Two students use the same stock solution of ZnSO₄ and solution of CuSO₄. The emf of one cell is 0.03 V higher than other. The conc. of CuSO₄ in the cell with higher emf value is 0.5 M. Find out the conc. of CuSO₄ in the other cell (2.203 RT/F = 0.06)

1 gm of charcoal adsorbs 100 ml 0.5 M CH₃COOH to form a monolayer, and thereby the molarity of CH₃COOH reduces to 0.49. Calculate the surface area of the charcoal adsorbed by each molecule of acetic acid. Surface area of charcoal = 3.01 $$\times$$ 10² m²/gm.

Using VSEPR theory deduce the structures of PCl₅ and BrF₅

Wavelength of high energy transition of H-atoms is 91.2 nm. Calculate the corresponding wavelength of He atoms.

Calculate the molarity of water if it's density is 1000 kg/m³

If $$f$$ is an even function then prove that
$$\int\limits_0^{\pi /2} {f\left( {\cos 2x} \right)\cos x\,dx = \sqrt 2 } \int\limits_0^{\pi /4} {f\left( {\sin 2x} \right)\cos x\,dx.} $$

(i) Find the equation of the plane passing through the points $$(2, 1, 0), (5, 0, 1)$$ and $$(4, 1, 1).$$
(ii) If $$P$$ is the point $$(2, 1, 6)$$ then find the point $$Q$$ such that $$PQ$$ is perpendicular to the plane in (i) and the mid point of $$PQ$$ lies on it.

If $$\overrightarrow u ,\overrightarrow v ,\overrightarrow w ,$$ are three non-coplanar unit vectors and $$\alpha ,\beta ,\gamma $$ are the angles between $$\overrightarrow u $$ and $$\overrightarrow v $$ and $$\overrightarrow w ,$$ $$\overrightarrow w $$ and $$\overrightarrow u $$ respectively and $$\overrightarrow x ,\overrightarrow y ,\overrightarrow z ,$$ are unit vectors along the bisectors of the angles $$\alpha ,\,\,\beta ,\,\,\gamma $$ respectively. Prove that $$\,\left[ {\overrightarrow x \times \overrightarrow y \,\,\overrightarrow y \times \overrightarrow z \,\,\overrightarrow z \times \overrightarrow x } \right] = {1 \over {16}}{\left[ {\overrightarrow u \,\,\overrightarrow v \,\,\overrightarrow w } \right]^2}\,{\sec ^2}{\alpha \over 2}{\sec ^2}{\beta \over 2}{\sec ^2}{\gamma \over 2}.$$

For a student to qualify, he must pass at least two out of three exams. The probability that he will pass the 1^st exam is $$p.$$ If he fails in one of the exams then the probability of his passing in the next exam is $${p \over 2}$$ otherwise it remains the same. Find the probability that he will qualify.

$$A$$ is targeting to $$B, B$$ and $$C$$ are targeting to $$A.$$ Probability of hitting the target by $$A,B$$ and $$C$$ are $${2 \over 3},{1 \over 2}$$ and $${1 \over 3}$$ respectively. If $$A$$ is hit then find the probability that $$B$$ hits the target and $$C$$ does not.

A right circular cone with radius $$R$$ and height $$H$$ contains a liquid which eveporates at a rate proportional to its surface area in contact with air (proportionality constant $$ = k > 0$$. Find the time after which the come is empty.

If $${z_1}$$ and $${z_2}$$ are two complex numbers such that $$\,\left| {{z_1}} \right| < 1 < \left| {{z_2}} \right|\,$$ then prove that $$\,\left| {{{1 - {z_1}\overline {{z_2}} } \over {{z_1} - {z_2}}}} \right| < 1$$.

If $$P(1)=0$$ and $${{dp\left( x \right)} \over {dx}} > P\left( x \right)$$ for all $$x \ge 1$$ then prove that
$$P(x)>0$$ for all $$x>1$$.

If the function $$f:\left[ {0,4} \right] \to R$$ is differentiable then show that
(i)$$\,\,\,\,\,$$ For $$a, b$$$$\,\,$$$$ \in \left( {0,4} \right),{\left( {f\left( 4 \right)} \right)^2} - {\left( {f\left( 0 \right)} \right)^2} = gf'\left( a \right)f\left( b \right)$$
(ii)$$\,\,\,\,\,$$ $$\int\limits_0^4 {f\left( t \right)dt = 2\left[ {\alpha f\left( {{\alpha ^2}} \right) + \beta \left( {{\beta ^2}} \right)} \right]\forall 0 < \alpha ,\beta < 2} $$

Find a point on the curve $${x^2} + 2{y^2} = 6$$ whose distance from
the line $$x+y=7$$, is minimum.

Using the relation $$2\left( {1 - \cos x} \right) < {x^2},\,x \ne 0$$ or otherwise,
prove that $$\sin \left( {\tan x} \right) \ge x,\,\forall x \in \left[ {0,{\pi \over 4}} \right]$$

If $${I_n}$$ is the area of $$n$$ sided regular polygon inscribed in a circle of unit radius and $${O_n}$$ be the area of the polygon circumscribing the given circle, prove that $$${I_n} = {{{O_n}} \over 2}\left( {1 + \sqrt {1 - {{\left( {{{2{I_n}} \over n}} \right)}^2}} } \right)$$$

Normals are drawn from the point $$P$$ with slopes $${m_1}$$, $${m_2}$$, $${m_3}$$ to the parabola $${y^2} = 4x$$. If locus of $$P$$ with $${m_1}$$ $${m_2}$$$$ = \alpha $$ is a part of the parabola itself then find $$\alpha $$.

For the circle $${x^2}\, + \,{y^2} = {r^2}$$, find the value of r for which the area enclosed by the tangents drawn from the point P (6, 8) to the circle and the chord of contact is maximum.

If a, b, c are in A.P., $${a^2}$$, $${b^2}$$, $${c^2}$$ are in H.P., then prove that either a = b = c or a, b, $${ - {c \over 2}}$$ form a G.P.

Prove that
$${2^k}\left( {\matrix{ n \cr 0 \cr } } \right)\left( {\matrix{ n \cr k \cr } } \right) - {2^{^{k - 1}\left( {\matrix{ n \cr 2 \cr } } \right)}}\left( {\matrix{ n \cr 1 \cr } } \right)\left( {\matrix{ {n - 1} \cr {k - 1} \cr } } \right)$$
$$ + {2^{k - 2}}\left( {\matrix{ {n - 2} \cr {k - 2} \cr } } \right) - .....{\left( { - 1} \right)^k}\left( {\matrix{ n \cr k \cr } } \right)\left( {\matrix{ {n - k} \cr 0 \cr } } \right) = {\left( {\matrix{ n \cr k \cr } } \right)^ \cdot }$$

If $${x^2} + \left( {a - b} \right)x + \left( {1 - a - b} \right) = 0$$ where $$a,\,b\, \in \,R$$ then find the values of a for which equation has unequal real roots for all values of $$b$$.

Prove that there exists no complex number z such that $$\left| z \right| < {1 \over 3}\,and\,\sum\limits_{r = 1}^n {{a_r}{z^r}} = 1$$ where $$\left| {{a_r}} \right| < 2$$.

If n^th divisions of the main scale coincide with (n+1)^th divisions of vernier scale. Given one main scale division is equal to 'a' units. Find the least count of the vernier.