If \(f(x)=\frac{\left(\tan 1^{\circ}\right) x+\log _{\varepsilon}(123)}{x \log _{\varepsilon}(1234)-\left(\tan 1^{\circ}\right)}, x > 0\), then the least value of \(f(f(x))+f\left(f\left(\frac{4}{x}\right)\right)\) is \(...........\).

Let the tangent to the parabola \(y^2=12 x\) at the point \((3, \alpha)\) be perpendicular to the line \(2 x+2 y=3\).Then the square of distance of the point \((6,-4)\)from the normal to the hyperbola \(\alpha^2 x^2-9 y^2=9 \alpha^2\)at its point \((\alpha-1, \alpha+2)\) is equal to \(........\).

Let \(f ( x )= x \cdot\left[\frac{ x }{2}\right],\) for \(-10< x <10,\) where \([ t ]\) denotes the greatest integer function. Then the number of points of discontinuity of \(f\) is equal to

In a triangle \(\mathrm{ABC}\), if \(|\overrightarrow{\mathrm{BC}}|=3,|\overrightarrow{\mathrm{C}}|=5\) and \(|\overrightarrow{\mathrm{BA}}|=7\), then the projection of the vector \(\overline{\mathrm{BA}}\) on \(\overline{\mathrm{BC}}\) is equal to:

The sum of all rational terms in the expansion of \(\left(2^{\frac{1}{5}}+5^{\frac{1}{3}}\right)^{15}\) is equal to :

If \({\Delta _1} = \left| {\begin{array}{*{20}{c}}
  x&{\sin \,\theta }&{\cos \,\theta } \\ 
  {\sin \,\theta }&{ - x}&1 \\ 
  {\cos \,\theta }&1&x 
\end{array}} \right|\) and \({\Delta _1} = \left| {\begin{array}{*{20}{c}}
  x&{\sin \,2\theta }&{\cos \,\,2\theta } \\ 
  {\sin \,2\theta }&{ - x}&1 \\ 
  {\cos \,\,2\theta }&1&x 
\end{array}} \right|\), \(x \ne 0\) ; then for all \(\theta  \in \left( {0,\frac{\pi }{2}} \right)\)

Let \(\vec{a}, \vec{b}, \vec{c}\) be three vectors mutually perpendicular to each other and have same magnitude. If a vector \(\overrightarrow{\mathrm{r}}\) satisfies. \(\overrightarrow{\mathrm{a}} \times\{(\overrightarrow{\mathrm{r}}-\overrightarrow{\mathrm{b}}) \times \overrightarrow{\mathrm{a}}\}+\overrightarrow{\mathrm{b}} \times\{(\overrightarrow{\mathrm{r}}-\overrightarrow{\mathrm{c}}) \times \overrightarrow{\mathrm{b}}\}+\overrightarrow{\mathrm{c}} \times\{(\overrightarrow{\mathrm{r}}-\overrightarrow{\mathrm{a}}) \times \overrightarrow{\mathrm{c}}\}=\overrightarrow{0}\) then \(\overrightarrow{\mathrm{r}}\) is equal to:

Let \(|\cos \theta \cos (60-\theta) \cos (60-\theta)| \leq \frac{1}{8}, \theta \in[0,2 \pi]\) Then, the sum of all \(\theta \in[0,2 \pi]\), where \(\cos 3 \theta\) attains its maximum value, is :

If the shortest distance of the parabola \(y^2=4 x\) from document security settings le \(x^2+y^2-4 x-16 y+64=0\) is \(\mathrm{d}\), then \(\mathrm{d}^2\) is equal to :

If \(a_n=\frac{-2}{4 n^2-16 n+15}\), then \(a_1+a_2+\ldots \ldots+a_{25}\) is equal to :

If the sum of the second, fourth and sixth terms of a G.P. of positive terms is 21 and the sum of its eighth, tenth and twelfth terms is 15309 , then the sum of its first nine terms is :

If \(\cos ^{-1}\left(\frac{y}{2}\right)=\log _{e}\left(\frac{x}{5}\right)^{5},|y|<2\), then

If \(z = x + iy\) satisfies \(|z|-2=0\) and \(|z-i|-|z+5 i|=0\), then