From a point \(P\) on the line \(4 x-3 y=6\) two tangents are drawn to the circle \(x^2+y^2-6 x-4 y+4=0\). If the angle between these tangents is \(\tan ^{-1}\left(\frac{24}{7}\right)\), then \(P=\)

If $f\left(x\right)=\left\{\begin{array}{cc}\frac{x^2-16}{x-4}& \mathrm{if} x>4\\ 2x& \mathrm{if} x\le 4\end{array}\right.$ then $f^{\text{'}}\left(4^{-}\right)+f^{\text{'}}\left(4^{+}\right)=$

A number \(n\) is chosen at random from \(S=\{1,2,3, \ldots, 50\}\). Let \(A=\left\{n \in S: n+\frac{50}{n}>27\right\}, B=\{n \in S: n\) is a prime) and \(C=\{n \in S: n\) is a square). Then, correct order of their probabilities is

Let \(A\) and \(B\) be not mutually exclusive events. If \(P(A)=\frac{4}{9}, P(A \cap \bar{B})=\frac{3}{7}\) then \(P\left(\frac{B}{A}\right)=\)

The function \(f(x)=2 x^3-9 a x^2+12 a^2 x+1\) where \(\mathrm{a}>0\) attains its local maximum and local minimum at \(p\) and \(q\) respectively. If \(p^2=q\) then \(a=\)

If ${\sin }^{-1}\left(\frac{x}{5}\right)+{\mathrm{cosec}}^{-1}\left(\frac{5}{4}\right)=\frac{\pi }{2},$ then $5+x=$

If \(1, \omega, \omega^2\) are the cube roots of unity and \(1, \alpha, \alpha^2, \alpha^3\) are the fourth roots of unity in usual notation then \(\alpha+\alpha \omega-\alpha^3 \omega^2=\)

If \(f:[a, b] \rightarrow[c, d]\) is a continuous and strictly increasing function, then \(\frac{d-c}{b-a}\) is

Henry's law constant for \(\mathrm{CO}_2\) in water is \(1.67 \mathrm{~K} \mathrm{bar}\) at \(25^{\circ} \mathrm{C}\). The quantity of \(\mathrm{CO}_2\) in \(1000 \mathrm{~mL}\) of soda water when packed under 5 bar \(\mathrm{CO}_2\) pressure at \(25^{\circ} \mathrm{C}\) is

With reference to the redox properties of hydrogen peroxide \(\left(\mathrm{H}_2 \mathrm{O}_2\right)\), which of these reactions are feasible? (i) \(2 \mathrm{Fe}^{2+} 2 \mathrm{H}^{+}+\mathrm{H}_2 \mathrm{O}_2 \longrightarrow 2 \mathrm{Fe}^{3+}+2 \mathrm{H}_2 \mathrm{O}\) (ii) \(2 \mathrm{MnO}_4^{-}+6 \mathrm{H}^{+}+5 \mathrm{H}_2 \mathrm{O}_2 \longrightarrow 2 \mathrm{Mn}^{2+}\) \(+8 \mathrm{H}_2 \mathrm{O}+5 \mathrm{O}_2\) (iii) \(2 \mathrm{Fe}^{2+}+\mathrm{H}_2 \mathrm{O}_2 \longrightarrow 2 \mathrm{Fe}^{3+}+2 \mathrm{OH}^{-}\) (iv)\(2 \mathrm{MnO}_4^{-}+3 \mathrm{H}_2 \mathrm{O}_2 \longrightarrow 2 \mathrm{MnO}_2+2 \mathrm{H}_2 \mathrm{O}\) \(+3 \mathrm{O}_2+2 \mathrm{OH}^{-}\)

Find the voltage \(V_2\) across \(R_2\) for the given circuit

\(\int \sqrt{e^x-4} d x\) equals

Let \(A_r=\left(x+\frac{1}{x}\right)^3 \cdot\left(x^2+\frac{1}{x^2}\right)^3 \cdot\left(x^3+\frac{1}{x^3}\right)^3 \ldots\). \(\left(x^r+\frac{1}{x^r}\right)^3\). If \(x^2+x+1=0\), then \(\frac{1}{A_3}+\frac{1}{A_6}+\frac{1}{A_9}+\frac{1}{A_{12}}+\ldots . . \infty=\)