The number of solutions of the equation \(\sin ^{-1}\left[x^{2}+\frac{1}{3}\right]+\cos ^{-1}\left[x^{2}-\frac{2}{3}\right]=x^{2}\) for \(x \in[-1,1],\) and \([x]\) denotes the greatest integer less than or equal to \(x\), is ...... .

For two non-zero complex number \(z_1\) and \(z_2\), if \(\operatorname{Re}\left(z_1 z_2\right)=0\) and \(\operatorname{Re}\left(z_1+z_2\right)=0\), then which of the following are possible ? \((A)\) \(\operatorname{Im}\left(z_1\right) > 0\) and \(\operatorname{Im}\left(z_2\right) > 0\) \((B)\) \(\operatorname{Im}\left(z_1\right) < 0\) and \(\operatorname{Im}\left(z_2\right) > 0\) \((C)\) \(\operatorname{Im}\left(z_1\right) > 0\) and \(\operatorname{Im}\left(z_2\right) < 0\) \((D)\) \(\operatorname{Im}\left( z _1\right) < 0\) and \(\operatorname{Im}\left( z _2\right) < 0\) Choose the correct answer from the options given below :

The sum of possible values of \(x\) for \(\tan ^{-1}( x +1)+\cot ^{-1}\left(\frac{1}{ x -1}\right)=\tan ^{-1}\left(\frac{8}{31}\right)\) is

The integral \(\int_{0}^{1} \frac{1}{{ }_{7}^{\left[\frac{1}{x}\right]}} d x=\) where [.] denotes the greatest integer function is equal to

Let \( A=\{2,3,5,7,9\} \). Let R be the relation on A defined by \(x\) Ry if and only if \( 2x\le3y \). Let \(l\) be the number of elements in R, and m be the minimum number of elements required to be added in R to make it a symmetric relation. Then \( l+m \) is equal to :

Let \(z_1\) and \(z_2\) be two complex number such that \(z_1\) \(+z_2=5\) and \(z_1^3+z_2^3=20+15 i\). Then \(\left|z_1^4+z_2^4\right|\) equals-

Let \(A\) be a \(2 \times 2\) matrix with \(\operatorname{det}(A)=-1\) and det \((( A + I )(\operatorname{Adj}( A )+ I ))=4\). Then the sum of the diagonal elements of \(A\) can be.

Statement \(1\) : The only circle having radius \(\sqrt {10} \) and a diameter along line \(2x + y = 5\) is \(x^2 + y^2 - 6x +2y = 0\).
Statement \(2\) : \(2x + y = 5\) is a normal to the circle \(x^2 + y^2 -6x+2y = 0\).

If \(\int \operatorname{cosec}^5 x d x=\alpha \cot x \operatorname{cosec} x\left(\operatorname{cosec}^2 x+\frac{3}{2}\right)+\beta \log _e\left|\tan \frac{x}{2}\right|+C\) where \(\alpha, \beta \in \mathbb{R}\) and \(\mathrm{C}\) is constant of integration , then the value of \(8(\alpha+\beta)\) equals ...........

If \(\mathrm{y}=\mathrm{y}(\mathrm{x})\) is the solution of the differential equation, \(\mathrm{e}^{\mathrm{y}}\left(\frac{\mathrm{dy}}{\mathrm{dx}}-1\right)=\mathrm{e}^{\mathrm{x}}\) such that \(\mathrm{y}(0)=0,\) then \(\mathrm{y}(1)\) is equal to 

The number of solutions of \(\sin ^{7} x+\cos ^{7}=1, x \in[0,4 \pi]\) is equal to :

The value of \(\left|\begin{array}{lll}(a+1)(a+2) & a+2 & 1 \\ (a+2)(a+3) & a+3 & 1 \\ (a+3)(a+4) & a+4 & 1\end{array}\right|\) is 

The acute angle between the planes \(P_{1}\) and \(P_{2}\), when \(P_{1}\) and \(P_{2}\) are the planes passing through the intersection of the planes \(5 x+8 y+13 z-29=0\) and \(8 x-7 y+z-20=0\) and the points \((2,1,3)\) and \((0,1,2)\), respectively, is