Equation of the line of the shortest distance between the lines x ⁡ 1 = y ⁡ - 1 = z ⁡ 1 and x ⁡ - 1 0 = y ⁡ + 1 - 2 = z ⁡ 1 is

x ⁡ - 1 1 = y ⁡ + 1 - 1 = z ⁡ - 2

x ⁡ 1 = y ⁡ - 1 = z ⁡ - 2

x ⁡ - 1 1 = y ⁡ + 1 - 1 = z ⁡ 1

If x = a t 2 , y = 2 a t , then find d 2 y d x 2 .

Given that we have, x = a t 2 , y = 2 a t Now, differentiate x with respect to t , then we get: d x d t = 2 a t Similarly, differentiate y with respect to t , then: d y d t = 2 a ∴ d y d x = d y d t d x d t = 2 a 2 a t = 1 t Now, differentiate again with respect to x , then: d 2 y d x 2 = - 1 t 2 · d t d x ⇒ d 2 y d x 2 = - 1 t 2 · 1 d x d t ⇒ d 2 y d x 2 = - 1 t 2 · 1 2 a t ∴ d 2 y d x 2 = - 1 2 a t 3 .

Find the shortest distance between the lines r → = ( 4 i ^ - j ^ ) + λ ( i ^ + 2 j ^ - 3 k ^ ) , and r → = ( i ^ - j ^ + 2 k ^ ) + μ ( 2 i ^ + 4 j ^ - 5 k ^ )

We know that the shortest distance between the lines r → = a → 1 + λ b → 1 and r → = a → 2 + λ b → 2 is given by d = a → 2 - a → 1 · b → 1 × b → 2 b → 1 × b → 2 Here, we have, a → 1 = 4 i ^ - j ^ , a → 2 = i ^ - j ^ + 2 k ^ , b → 1 = i + 2 j - 3 k ^ and b → 2 = 2 i ^ + 4 j ^ - 5 k ^ Now, a → 2 - a → 1 = i ^ - j ^ + 2 k ^ - 4 i ^ + j ^ = - 3 i ^ + 0 j ^ + 2 k ^ And b → 1 × b → 2 = i ^ j ^ k ^ 1 2 - 3 2 4 - 5 = - 10 + 12 i ^ - - 5 + 6 j ^ + 4 - 4 k ^ = 2 i ^ - j ^ + 0 k ^ Therefore, a → 2 - a → 1 · b → 1 × b → 2 = ( - 3 i ^ + 0 j ^ + 2 k ^ ) · ( 2 i ^ - j ^ + 0 k ^ ) = - 6 + 0 + 0 = - 6 And b → 1 × b → 2 = 4 + 1 + 0 = 5 Therefore, d = a → 2 - a → 1 · b → 2 × b → 1 b → 2 × b → 1 = - 6 5 = 6 5 5 units. Hence, the shortest distance between the lines is 6 5 5 units.

If x = a ( 2 θ - sin 2 θ ) and y = a ( 1 - cos 2 θ ) , find d y d x when θ = π 3 .

Given that we have, x = a ( 2 θ - sin 2 θ ) and y = a ( 1 - cos 2 θ ) Now, differentiate x with respect to θ , then we get: d x d θ = a ( 2 - cos 2 θ · 2 ) ⇒ d x d θ = 2 a ( 1 - cos 2 θ ) Similarly, differentiate y with respect to θ , then d y d θ = a ( sin 2 θ · 2 ) ⇒ d y d θ = 2 a sin 2 θ ∴ d y d x = d y d θ d x d θ = 2 a sin 2 θ 2 a ( 1 - cos 2 θ ) ⇒ d y d x = sin 2 θ 1 - cos 2 θ ⇒ d y d x = 2 sin θ cos θ 2 sin 2 θ ⇒ d y d x = cot θ ∴ d y d x θ = π 3 = cot π 3 = 1 3 .

The nuclear radius of Al 13 27 is 3 . 6 fermi. Find the nuclear radius of Cu 29 64 .

Radius of Al is r Al = r ∘ A 1 3 = r ∘ 27 1 3 = 3 . 6 fermi, here A is atomic mass number corresponding to Al . For the radius of Cu is r Cu = r ∘ A 1 3 = r ∘ 64 1 3 . Now, r Cu r Al = 64 1 3 27 1 3 = 4 3 ⇒ r Cu = r Al × 4 3 = 4 . 8 fermi. Hence, the nuclear radius of Cu 29 64 is 4 . 8 fermi.

