Monomer A of a polymer on ozonolysis yields two moles of HCHO and one mole of CH$$_3$$COCHO.

(A) Deduce the structure of A.

(B) Write the structure of "all $$cis$$" forms of polymer of compound A.

Fill in the blanks:

(A) $$_{92}^{235}$$U + $$_{0}^{1}$$n $$\to$$ $$_{52}^{137}$$A + $$_{40}^{97}$$B + ____________.

(B) $$_{34}^{82}$$Se $$\to$$ 2 $${}_{ - 1}{e^0}$$ + __________.

(A) Calculate the amount of calcium oxide required to react with 852 g of P$$_4$$O$$_{10}$$.

(B) Write the structure of P$$_4$$O$$_{10}$$.

An element crystallises in fcc lattice having edge length 400 pm. Calculate the maximum diameter of the atom which can be placed in interstitial site such that the structure is not distorted.

Nitrogen gas is absorbed on 20% surface sites. On heating N$$_2$$ gas evolved from sites and was collected at 0.01 atm and 298 K in a container of volume 2.46 cm$$^3$$. Find out the number of surface sites occupied per molecule of N, if the density of surface sites is $$6.023\times10^{14}/\mathrm{cm^3}$$ and surface area is 1000 cm$$^2$$.

Predict whether the following molecules are iso-structural or not. Justify your answer.

(A) NMe$$_3$$

(B) N(SiMe$$_3$$)$$_3$$

In the following reaction

Identify X and Y.

Which of the following disaccharide will not reduce Tollen's reagent?

Write the balanced chemical equation for developing a back and white photographic film. Also explain why the solution of sodium thiosulphate on acidification turns milky white.

$$F{e^{3 + }}\buildrel {SN{C^ - }(excess)} \over \longrightarrow $$ blood red $$F{e^{3 + }}\buildrel {SN{C^ - }(excess)} \over \longrightarrow $$ colourless (B):

(A) Identify compound A and B and write their IUPAC names.

(B) Determine the spin only magnetic moment of B.

For the following reaction

2X(g) $$\to$$ 3Y(g) + 2Z(g)

assuming ideal gas conditions, the data for change of partial pressure with time is as follows:

$$ \begin{array}{llll} \hline \text { Time (in min) } & 0 & 100 & 200 \\ \hline \begin{array}{l} \text { Partial pressure of X } \\ \text { (in mm of Hg) } \end{array} & 800 & 400 & 200 \end{array} $$

Calculate

(A) Order of reaction.

(B) Rate constant.

(C) Time taken for 75% completion of reaction.

(D) Total pressure of the reaction mixture when $$p_x=700$$ mm.

(A) Calculate velocity of electron in the first orbit of hydrogen atom (Given : $$r=a_0=0.529$$ $$\mathop A\limits^o $$).

(B) Calculate the de Broglie's wavelength of the electron in first Bohr orbit.

(C) Calculate the orbital angular momentum of 2p orbital in terms of $$h/2\pi$$ units.

In the following reaction, X is optically active.

$\underset{\mathbf{X}}{\mathrm{C}_5 \mathrm{H}_{13}} \mathrm{~N} \xrightarrow[\mathrm{~N}_2]{\mathrm{NaNO}_2 / \mathrm{HCl}} \mathbf{Y}($ Tertiary alcohol $)+$ Other products

Find X and Y. Is y optically active? Write all the intermediate steps.

Give reasons :

In the following reaction sequence :

Identify A, B, C and D. Also write chemical equations for cons version of A to B and A to C.

In the following reaction sequence, M is a transition metal.

Identify the metal M and MCl$$_4$$. Explain the difference in colours of MCl$$_4$$ and A.

Given : that $${\mu _{obs}} = \sum {{\mu _i}\,{x_i}} $$ where $${\mu _i}$$ is the dipole moment of stable conformer and $${x_i}$$ is the mole fraction of that conformer.

(A) Write stable conformer for Z-CH$$_2$$-CH$$_2$$-Z in Newman's projection.

If $${\mu _{solution}}$$ = 1.0 D and mole fraction of anti-form = 0.82, find $${\mu _{Gauche}}$$.

