Identify the product A and product B in the following set of reactions.

The molar conductivity for electrolytes $$A$$ and $$B$$ are plotted against $$C^{3 / 2}$$ as shown below. Electrolytes $$A$$ and $$B$$ respectively are:

Given below are two statements :

Statement (I) : The oxidation state of an element in a particular compound is the charge acquired by its atom on the basis of electron gain enthalpy consideration from other atoms in the molecule.

Statement (II) : $$\mathrm{p} \pi-\mathrm{p} \pi$$ bond formation is more prevalent in second period elements over other periods.

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

$$0.05 \mathrm{M} \mathrm{~CuSO}_4$$ when treated with $$0.01 \mathrm{M} \mathrm{~K}_2 \mathrm{Cr}_2 \mathrm{O}_7$$ gives green colour solution of $$\mathrm{Cu}_2 \mathrm{Cr}_2 \mathrm{O}_7$$. The two solutions are separated as shown below : [SPM : Semi Permeable Membrane]

Due to osmosis :

Identify major product "X" formed in the following reaction :

The $$\mathrm{F}^{-}$$ ions make the enamel on teeth much harder by converting hydroxyapatite (the enamel on the surface of teeth) into much harder fluoroapatite having the formula.

Identify the incorrect statements regarding primary standard of titrimetric analysis.

(A) It should be purely available in dry form.

(B) It should not undergo chemical change in air.

(C) It should be hygroscopic and should react with another chemical instantaneously and stoichiometrically.

(D) It should be readily soluble in water.

(E) $$\mathrm{KMnO}_4$$ & $$\mathrm{NaOH}$$ can be used as primary standard.

Choose the correct answer from the options given below :

What is the structure of C?

The electronic configuration of $$\mathrm{Cu}(\mathrm{II})$$ is $$3 \mathrm{~d}^9$$ whereas that of $$\mathrm{Cu}(\mathrm{I})$$ is $$3 \mathrm{~d}^{10}$$. Which of the following is correct?

In which one of the following pairs the central atoms exhibit $$\mathrm{sp}^2$$ hybridization ?

Relative stability of the contributing structures is :

Correct order of basic strength of Pyrrole , Pyridine and Piperidine is :

For the given compounds, the correct order of increasing $$\mathrm{pK}_{\mathrm{a}}$$ value :

Choose the correct answer from the options given below :

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

Assertion (A) : Both rhombic and monoclinic sulphur exist as $$\mathrm{S}_8$$ while oxygen exists as $$\mathrm{O}_2$$.

Reason (R) : Oxygen forms $$p \pi-p \pi$$ multiple bonds with itself and other elements having small size and high electronegativity like $$\mathrm{C}, \mathrm{N}$$, which is not possible for sulphur.

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

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

Assertion (A): The total number of geometrical isomers shown by $$[\mathrm{Co}(\mathrm{en})_2 \mathrm{Cl}_2]^{+}$$ complex ion is three.

Reason (R): $$[\mathrm{Co}(\mathrm{en})_2 \mathrm{Cl}_2]^{+}$$ complex ion has an octahedral geometry.

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

Methods used for purification of organic compounds are based on :

Compare the energies of following sets of quantum numbers for multielectron system.

(A) $$\mathrm{n}=4,1=1$$

(B) $$\mathrm{n}=4,1=2$$

(C) $$\mathrm{n}=3, \mathrm{l}=1$$

(D) $$\mathrm{n}=3,1=2$$

(E) $$\mathrm{n}=4,1=0$$

Choose the correct answer from the options given below :

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

Assertion (A) : $$\mathrm{S}_{\mathrm{N}} 2$$ reaction of $$\mathrm{C}_6 \mathrm{H}_5 \mathrm{CH}_2 \mathrm{Br}$$ occurs more readily than the $$\mathrm{S}_{\mathrm{N}} 2$$ reaction of $$\mathrm{CH}_3 \mathrm{CH}_2 \mathrm{Br}$$.

Reason (R) : The partially bonded unhybridized p-orbital that develops in the trigonal bipyramidal transition state is stabilized by conjugation with the phenyl ring.

