Let \(P(4, -4)\) and \(Q(9, 6)\) be two points on the parabola, \(y^2 = 4x\) and let \(X\) be any point on the arc \(POQ\) of this parabola, where \(O\) is the vertex of this parabola, such that the area of \(\Delta PXQ\) is maximum. Then this maximum Area (in sq. units) is

If \(\mathrm{x}=\sum\limits_{\mathrm{n}=0}^{\infty}(-1)^{\mathrm{n}} \tan ^{2 \mathrm{n}} \theta\) and \(\mathrm{y}=\sum\limits_{\mathrm{n}=0}^{\infty} \cos ^{2 \mathrm{n}} \theta,\) for \(0<\theta<\frac{\pi}{4},\) then

Let  \( \lim _{n \rightarrow \infty}\left(\frac{n}{\sqrt{n^4+1}}-\frac{2 n}{\left(n^2+1\right) \sqrt{n^4+1}}+\frac{n}{\sqrt{n^4+16}}-\frac{8 n}{\left(n^2+4\right) \sqrt{n^4+16}}\right. \) \( \left.+\ldots \ldots+\frac{n}{\sqrt{n^4+n^4}}-\frac{2 n \cdot n^2}{\left(n^2+n^2\right) \sqrt{n^4+n^4}}\right) \text { be } \frac{\pi}{k},\) using only the principal values of the inverse trigonometric functions. Then \(\mathrm{k}^2\) is equal to ..............

Let the line of the shortest distance between the lines \(L_1: \vec{r}=(\hat{i}+2 \hat{j}+3 \hat{k})+\lambda(\hat{i}-\hat{j}+\hat{k})\) and  \(L_2: \vec{r}=(4 \hat{i}+5 \hat{j}+6 \hat{k})+\mu(\hat{i}+\hat{j}-\hat{k})\)  intersect \(\mathrm{L}_1\) and \(\mathrm{L}_2\) at \(\mathrm{P}\) and \(\mathrm{Q}\)  respectively. If \((\alpha, \beta, \gamma)\) is the midpoint  of the line segment \(PQ\), then \(2(\alpha+\beta+\gamma)\) is equal to ...........

Let \(f\) be a differentiable function such that \(x ^2 f ( x )- x =4 \int \limits_0^x t f(t) d t, f(1)=\frac{2}{3}\).Then \(18 f(3)\) is equal to \(......\).

If the tangent to the parabola \(y^2 = x\) at a point \(\left( {\alpha ,\beta } \right)\,,\,\left( {\beta  > 0} \right)\) is also a tangent to the ellipse, \(x^2 + 2y^2 = 1\), then \(a\) is equal to

If the domain of the function \( f(x)=\sin^{-1}(\frac{2}{x^{2}-2x-2}) \) is \( (-\infty,\alpha]\cup[\beta,\gamma]\cup[\delta,\infty) \), then \( \alpha+\beta+\gamma+\delta \) is equal to

The sum of the roots of the equation, \({x^2}\, + \,\left| {2x - 3} \right|\, - \,4\, = \,0,\) is

Statement \(-1\) : The system of linear equations \(x + \left( {\sin \,\alpha } \right)y + \left( {\cos \,\alpha } \right)z = 0\) \(x + \left( {\cos \,\alpha } \right)y + \left( {\sin \alpha } \right)z = 0\) \(x - \left( {\sin \,\alpha } \right)y - \left( {\cos \alpha } \right)z = 0\) has a non-trivial solution for only one value of \(\alpha \) lying in the interval \(\left( {0\,,\,\frac{\pi }{2}} \right)\)  Statement \(-2\) : The equation in \(\alpha \) \(\left| {\begin{array}{*{20}{c}}
  {\cos {\mkern 1mu} \alpha }&{\sin {\mkern 1mu} \alpha }&{\cos {\mkern 1mu} \alpha } \\ 
  {\sin {\mkern 1mu} \alpha }&{\cos {\mkern 1mu} \alpha }&{\sin {\mkern 1mu} \alpha } \\ 
  {\cos {\mkern 1mu} \alpha }&{ - \sin {\mkern 1mu} \alpha }&{ - \cos {\mkern 1mu} \alpha } 
\end{array}} \right| = 0\) has only one solution lying in the interval \(\left( {0\,,\,\frac{\pi }{2}} \right)\)

The locus of the centroid of the triangle formed by any point \(\mathrm{P}\) on the hyperbola \(16 \mathrm{x}^{2}-9 \mathrm{y}^{2}+\) \(32 x+36 y-164=0\), and its foci is:

Number of solutions of \( \sqrt{3}\cos 2\theta+8\cos \theta+3\sqrt{3}=0 \), \( \theta \in [-3\pi, 2\pi] \) is:

Let \(P = \left[ {\begin{array}{*{20}{c}}
  1&0&0 \\ 
  3&1&0 \\ 
  9&3&1 
\end{array}} \right]\) and \(Q = [q_{ij}]\) be two \(3\times3\) matrices such that \(Q -P^5 = I_3\). Then \(\frac{{{q_{21}} + {q_{31}}}}{{{q_{32}}}}\) is equal to

The tangent to the circle \(C_1 : x^2 + y^2 - 2x- 1\, = 0\) at the point \((2, 1)\) cuts off a chord of length \(4\) from a circle \(C_2\) whose centre is \((3, - 2)\). The radius of \(C_2\) is