(B) Write most stable meso conformer of

If (i) Y = CH$$_3$$ about C$$_2$$ - C$$_3$$ rotation and (ii) Y = OH about C$$_1$$ - C$$_2$$ rotation.

(A) Calculate $$\Delta_r G^\circ$$ of the following reaction

$$A{g^ + }(aq.) + C{l^ - }(aq.) \to AgCl(s)$$

Given :

$$\mathrm{\Delta_r G^\circ(AgCl)\quad-109~kJ/mole}$$

$$\mathrm{\Delta_r G^\circ(Cl^-)\quad-129~kJ/mole}$$

$$\mathrm{\Delta_r G^\circ(Ag^+)\quad-77~kJ/mole}$$

(i) Represent the above reaction in form of a cell.

(ii) Calculate E$$^\circ$$ of the cell.

(iii) Find $${\log _{10}}{K_{sp}}$$ of AgCl.

(B) If $$6.539\times10^{-2}$$ g of metallic Zn (amu = 65.39) was added to 100 mL of saturated solution of AgCl, then calculate $${\log _{10}} = {{[Z{n^{2 + }}]} \over {{{[A{g^ + }]}^2}}}$$. Also find how many moles of Ag will be formed.

Given that :

$$\mathrm{Ag^++e^-\to Ag\quad E^\circ=0.80~V}$$

$$\mathrm{Zn^{2+}+2e^-\to Zn\quad E^\circ=-0.76~V}$$

A person goes office either by car, scooter, bus or train, proability of which being $$\frac{1}{7}, \frac{3}{2}, \frac{2}{7}$$ and $$\frac{1}{7}$$, respectively. Probability that he reaches office late, if he takes car, scooter, bus or train is $$\frac{2}{9}, \frac{1}{9}, \frac{4}{9}$$ and $$\frac{1}{9}$$, respectively. Given that he reached office in time, then what is the probability that he travelled by a car?

Find the range of value of $$t$$ for which

$$2 \sin t=\frac{1-2 x+5 x^{2}}{3 x^{2}-2 x-1}, t \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$$

Circles with radii 3, 4 and 5 touch each other externally if P is the point of intersection of tangents to these circles at their points of contact. Find the distance of P from the point of contact.

Find the equation of the plane containing the line $$2 x-y+z-3=0,3 x+y+z=5$$ and at a distance of $$\frac{1}{\sqrt{6}}$$ from the point $$(2,1,-1)$$.

If $$\left|f\left(x_{1}\right)-f\left(x_{2}\right)\right| \leq\left(x_{1}-x_{2}\right)^{2}$$, for all $$x_{1}, x_{2} \in$$ $$\mathbb{R}$$. Find the equation of tangent to the curve $$y=f(x)$$ at the point $$(1,2)$$.

If total number of runs scored in $$n$$ matches is $$\left(\frac{n+1}{4}\right)\left(2^{n+1}-n-2\right)$$ where $$n > 1$$, and the runs scored in the $$k^{\text {th }}$$ match are given by $$k .2^{n+1-k}$$, where $$1 \leq k \leq n$$. Find, $$n$$.

The area of the triangle formed by the intersection of a line parallel to X-axis and passing through $$(h, k)$$ with the lines $$y=x$$ and $$x+y=2$$ is $$4 h^{2}$$. Find the locus of point $$P$$.

Evatuate:

$$\int_\limits{0}^{\pi} e^{|\cos x|}\left[2 \sin \left(\frac{1}{2} \cos x\right)+3 \cos \left(\frac{1}{2} \cos x\right)\right] \sin x ~d x$$

Incident ray is along the unit vector $$\hat{v}$$ and the reflected ray is along the unit vector $$\widehat{w}$$. The normal is along unit vector $$\hat{a}$$ outwards. Express $$\hat{w}$$, in terms of $$\hat{a}$$ and $$\hat{v}$$.

Tangents are drawn from any point on the hyperbola $$\frac{x^{2}}{9}-\frac{y^{2}}{4}=1$$ to the circle $$x^{2}+y^{2}=9$$. Find the locus of mid-point of the chord of contact.

Find the equation of the common tangent in the first quadrant to the circle $$x^{2}+y^{2}=16$$ and the ellipse $$\frac{x^{2}}{25}+\frac{y^{2}}{4}=1$$. Also find the length of the intercept of the tangent between the coordinate axes.