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

On reaction of Lead Sulphide with dilute nitric acid which of the following is not formed?

In the following sequence of reaction, the major products B and C respectively are :

The standard reduction potentials at $$298 \mathrm{~K}$$ for the following half cells are given below :

$$\mathrm{Cr}_2 \mathrm{O}_7^{2-}+14 \mathrm{H}^{+}+6 \mathrm{e}^{-} \rightarrow 2 \mathrm{Cr}^{3+}+7 \mathrm{H}_2 \mathrm{O}, \quad \mathrm{E}^{\circ}=1.33 \mathrm{~V}$$

$$\begin{array}{ll} \mathrm{Fe}^{3+}(\mathrm{aq})+3 \mathrm{e}^{-} \rightarrow \mathrm{Fe} & \mathrm{E}^{\circ}=-0.04 \mathrm{~V} \\ \mathrm{Ni}^{2+}(\mathrm{aq})+2 \mathrm{e}^{-} \rightarrow \mathrm{Ni} & \mathrm{E}^{\circ}=-0.25 \mathrm{~V} \\ \mathrm{Ag}^{+}(\mathrm{aq})+\mathrm{e}^{-} \rightarrow \mathrm{Ag} & \mathrm{E}^{\circ}=0.80 \mathrm{~V} \\ \mathrm{Au}^{3+}(\mathrm{aq})+3 \mathrm{e}^{-} \rightarrow \mathrm{Au} & \mathrm{E}^{\circ}=1.40 \mathrm{~V} \end{array}$$

Consider the given electrochemical reactions,

The number of metal(s) which will be oxidized be $$\mathrm{Cr}_2 \mathrm{O}_7^{2-}$$, in aqueous solution is _________.

Total number of essential amino acid among the given list of amino acids is ________.

Arginine, Phenylalanine, Aspartic acid, Cysteine, Histidine, Valine, Proline

Number of ambidentate ligands among the following is _________.

$$\mathrm{NO}_2^{-}, \mathrm{SCN}^{-}, \mathrm{C}_2 \mathrm{O}_4^{2-}, \mathrm{NH}_3, \mathrm{CN}^{-}, \mathrm{SO}_4^{2-}, \mathrm{H}_2 \mathrm{O} \text {. }$$

Number of colourless lanthanoid ions among the following is __________.

$$\mathrm{Eu}^{3+}, \mathrm{Lu}^{3+}, \mathrm{Nd}^{3+}, \mathrm{La}^{3+}, \mathrm{Sm}^{3+}$$

The total number of species from the following in which one unpaired electron is present, is _______.

$$\mathrm{N}_2, \mathrm{O}_2, \mathrm{C}_2^{-}, \mathrm{O}_2^{-}, \mathrm{O}_2^{2-}, \mathrm{H}_2^{+}, \mathrm{CN}^{-}, \mathrm{He}_2^{+}$$

Given below are two statements :

Statement I : The rate law for the reaction $$A+B \rightarrow C$$ is rate $$(r)=k[A]^2[B]$$. When the concentration of both $$\mathrm{A}$$ and $$\mathrm{B}$$ is doubled, the reaction rate is increased "$$x$$" times.

Statement II :

The figure is showing "the variation in concentration against time plot" for a "$$y$$" order reaction.

How many compounds among the following compounds show inductive, mesomeric as well as hyperconjugation effects?

When equal volume of $$1 \mathrm{~M} \mathrm{~HCl}$$ and $$1 \mathrm{~M} \mathrm{~H}_2 \mathrm{SO}_4$$ are separately neutralised by excess volume of $$1 \mathrm{M}$$ $$\mathrm{NaOH}$$ solution. $$x$$ and $$y \mathrm{~kJ}$$ of heat is liberated respectively. The value of $$y / x$$ is __________.

The heat of solution of anhydrous $$\mathrm{CuSO}_4$$ and $$\mathrm{CuSO}_4 \cdot 5 \mathrm{H}_2 \mathrm{O}$$ are $$-70 \mathrm{~kJ} \mathrm{~mol}^{-1}$$ and $$+12 \mathrm{~kJ} \mathrm{~mol}^{-1}$$ respectively.

