\(\left(\frac{1+i}{1-i}\right)^4+\left(\frac{1-i}{1+i}\right)^4\) is equal to

If the direction ratio of two lines are given by \(l+m+n=0, m n-2 l n+l m=0, \quad\) then the angle between the lines is

If $10$ is the mean of the data $2, 3, 5, 18, 17, 15, 13, x, 9$ and $7,$ then the mean deviation of this data about its mean is

Two tangents are drawn from the point \((-1,-2)\) to the parabola \(\mathrm{y}^2=4 \mathrm{x}\). If \(\theta\) is the angle between these tangents, then \(\tan \theta=\)

If at any point \(\left(x_1, y_1\right)\) on the curve \(y=f(x)\) the lengths of the subtangent and subnormal are equal, then the length of the tangent drawn to that curve at that point is

If the circles \(x^2+y^2-4 x+6 y+13-a^2=0\) and \(x^2+y^2-10 x-2 y+17=0\) intersect in two distinct points, then ' \(a\) ' is

\(\sqrt{12-\sqrt{68+48 \sqrt{2}}}\) is equal to :

Let \(z=x+i y\) and \(\mathrm{P}(x, y)\) be a point on the Argand plane. If \(z\) satisfies the condition \(\operatorname{Arg}\left(\frac{z-3 i}{z+2 i}\right)=\frac{\pi}{4}\) then the locus of P is

A straight uniform wire of resistance \(36 \Omega\) is bent in the form of a semi-circular loop. The effective resistance between the ends of the diameter of the semi-circular loop is

The equation of a curve \(C\) is transformed to \(X^2+Y^2-6 X+8 Y+21=0\) by the rotation of coordinate axes about the origin through an angle of \(\frac{\pi}{4}\) in the positive direction of \(X\)-axis. If \(a x^2+b y^2+c x+d y+e=0\) is the equation of the curve \(C\) before the transformation, then \(\left(a+b+c^2+d^2-5 e\right)^2=0\).

Match the following methods for treating hardness of water with their corresponding reagents used. \begin{array}{|c|c|c|} \hline List-I & & List II \ \hline A. Treatment with washing soda & \mathrm{I}. & \mathrm{Ca}(\mathrm{OH})_2 \ \hline B. Calgon's method & II. & \mathrm{Na}_2 \mathrm{CO}_3 \ \hline C. Clark's method & III & \mathrm{NaAlSiO}_4 \ \hline D. Zeolite/permutit process & IV & \mathrm{Na}_6 \mathrm{P}_6 \mathrm{O}_{18} \ \hline \end{array} The correct match is \(\begin{array}{llllllll}\text { A } & \text { B } & \text { C } & \text { D } & \text { A } & \text { B } & \text { C } & \text { D }\end{array}\)

Which of the following reagents when heated with ethyl chloride, forms ethylene ?

\(f(x)\) is a twrice differentiable function such that \(f^{\prime \prime}(x)=-f(x)\) and \(f^{\prime}(x)=g(x)\). If \(\left.h(x)=(f(x))^2+g(x)\right)^2\) and \(h(l)=2\), then \(h(2)=\)