Let \(u=\frac{2 z+i}{z-k i}, z=x+i y\) and \(k>0 .\) If the curve represented by \(\operatorname{Re}( u )+\operatorname{Im}( u )=1\) intersects the \(y\)-axis at the points \(P\) and \(Q\) where \(P Q=5,\) then the value of \(k\) is 

Let the function \(f(x)=\left(x^2+1\right)\left|x^2-a x+2\right|+\cos |x|\) be not differentiable at the two points \(x=\alpha=2\) and \(x=\beta\). Then the distance of the point \((\alpha, \beta)\) from the line \(12 x+5 y+10=0\) is equal to :

Let \(P\) be a variable point on the parabola \(y=4 x^{2}+1\). Then the locus of the mid-point of the point \(P\) and the foot of the perpendicular drawn from the point \(P\) to the line \(y=x\) is:

Let \(f, g\) and \(h\) be the real valued functions defined on \(R\) as \(f(x)=\left\{\begin{array}{cc}\frac{x}{|x|}, & x \neq 0 \\ 1, & x=0\end{array}\right.\), \(g(x)=\left\{\begin{array}{cl}\frac{\sin (x+1)}{(x+1)}, & x \neq-1 \\1, & x=-1\end{array} \text { and } h(x)=2[x]-f(x),\right.\) where \([x]\) is the greatest integer \(\leq x\). Then the value of \(\lim _{x \rightarrow 1} g(h(x-1))\) is

The value of the integral \(\displaystyle\int_{0}^{2} \dfrac{\sqrt{x(x^2+x+1)}}{(\sqrt{x+1})(\sqrt{x^4+x^2+1})} \, dx\) is equal to:

Suppose the vectors \(x_{1}, x_{2}\) and \(x_{3}\) are the solutions of the system of linear equations, \(Ax = b\) when the vector \(b\) on the right side is equal to \(b _{1}, b _{2}\) and \(b _{3}\) respectively. If \(x =\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right], x _{2}=\left[\begin{array}{l}0 \\ 2 \\ 1\end{array}\right], x _{3}=\left[\begin{array}{l}0 \\ 0 \\ 1\end{array}\right], b _{1}=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]\) \(b _{2}=\left[\begin{array}{l}0 \\ 2 \\ 0\end{array}\right]\) and \(b _{3}=\left[\begin{array}{l}0 \\ 0 \\ 2\end{array}\right],\) then the determinant of \(A\) is equal to

Let \(a_{n}\) be the \(n^{\text {th }}\) term of a G.P. of positive terms. If \(\sum\limits_{n=1}^{100} a_{2 n+1}=200\) and \(\sum\limits_{n=1}^{100} a_{2 n}=100,\) then \(\sum\limits_{n=1}^{200} a_{n}\) is equal to 

\(\text { If } \int\limits_{0}^{2}\left(\sqrt{2 x}-\sqrt{2 x-x^{2}}\right) d x=\) \(\int\limits_{0}^{1}\left(1-\sqrt{1-y^{2}}-\frac{y^{2}}{2}\right) d y+\int\limits_{1}^{2}\left(2-\frac{y^{2}}{2}\right) d y+I\)  then \(I=\dots\dots\dots\)

If \(\frac{1}{n+1}{ }^n C_n+\frac{1}{n}{ }^n C_{n-1}+\ldots+\frac{1}{2}{ }^{ n } C _1+{ }^{ n } C _0=\frac{1023}{10}\) then \(n\) is equal to

\(\int {\frac{{2x + 5}}{{\sqrt {7 - 6x - {x^2}} }}dx}  = A\sqrt {7 - 6x - {x^2}}  + B\,{\sin ^{ - 1}}\left( {\frac{{x + 3}}{4}} \right) + C\) (where \(C\) is a constant of integration), then the ordered pair \((A, B)\) is equal to

If three of the six vertices of a regular hexagon are chosen at random, then the probability that the triangle formed with these chosen vertices is equilateral is

The area of the region bounded by the parabola \((y-2)^{2}=(x-1)\), the tangent to it at the point whose ordinate is \(3\) and the \(\mathrm{x}\)-axis is :

Let \(P = \{\theta \in [0, 4\pi] : \tan^2\theta \neq 1\}\) and \(S = \{a \in \mathbb{Z} : 2(\cos^8\theta - \sin^8\theta)\sec 2\theta = a^2, \theta \in P\}\). Then \(n(S)\) is: