The middle term in the expansion of \((1+x)^{2 n}\) is

The circles \(a x^{2}+a y^{2}+2 g_{1} x+2 f_{1} y+c_{1}=0\) and \(b x^{2}+b y^{2}+2 g_{2} x+2 f_{2} y+c_{2}=0\) \((a \neq 0\) and \(b \neq 0)\) cut orthogonally if

If \(A=\{3,5,7\}\) and \(B=\{1,2,3,5\}\), then \(A \times B \cap B \times A\) is equal to

\(\sin \left(2 \sin ^{-1} \sqrt{\frac{63}{65}}\right)\) is equal to

The characteristic roots of the matrix \(\left[\begin{array}{ccc}1 & 0 & 0 \\ 2 & 3 & 0 \\ 4 & 5 & 6\end{array}\right]\) are

The probability of choosing randomly a number \(c\) from the set \(\{1,2,3, \ldots 9\}\) such that the quadratic equation \(x^2+4 x+c=0\) has real roots, is

If \(y = \tan^{-1}\left(\dfrac{\sqrt{1+x^3} + \sqrt{1-x^3}}{\sqrt{1+x^3} - \sqrt{1-x^3}}\right)\) then \(\dfrac{dy}{dx} =\)

The critical angle for a typical glass air interface is \(42^\circ\). If a ray of light falls normally on one of the faces of the prism of angle \(45^\circ\). The emergent ray will:

Which of the following species do not show disproportionation reaction?

Match \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) and D with the appropriate functions given.
\(\begin{array}{|l|l|l|l|} \hline & \text { Column I } & & \text { Functions } \\ \hline \text { A } & \text { Oxidoreductase } & \text { P } & \text { Malfunctioning leads to Addison's disease. } \\ \hline \text { B } & \text { Phosphodiester bonds } & \text { Q } & \text { Regulates responses to external stimuli. } \\ \hline \text { C } & \text { Adrenal cortex } & \text { R } & \text { Catalyses glycolysis. } \\ \hline \text { D } & \text { Epinephrine. } & \text { S } & \text { Links nucleotides together. } \\ \hline \end{array}\)

The resistor of resistance \(R\) is connected to 25 V supply and heat produced in it is \(25 \mathrm{Js}^{-1}\). The value of \(R\) is

Choose the correct statement from the options given

If the sum of the slopes of the lines given by \(x^{2}-4 p x y+8 y^{2}=0\) is three times their product, then \(p\) has the value