If a chord, which is not a tangent, of the parabola $y^2=16x$ has the equation $2x+y=p$, and midpoint (h, k), then which of the following is (are) possible value(s) of $p,h$ and $k$?
 

Let $f:\mathrm{ℝ}\to \mathrm{ℝ}$ be a differentiable function with $f\left(0\right)=0$. If $y=f\left(x\right)$ satisfies the differential equation $\frac{dy}{dx}=\left(2+5y\right)\left(5y-2\right)$, then the value of  $\underset{x\to -\infty }{\lim }f\left(x\right)$ is 

Let \(p(x)\) be a real polynomial of least degree which has a local maximum at \(x=1\) and a local minimum at \(x=3\). If \(p(1)=6\) and \(p(3)=2\), then \(p^{\prime}(0)\) is

Let $L_1$ and $L_2$ be the following straight lines.
$L_1:\frac{x-1}{1}=\frac{y}{-1}=\frac{z-1}{3}$ and $L_2:\frac{x-1}{-3}=\frac{y}{-1}=\frac{z-1}{1}$
Suppose the straight line $L:\frac{x-\alpha }{l}=\frac{y-1}{m}=\frac{z-\gamma }{-2}$ lies in the plane containing $L_1$ and $L_2$, and passes through the point of intersection of $L_1$ and $L_2$. If the line $L$ bisects the acute angle between the lines $L_1$ and $L_2$, then which of the following statements is/are TRUE?

Statement I The curve \(y=-\frac{x^2}{2}+x+1\) is symmetric with respect to the line \(x=1\)
Statement II A parabola is symmetric about its axis.

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Let \(a, r, s, t\) be nonzero real numbers. Let \(P\left(a t^{2}, 2 a t\right), Q, R\left(a r^{2}, 2 a r\right)\) and \(S\left(a s^{2}, 2 a s\right)\) be distinct points on the parabola \(y^{2}=4 a x\). Suppose that \(P Q\) is the focal chord and lines \(Q R\) and \(P K\) are parallel, where \(K\) is the point \((2 a, 0)\).


Question:

If \(s t=1\), then the tangent at \(P\) and the normal at \(S\) to the parabola meet at a point whose ordinate is

The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line \(4 x-\) \(5 y=20\) to the circle \(x^{2}+y^{2}=9\) is

Match the compounds in LIST-$\mathrm{I}$ with the observations in LIST-$\mathrm{II}$, and choose the correct option.

	LIST-$\mathrm{I}$		LIST-$\mathrm{II}$
$\left(\mathrm{I}\right)$	Aniline	$\left(\mathrm{P}\right)$	Sodium fusion extract of the compound on boiling with ${\mathrm{FeSO}}_4$, followed by acidification with conc. ${\mathrm{H}}_2{\mathrm{SO}}_4$, gives Prussian blue color.
$\left(\mathrm{II}\right)$	$\mathrm{o}-$Cresol	$\left(\mathrm{Q}\right)$	Sodium fusion extract of the compound on treatment with sodium nitroprusside gives blood red color.
$\left(\mathrm{III}\right)$	Cysteine	$\left(\mathrm{R}\right)$	Addition of the compound to a saturated solution of ${\mathrm{NaHCO}}_3$ results in effervescence.
$\left(\mathrm{IV}\right)$	Caprolactam	$\left(\mathrm{S}\right)$	The compound reacts with bromine water to give a white precipitate.
		$\left(\mathrm{T}\right)$	Treating the compound with neutral ${\mathrm{FeCl}}_3$ solution produces violet color.

As shown schematically in the figure, two vessels contain water solutions (at temperature $T$) of potassium permanganate $\left({\mathrm{KMnO}}_4\right)$ of different concentrations $n_1$ and $n_2\left(n_1>n_2\right)$ molecules per unit volume with $\mathrm{\Delta }n=\left(n_1-n_2\right)\ll n_1.$ When they are connected by a tube of small length $l$ and cross-sectional area $S,{\mathrm{KMnO}}_4$ starts to diffuse from the left to the right vessel through the tube. Consider the collection of molecules to behave as dilute ideal gases and the difference in their partial pressure in the two vessels causing the diffusion. The speed $v$ of the molecules is limited by the viscous force $-\beta v$ on each molecule, where $\beta$ is a constant. Neglecting all terms of the order ${\left(\mathrm{\Delta }n\right)}^2$. Which of the following is/are correct? ($k_B$ is the Boltzmann constant)

In the circuit shown in the figure, there are two parallel plate capacitors each of capacitance C. The switch S₁ is pressed first to fully charge the capacitor C₁ and the released. The switch S₂ is then pressed to charge the capacitor C₂. After some time, S₂ is released and then is S₃ pressed. After some time,

The correct option(s) related to adsorption processes is(are)

The option(s) with the value of $a$ and $L$ that satisfy the following equation is(are) $\frac{\underset{0}{\overset{4\pi }{\int }}{e}^t\left({\sin }^{6}at+{\cos }^{4}at\right) dt}{\underset{0}{\overset{\pi }{\int }}\left({\sin }^{6}at+{\cos }^{4}at\right) dt}=L ?$

The radius of the orbit of an electron in a Hydrogen-like atom is \(4.5 a _0\), where \(a _0\) is the Bohr radius. Its orbital angular momentum is \(\frac{3 h}{2 \pi}\). It is given that \(h\) is Planck's constant and R is Rydberg constant. the possible wavelength \((s)\), when the atom de-excites, is (are)