The value of the integral $$\int\limits_0^1 {x{{\cot }^{ - 1}}(1 - {x^2} + {x^4})dx} $$ is :-

If the system of equations 2x + 3y – z = 0, x + ky – 2z = 0 and 2x – y + z = 0 has a non-trival solution (x, y, z), then $${x \over y} + {y \over z} + {z \over x} + k$$ is equal to :-

The value of sin 10º sin30º sin50º sin70º is :-

If $$\cos x{{dy} \over {dx}} - y\sin x = 6x$$, (0 < x < $${\pi \over 2}$$)
and $$y\left( {{\pi \over 3}} \right)$$ = 0 then $$y\left( {{\pi \over 6}} \right)$$ is equal to :-

Let z $$ \in $$ C be such that |z| < 1.

If $$\omega = {{5 + 3z} \over {5(1 - z)}}$$z, then :

A rectangle is inscribed in a circle with a diameter lying along the line 3y = x + 7. If the two adjacent vertices of the rectangle are (–8, 5) and (6, 5), then the area of the rectangle (in sq. units) is :

If a unit vector $$\overrightarrow a $$ makes angles $$\pi $$/3 with $$\widehat i$$ , $$\pi $$/ 4 with $$\widehat j$$ and $$\theta $$$$ \in $$(0, $$\pi $$) with $$\widehat k$$, then a value of $$\theta $$ is :-

The mean and the median of the following ten numbers in increasing order 10, 22, 26, 29, 34, x, 42, 67, 70, y are 42 and 35 respectively, then $${y \over x}$$ is equal to

The area (in sq. units) of the region
A = {(x, y) : $${{y{}^2} \over 2}$$ $$ \le $$ x $$ \le $$ y + 4} is :-

The domain of the definition of the function

$$f(x) = {1 \over {4 - {x^2}}} + {\log _{10}}({x^3} - x)$$ is

Two newspapers A and B are published in a city. It is known that 25% of the city populations reads A and 20% reads B while 8% reads both A and B. Further, 30% of those who read A but not B look into advertisements and 40% of those who read B but not A also look into advertisements, while 50% of those who read both A and B look into advertisements. Then the percentage of the population who look into advertisement is :-

If m is chosen in the quadratic equation

(m² + 1) x² – 3x + (m² + 1)² = 0

such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is :-

$$\int {{e^{\sec x}}}$$ $$(\sec x\tan xf(x) + \sec x\tan x + se{x^2}x)dx$$
= e^secxf(x) + C then a possible choice of f(x) is :-

If the function $$f(x) = \left\{ {\matrix{ {a|\pi - x| + 1,x \le 5} \cr {b|x - \pi | + 3,x > 5} \cr } } \right.$$
is continuous at x = 5, then the value of a – b is :-

If $$f(x) = [x] - \left[ {{x \over 4}} \right]$$ ,x $$ \in $$ 4 , where [x] denotes the greatest integer function, then

The total number of matrices
$$A = \left( {\matrix{ 0 & {2y} & 1 \cr {2x} & y & { - 1} \cr {2x} & { - y} & 1 \cr } } \right)$$
(x, y $$ \in $$ R,x $$ \ne $$ y) for which A^TA = 3I₃ is :-

The vertices B and C of a $$\Delta $$ABC lie on the line,

$${{x + 2} \over 3} = {{y - 1} \over 0} = {z \over 4}$$ such that BC = 5 units.

Then the area (in sq. units) of this triangle, given that the point A(1, –1, 2), is :

A water tank has the shape of an inverted right circular cone, whose semi-vertical angle is $${\tan ^{ - 1}}\left( {{1 \over 2}} \right)$$. Water is poured into it at a constant rate of 5 cubic meter per minute. The the rate (in m/min.), at which the level of water is rising at the instant when the depth of water in the tank is 10m; is :-

If the two lines x + (a – 1) y = 1 and 2x + a²y = 1 (a$$ \in $$R – {0, 1}) are perpendicular, then the distance of their point of intersection from the origin is :

If f : R $$ \to $$ R is a differentiable function and f(2) = 6,
then $$\mathop {\lim }\limits_{x \to 2} {{\int\limits_6^{f\left( x \right)} {2tdt} } \over {\left( {x - 2} \right)}}$$ is :-

If the sum and product of the first three term in an A.P. are 33 and 1155, respectively, then a value of its 11^th term is :-