Arrange of the following in :
Arrange the following ions in order of their increasing radii:
Li⁺, Mg²⁺, K⁺, Al³⁺

The weight of $$1 \times 10^{22}$$ molecules of CuSO₄.5H₂O is ____.

The oxidation state of the most electronegative element in the products of the reaction. BaO₂ with dil. H₂SO₄ are

A 1.0 g sample of Fe₂O₃ solid of 55.2% purity is dissolved in acid and reduced by heating the solution with zinc dust. The resultant solution is cooled and made upto 100.0 ml. An aliquot of 25.0 ml of this solution requires 17.0 ml of 0.0167M solution of an oxidant for titration. Calculate the number of electrons taken by the oxidant in the reaction of the above titration.

A solution of 0.2 g of a compound containing Cu²⁺ and $$C_2O_4^{2-}$$ ions on titration with 0.02M $$KMnO_4$$ in presence of $$H_2SO_4$$ consumes 22.6 ml. of the oxidant. The resultant solution in neutralized with Na₂CO₃, acidified with dil. acetic acid and treated with excess KI. The liberated iodine requires 11.3 ml of 0.05M Na₂S₂O₃ solution for complete reduction.

Find out the molar ratio of Cu²⁺ to $$C_2O_4^{2-}$$ in the compound.Write down the balanced redox reactions involved in the above titrations.

Read the following statement and explanation and answer as per the options given below:

STATEMENT (S): In the titration of Na₂CO₃ with HCl using methyl orange indicator, the volume required at the equivalence point is twice that of the acid required using phenolphthalein indicator.

EXPLANATION (E): Two moles of HCl are required for the complete neutralization of one mole of Na₂CO₃

Arrange of the following in :
Increasing order of ionic size:
N^3-, Na⁺, F^-, O^2-, Mg²⁺

Arrange of the following in :
Increasing order of basic character
MgO, SrO, K₂O, NiO, Cs₂O

The linear structure is assumed by :

Arrange the following :

Increasing strength of hydrogen bonding (X-H-X):

O, S, F, Cl, N

Calculate the volume occupied by 5.0 g of acetylene gas at 50^oC and 740 mm pressure.

The volume strength of 1.5 N H₂O₂ solution is

Write down the balanced equation for the reaction when:
Carbon dioxide is passed through a suspension of lime stone in water.

The hybridization of carbon atoms in C-C single bond of HC $$ \equiv $$ C - CH = CH₂ is

The degree of dissociation of calcium nitrate in a dilute aqueous solution, containing 7.0 g. of the salt per 100 gm of water at 100^oC is 70%. If the vapour pressure of water at 100^oC is 760 mm, calculate the vapour pressure of the solution.

Zinc granules are added in excess to a 500 ml. of 1.0 M nickel nitrate solution at 25^oC until the equilibrium is reached. If the standard reduction potential of Zn²⁺ | Zn and Ni²⁺ | Ni are -0.75 V and -0.24 V respectively, find out the concentration of Ni²⁺ in solution at equilibrium.

The current of 1.70 A is passed through 300.0 ml of 0.160 M solution of a ZnSO₄ for 230 sec. with a current efficiency of 90%. Find out the molarity of Zn²⁺ after the deposition of Zn. Assume the volume of the solution to remain constant during electrolysis.

The decomposition of N₂O₅ according to the equation:
2N₂O₅ (g) $$\to$$ 4NO₂(g) + O₂(g)
is a first order reaction. After 30 min. from the start of the decomposition in a closed vessel, the total pressure developed is found to be 284.5 mm of Hg and on complete decomposition, the total pressure is 584.5 mm of Hg. Calculate the rate constant of the reaction.

Write the IUPAC name for the following :

In a test an examine either guesses or copies or knows the answer to a multiple choice question with four choices. The probability that he make a guess is $$1/3$$ and the probability that he copies the answer is $$1/6$$. The probability that his answer is correct given that he copied it, is $$1/8$$. Find the probability that he knew the answer to the questions given that he correctly answered it.

Given that $$\overrightarrow a = \left( {1,1,1} \right),\,\,\overrightarrow c = \left( {0,1, - 1} \right),\,\overrightarrow a .\overrightarrow b = 3$$ and $$\overrightarrow a \times \overrightarrow b = \overrightarrow c ,$$ then $$\overrightarrow b \, = $$.........

Determine the value of $$'c'$$ so that for all real $$x,$$ the vector
$$cx\widehat i - 6\widehat j - 3\widehat k$$ and $$x\widehat i + 2\widehat j + 2cx\widehat k$$ make an obtuse angle with each other.

If $$\exp \,\,\,\left\{ {\left( {\left( {{{\sin }^2}x + {{\sin }^4}x + {{\sin }^6}x + \,\,\,..............\infty } \right)\,In\,\,2} \right)} \right\}$$ satiesfies the equation $${x^2} - 9x + 8 = 0,$$ find the value of $${{\cos x} \over {\cos x + \sin x}},\,0 < x < {\pi \over 2}.$$

If the mean and the variance of binomial variate $$X$$ are $$2$$ and $$1$$ respectively, then the probability that $$X$$ takes a value greater than one is equal to ...............

