At constant temperature and volume, X decomposes as 2X(g) $$\to$$ 3Y(g) + 2Z(g); P_x is the partial pressure of X.
(i) What is the order of the reaction to X?
(ii) Find the rate constant
(iii) Find the time for 75% completion of the reaction.
(iv) Find the total pressure when pressure of X is 700 mm of Hg

(a). For the reaction
Ag⁺ (aq) + Cl^- (aq) $$\leftrightharpoons$$ AgCl (s)
Given:

Species	$$\Delta G_f^o$$ (kJ/mol)
Ag+ (aq)	+77
Cl- (aq)	-129
AgCl (s)	-109

Write the cell representation of above reaction and calculate $$E_{cell}^o$$ at 298 K. Also find the solubility product if AgCl.
(b) If 6.539 $$\times$$ 10^-2 g of metallic zinc is added to 100 ml saturated solution of AgCl. Find the value of $${\log _{10}}{{\left[ {Z{n^{2 + }}} \right]} \over {{{\left[ {A{g^ + }} \right]}^2}}}$$. How many moles of Ag will be precipitated in the above reaction. Given that
Ag⁺ + e^- $$\to$$ Ag; E^o = 0.80 V;
Zn²⁺ + 2e^- $$\to$$ Zn; E^o = -0.76 V;
(It was given that atomic mass of Zn = 65.39)

Find the velocity (ms^-1) of electron in first Bohr's orbit of radius a₀. Also find the de Broglie's wavelength (in m). Find the orbital angular momentum of 2p orbital of hydrogen atom in units of $$h/2 \pi$$.

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

If the incident ray on a surface is along the unit vector $$\widehat v\,\,,$$ the reflected ray is along the unit vector $$\widehat w\,\,$$ and the normal is along unit vector $$\widehat a\,\,$$ outwards. Express $$\widehat w\,\,$$ in terms of $$\widehat a\,\,$$ and $$\widehat v\,\,.$$

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

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

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

$$f(x)$$ is a differentiable function and $$g(x)$$ is a double differentiable
function such that $$\left| {f\left( x \right)} \right| \le 1$$ and $$f'(x)=g(x).$$
If $${f^2}\left( 0 \right) + {g^2}\left( 0 \right) = 9.$$ Prove that there exists some $$c \in \left( { - 3,3} \right)$$
such that $$g(c).g''(c)<0.$$

If $$\left[ {\matrix{ {4{a^2}} & {4a} & 1 \cr {4{b^2}} & {4b} & 1 \cr {4{c^2}} & {4c} & 1 \cr } } \right]\left[ {\matrix{ {f\left( { - 1} \right)} \cr {f\left( 1 \right)} \cr {f\left( 2 \right)} \cr } } \right] = \left[ {\matrix{ {3{a^2} + 3a} \cr {3{b^2} + 3b} \cr {3{c^2} + 3c} \cr } } \right],\,\,f\left( x \right)$$ is a quadratic
function and its maximum value occurs at a point $$V$$. $$A$$ 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 $$V$$. Find the area enclosed by $$f(x)$$ and chord $$AB$$.

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

Evaluate $$\,\int\limits_0^\pi {{e^{\left| {\cos x} \right|}}} \left( {2\sin \left( {{1 \over 2}\cos x} \right) + 3\cos \left( {{1 \over 2}\cos x} \right)} \right)\sin x\,\,dx$$

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

If $$\left| {f\left( {{x_1}} \right) - f\left( {{x_2}} \right)} \right| < {\left( {{x_1} - {x_2}} \right)^2},$$ for all $${x_1},{x_2} \in R$$. Find the equation of tangent to the cuve $$y = f\left( x \right)$$ at the point $$(1, 2)$$.

In an equilateral triangle, $$3$$ coins of radii $$1$$ unit each are kept so that they touch each other and also the sides of the triangle. Area of the triangle is

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

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

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

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

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

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.