If length of tangent at any point on the curve $$y = f(x)$$ intercepted between the point and the X-axis is of length 1. Find the equation of the curve.

Find the area bounded by the curves $$x^{2}=y, x^{2}=-y$$ and $$y^{2}=4 x-3$$.

If one of the vertices of the square circumscribing the circle $$|z-1|=\sqrt{2}$$ is $$(2+\sqrt{3 i})$$. Find the other vertices of square.

If $$f(x-y)=f(x) \circ g(y)-f(y) \circ g(x)$$ And $$g(x-y) =g(x) \circ g(y)+f(x) \circ f(y)$$ for all $$x, y \in \mathrm{R}$$. If right-hand derivative at $$x=0$$ exists for $$f(x)$$, find the derivative of $$g(x)$$ at $$x=0$$

If $$p(x)$$ be a polynomial of degree 3 satisfying $$p(-1)=10, p(1)=-6$$ and $$p(x)$$ has maximum at $$x=-1$$ and $$p'(x)$$ has minima at $$x=1$$. Find the distance between the local maximum and local minimum of the curve.

If $$f(x)$$ is a differentiable function and $$g(x)$$ is a double differentiable function such that $$|f(x)| \leq 1$$ and $$f'(x)=g(x)$$, where,$$f^{2}(0)+g^{2}(0)=9$$ then prove that there exists some $$c \in(-3,3)$$ such that $$g(c) \circ g^{n}(c) < 0$$.

If $$\left[\begin{array}{lll}4 a^{2} & 4 a & 1 \\ 4 b^{2} & 4 b & 1 \\ 4 c^{2} & 4 c & 1\end{array}\right]\left[\begin{array}{c}f(-1) \\ f(1) \\ f(2)\end{array}\right]=\left[\begin{array}{c}3 a^{2}+3 a \\ 3 b^{2}+3 b \\ 3 c^{2}+3 c\end{array}\right], \quad f(x)$$

is a quadratic function and its maximum value occurs at a point $$\mathrm{V}$$. If A is a point of intersection of $$y=f(x)$$ with $$x$$-axis and point B is such that chord AB subtends a right angle at point $$\mathrm{V}$$. Find the area enclosed by $$f(x)$$ and chord AB.

A whistling train approaches a junction. An observer standing at the junction observes the frequency to be 2.2 kHz and 1.8 kHz of the approaching and the receding train. Find the speed of the train (speed of sound = 300 m/s).

A conducting liquid bubble of radius $$a$$ and thickness $$t(t < < a)$$ is charged to potential V. If the bubble collapses to a droplet, find the potential on the droplet.

The potential energy of a particle of mass m is given by

$$\mathrm{U}(x)=\left\{\begin{array}{cc}\mathrm{E}_{0} & 0 \leq x \leq 1 \\ 0 & x>1\end{array}\right.$$

$$\lambda_{1}$$ and $$\lambda_{2}$$ are the de Broglie wavelengths of the particle, when $$0 \leq x \leq 1$$ and $$x > 1$$, respectively. If the total energy of particle is $$2 \mathrm{E}_{0}$$, find $$\frac{\lambda_{1}}{\lambda_{2}}$$.

A U-tube is rotated about one of its limbs with an angular velocity $$\omega$$. Find the difference in height $$\mathrm{H}$$ of the liquid (density $$\rho$$ ) level, where the diameter of the tube is $$d < <\mathrm{L}$$.

A wooden log of mass M and length L is hinged by a frictionless nail at O; a bullet of mass m strikes with velocity $$v$$ and sticks to it. Find angular velocity of the system immediately after the collision about O.

What will be the minimum angle of incidence such that the total internal reflection occurs on both the surfaces?

The side of a cube is measured by vernier callipers (10 divisions of a vernier scale coincide with 9 divisions of main scale, where 1 division of main scale is 1 mm). The main scale reads 10 mm and first division of vernier scale coincides with the main scale; Mass of the cube is 2.736 g. Find the density of the cube in appropriate significant figures.

An unknown resistance X is to be determined using resistances R$$_1$$, R$$_2$$ or R$$_3$$. Their corresponding null points are A, B and C. Find which of the above will give the most accurate reading and why ?