The heat of hydration of $$\mathrm{CuSO}_4$$ to $$\mathrm{CuSO}_4 \cdot 5 \mathrm{H}_2 \mathrm{O}$$ is $$-x \mathrm{~kJ}$$. The value of $$x$$ is ________. (nearest integer).

Molarity $$(\mathrm{M})$$ of an aqueous solution containing $$x \mathrm{~g}$$ of anhyd. $$\mathrm{CuSO}_4$$ in $$500 \mathrm{~mL}$$ solution at $$32^{\circ} \mathrm{C}$$ is $$2 \times 10^{-1} \mathrm{M}$$. Its molality will be _________ $$\times 10^{-3} \mathrm{~m}$$. (nearest integer). [Given density of the solution $$=1.25 \mathrm{~g} / \mathrm{mL}$$]

If the domain of the function $$f(x)=\sin ^{-1}\left(\frac{x-1}{2 x+3}\right)$$ is $$\mathbf{R}-(\alpha, \beta)$$, then $$12 \alpha \beta$$ is equal to :

Let three vectors ,$$\overrightarrow{\mathrm{a}}=\alpha \hat{i}+4 \hat{j}+2 \hat{k}, \overrightarrow{\mathrm{b}}=5 \hat{i}+3 \hat{j}+4 \hat{k}, \overrightarrow{\mathrm{c}}=x \hat{i}+y \hat{j}+z \hat{k}$$ form a triangle such that $$\vec{c}=\vec{a}-\vec{b}$$ and the area of the triangle is $$5 \sqrt{6}$$. If $$\alpha$$ is a positive real number, then $$|\vec{c}|^2$$ is equal to:

Let $$f(x)=a x^3+b x^2+c x+41$$ be such that $$f(1)=40, f^{\prime}(1)=2$$ and $$f^{\prime \prime}(1)=4$$. Then $$a^2+b^2+c^2$$ is equal to:

The parabola $$y^2=4 x$$ divides the area of the circle $$x^2+y^2=5$$ in two parts. The area of the smaller part is equal to :

The solution of the differential equation $$(x^2+y^2) \mathrm{d} x-5 x y \mathrm{~d} y=0, y(1)=0$$, is :

Let $$|\cos \theta \cos (60-\theta) \cos (60+\theta)| \leq \frac{1}{8}, \theta \epsilon[0,2 \pi]$$. Then, the sum of all $$\theta \in[0,2 \pi]$$, where $$\cos 3 \theta$$ attains its maximum value, is :

If the sum of the series $$\frac{1}{1 \cdot(1+\mathrm{d})}+\frac{1}{(1+\mathrm{d})(1+2 \mathrm{~d})}+\ldots+\frac{1}{(1+9 \mathrm{~d})(1+10 \mathrm{~d})}$$ is equal to 5, then $$50 \mathrm{~d}$$ is equal to :

The coefficient of $$x^{70}$$ in $$x^2(1+x)^{98}+x^3(1+x)^{97}+x^4(1+x)^{96}+\ldots+x^{54}(1+x)^{46}$$ is $${ }^{99} \mathrm{C}_{\mathrm{p}}-{ }^{46} \mathrm{C}_{\mathrm{q}}$$. Then a possible value of $$\mathrm{p}+\mathrm{q}$$ is :

The frequency distribution of the age of students in a class of 40 students is given below.

If the mean deviation about the median is 1.25, then $$4x+5y$$ is equal to :

Let a circle passing through $$(2,0)$$ have its centre at the point $$(\mathrm{h}, \mathrm{k})$$. Let $$(x_{\mathrm{c}}, y_{\mathrm{c}})$$ be the point of intersection of the lines $$3 x+5 y=1$$ and $$(2+\mathrm{c}) x+5 \mathrm{c}^2 y=1$$. If $$\mathrm{h}=\lim _\limits{\mathrm{c} \rightarrow 1} x_{\mathrm{c}}$$ and $$\mathrm{k}=\lim _\limits{\mathrm{c} \rightarrow 1} y_{\mathrm{c}}$$, then the equation of the circle is :

