Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a function defined by \(f(x)=\left\{\begin{array}{cc}x^2 \sin \left(\frac{\pi}{x^2}\right), & \text { if } x \neq 0 \\ 0, & \text { if } x=0\end{array}\right.\) Then which of the following statements is TRUE?

Equation of the plane containing the straight line \(\frac{x}{2}=\frac{y}{3}=\frac{z}{4}\) and perpendicular to the plane containing the straight lines \(\frac{x}{3}=\frac{y}{4}=\frac{z}{2}\) and \(\frac{x}{4}=\frac{y}{2}=\frac{z}{3}\) is

Let $z$ be a complex number satisfying ${\left|z\right|}^3+2z^2+4\overline{z}-8=0$, where $\overline{z}$ denotes the complex conjugate of $z$. Let the imaginary part of $z$ be nonzero.
Match each entry in List-I to the correct entries in List-II.

List - I	List - II
(P) \(|z|^2\) is equal to	(1) 12
(Q) \(|z-\bar{z}|^2\) is equal to	(2) 4
(R) \(|z|^2+|z+\bar{z}|^2\) is equal to	(3) 8
(S) \(|z+1|^2\) is equal to	(4) 10
	(5) 7

The correct option is

Let $T_1$ and $T_2$ be two distinct common tangents to the ellipse $E:\frac{x^2}{6}+\frac{y^2}{3}=1$ and the parabola $P:y^2=12x$. Suppose that the tangent $T_1$ touches $P$ and $E$ at the points $A_1$ and $A_2$, respectively and the tangent $T_2$ touches $P$ and $E$ at the points $A_4$ and $A_3$, respectively. Then which of the following statements is(are) true?

If \(f(x)=\left\{\begin{array}{cc}-x-\frac{\pi}{2}, & x \leq-\frac{\pi}{2} \\ -\cos x, & -\frac{\pi}{2} < x \leq 0 \\ x-1, & 0 < x \leq 1 \\ \log x, & x>1\end{array}\right.\), then

Let \(y^{\prime}(x)+y(x) g^{\prime}(x)=g(x) g^{\prime}(x) y(0)=0, x \in R\), where \(f^{\prime}(x)\) denotes \(\frac{d f(x)}{d x}\) and \(g(x)\) is a given non-constant differentiable function on \(R\) with \(g(0)=g(2)=0\). Then, the value of \(y\) (2) is...

Let \(\mathbb{R}\) denote the set of all real numbers. Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) and \(g: \mathbb{R} \rightarrow(0,4)\) be functions defined by \(f(x)=\log _e\left(x^2+2 x+4\right)\), and \(g(x)=\frac{4}{1+e^{-2 x}}\)
Define the composite function \(f \circ g^{-1}\) by \(\left(f \circ g^{-1}\right)(x)=\left(g^{-1}(x)\right)\), where \(g^{-1}\) is the inverse of the function \(g\).
Then the value of the derivative of the composite function \(f \circ g^{-1}\) at \(x=2\) is ________ .

The \(\beta\)-decay process, discovered around 1900 , is basically the decay of a neutron \((n)\). In the laboratory, a proton \((p)\) and an electron \(\left(e^{-}\right)\)are observed as the decay products of the neutron. Therefore, considering the decay of a neutron as a two-body decay process, it was predicted theoretically that the kinetic energy of the electron should be a constant. But experimentally, it was observed that the electron kinetic energy has continuous spectrum. Considering a three-body decay process, i.e.
\(n \rightarrow p+e^{-}+\bar{v}_{e}\), around 1930, Pauli explained the observed electron energy spectrum. Assuming the anti-neutrino \(\left(\bar{v}_{e}\right)\) to be massless and possessing negligible energy, and the neutron to be at rest, momentum and energy conservation principles are applied. From this calculation, the maximum kinetic energy of the electron is \(0.8 \times 10^{6} \mathrm{eV}\). The kinetic energy carried by the proton is only the recoil energy.

Question:
If the anti-neutrino had a mass of \(3 \mathrm{eV} / \mathrm{c}^{2}\) (where \(\mathrm{c}\) is the speed of light) instead of zero mass, what should be the range of the kinetic energy, \(K\), of the electron?

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Upon heating \(\mathrm{KClO}_{3}\) in the presence of catalytic amount of \(\mathrm{MnO}_{2}\), a gas \(\mathbf{W}\) is formed. Excess amount of \(\mathbf{W}\) reacts with white phosphorus to give \(\mathbf{X}\). The reaction of \(\mathbf{X}\) with pure \(\mathrm{HNO}_{3}\) gives \(\mathbf{Y}\) and \(\mathbf{Z}\).
Question:
\(\mathbf{Y}\) and \(\mathbf{Z}\) are, respectively

Assuming that Hund's rule is violated, the bond order and magnetic nature of the diatomic molecule \(\mathrm{B}_2\) is

During an experiment with a metre bridge, the galvanometer shows a null point when the jockey is pressed at $40.0\mathrm{cm}$using a standard resistance of $90\mathrm{}\mathrm{\Omega }$ ,as shown in the figure. The least count of the scale used in the metre bridge is $1\mathrm{mm}$. The unknown resistance is

A monochromatic beam of light is incident at ${60}^{\mathrm{o}}$ on one face of an equilateral prism of refractive index $\mathrm{n}$ and emerges from the opposite face making an angle $\mathrm{\theta }\left(\mathrm{n}\right)$ with the normal (see the figure). For $\mathrm{n}=\sqrt{3}$ the value of $\mathrm{\theta }$ is ${60}^{\mathrm{o}}$ and $\frac{\mathrm{d}\mathrm{\theta }}{\mathrm{d}\mathrm{n}}=\mathrm{m}$ . The value of $\mathrm{m}\mathrm{}$ is

A small object is placed 50 cm to the left of thin convex lens of focal length 30 cm. A convex spherical mirror of radius of curvature 100 cm is placed to the right of the lens at a distance of 50 cm. The mirror is tilted such that the axis of the mirror is at an angle $\theta ={30}^o$ to the axis of the lens, as shown in the figure. If the origin of the coordinate system is taken to be at the centre of the lens, the coordinates (in cm) of the point (x, y) at which the image is formed are: