Let \(a = lm\left( {\frac{{1 + {z^2}}}{{2iz}}} \right)\), where \(z\) is any non-zero complex number. The set \(A = \{ a:\left| z \right| = 1\,and\,z \ne  \pm 1\} \) is equal to

Let \(m\) and \(n\) be the number of points at which the function \(f(\mathrm{x})=\max \left\{\mathrm{x}, \mathrm{x}^3, \mathrm{x}^5, \ldots ., \mathrm{x}^{21}\right\}, \mathrm{x} \in \mathbb{R}\), is not differentiable and not continuous, respectively. Then \(\mathrm{m}+\mathrm{n}\) is equal to ________ .

Consider the following two statements : Statement \(I\) : For any two non-zero complex numbers \(\mathrm{z}_1, \mathrm{z}_2\) \(\left(\left|z_1\right|+\left|z_2\right|\right)\left|\frac{z_1}{\left|z_1\right|}+\frac{z_2}{\left|z_2\right|}\right| \leq 2\left(\left|z_1\right|+\left|z_2\right|\right)\) and Statement \(II\) : If \(\mathrm{x}, \mathrm{y}, \mathrm{z}\) are three distinct complex numbers and a, b, c are three positive real numbers such that \(\frac{a}{|y-z|}=\frac{b}{|z-x|}=\frac{c}{|x-y|}\), then \(\frac{\mathrm{a}^2}{\mathrm{y}-\mathrm{z}}+\frac{\mathrm{b}^2}{\mathrm{z}-\mathrm{x}}+\frac{\mathrm{c}^2}{\mathrm{x}-\mathrm{y}}=1\) Between the above two statements,

Let \(\alpha, \beta\) be the roots of the equation \(x^2+2 \sqrt{2} x-1=0\). The quadratic equation, whose roots are \(\alpha^4+\beta^4\) and \(\frac{1}{10}\left(\alpha^6+\beta^6\right)\), is :

If \(f(x)\) is a quadratic expression such that \(f(1) + f (2)\, = 0\) , and \(-1\) is a root of \(f(x)\, = 0\), then the other root of \(f(x)\, = 0\) is

Let \(P\) be a plane \(l x+m y+n z=0\) containing the line, \(\frac{1-x}{1}=\frac{y+4}{2}=\frac{z+2}{3} .\) If plane \(P\) divides the line segment \(AB\) joining points \(A (-3,-6,1)\) and \(B (2,4,-3)\) in ratio \(k : 1\) then the value of \(k\) is equal to

Let \(\alpha\) be the angle between the lines whose direction cosines satisfy the equations \(l+m-n=0\) and \(l^{2}+m^{2}-n^{2}=0 .\) Then the value of \(\sin ^{4} \alpha+\cos ^{4} \alpha\) is

If the solution of the differential equation \(\frac{d y}{d x}+e^{x}\left(x^{2}-2\right) y=\left(x^{2}-2 x\right)\left(x^{2}-2\right) e^{2 x} \quad\) satisfies \(y(0)=0\), then the value of \(y(2)\) is

For \(a>0,\) let the curves \(C_{1}: y^{2}=a x\) and \(\mathrm{C}_{2}: \mathrm{x}^{2}=\) ay intersect at origin \(\mathrm{O}\) and a point \(\mathrm{P}\) Let the line \(\mathrm{x}=\mathrm{b}(0<\mathrm{b}<\mathrm{a})\) intersect the chord \(OP\) and the \(\mathrm{x}\) -axis at points \(\mathrm{Q}\) and \(\mathrm{R}\), respectively. If the line \(x=b\) bisects the area bounded by the curves, \(\mathrm{C}_{1}\) and \(\mathrm{C}_{2},\) and the area of \(\Delta \mathrm{OQR}=\frac{1}{2},\) then '\(a\)' satisfies the equation

Let the matrix \(A =\left[\begin{array}{lll}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0\end{array}\right]\) and the matrix \(B _{0}= A ^{49}+2 A ^{98}\). If \(B _{ n }= Adj \left( B _{ n -1}\right)\) for all \(n \geq 1\),then \(\operatorname{det}\left( B _{4}\right)\) is equal to.

If the area enclosed by the parabolas \(P_1: 2 y=5 x^2\) and \(P_2: x^2-y+6=0\) is equal to the area enclosed by \(P_1\) and \(y=\alpha x, \alpha > 0\), then \(\alpha^3\) is equal to \(......\).

The area (in sq. units) of the region described by \(\{(x,y):\)\({y^2} \le 2x \,and\,y \ge 4x - 1\)\(\}\) is

If \(\sum_{k=1}^{10} \frac{k}{k^{4}+k^{2}+1}=\frac{m}{n}\), where \(m\) and \(n\) are coprime, then \(m+n\) is equal to.