The value of
$$\sin {\pi \over {14}}\sin {{3\pi } \over {14}}\sin {{5\pi } \over {14}}\sin {{7\pi } \over {14}}\sin {{9\pi } \over {14}}\sin {{11\pi } \over {14}}\sin {{13\pi } \over {14}}$$ is equal to ______.

The product of $$n$$ positive numbers is unity. Then their sum is

Using induction or otherwise, prove that for any non-negative integers $$m$$, $$n$$, $$r$$ and $$k$$ ,
$$\sum\limits_{m = 0}^k {\left( {n - m} \right)} {{\left( {r + m} \right)!} \over {m!}} = {{\left( {r + k + 1} \right)!} \over {k!}}\left[ {{n \over {r + 1}} - {k \over {r + 2}}} \right]$$

Eighteen guests have to be seated, half on each side of a long table. Four particular guests desire to sit on one particular side and three others on the other side. Determine the number of ways in which the sitting arrangements can be made.

Let p be the first of the n arithmetic means between two numbers and q the first of n harmonic means between the same numbers. Show that q does not lie between p and $$\,{\left( {{{n + 1} \over {n - 1}}} \right)^2}\,p$$.

If $${S_1}$$, $${S_2}$$, $${S_3}$$,.............,$${S_n}$$ are the sums of infinite geometric series whose first terms are 1, 2, 3, ...................,n and whose common ratios are $${1 \over 2}$$, $${1 \over 3}$$, $${1 \over 4}$$,....................$$\,{1 \over {n + 1}}$$ respectively, then find the values of $${S_1}^2 + {S_2}^2 + {S_3}^2 + ....... + {S^2}_{2n - 1}$$

Let the algebraic sum of the perpendicular distances from the points $$\left( {2,0} \right),\,\left( {0,\,2} \right)$$ $$\left( {1,\,1} \right)$$ to a variable straight line be zero; then the line passes through a fixed point whose cordinates are ...............

Show that all chords of the curve $$3{x^2} - {y^2} - 2x + 4y = 0,$$ which subtend a right angle at the origin, pass through a fixed point. Find the coordinates of the point.

If a circle passes through the points of intersection of the coordinate axes with the lines $$\lambda \,x - y + 1 = 0$$ and x - 2y + 3 = 0, then the value of $$\lambda $$ = .........

Find the equation of the line passing through the point $$(2, 3)$$ and making intercept of length 2 units between the lines $$y + 2x = 3$$ and $$y + 2x = 5$$.

Two circles, each of radius 5 units, touch each other at (1, 2). If the equation of their common tangent is 4x + 3y = 10, find the equation of the circles.

Three normals are drawn from the point $$(c, 0)$$ to the curve $${y^2} = x.$$ Show that $$c$$ must be greater than $$1/2$$. One normal is always the $$x$$-axis. Find $$c$$ for which the other two normals are perpendicular to each other.

Find $${{{dy} \over {dx}}}$$ at $$x=-1$$, when
$${\left( {\sin y} \right)^{\sin \left( {{\pi \over 2}x} \right)}} + {{\sqrt 3 } \over 2}{\sec ^{ - 1}}\left( {2x} \right) + {2^x}\tan \left( {In\left( {x + 2} \right)} \right) = 0$$

A man notices two objects in a straight line due west. After walking a distance $$c$$ due north he observes that the objects subtend an angle $$\alpha $$ at his eye; and, after walking a further distance $$2c$$ due north, an angle $$\beta $$. Show that the distance between the objects is $${{8c} \over {3\cot \beta - \cot \alpha }}$$; the height of the man is being ignored.

In a triangle of base a the ratio of the other two sides is $$r\left( { < 1} \right)$$. Show that the altitude of the triangle is less than of equal to $${{ar} \over {1 - {r^2}}}$$

The sides of a triangle are three consecutive natural numbers and its largest angle is twice the smallest one. Determine the sides of the triangle.

A window of perimeter $$P$$ (including the base of the arch) is in the form of a rectangle surmounded by a semi circle. The semi-circular portion is fitted with coloured glass while the rectangular part is fitted with clear glass transmits three times as such light per square meter as the coloured glass does.

What is the ratio for the sides of the rectangle so that the window transmits the maximum light ?

Sketch the curves and identify the region bounded by
$$x = {1 \over 2},x = 2,y = \ln \,x$$ and $$y = {2^x}.$$ Find the area of this region.

Evaluate $$\,\int\limits_0^\pi {{{x\,\sin \,2x\,\sin \left( {{\pi \over 2}\cos x} \right)} \over {2x - \pi }}dx} $$

If $$'f$$ is a continuous function with $$\int\limits_0^x {f\left( t \right)dt \to \infty } $$ as $$\left| x \right| \to \infty ,$$ then show that every line $$y=mx$$
intersects the curve $${y^2} + \int\limits_0^x {f\left( t \right)dt = 2!} $$

For any two events $$A$$ and $$B$$ in a simple space