The triad of nuclei that is isotonic is

The equivalent weight of MnSO₄ is half of its molecular weight, when it converts to

In which mode of expression, the concentration of a solution remains independent of temperature?

A sample of hydrazine sulphate (N₂H₆SO₄) was dissolved in 100 ml. of water, 10 ml of this solution was reacted with excess of ferric chloride solution and warmed to complete the reaction. Ferrous ion formed was estimated and it required 20 ml of M/50 potassium permanganate solution. Estimate the amount of hydrazine sulphate in one litre of the solution.

Reaction:

4Fe³⁺ + N₂H₄ $$\to$$ N₂ + 4Fe²⁺ + 4H⁺

$$MnO_4^-$$ + 4Fe²⁺ + 8H⁺ $$\to$$ Mn²⁺ + 5Fe³⁺ + 4H₂O

A sugar syrup of weight 214.2 g contains 34.2 g of sugar (C₁₂H₂₂O₁₁). Calculate (i) molal concentration and (ii) mole fraction of sugar in the syrup

The uncertainty principle and the concept of wave nature of matter were proposed by ______ and ______ respectively. (Heisenberg Schrodinger, Maxwell, de Broglie)

The wavelength of a spectral line for an electronic transition is inversely related to :

The outermost electronic configuration of the most electronegative element is

The first ionisation potential of Na, Mg, Al and Si are in the order

The statements that are true for the long form of the periodic table are:

The molecule that has linear structure is

The species in which the central atom uses sp² hybrid orbitals in its bonding is

Arrange the following :

N₂, O₂, F₂, Cl₂ in increasing order of bond dissociation energy

Write down the balanced equation for the reaction when:
Carbon dioxide is passed through a concentrated aqueous solution of sodium chloride saturated with ammonia.

Urn $$A$$ contains $$6$$ red and $$4$$ black balls and urn $$B$$ contains $$4$$ red and $$6$$ black balls. One ball is drawn at random from urn $$A$$ and placed in urn $$B$$. The one ball is drawn at random from urn $$B$$ and placed in urn $$A$$. If one ball is now drawn at random from urn $$A$$, the probability that it is found to be red is ................

One hundred identical coins, each with probability, $$p,$$ of showing up heads are tossed once. If $$0 < p < 1$$ and the probability of heads showing on $$50$$ coins is equal to that of heads showing on $$51$$ coins, then the value of $$p$$ is

For two given events $$A$$ and $$B,$$ $$P\left( {A \cap B} \right)$$

A box contains $$2$$ fifty paise coins, $$5$$ twenty five paise coins and a certain fixed number $$N\,\,\left( { \ge 2} \right)$$ of ten and five paise coins. Five coins are taken out of the box at random. Find the probability that the total value of these $$5$$ coins is less than one rupee and fifty paise.

The components of a vector $$\overrightarrow a $$ along and perpendicular to a non-zero vector $$\overrightarrow b $$ are ......and .....respectively.

Let $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c ,$$ be three non-coplanar vectors and $$\overrightarrow p ,\overrightarrow q ,\overrightarrow r,$$ are vectors defined by the relations $$\overrightarrow p = {{\overrightarrow b \times \overrightarrow c } \over {\left[ {\overrightarrow a \overrightarrow b \overrightarrow c } \right]}},\,\,\overrightarrow q = {{\overrightarrow c \times \overrightarrow a } \over {\left[ {\overrightarrow a \overrightarrow b \overrightarrow c } \right]}},\,\,\overrightarrow r = {{\overrightarrow a \times \overrightarrow b } \over {\left[ {\overrightarrow a \overrightarrow b \overrightarrow c } \right]}}$$ then the value of the expression $$\left( {\overrightarrow a + \overrightarrow b } \right).\overrightarrow p + \left( {\overrightarrow b + \overrightarrow c } \right).\overrightarrow q + \left( {\overrightarrow c + \overrightarrow a } \right),\overrightarrow r $$ is equal to

Let $$OA$$ $$CB$$ be a parallelogram with $$O$$ at the origin and $$OC$$ a diagonal. Let $$D$$ be the midpoint of $$OA.$$ Using vector methods prove that $$BD$$ and $$CO$$ intersect in the same ratio. Determine this ratio.

Evaluate $$\int\limits_0^1 {\log \left[ {\sqrt {1 - x} + \sqrt {1 + x} } \right]dx} $$

If $$P=(1, 0),$$ $$Q=(-1, 0)$$ and $$R=(2, 0)$$ are three given points, then locus of the point $$S$$ satisfying the relation $$S{Q^2} + S{R^2} = 2S{P^2},$$ is

The cube roots of unity when represented on Argand diagram form the vertices of an equilateral triangle.