A transverse harmonic disturbance is produced in a string. The maximum transverse velocity is 3 m/s and the maximum transverse acceleration is 90 m/s$$^2$$ . If the wave velocity is 20 m/s, then find the waveform.

A cylinder of mass m and radius R rolls down on an inclined plane of inclination $$\theta$$. Calculate the linear acceleration of axis of cylinder.

A long solenoid of radius a and number of turns per unit length $$n$$ is enclosed by cylindrical shell of radius R, thickness $$d$$ $$(d < < R)$$ and length L. A variable current $$\mathrm{I}=\mathrm{I}_{0} \sin \omega t$$ flows through the coil. If the resistivity of the material of cylindrical shell is $$\mathrm{P}$$, find the induced current in the shell.

Two identical ladders, each of mass M and length L are resting on the rough horizontal surface as shown in the figure. A block of mass $$m$$ hangs from P. If the system is in equilibrium, find the magnitude and the direction of frictional force at A and B.

Highly energetic electrons are bombarded on a target of an element containing 30 neutrons. The ratio of radii of nucleus to that of Helium nucleus is $$(14)^{\frac{1}{3}}$$. Find

(A) Atomic number of the nucleus;

(B) the frequency of $$\mathrm{K}_{\alpha}$$ line of the X-ray produced.

$$\left(\mathrm{R}=1.1 \times 10^{7} \mathrm{~m}^{-1}\right.$$ and $$\left.c=3 \times 10^{8} \mathrm{~m} / \mathrm{s}\right)$$

A small body attached to one end of a vertically hanging spring is performing SHM about its mean position with angular frequency $$\omega$$ and amplitude $$a$$. If at a height $$y^{\prime}$$ from the mean position, the body gets detached from the spring, calculate the value of $$y^{\prime}$$ so that the height $$\mathrm{H}$$ attained by the mass is maximum. The body does not interact with the spring during its subsequent motion after detachment $$\left(a \omega^{2}>g\right)$$

In the given circuit, the switch S is closed at time $$t=0$$. The charge Q on the capacitor at any instant t is given by $$Q(t)=Q_0(1-e^{-\alpha t})$$. Find the value of Q$$_0$$ and $$\alpha$$ in terms of given parameters shown in the circuit.

Two identical prisms of refractive index $$\sqrt{3}$$ are kept as shown in the figure. A light ray strikes the first prism at face AB. Find

(A) the angle of incidence so that the emergent ray from the first prism has minimum deviation;

(B) through what angle, the prism DCE should be rotated about C so that the final emergent ray also has minimum deviation?

A cylinder of mass $$1 \mathrm{~kg}$$ is given heat of $$20000 \mathrm{~J}$$ at atmospheric pressure. If initially temperature of cylinder is $$20^{\circ} \mathrm{C}$$, find

(A) The final temperature of the cylinder;

(B) The work done by the cylinder;

(C) The change in internal energy of the cylinder.

Given :

The specific heat of cylinder

$$=400 \mathrm{~J} \mathrm{~kg}^{-1 \circ} \mathrm{C}^{-1}$$

Coefficient of volume expansion

$$=9 \times 10^{-5}{ }^{\circ} \mathrm{C}^{-1} \text {; }$$

Atmospheric pressure $$=10^{5} \mathrm{~N} / \mathrm{m}^{2}$$ Density of cylinder $$=9000 \mathrm{~kg} / \mathrm{m}^{3}$$ )

In a moving coil galvanometer, torque on the coil can be expressed as $$\tau=k i$$, where $$i$$ is current through the wire and $$k$$ is constant. The rectangular coil of the galvanometer having numbers of turns $$\mathrm{N}$$, area $$\mathrm{A}$$ and moment of inertia I is placed in magnetic field B. Find

(A) $$k$$ in terms of given parameters $$\mathrm{N}, \mathrm{I}, \mathrm{A}$$ and B;

(B) The torsional constant of the spring, if a current $$i_{0}$$ produces a deflection of $$\frac{\pi}{2}$$ in the coil;

(C) The maximum angle through which coil is deflected, if charge $$\mathrm{Q}$$ is passed through the coil almost instantaneously. (Ignore the damping in mechanical oscillations.)