If the tangent at the point \((1,2)\) on the ellipse \(3 x^2+4 y^2=19\) is also a tangent to the parabola \(y^2-k x=0\), then \(k=\)

\(\int_0^{\pi / 2} \frac{d x}{1+\tan ^3 x}\) is equal to :

A basket contains 12 apples in which 3 are rotten. If 3 apples are drawn at random simultaneously from it, then the probability of getting atmost one rotten apple is

If a straight line passing through the point \(\mathrm{P}(3,4)\) makes an angle \(\frac{\pi}{6}\) with the \(\mathrm{X}\)-axis and meets the line \(12 x+5 y+10=0\) at \(\mathrm{Q}\), then the length of \(\mathrm{PQ}\) is

If \({ }^{27} \mathrm{P}_{\mathrm{r}+7}=7722{ }^{25} \mathrm{P}_{(\mathrm{r}+4)}\), then \(\mathrm{r}=\)

If \(\alpha, \beta\) are the eccentric angles of the extremities of a focal chord (other than the major axis) of the ellipse \(x^2+4 y^2=4\) then \(\sqrt{3} \cos \frac{\alpha+\beta}{2}=\)

If \(x \sqrt{1+y}+y \sqrt{1+x}=0\), then \(\frac{d y}{d x}\) is equal to

If $6$ is the mean of a Poisson distribution, then $P\left(X\ge 3\right)=$

Which of the following statements is correct about "shielding effect"?
1. It is effective, when the orbitals in the inner shells are completely filled.
2. Inner shells do not show any effect.
3. With increase of shielding down the group, ionisation energy increases.
4. With increasing nuclear charge, the shielding effect increases.

What are \(X\) and \(Y\) in the following reaction?
\(\mathrm{CF}_2 \mathrm{Cl}_2 \stackrel{\mathrm{UV}}{\longrightarrow} X+Y\)

In the random experiment of tossing two unbiased dice let \(E\) be the event of getting the sum 8 and \(F\) be the event of getting even numbers on both the dice. Then :
I. \(P(E)=\frac{7}{36}\)
II. \(P(F)=\frac{1}{3}\)
Which of the following is a correct statement?

\(\mathrm{K}_{\mathrm{c}}\) for the following reaction is 99.0
\(\mathrm{A}_2(\mathrm{~g}) \stackrel{\mathrm{T}(\mathrm{~K})}{\rightleftharpoons} \mathrm{B}_2(\mathrm{~g})\)
In a one litre flask, 2 moles of \(\mathrm{A}_2\) was heated to \(\mathrm{T}(\mathrm{K})\) and the above equilibrium is reached. The concentrations at equilibrium of \(\mathrm{A}_2\) and \(\mathrm{B}_2\) are \(\mathrm{C}_1\left(\mathrm{~A}_2\right)\) and \(\mathrm{C}_2\left(\mathrm{~B}_2\right)\) respectively. Now, one mole of \(\mathrm{A}_2\) was added to flask and heated to \(\mathrm{T}(\mathrm{K})\) to establish the equilibrium again. The concentrations of \(\mathrm{A}_2\) and \(\mathrm{B}_2\) are \(\mathrm{C}_3\left(\mathrm{~A}_2\right)\) and \(\mathrm{C}_4\left(\mathrm{~B}_2\right)\) respectively. What is the value of \(\mathrm{C}_3\left(\mathrm{~A}_2\right)\) in \(\mathrm{mol}^{-1}\) ?

A value of \(c\) according to the Lagrange's mean value theorem for \(f(x)=(x-1)(x-2)(x-3)\) in \([0,4]\) is