Let $$\int \frac{2-\tan x}{3+\tan x} \mathrm{~d} x=\frac{1}{2}\left(\alpha x+\log _e|\beta \sin x+\gamma \cos x|\right)+C$$, where $$C$$ is the constant of integration. Then $$\alpha+\frac{\gamma}{\beta}$$ is equal to :

Let $$\alpha, \beta$$ be the roots of the equation $$x^2+2 \sqrt{2} x-1=0$$. The quadratic equation, whose roots are $$\alpha^4+\beta^4$$ and $$\frac{1}{10}(\alpha^6+\beta^6)$$, is:

Let $$f(x)=x^2+9, g(x)=\frac{x}{x-9}$$ and $$\mathrm{a}=f \circ g(10), \mathrm{b}=g \circ f(3)$$. If $$\mathrm{e}$$ and $$l$$ denote the eccentricity and the length of the latus rectum of the ellipse $$\frac{x^2}{\mathrm{a}}+\frac{y^2}{\mathrm{~b}}=1$$, then $$8 \mathrm{e}^2+l^2$$ is equal to.

The shortest distance between the lines $$\frac{x-3}{4}=\frac{y+7}{-11}=\frac{z-1}{5}$$ and $$\frac{x-5}{3}=\frac{y-9}{-6}=\frac{z+2}{1}$$ is:

A variable line $$\mathrm{L}$$ passes through the point $$(3,5)$$ and intersects the positive coordinate axes at the points $$\mathrm{A}$$ and $$\mathrm{B}$$. The minimum area of the triangle $$\mathrm{OAB}$$, where $$\mathrm{O}$$ is the origin, is :

The solution curve, of the differential equation $$2 y \frac{\mathrm{d} y}{\mathrm{~d} x}+3=5 \frac{\mathrm{d} y}{\mathrm{~d} x}$$, passing through the point $$(0,1)$$ is a conic, whose vertex lies on the line :

Let $$\lambda, \mu \in \mathbf{R}$$. If the system of equations

$$\begin{aligned} & 3 x+5 y+\lambda z=3 \\ & 7 x+11 y-9 z=2 \\ & 97 x+155 y-189 z=\mu \end{aligned}$$

has infinitely many solutions, then $$\mu+2 \lambda$$ is equal to :

A ray of light coming from the point $$\mathrm{P}(1,2)$$ gets reflected from the point $$\mathrm{Q}$$ on the $$x$$-axis and then passes through the point $$R(4,3)$$. If the point $$S(h, k)$$ is such that $$P Q R S$$ is a parallelogram, then $$hk^2$$ is equal to:

Let the line $$\mathrm{L}$$ intersect the lines $$x-2=-y=z-1,2(x+1)=2(y-1)=z+1$$ and be parallel to the line $$\frac{x-2}{3}=\frac{y-1}{1}=\frac{z-2}{2}$$. Then which of the following points lies on $$\mathrm{L}$$ ?

Let $$\overrightarrow{O A}=2 \vec{a}, \overrightarrow{O B}=6 \vec{a}+5 \vec{b}$$ and $$\overrightarrow{O C}=3 \vec{b}$$, where $$O$$ is the origin. If the area of the parallelogram with adjacent sides $$\overrightarrow{O A}$$ and $$\overrightarrow{O C}$$ is 15 sq. units, then the area (in sq. units) of the quadrilateral $$O A B C$$ is equal to:

Let $$A=\{2,3,6,7\}$$ and $$B=\{4,5,6,8\}$$. Let $$R$$ be a relation defined on $$A \times B$$ by $$(a_1, b_1) R(a_2, b_2)$$ if and only if $$a_1+a_2=b_1+b_2$$. Then the number of elements in $$R$$ is __________.

