The quadratic equation whose roots are \(\sin ^2 18^{\circ}\) and \(\cos ^2 36^{\circ}\) is

If $\Delta =\left|\begin{array}{lll}1& a& a^2\\ 1& b& b^2\\ 1& c& c^2\end{array}\right|=K\left(a-b\right)\left(b-c\right)\left(c-a\right),$ then $K=$

If the circles \(x^2+y^2+2 k x-4 y+1=0\) and \(x^2+y^2-8 x-12 y+43=0\) touch each other then \(k=\)

If \(f(x)=\lim _{n \rightarrow \infty}\left(\frac{\log (2+x)-x^{2 n} \sin x}{1+x^{2 n}}\right), 0 \leq x \leq \frac{\pi}{2}\) \(x \in \mathbf{R}\), then at \(x=1, f(x)\) is

The equation of the parabola with the focus \((3,0)\) and the directrix \(x+3=0\), is.

If \(\sin \theta+\sin ^2 \theta=1\) and \(\cos ^{12} \theta+a \cos ^{10} \theta+b \cos ^8 \theta+c \cos ^6 \theta+d=0\), then

For \(x \in\left(\frac{3 \pi}{4}, \pi\right), \int(\sqrt{1+\sin 2 x}+\sqrt{1-\sin 2 x})\) \(d x=\)

If the line \(x+3 y=0\) is the tangent at \((0,0)\) to the circle of radius 1 , then the centre of one such circle is

If $\left(0,\frac{3}{4}\right)$ is the radical centre of the circles $S\equiv x^2+y^2+\alpha x+6y=0,S^{\text{'}}\equiv x^2+y^2+2\alpha x+\alpha y+6=0$ and $S^{"}\equiv x^2+y^2+6\alpha x-\alpha y+3=0$ then the distance between the radical centre and the centre of the circle $S^{\text{'}}=0$ is

The equation of a straight line, perpendicular to \(3 x-4 y=6\) and forming a triangle of area 6 sq. units with coordinate axes, is

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 \(\begin{array}{llllllll}\text { A } & \text { B } & \text { C } & \text { D } & \text { A } & \text { B } & \text { C } & \text { D }\end{array}\)

For a real variable \(a>1\), consider the points \(A_k=\left(k a, a^k\right), k=1,2, \ldots ., n\) in the Cartesian plane. If \(\alpha\) and \(\beta\) represent respectively the arithmetic mean of \(x\)-coordinates and the geometric mean of \(y\) coordinates of \(A_k\), then the locus of the point \(P(\alpha, \beta)\) is

If the lines \(L_1 \equiv x-2 y+3=0, L_2 \equiv 2 x+y+1=0\) and \(L_3 \equiv 3 x+y+c=0\) are concurrent and \(\theta\) is the acute angle between the lines \(L_1=0\) and \(L_3=0\), then \(\tan \theta=\)