The values of $$\theta $$ lying between $$\theta = \theta $$ and $$\theta = \pi /2$$ and satisfying the equation

$$\left| {\matrix{ {1 + {{\sin }^2}\theta } & {{{\cos }^2}\theta } & {4\sin 4\theta } \cr {{{\sin }^2}\theta } & {1 + {{\cos }^2}\theta } & {4\sin 4\theta } \cr {{{\sin }^2}\theta } & {{{\cos }^2}\theta } & {1 + 4\sin 4\theta } \cr } } \right| = 0$$ are

Solve $$\left| {{x^2} + 4x + 3} \right| + 2x + 5 = 0$$

Total number of ways in which six ' + ' and four ' - ' signs can be arranged in a line such that no two ' - ' signs occur together is.....................................

There are four balls of different colours and four boxes of colours, same as those of the balls. The number of ways in which the balls, one each in a box, could be placed such that a ball does not go to a box of its own colour is.......................

Let $$R$$ $$ = {\left( {5\sqrt 5 + 11} \right)^{2n + 1}}$$ and $$f = R - \left[ R \right],$$ where [ ] denotes the greatest integer function. Prove that $$Rf = {4^{2n + 4}}$$

The sum of the first n terms of the series $${1^2} + {2.2^2} + {3^2} + {2.4^2} + {5^2} + {2.6^2} + .........$$ is
$$n\,\,{\left( {n + 1} \right)^2}/2,$$ when $$n$$ is even. When $$n$$ is odd, the sum is .............

Sum of the first n terms of the series $${1 \over 2} + {3 \over 4} + {7 \over 8} + {{15} \over {16}} + ............$$ is equal to

If the first and the $$(2n-1)$$st terms of an A.P., a G.P. and an H.P. are equal and their $$n$$-th terms are $$a,b$$ and $$c$$ respectively, then

The lines $$2x + 3y + 19 = 0$$ and $$9x + 6y - 17 = 0$$ cut the coordinates axes in concyclic points.

The value of the expression $$\sqrt 3 \,\cos \,ec\,{20^0} - \sec \,{20^0}$$ is equal to

Lines$${L_1} = ax + by + c = 0$$ and $${L_2} = lx + my + n = 0$$ intersect at the point $$P$$ and make an angle $$\theta $$ with each other. Find the equation of a line $$L$$ different from $${L_2}$$ which passes through $$P$$ and makes the same angle $$\theta $$ with $${L_1}$$.

If the circle $${C_1}:{x^2} + {y^2} = 16$$ intersects another circle $${C_2}$$ of radius 5 in such a manner that common chord is of maximum lenght and has a slope equal to 3/4, then the coordinates of the centre of $${C_2}$$ are.............................

If a circle passes through the point (a, b) and cuts the circle $${x^2}\, + \,{y^2}\, = \,{k^2}$$ orthogonally, then the equation of the locus of its centre is

The equations of the tangents drawn from the origin to the circle $${x^2}\, + \,{y^2}\, - \,2rx\,\, - 2hy\, + {h^2} = 0$$, are

If $${y^2} = P\left( x \right)$$, a polynomial of degree $$3$$, then $$2{d \over {dx}}\left( {{y^3}{{{d^2}y} \over {d{x^2}}}} \right)$$ equals

If the angles of a triangle are $${30^ \circ }$$ and $${45^ \circ }$$ and the included side is $$\left( {\sqrt 3 + 1} \right)$$ cms, then the area of the triangle is ...............

A sign -post in the form of an isosceles triangle $$ABC$$ is mounted on a pole of height $$h$$ fixed to the ground. The base $$BC$$ of the triangle is parallel to the ground. A man standing on the ground at a distance $$d$$ from the sign-post finds that the top vertex $$A$$ of the triangle subtends an angle $$\beta $$ and either of the other two vertices subtends the same angle $$\alpha $$ at his feet. Find the area of the triangle.

Investigate for maxima and minimum the function $$$f\left( x \right) = \int\limits_1^x {\left[ {2\left( {t - 1} \right){{\left( {t - 2} \right)}^3} + 3{{\left( {t - 1} \right)}^2}{{\left( {t - 2} \right)}^2}} \right]} dt$$$

The integral $$\int\limits_0^{1.5} {\left[ {{x^2}} \right]dx,} $$

Where [ ] denotes the greatest integer function, equals .............

The value of the integral $$\int\limits_0^{2a} {[{{f\left( x \right)} \over {\left\{ {f\left( x \right) + f\left( {2a - x} \right)} \right\}}}]\,dx} $$ is equal to $$a$$.

Find the area of the region bounded by the curve $$C:y=$$
$$\tan x,$$ tangent drawn to $$C$$ at $$x = {\pi \over 4}$$ and the $$x$$-axis.

In the formula X = 3YZ², X and Z have dimensions of capacitance and magnetic induction respectively. The dimensions of Y in MKSQ system are _____________

A boat which has a speed of 5 km/hr in still water crosses a river of width 1 km along the shortest possible path in 15 minutes. The velocity of the river water in km/hr is

Two bodies M and N of equal masses are suspended from two separate massless springs of spring constant k₁ and k₂ respectively. If the two bodies oscillate vertically such that their maximum velocities are equal, the ratio of the amplitude of vibration of M to that of N is