The number of solutions of \(\operatorname{Tan}^{-1} 1+\frac{1}{2} \operatorname{Cos}^{-1} x^2-\operatorname{Tan}^{-1}\left(\frac{\sqrt{1+x^2}+\sqrt{1-x^2}}{\sqrt{1+x^2}-\sqrt{1-x^2}}\right)=0\) is

If the ends of the hypotenuse of a right angled triangle are \((0, a)\) and \((a, 0)\) then the locus of the third vertex is

If \(a, b, c\) and \(d\) are real numbers such that \(a^2+b^2+c^2+d^2=1 \quad\) and \(A=\left[\begin{array}{c}a+i b c+i d \\ -c+i d a-i b\end{array}\right]\), then \(A^{-1}\) equals to

If \(S\) and \(S^{\prime}\) are the foci of an ellipse, \(B\) is one end of the minor axis and \(\angle S B S^{\prime}=90^{\circ}\), then the eccentricity of that ellipse is

Solve \((8-t)^2 < \left(t^2-3 t-10\right)\)

The vector equation of any plane passing through the line of intersection of the planes \(\overrightarrow{\mathrm{r}} \cdot \overrightarrow{\mathrm{m}}_1=\mathrm{q}_1\) and \(\overrightarrow{\mathrm{r}} \cdot \overrightarrow{\mathrm{m}}_2=\mathrm{q}_2\) is given by \(\overrightarrow{\mathrm{r}}\left(\overrightarrow{\mathrm{m}}_1+\lambda \overrightarrow{\mathrm{m}}_2\right)=\mathrm{q}_1+\lambda \mathrm{q}_2\) for \(\lambda \in \mathbb{R}\). The vector equation of a plane passing through the point \(2 \hat{i}-3 \hat{j}+\hat{k}\) and the line of intersection of the planes r. \((\hat{i}-2 \hat{j}+3 \hat{k})=5\) and \(\vec{r} \cdot(3 \hat{i}+\hat{j}-2 \hat{k})=7\)

The sum of the distinct values of \(x\) for which the matrix \(A=\left[\begin{array}{lll}1 & 1 & x \\ 1 & x & 1 \\ x & 1 & 1\end{array}\right]\) has no inverse, is

Three very large plates of same area are kept parallel and close to each other. They are considered as ideal black surfaces and have very high thermal conductivity. First and third plates are maintained at absolute temperatures \(2 T\) and \(3 T\) respectively.
Temperature of the middle plate in steady state is

The equations \(x-y=4\) and \(x^2+4 x y+y^2=0\) represent the sides of a/an

\(\int \frac{x^5 d x}{\left(x^2+x+1\right)\left(x^6+1\right)\left(x^4-x^3+x-1\right)}=\)

The number of possible straight lines passing through the point \((2,3)\), while forming a triangle with coordinate axes enclosing an area 12 sq. units is

If the mean deviation about the mean is \(m\) and variance is \(\sigma^2\) for the following data, then \(m+\sigma^2=\)
\begin{array}{|c|c|c|c|c|c|}\hlinex & 1 & 3 & 5 & 7 & 9 \\\hlinef & 4 & 24 & 28 & 16 & 8 \\\hline\end{array}

If \(m\) and M denote the mean deviations about mean and about median respectively of the data \(20,5.15,2,7,3,11\) then the mean deviation about the mean of \(m\) and \(M\) is