At 380^oC, the half-life period for the first order decomposition of H₂O₂ is 360 min. Calculate the time required for 75% decomposition at 450^oC.

An excess of liquid mercury is added to an acidified solution of 1.0 $$\times$$ 10^-3 M Fe³⁺. It is found that 5% of Fe³⁺ remains at equilibrium at 25^oC. Calculate $$E_{Hg_2^{2 + }|\,Hg}^o$$, assuming that only reaction that occurs is
2Hg + 2Fe³⁺ $$\to$$ $$Hg_2^{2+}$$ + 2Fe²⁺
(Given $$E_{F{e^{3 + }}|\,F{e^{2 + }}}^o$$ = 0.77 V)

A 5.0 cm³ solution of H₂O₂ liberates 0.508 g of iodine from an acidified KI solution. Calculate the strength of H₂O₂ solution in terms of volume strength at STP.

Iodine molecule dissociates into atoms after absorbing light of 4500 Å. If one quantum of radiation is absorbed by each molecule, calculate the kinetic energy of iodine atoms. (Bond energy of I₂ = 240 kJ mol^-1)

Let $$(h, k)$$ be a fixed point, where $$h > 0,k > 0.$$. A straight line passing through this point cuts the possitive direction of the coordinate axes at the points $$P$$ and $$Q$$. Find the minimum area of the triangle $$OPQ$$, $$O$$ being the origin.

Let $$y=f(x)$$ be a curve passing through $$(1,1)$$ such that the triangle formed by the coordinate axes and the tangent at any point of the curve lies in the first quadrant and has area $$2.$$ From the differential equation and determine all such possible curves.

Evaluate the definite integral : $$$\int\limits_{ - 1/\sqrt 3 }^{1/\sqrt 3 } {\left( {{{{x^4}} \over {1 - {x^4}}}} \right){{\cos }^{ - 1}}\left( {{{2x} \over {1 + {x^2}}}} \right)} dx$$$

Consider a square with vertices at $$(1,1), (-1,1), (-1,-1)$$ and $$(1, -1)$$. Let $$S$$ be the region consisting of all points inside the square which are nearer to the origin than to any edge. Sketch the region $$S$$ and find its area.

Let $${I_m} = \int\limits_0^\pi {{{1 - \cos mx} \over {1 - \cos x}}} dx.$$ Use mathematical induction to prove that $${I_m} = m\,\pi ,m = 0,1,2,........$$

The minimum value of the expression $$\sin \,\alpha + \sin \,\beta \, + \sin \,\gamma ,\,$$ where $$\alpha ,\,\beta ,\,\gamma $$ are real numbers satisfying $$\alpha + \beta + \gamma = \pi $$ is

Let '$$d$$' be the perpendicular distance from the centre of the ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$ to the tangent drawn at a point $$P$$ on the ellipse. If $${F_1}$$ and $${F_2}$$ are the two foci of the ellipse, then show that $${\left( {P{F_1} - P{F_2}} \right)^2} = 4{a^2}\left( {1 - {{{b^2}} \over {{d^2}}}} \right)$$.

Show that the locus of a point that divides a chord of slope $$2$$ of the parabola $${y^2} = 4x$$ internally in the ratio $$1:2$$ is a parabola. Find the vertex of this parabola.

The orthocentre of the triangle formed by the lines $$xy=0$$ and $$x+y=1$$ is

Let $$a,\,b,\,c$$ be real. If $$a{x^2} + bx + c = 0$$ has two real roots $$\alpha $$ and $$\beta ,$$ where $$\alpha < - 1$$ and $$\beta > 1,$$ then show that $$1 + {c \over a} + \left| {{b \over a}} \right| < 0.$$

Find the smallest positive number $$p$$ for which the equation $$\cos \left( {p\,\sin x} \right) = \sin \left( {p\cos x} \right)$$ has a solution $$x\, \in \,\left[ {0,2\pi } \right]$$.

If $$\left| {Z - W} \right| \le 1,\left| W \right| \le 1$$, show that $${\left| {Z - W} \right|^2} \le {(\left| Z \right| - \left| W \right|)^2} + {(ArgZ - Arg\,W)^2}$$

If $$i{z^3} + {z^2} - z + i = 0$$ , then show that $$\left| z \right| = 1$$.