An aqueous solution of 6.3 g oxalic acid dihydrate is made up to 250ml. the volume of 0.1N NaOH required to completely neutralize 10ml of this solution is :

The reaction, $$3Cl{O^ - }(aq)$$$$ \to ClO_3^ - (aq) + 2C{l^ - }(aq)$$ is an example of :

In the standardization of Na₂S₂O₃ using K₂Cr₂O₇ by iodometry, the equivalent weight of K₂Cr₂O₇ is:

The wavelength associated with a golf ball weighing 200 g and moving at speed of 5 m/h is of the order

The quantum numbers +1/2 and -1/2 for the electron spin represent

The common features among the species CN^-, CO and NO⁺ are :

The set representing the correct order of first ionization potential is

If $${\sin ^{ - 1}}\left( {x - {{{x^2}} \over 2} + {{{x^3}} \over 4} - ....} \right)$$ $$$ + {\cos ^{ - 1}}\left( {{x^2} - {{{x^4}} \over 2} + {{{x^6}} \over 4} - ....} \right) = {\pi \over 2}$$$
for $$0 < \left| x \right| < \sqrt 2 ,$$ then $$x$$ equals

If $$f\left( x \right) = x{e^{x\left( {1 - x} \right)}},$$ then $$f(x)$$ is

The triangle formed by the tangent to the curve $$f\left( x \right) = {x^2} + bx - b$$ at the point $$(1, 1)$$ and the coordinate axex, lies in the first quadrant. If its area is $$2$$, then the value of $$b$$ is

Let $$f\left( x \right) = \left( {1 + {b^2}} \right){x^2} + 2bx + 1$$ and let $$m(b)$$ be the minimum value of $$f(x)$$. As $$b$$ varies, the range of $$m(b)$$ is

The value of $$\int\limits_{ - \pi }^\pi {{{{{\cos }^2}x} \over {1 + {a^x}}}dx,\,a > 0,} $$ is

Let $$\overrightarrow a = \overrightarrow i - \overrightarrow k ,\overrightarrow b = x\overrightarrow i + \overrightarrow j + \left( {1 - x} \right)\overrightarrow k $$ and
$$\overrightarrow c = y\overrightarrow i - x\overrightarrow j + \left( {1 + x - y} \right)\overrightarrow k .$$ Then $$\left[ {\overrightarrow a \,\overrightarrow b \,\overrightarrow c } \right]$$ depends on

If $$\overrightarrow a \,,\,\overrightarrow b $$ and $$\overrightarrow c $$ are unit vectors, then $${\left| {\overrightarrow a - \overrightarrow b } \right|^2} + {\left| {\overrightarrow b - \overrightarrow c } \right|^2} + {\left| {\overrightarrow c - \overrightarrow a } \right|^2}$$ does NOT exceed

A man from the top of a $$100$$ metres high tower sees a car moving towards the tower at an angle of depression of $${30^ \circ }$$. After some time,the angle of depression becomes $${60^ \circ }$$. The distance (in metres) travelled by the car during this time is

If the sum of the first $$2n$$ terms of the A.P.$$2,5,8,......,$$ is equal to the sum of the first $$n$$ terms of the A.P.$$57,59,61,.....,$$ then $$n$$ equals

The maximum value of $$\left( {\cos {\alpha _1}} \right).\left( {\cos {\alpha _2}} \right).....\left( {\cos {\alpha _n}} \right),$$ under the restrictions
$$0 \le {\alpha _1},{\alpha _2},....,{\alpha _n} \le {\pi \over 2}$$ vand $$\left( {\cot {\alpha _1}} \right).\left( {\cot {\alpha _2}} \right)....\left( {\cot {\alpha _n}} \right) = 1$$ is

If $$\alpha + \beta = \pi /2$$ and $$\beta + \gamma = \alpha ,$$ then $$\tan \,\alpha \,$$ equals

The complex numbers $${z_1},\,{z_2}$$ and $${z_3}$$ satisfying $${{{z_1} - {z_3}} \over {{z_2} - {z_3}}} = {{1 - i\sqrt 3 } \over 2}\,$$ are the vertices of a triangle which is

Let $${z_1}$$ and $${z_2}$$ be $${n^{th}}$$ roots of unity which subtend a right angle at the origin. Then $$n$$ must be of the form

In the binomial expansion of $${\left( {a - b} \right)^n},\,n \ge 5,$$ the sum of the $${5^{th}}$$ and $${6^{th}}$$ terms is zero. Then $$a/b$$ equals

Let $${T_n}$$ denote the number of triangles which can be formed using the vertices of a regular polygon of n sides. If $${T_{n + 1}} - {T_n} = 21$$, then n equals

Let $$\alpha $$, $$\beta $$ be the roots of $${x^2} - x + p = 0$$ and $$\gamma ,\delta $$ be the roots of $${x^2} - 4x + q = 0.$$ If $$\alpha ,\beta ,\gamma ,\delta $$ are in G.P., then the integral values of $$p$$ and $$q$$ respectively, are

Let the positive numbers $$a,b,c,d$$ be in A.P. Then $$abc,$$ $$abd,$$ $$acd,$$ $$bcd,$$ are

The number of distinct real roots of $$\left| {\matrix{ {\sin x} & {\cos x} & {\cos x} \cr {\cos x} & {\sin x} & {\cos x} \cr {\cos x} & {\cos x} & {\sin x} \cr } } \right|\,$$
$$\, = 0$$ in the interval $$ - {\pi \over 4} \le x \le {\pi \over 4}$$ is

The number of integer values of $$m$$, for which the $$x$$-coordinate of the point of intersection of the lines $$3x + 4y = 9$$ and $$y = mx + 1$$ is also an integer, is

Area of the parallelogram formed by the lines $$y = mx$$, $$y = mx + 1$$, $$y = nx$$ and $$y = nx + 1$$ equals

Let A B be a chord of the circle $${x^2} + {y^2} = {r^2}$$ subtending a right angle at the centre. Then the locus of the centriod of the triangle PAB as P moves on the circle is

Let PQ and RS be tangents at the extremities of the diameter PR of a circle of radius r. If PS and RQ intersect at a point X on the circumference of the circle, then 2r equals

The equation of the directrix of the parabola $${y^2} + 4y + 4x + 2 = 0$$

The equation of the common tangent touching the circle $${\left( {x - 3} \right)^2} + {y^2} = 9$$ and the parabola $${y^2} = 4x$$ above the $$x$$-axis is

Let $$f:\left( {0,\infty } \right) \to R$$ and $$F\left( x \right) = \int\limits_0^x {f\left( t \right)dt.} $$ If $$F\left( {{x^2}} \right) = {x^2}\left( {1 + x} \right)$$, then $$f(4)$$ equals

A quantity X is given by $${\varepsilon _0}L{{\Delta V} \over {\Delta t}}$$ where $${\varepsilon _0}$$ is the permittivity of the free space, L is a length, $${\Delta V}$$ is a potential difference and $${\Delta t}$$ is a time interval. The dimentional formula for X is the same as that of

The pulleys and strings shown in the figure are smooth and of negligible mass. For the system to remain in equilibrium, the angle $$\theta $$ should be

A particle executes simple harmonic motion between x = - A to x = + A. The time taken for it to go from 0 to $${A \over 2}$$ is T₁ and to go from $${A \over 2}$$ to A is T₂. Then