Let $$\lim _\limits{n \rightarrow \infty}\left(\frac{n}{\sqrt{n^4+1}}-\frac{2 n}{\left(n^2+1\right) \sqrt{n^4+1}}+\frac{n}{\sqrt{n^4+16}}-\frac{8 n}{\left(n^2+4\right) \sqrt{n^4+16}}\right.$$ $$\left.+\ldots+\frac{n}{\sqrt{n^4+n^4}}-\frac{2 n \cdot n^2}{\left(n^2+n^2\right) \sqrt{n^4+n^4}}\right)$$ be $$\frac{\pi}{k}$$, using only the principal values of the inverse trigonometric functions. Then $$\mathrm{k}^2$$ is equal to _________.

Let $$f:(0, \pi) \rightarrow \mathbf{R}$$ be a function given by $$f(x)=\left\{\begin{array}{cc}\left(\frac{8}{7}\right)^{\frac{\tan 8 x}{\tan 7 x}}, & 0< x<\frac{\pi}{2} \\ \mathrm{a}-8, & x=\frac{\pi}{2} \\ (1+\mid \cot x)^{\frac{\mathrm{b}}{\mathrm{a}}|\tan x|}, & \frac{\pi}{2} < x < \pi\end{array}\right.$$

where $$\mathrm{a}, \mathrm{b} \in \mathbf{Z}$$. If $$f$$ is continuous at $$x=\frac{\pi}{2}$$, then $$\mathrm{a}^2+\mathrm{b}^2$$ is equal to _________.

The remainder when $$428^{2024}$$ is divided by 21 is __________.

If a function $$f$$ satisfies $$f(\mathrm{~m}+\mathrm{n})=f(\mathrm{~m})+f(\mathrm{n})$$ for all $$\mathrm{m}, \mathrm{n} \in \mathbf{N}$$ and $$f(1)=1$$, then the largest natural number $$\lambda$$ such that $$\sum_\limits{\mathrm{k}=1}^{2022} f(\lambda+\mathrm{k}) \leq(2022)^2$$ is equal to _________.

Let the centre of a circle, passing through the points $$(0,0),(1,0)$$ and touching the circle $$x^2+y^2=9$$, be $$(h, k)$$. Then for all possible values of the coordinates of the centre $$(h, k), 4\left(h^2+k^2\right)$$ is equal to __________.

The sum of the square of the modulus of the elements in the set $$\{z=\mathrm{a}+\mathrm{ib}: \mathrm{a}, \mathrm{b} \in \mathbf{Z}, z \in \mathbf{C},|z-1| \leq 1,|z-5| \leq|z-5 \mathrm{i}|\}$$ is __________.

Let the set of all positive values of $$\lambda$$, for which the point of local minimum of the function $$(1+x(\lambda^2-x^2))$$ satisfies $$\frac{x^2+x+2}{x^2+5 x+6}<0$$, be $$(\alpha, \beta)$$. Then $$\alpha^2+\beta^2$$ is equal to _________.

Let $$A$$ be a non-singular matrix of order 3. If $$\operatorname{det}(3 \operatorname{adj}(2 \operatorname{adj}((\operatorname{det} A) A)))=3^{-13} \cdot 2^{-10}$$ and $$\operatorname{det}(3\operatorname{adj}(2 \mathrm{A}))=2^{\mathrm{m}} \cdot 3^{\mathrm{n}}$$, then $$|3 \mathrm{~m}+2 \mathrm{n}|$$ is equal to _________.

Let $$\mathrm{a}, \mathrm{b}$$ and $$\mathrm{c}$$ denote the outcome of three independent rolls of a fair tetrahedral die, whose four faces are marked $$1,2,3,4$$. If the probability that $$a x^2+b x+c=0$$ has all real roots is $$\frac{m}{n}, \operatorname{gcd}(\mathrm{m}, \mathrm{n})=1$$, then $$\mathrm{m}+\mathrm{n}$$ is equal to _________.

