If \(k=tan(\frac{\pi}{4}+\frac{1}{2}cos^{-1}(\frac{2}{3}))+tan(\frac{1}{2}sin^{-1}(\frac{2}{3}))\) then the number of solutions of the equation \(sin^{-1}(kx-1)=sin^{-1}x-cos^{-1}x\) is ___ .

The equation of a tangent to the hyperbola \(4x^2 -5y^2 = 20\) parallel to the line \(x -y = 2\) is

 If the shortest distance between the lines \(\frac{{x - 1}}{\alpha } = \frac{{y + 1}}{{ - 1}} = \frac{z}{1},(\alpha  \ne \, - 1)\) and \(x+y+z +1=0= 2x - y + z + 3\) is \(\frac {1}{\sqrt 3},\) then a value \(\alpha \) 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 

Let \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}\) be three points on the parabola \(y^2=6 x\) and let the line segment \(A B\) meet the line \(L\) through \(\mathrm{C}\) parallel to the \(\mathrm{x}\)-axis at the point \(\mathrm{D}\). Let \(\mathrm{M}\) and \(\mathrm{N}\) respectively be the feet of the perpendiculars from \(\mathrm{A}\) and \(\mathrm{B}\) on \(\mathrm{L}\). Then \(\left(\frac{\mathrm{AM} \cdot \mathrm{BN}}{\mathrm{CD}}\right)^2\) is equal to ...........

If the distance of the point \(P(43, \alpha, \beta), \beta<0\), from the line \(\vec{r}=4\hat{i}-\hat{k}+\mu(2\hat{i}+3\hat{k}), \mu\in R\) along a line with direction ratios \(3, -1, 0\) is \(13\sqrt{10}\), then \(\alpha^{2}+\beta^{2}\) is equal to ___ .

The value of \(\cot \left(\sum\limits_{n=1}^{50} \tan ^{-1}\left(\frac{1}{1+n+n^{2}}\right)\right)\) is

If a point \(A ( x , y )\) lies in the region bounded by the \(y\)-axis, straight lines \(2 y+x=6\) and \(5 x-6 y=30\), then the probability that \(y <1\) is

If the solution curve \( y=f(x) \) of the differential equation \( (x^{2}-4)y^{\prime}-2xy+2x(4-x^{2})^{2}=0, x>2 \) passes through the point (3, 15), then the local maximum value of f is:

The number of solutions of the equation \(sin\, 2x - 2\,cos\,x+ 4\,sin\, x\, = 4\) in the interval \([0, 5\pi ]\) is

If four distinct points with position vectors \(\vec{a}, \vec{b}, \vec{c}\) and \(\vec{d}\) are coplanar; then \([\vec{a} \vec{b} \vec{c}]\) is equal to

The integral \(\int \frac{(2 x-1) \cos \sqrt{(2 x-1)^{2}+5}}{\sqrt{4 x^{2}-4 x+6}} d x\) is equal to (where \(c\) is a constant of integration)

Let \(\mathrm{ABC}\) be a triangle with \(\mathrm{A}(-3,1)\) and \(\angle \mathrm{ACB}=\theta, 0<\theta<\frac{\pi}{2} .\) If the equation of the median through \(\mathrm{B}\) is \(2 \mathrm{x}+\mathrm{y}-3=0\) and the equation of angle bisector of \(\mathrm{C}\) is \(7 \mathrm{x}-4 \mathrm{y}-1=0\) then \(\tan\, \theta\) is equal to: