Define the collections $\left\{{\mathrm{E}}_1,{\mathrm{E}}_2,{\mathrm{E}}_3,....\right\}$ of ellipses and $\left\{{\mathrm{R}}_1,{\mathrm{R}}_2,{\mathrm{R}}_3,....\right\}$ of rectangles as follows:
${\mathrm{E}}_1:\frac{{\mathrm{x}}^2}{9}+\frac{{\mathrm{y}}^2}{4}=1;$
${\mathrm{R}}_1:$ rectangle of largest area, with sides parallel to the axes, inscribed in ${\mathrm{E}}_1;$
${\mathrm{E}}_{\mathrm{n}}:$ ellipse $\frac{{\mathrm{x}}^2}{{\mathrm{a}}_{\mathrm{n}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}_{\mathrm{n}}^2}=1$ of largest area inscribed in ${\mathrm{R}}_{\mathrm{n}-1},\mathrm{n}>1;$
${\mathrm{R}}_{\mathrm{n}}:$ rectangle of largest area, with sides parallel to the axes, inscribed in ${\mathrm{E}}_{\mathrm{n}},\mathrm{n}>1.$
Then which of the following options is/are correct?

Let $f\left(x\right)=7{\tan }^{8}x+7{\tan }^{6}x-3{\tan }^{4}x-3{\tan }^{2}x$ for all $x\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right).$ Then the correct expression(S) is(are)

The minimum number of times a fair coin needs to be tossed so that the probability of getting at least two heads is at least $0.96$, is

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Let \(A\) be the set of all \(3 \times 3\) symmetric matrices all of whose entries are either 0 or 1 . Five of these entries are 1 and four of them are 0 .Question:
The number of matrices \(A\) in \(A\) for which the system of linear equations \(A\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]\) is is inconsistent, is

Let \(f\) be a real-valued function defined on the interval \((-1,1)\) such that \(e^{-x} f(x)=2+\int_0^x \sqrt{t^4+1} d t, \quad\) for all \(x \in(-1,1)\) and let \(f^{-1}\) be the inverse function of \(f\). Then \(\left(f^{-1}\right)^{\prime}(2)\) is equal to

Let \(A B C\) be a triangle such that \(\angle A C B=\frac{\pi}{6}\) and let \(a, b\) and \(c\) denote the lengths of the sides opposite to \(A, B\) and \(C\) respectively. The value(s) of \(x\) for which \(a=x^2+x+1, b=x^2-1\) and \(c=2 x+1\) is (are)

Let $P$ be a point in the first octant, whose image $Q$ in the plane $x+y=3$ (that is, the line segment $PQ$ is perpendicular to the plane $x+y=3$ and the mid-point of $PQ$ lies in the plane $x+y=3$) lies on the z-axis. Let the distance of $P$ from the x-axis be $5$. If $R$ is the image of $P$ in the xy-plane, then the length of $PR$ is ______.

A transparent slab of thickness d has a refractive index n (z) that increases with z. Here z is the vertical distance inside the slab, measured from the top. The slab is placed between two media with uniform refractive indices $n_{1}and{n}_2\left(>n_1\right),$ as shown in the figure. A ray of light is incident with angle ${\theta }_i$ from medium 1 and emerges in medium 2 with refraction angle ${\theta }_f$ with a lateral displacement $\mathcal{l}$ . Which of the following statement(s) is (are) true?

A disk of radius $R$ with uniform positive charge density $\sigma$ is placed on the $xy$ plane with its center at the origin. The Coulomb potential along the $z$-axis is
$V\left(z\right)=\frac{\sigma }{2{ϵ}_0}\left(\sqrt{R^2+z^2}-z\right)$ A particle of positive charge $q$ is placed initially at rest at a point on the $z$-axis with $z=z_0$ and $z_0>0$. In addition to the Coulomb force, the particle experiences a vertical force $\overrightarrow{F}=-c\hat{k}$ with $c>0$. Let $\beta =\frac{2c{\epsilon }_0}{q\sigma }$. Which of the following statement(s) is(are) correct?

An optical component and an object \(S\) placed along its optic axis are given in Column I. The distance between the object and the component can be varied. The properties of images are given in Column II. Match all the properties of images from Column II with the appropriate components given in Column I. Indicate your answer by darkening the appropriate bubble of the \(4 \times 4\) matrix given in the ORS.

In the circuit shown, $L=1\text{μ}H,C=1\text{μ}FandR=1k\mathrm{\Omega }.$ They are connected in series with an a.c. source $V=V_0\sin \text{ω}t$ as shown. Which of the following options is/are correct?

If \(f(x)\) is twice differentiable function such that \(f(a)=0, f(b)=2, f(c)=1, f(d)=2\), \(f(e)=0\), where \(a < b < c < d < e\), then the minimum number of zeros of \(g(x)=\left\{f^{\prime}(x)\right\}^2+f^{\prime \prime}(x) \cdot f(x)\) in the interval \([a, e]\) is

A pendulum consists of a bob of mass \(m=0.1 \mathrm{~kg}\) and a massless inextensible string of length \(L=1.0 \mathrm{~m}\). It is suspended from a fixed point at height \(H=0.9 \mathrm{~m}\) above a frictionless horizontal floor. Initially, the bob of the pendulum is lying on the floor at rest vertically below the point of suspension. A horizontal impulse \(P=0.2 \mathrm{~kg}-\mathrm{m} / \mathrm{s}\) is imparted to the bob at some instant. After the bob slides for some distance, the string becomes taut and the bob lifts off the floor. The magnitude of the angular momentum of the pendulum about the point of suspension just before the bob lifts off is \(J \mathrm{~kg}-\mathrm{m}^{2} / \mathrm{s}\). The kinetic energy of the pendulum just after the liftoff is \(K\) Joules.
The value of $K$ is ____.