A plane EM wave is propagating along $$x$$ direction. It has a wavelength of $$4 \mathrm{~mm}$$. If electric field is in $$y$$ direction with the maximum magnitude of $$60 \mathrm{~Vm}^{-1}$$, the equation for magnetic field is :

A galvanometer has a coil of resistance $$200 \Omega$$ with a full scale deflection at $$20 \mu \mathrm{A}$$. The value of resistance to be added to use it as an ammeter of range $$(0-20) \mathrm{mA}$$ is :

The dimensional formula of latent heat is :

A sphere of relative density $$\sigma$$ and diameter $$D$$ has concentric cavity of diameter $$d$$. The ratio of $$\frac{D}{d}$$, if it just floats on water in a tank is :

The volume of an ideal gas $$(\gamma=1.5)$$ is changed adiabatically from 5 litres to 4 litres. The ratio of initial pressure to final pressure is :

A light emitting diode (LED) is fabricated using GaAs semiconducting material whose band gap is $$1.42 \mathrm{~eV}$$. The wavelength of light emitted from the LED is :

A particle of mass $$m$$ moves on a straight line with its velocity increasing with distance according to the equation $$v=\alpha \sqrt{x}$$, where $$\alpha$$ is a constant. The total work done by all the forces applied on the particle during its displacement from $$x=0$$ to $$x=\mathrm{d}$$, will be :

Given below are two statements :

Statement (I) : When currents vary with time, Newton's third law is valid only if momentum carried by the electromagnetic field is taken into account

Statement (II) : Ampere's circuital law does not depend on Biot-Savart's law.

In the light of the above statements, choose the correct answer from the options given below :

A particle moving in a straight line covers half the distance with speed $$6 \mathrm{~m} / \mathrm{s}$$. The other half is covered in two equal time intervals with speeds $$9 \mathrm{~m} / \mathrm{s}$$ and $$15 \mathrm{~m} / \mathrm{s}$$ respectively. The average speed of the particle during the motion is :

Given below are two statements :

Statement (I) : When an object is placed at the centre of curvature of a concave lens, image is formed at the centre of curvature of the lens on the other side.

Statement (II) : Concave lens always forms a virtual and erect image.

In the light of the above statements, choose the correct answer from the options given below :

A sample of 1 mole gas at temperature $$T$$ is adiabatically expanded to double its volume. If adiab constant for the gas is $$\gamma=\frac{3}{2}$$, then the work done by the gas in the process is :

A capacitor is made of a flat plate of area A and a second plate having a stair-like structure as shown in figure. If the area of each stair is $$\frac{A}{3}$$ and the height is $$d$$, the capacitance of the arrangement is :

One main scale division of a vernier caliper is equal to $$\mathrm{m}$$ units. If $$\mathrm{n}^{\text {th }}$$ division of main scale coincides with $$(n+1)^{\text {th }}$$ division of vernier scale, the least count of the vernier caliper is :

A heavy iron bar, of weight $$W$$ is having its one end on the ground and the other on the shoulder of a person. The bar makes an angle $$\theta$$ with the horizontal. The weight experienced by the person is :

A light unstretchable string passing over a smooth light pulley connects two blocks of masses $$m_1$$ and $$m_2$$. If the acceleration of the system is $$\frac{g}{8}$$, then the ratio of the masses $$\frac{m_2}{m_1}$$ is :

The energy equivalent of $$1 \mathrm{~g}$$ of substance is :

The equivalent resistance between A and B is :

An astronaut takes a ball of mass $$m$$ from earth to space. He throws the ball into a circular orbit about earth at an altitude of $$318.5 \mathrm{~km}$$. From earth's surface to the orbit, the change in total mechanical energy of the ball is $$x \frac{\mathrm{GM}_{\mathrm{e}} \mathrm{m}}{21 \mathrm{R}_{\mathrm{e}}}$$. The value of $$x$$ is (take $$\mathrm{R}_{\mathrm{e}}=6370 \mathrm{~km})$$ :

A bulb and a capacitor are connected in series across an ac supply. A dielectric is then placed between the plates of the capacitor. The glow of the bulb :

A proton, an electron and an alpha particle have the same energies. Their de-Broglie wavelengths will be compared as :

A square loop of edge length $$2 \mathrm{~m}$$ carrying current of $$2 \mathrm{~A}$$ is placed with its edges parallel to the $$x$$-$$y$$ axis. A magnetic field is passing through the $$x$$-$$y$$ plane and expressed as $$\vec{B}=B_0(1+4 x) \hat{k}$$, where $$B_o=5 T$$. The net magnetic force experienced by the loop is _________ $$\mathrm{N}$$.

