be smallest for the light of

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Second Order Derivative

The angle of incidence for a ray of light at a refracting surface of a prism is

45^{o}

. The angle of prism is

60^{o}

. If the ray suffers minimum deviation through the prism, the angle of minimum deviation and refractive index of the material of the prism respectively, are:

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Second Order Derivative

A thin prism having refracting angle

10^{o}

is made of glass of refractive index 1.42. This prism is combined with another thin prism made of glass of refractive index 1.7. This combination produces dispersion without deviation. The refracting angle of second prism should be

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Second Order Derivative

A ray of light is incident at an angle of incidence i, on one face of a prism of angle A (assumed to be small) and emerges normally from the opposite face. If the refractive index of the prism is μ, the angle of incidence i, is nearly equal to

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Second Order Derivative

A beam of light consisting of red, green and blue colours is incident on a right-angled prism

A B C .

The refractive indices of the material of the prism for the above red, green and blue wavelengths are 1.39, 1.44 and 1.47 respectively. The colour/colours transmitted through the face

A C

of the prism will be

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Second Order Derivative

What is a non dispersive medium?

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Second Order Derivative

For a medium, refractive indices for violet, red and yellow are

1.62

1.52

and

1.55

respectively. The dispersive power of the medium will be

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Second Order Derivative

A thin prism

P

of refracting angle

3 °

and refractive index

1.5

is combined with another thin prism

Q

of refractive index

1.6

to produce dispersion without deviation. Then the angle of prism

Q

Which measure of dispersion is calculated as the square root of the average of the squared deviations from the mean?

Important Questions on Continuity and Differentiability

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