The integral \int\limits_2^4 {{{\log \,{x^2}} \over {\log {x^2} + \log \left( {36 - 12x + {x^2}} \right)}}dx} is equal to :

A complex number z is said to be unimodular if $$\left|z\right|=1$$. Suppose $$z_1$$ and $$z_2$$ are complex numbers such that {{{z_1} - 2{z_2}} \over {2 - {z_1}\overline {{z_2}} }} is unimodular and $$z_2$$ is not unimodular. Then the point $$z_1$$ lies on a :

Locus of the image of the point $$(2,3)$$ in the line $$\left(2x-3y+4\right)+k\left(x-2y+3\right)=0,k\in R,$$ is a :

The number of common tangents to the circles $$x^2+y^2-4x-6x-12=0$$ and $$x^2+y^2+6x+18y+26=0,$$ is :

The area (in sq. units) of the quadrilateral formed by the tangents at the end points of the latera recta to the ellipse {{{x^2}} \over 9} + {{{y^2}} \over 5} = 1, is :

Let A and B be two sets containing four and two elements respectively. Then, the number of subsets of the set A $\times$ B , each having atleast three elements are

\mathop {\lim }\limits_{x \to 0} {{\left( {1 - \cos 2x} \right)\left( {3 + \cos x} \right)} \over {x\tan 4x}} is equal to

If the function. g\left( x \right) = \left\{ {\matrix{ {k\sqrt {x + 1} ,} & {0 \le x \le 3} \cr {m\,x + 2,} & {3 < x \le 5} \cr } } \right. is differentiable, then the value of $$k+m$$ is :

Let {\tan ^{ - 1}}y = {\tan ^{ - 1}}x + {\tan ^{ - 1}}\left( {{{2x} \over {1 - {x^2}}}} \right), where \left| x \right| < {1 \over {\sqrt 3 }}. Then a value of $$y$$ is :

The number of integers greater than 6,000 that can be formed, using the digits 3, 5, 6, 7 and 8, without repetition, is:

Let $$\alpha$$ and $$\beta$$ be the roots of equation $$x^2-6x-2=0$$. If $$a_n={\alpha }^n-{\beta }^n,$$ for $$n\ge 1,$$ then the value of {{{a_{10}} - 2{a_8}} \over {2{a_9}}} is equal to :

The distance of the point $$(1,0,2)$$ from the point of intersection of the line {{x - 2} \over 3} = {{y + 1} \over 4} = {{z - 2} \over {12}} and the plane $$x-y+z=16,$$ is :

The equation of the plane containing the line $$2x-5y+z=3;x+y+4z=5,$$ and parallel to the plane, $$x+3y+6z=1,$$ is :

The mean of the data set comprising of 16 observations is 16. If one of the observation valued 16 is deleted and three new observations valued 3, 4 and 5 are added to the data, then the mean of the resultant data, is :

The sum of first 9 terms of the series. {{{1^3}} \over 1} + {{{1^3} + {2^3}} \over {1 + 3}} + {{{1^3} + {2^3} + {3^3}} \over {1 + 3 + 5}} + ......

If m is the A.M. of two distinct real numbers l and n $$(l,n>1)$$ and $$G_1,G_2$$ and $$G_3$$ are three geometric means between $$l$$ and n, then $$G_1^4+2G_2^4+G_3^4$$ equals:

Let $$f(x)$$ be a polynomial of degree four having extreme values at $$x=1$$ and $$x=2$$. If \mathop {\lim }\limits_{x \to 0} \left[ {1 + {{f\left( x \right)} \over {{x^2}}}} \right] = 3, then f$$(2)$$ is equal to :

The normal to the curve, $$x^2+2xy-3y^2=0$$, at $$(1,1)$$

The area (in sq. units) of the region described by $$\{\left(x,y\right):y^2\le 2x$$ and $$y\ge 4x-1\}$$ is :

The negation of $$\sim s\vee \left(\sim r\wedge s\right)$$ is equivalent to :