At the centre of a half ring of radius $$\mathrm{R}=10 \mathrm{~cm}$$ and linear charge density $$4 \mathrm{~nC} \mathrm{~m}^{-1}$$, the potential is $$x \pi \mathrm{V}$$. The value of $$x$$ is _________.

In a Young's double slit experiment, the intensity at a point is $$\left(\frac{1}{4}\right)^{\text {th }}$$ of the maximum intensity, the minimum distance of the point from the central maximum is _________ $$\mu \mathrm{m}$$. (Given : $$\lambda=600 \mathrm{~nm}, \mathrm{~d}=1.0 \mathrm{~mm}, \mathrm{D}=1.0 \mathrm{~m}$$)

If $$\vec{a}$$ and $$\vec{b}$$ makes an angle $$\cos ^{-1}\left(\frac{5}{9}\right)$$ with each other, then $$|\vec{a}+\vec{b}|=\sqrt{2}|\vec{a}-\vec{b}|$$ for $$|\vec{a}|=n|\vec{b}|$$ The integer value of $$\mathrm{n}$$ is _________.

Two persons pull a wire towards themselves. Each person exerts a force of $$200 \mathrm{~N}$$ on the wire. Young's modulus of the material of wire is $$1 \times 10^{11} \mathrm{~N} \mathrm{~m}^{-2}$$. Original length of the wire is $$2 \mathrm{~m}$$ and the area of cross section is $$2 \mathrm{~cm}^2$$. The wire will extend in length by _________ $$\mu \mathrm{m}$$.

The position, velocity and acceleration of a particle executing simple harmonic motion are found to have magnitudes of $$4 \mathrm{~m}, 2 \mathrm{~ms}^{-1}$$ and $$16 \mathrm{~ms}^{-2}$$ at a certain instant. The amplitude of the motion is $$\sqrt{x}, \mathrm{~m}$$ where $$x$$ is _________.

A star has $$100 \%$$ helium composition. It starts to convert three $${ }^4 \mathrm{He}$$ into one $${ }^{12} \mathrm{C}$$ via triple alpha process as $${ }^4 \mathrm{He}+{ }^4 \mathrm{He}+{ }^4 \mathrm{He} \rightarrow{ }^{12} \mathrm{C}+\mathrm{Q}$$. The mass of the star is $$2.0 \times 10^{32} \mathrm{~kg}$$ and it generates energy at the rate of $$5.808 \times 10^{30} \mathrm{~W}$$. The rate of converting these $${ }^4 \mathrm{He}$$ to $${ }^{12} \mathrm{C}$$ is $$\mathrm{n} \times 10^{42} \mathrm{~s}^{-1}$$, where $$\mathrm{n}$$ is _________. [ Take, mass of $${ }^4 \mathrm{He}=4.0026 \mathrm{u}$$, mass of $${ }^{12} \mathrm{C}=12 \mathrm{u}$$]

A string is wrapped around the rim of a wheel of moment of inertia $$0.40 \mathrm{~kgm}^2$$ and radius $$10 \mathrm{~cm}$$. The wheel is free to rotate about its axis. Initially the wheel is at rest. The string is now pulled by a force of $$40 \mathrm{~N}$$. The angular velocity of the wheel after $$10 \mathrm{~s}$$ is $$x \mathrm{~rad} / \mathrm{s}$$, where $$x$$ is __________.

When a coil is connected across a $$20 \mathrm{~V}$$ dc supply, it draws a current of $$5 \mathrm{~A}$$. When it is connected across $$20 \mathrm{~V}, 50 \mathrm{~Hz}$$ ac supply, it draws a current of $$4 \mathrm{~A}$$. The self inductance of the coil is __________ $$\mathrm{mH}$$. (Take $$\pi=3$$)

The current flowing through the $$1 \Omega$$ resistor is $$\frac{n}{10}$$ A. The value of $$n$$ is _______.