Let \(P\) be the point of intersection of the common tangents to the parabola \(y^2 = 12x\) and the hyperbola \(8x^2 -y^2 = 8\). If \(S\) and \(S'\) denote the foci of the hyperbola where \(S\) lies on the positive \(x-\) axis then \(P\) divides \(SS'\) in a ratio

The value of \(\cot \frac{\pi}{24}\) is :

Let \(\vec{a}, \vec{b}\) and \(\vec{c}\) be three units vectors such that \(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}=\overrightarrow{0} .\) If \(\lambda=\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{c}} \cdot \overrightarrow{\mathrm{a}} \) and \(\overrightarrow{\mathrm{d}}=\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}},\) then the ordered pair \((\lambda, {\mathrm{\vec d}})\) is equal to 

The number of distinct real roots of the equation \(3 x^{4}+4 x^{3}-12 x^{2}+4=0\) is ..... .

If \(m\) is the slope of a common tangent to the curves \(\frac{x^{2}}{16}+\frac{y^{2}}{9}=1\) and \(x^{2}+y^{2}=12\), then \(12\; m ^{2}\) is equal to

The length of the chord of the ellipse \(\frac{x^2}{4}+\frac{y^2}{2}=1\), whose mid-point is \(\left(1, \frac{1}{2}\right)\), is :

If the area (in \(sq. units\)) of the region \(\left\{ {\left( {x,y} \right):{y^2} \le 4x,x + y \le 1,x \ge 0,y \ge 0} \right\}\) is \(a\sqrt 2  + b\), then \(a -b\) is equal to

Let \(f\left( x \right) = \frac{x}{{\sqrt {{a^2} + {x^2}} }} - \frac{{d - x}}{{\sqrt {{b^2} + {{\left( {d - x} \right)}^2}} }},x \in R\,\), where \(a, b\) and \(d\) are non -zero real constant. Then

The domain of the function \(f(x)=\frac{1}{\sqrt{[x]^2-3[x]-10}}\) is (where \([x]\) denotes the greatest integer less than or equal to \(x\) )

The sum of all rational terms in the expansion of \((2+\sqrt{3})^8\) is

Let \(f\) be a function such that \(f(x)+3 f\left(\frac{24}{x}\right)\) \(=4 x, x \neq 0\). Then \(f(3)+f(8)\) is equal to

The sum of all those terms, of the anithmetic progression \(3,8,13, \ldots \ldots .373\), which are not divisible by \(3\),is equal to \(.......\).

Let \(\int_\alpha^{\log _e^4} \frac{\mathrm{dx}}{\sqrt{\mathrm{e}^{\mathrm{x}}-1}}=\frac{\pi}{6}\). Then \(\mathrm{e}^\alpha\) and \(\mathrm{e}^{-\alpha}\) are the roots of the equation :