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Important Questions on Amines

HARD

Chemistry>Organic Chemistry>Amines>Structure of Amines

Equation of the line of the shortest distance between the lines

\frac{x}{1} = \frac{y}{- 1} = \frac{z}{1}

and

\frac{x - 1}{0} = \frac{y + 1}{- 2} = \frac{z}{1}

EASY

Chemistry>Organic Chemistry>Amines>Structure of Amines

The ratio of mass densities of nuclei of

^{40} C a

and

^{16} O

is close to:

MEDIUM

Chemistry>Organic Chemistry>Amines>Structure of Amines

x = a t^{2}, y = 2 a t

, then find

\frac{d^{2} y}{d x^{2}}

MEDIUM

Chemistry>Organic Chemistry>Amines>Structure of Amines

Statement (a): Carboxylic acids with $C_{10}$ and above are insoluble in water.

Statement (b): Ethanoic acid is having less boiling point than $propan - 1 - ol$ .

The correct answer is

EASY

Chemistry>Organic Chemistry>Amines>Structure of Amines

Chemical formula of urotropine is

MEDIUM

Chemistry>Organic Chemistry>Amines>Structure of Amines

Find the shortest distance between the lines

\vec{r} = (4 \hat{i} - \hat{j}) + λ (\hat{i} + 2 \hat{j} - 3 \hat{k})

, and

\vec{r} = (\hat{i} - \hat{j} + 2 \hat{k}) + μ (2 \hat{i} + 4 \hat{j} - 5 \hat{k})

MEDIUM

Chemistry>Organic Chemistry>Amines>Structure of Amines

x = a (2 θ - \sin 2 θ) and y = a (1 - \cos 2 θ)

, find

\frac{d y}{d x}

when

θ = \frac{π}{3}

MEDIUM

Chemistry>Organic Chemistry>Amines>Structure of Amines

Given below are two statements: one is labelled as Assertion and the other is labelled as Reason.

Assertion: A mixture contains benzoic acid and napthalene. The pure benzoic acid can be separated out by the use of benzene.

Reason: Benzoic acid is soluble in hot water.

In the light of the above statements, choose the most appropriate answer from the options given below.

MEDIUM

Chemistry>Organic Chemistry>Amines>Structure of Amines

x = 3 t a n t

and

y = 3 s e c t,

then the value of

\frac{d^{2} y}{d x^{2}}

t = \frac{π}{4},

is:

MEDIUM

Chemistry>Organic Chemistry>Amines>Structure of Amines

x = \sqrt{2^{{cosec}^{- 1} t}}

and

y = \sqrt{2^{\sec^{- 1} t}}, (|t| \geq 1),

then

\frac{d y}{d x}

is equal to

MEDIUM

Chemistry>Organic Chemistry>Amines>Structure of Amines

The entropy of vaporization of acetic acid is:

EASY

Chemistry>Organic Chemistry>Amines>Structure of Amines

The number of structural isomers possible from the molecular formula

C_{3} H_{9} N

is:

HARD

Chemistry>Organic Chemistry>Amines>Structure of Amines

In vapour phase, acetic acid exists as

MEDIUM

Chemistry>Organic Chemistry>Amines>Structure of Amines

The shortest distance between the lines

\frac{x}{2} = \frac{y}{2} = \frac{z}{1}

and

\frac{x + 2}{- 1} = \frac{y - 4}{8} = \frac{z - 5}{4},

lies in the interval:

MEDIUM

Chemistry>Organic Chemistry>Amines>Structure of Amines

Two statements are given belwo :

Statement I : The melting point of monocarboxylic acid with even number of carbon atoms is higher than that of with odd number of carbon atoms acid immediately below and above it in the series.

Statement II : The solubility of monocarboxylic acids in water decreases with increase in molar mass.

Choose the most appropriate option :

EASY

Chemistry>Organic Chemistry>Amines>Structure of Amines

Carboxylic acids have higher boiling points than aldehydes, ketones and even alcohols of comparable molecular mass. It is due to their

HARD

Chemistry>Organic Chemistry>Amines>Structure of Amines

The derivative of

\cos^{- 1} (2 x^{2} - 1)

w.r.t.

\cos^{- 1} x

EASY

Chemistry>Organic Chemistry>Amines>Structure of Amines

How many primary amines with empirical formula

C_{4} H_{11} N

are possible?

MEDIUM

Chemistry>Organic Chemistry>Amines>Structure of Amines

The nuclear radius of

{}_{13}^{27}{Al}

3.6

fermi. Find the nuclear radius of

{}_{29}^{64}{Cu}

MEDIUM

Chemistry>Organic Chemistry>Amines>Structure of Amines

A nucleus with

A = 235

splits into two nuclei whose mass numbers are in the ratio

2 : 1

. If

R_{0} = 1.4 fm

, find the radii of the new nuclei.

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Important Questions on Amines

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