The points of intersection of two ellipses \(x^{2}+2 y^{2}-6 x-12 y+20=0\) and \(2 x^{2}+y^{2}-10 x-6 y+15=0\) lie on a circle. The centre of the circle is

The value of \(\sum_{n=1}^{13}\left(i^{n}+i^{n+1}\right), i=\sqrt{-1}\) is

Time period \(T\) of a simple pendulum of length \(l\) is given by \(T=2 \pi \sqrt{\frac{l}{g}}\). If the length is increased by \(2 \%\), then an approximate change in the time period is

The point \(P(3,6)\) is first reflected on the line \(y=x\) and then the image point \(Q\) is again reflected on the line \(y=-x\) to get the image point \(Q^{\prime} .\) Then, the circumcentre of the \(\Delta P Q Q^{\prime}\) is

A moving line intersects the lines \(x+y=0\) and \(x-y=0\) at the points A, B respectively such that the area of the triangle with vertices \((0,0), A \& B\) has a constant area \(C\). The locus of the mid-point \(A B\) is given by the equation

If the sum of two unit vectors is a unit vector, then the magnitude of their difference is

Let \(f: R \rightarrow R\) be a twice continuously differentiable function such that \(f(0)=f(1)=f^{\prime}(0)=0 .\) Then

The melting points of (i) BeCl\(_{2}\) (ii) \(\mathrm{CaCl}_{2}\) and (iii) \(\mathrm{HgCl}_{2}\) follows the order

In aluminium extraction by the Bayer process, alumina is extracted from bauxite by sodium hydroxide at high temperature and pressures :
\(\mathrm{Al}_2 \mathrm{O}_3(\mathrm{~s})+2 \mathrm{OH}^{-}(\mathrm{aq}) \rightarrow 2 \mathrm{Al}_2 \mathrm{O}_2(\mathrm{aq})+\mathrm{H}_2 \mathrm{O}(1)\)

Solid impurities such as \(\mathrm{Fe}_2 \mathrm{O}_3\) and \(\mathrm{SiO}_2\) are removed and then \(\mathrm{Al}(\mathrm{OH})_4^{-}\)is reprecipitated : \(2 \mathrm{Al}(\mathrm{OH})_4^{-} \rightarrow \mathrm{Al}_2 \mathrm{O}_3 \cdot 3 \mathrm{H}_2 \mathrm{O}(\mathrm{s})+2 \mathrm{OH}^{-}\)(aq) . In the industrial world :

In order to get a head at least once with probability \(\geq 0.9\), the minimum number of times a unbiased coin needs to be tossed is

Which type of Gate the following truth table represents?

Which of the following is not always true?

\(\mathrm{Na}_2 \mathrm{CO}_3\) is prepared by Solvay process but \(\mathrm{K}_2 \mathrm{CO}_3\) cannot be prepared by the same because