If \(\alpha, \beta\) and \(\gamma\) are the roots of the equation \(x^3+p x^2+q x+r=0\), then the coefficient of \(x\) in the cubic equation whose roots are \(\alpha(\beta+\gamma), \beta(\gamma+\alpha)\) and \(\gamma(\alpha+\beta)\) is

In the triangle with vertices at \(A(6,3), B(-6,3)\) and \(C(-6,-3)\), the median through \(A\) meets
\(B C\) at \(P\), the line \(A C\) meets the \(x\)-axis at \(Q\), while \(R\) and \(S\) respectively denote the orthocentre and centroid of the triangle. Then the correct matching of the coordinates of points in List-I to List-II is
\(\begin{array}{llll} & \text { List-I } & & \text { List-II } \\ \text { (i) } & P & \text { (A) } & (0,0) \\ \text { (ii) } & Q & \text { (B) } & (6,0) \\ \text { (iii) } & R & \text { (C) } & (-2,1) \\ \text { (iv) } & S & \text { (D) }(-6,0) \\ & & \text { (E) }(-6,-3) \\ & & \text { (F) }(-6,3)\end{array}\)
(i)
(ii)
(iii)
(iv)

If the function \(y=\sin ^{-1} x\), then \(\left(1-x^2\right) \frac{d^2 y}{d x^2}\) is equal to

The solution of the differential equation $2x\left(\frac{dy}{dx}\right)-y=4$ represents a family of

In \(\triangle \grave{A B C}\), if \(4 r_1=5 r_2=6 r_3\), then \(\sin ^2 \frac{A}{2}+\sin ^2 \frac{B}{2}+\sin ^2 \frac{C}{2}=\)

If \(\alpha \in \mathbb{R}-\{-1\}\) and \(\mathrm{f}(\mathrm{x})=|(|\mathrm{x}|+\alpha)(|\mathrm{x}|-1)|\), then the number of points at which \(f(x)\) is not differentiable, is

If \(A(4,0)\) and \(B(-4,0)\) are two points, then the locus of a point \(\mathrm{P}\) such that \(\mathrm{PA}-\mathrm{PB}=4\) is

The rms speed of oxygen and room temperature is about \(500 \mathrm{~ms}^{-1}\). The rms speed of hydrogen at the same temperature is about

A metal has work function \(2.5 \mathrm{eV}\). If a radiation of frequency \(32 \times 10^{15} \mathrm{~Hz}\) is incident on this metal surface, then the maximum kinetic energy of ejected photoelectrons is (Planck's constant, \(h=6.6 \times 10^{-34} \mathrm{~J}-\mathrm{s}\) )

The equilibrium constant for the given reaction is 100 .
\(\mathrm{N}_2(g)+2 \mathrm{O}_2(g) \rightleftharpoons 2 \mathrm{NO}_2(g)\)
What is the equilibrium constant for the reaction given below?
\(\mathrm{NO}_2(g) \rightleftharpoons \frac{1}{2} \mathrm{~N}_2(g)+\mathrm{O}_2(g)\)

If \(y=\sin ^{-1} x\) then \(\left(1-x^2\right) y_2-x y_1=\)

Among the given configurations, identify the element which doesn't belong to the same family as the others?

Identify the correct statements from the following:
i. In the presence of UV light, \(\mathrm{CF}_2 \mathrm{Cl}_2(g)\) gives chlorine free radicals which will react with \(\mathrm{O}_3(g)\) to form \(\mathrm{O}_2(g)\).
ii. Drinking water with \(10 \mathrm{ppm}\) fluoride is better than drinking water with \(1 \mathrm{ppm}\) fluoride.
iii. The maximum permissible concentration of lead in drinking water is \(50 \mathrm{ppb}\).