If the function $f:R\to R$ is defined by $f\left(x\right)=\left|x\right|\left(x-\sin x\right)$, then which of the following statements is TRUE?

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a function such that \(f(x+y)=f(x)+f(y)\) for all \(x, y \in \mathbb{R}\), and \(g: \mathbb{R} \rightarrow(0, \infty)\) be a function such that \(g(x+y)=g(x) g(y)\) for all \(x, y \in \mathbb{R}\). If \(f\left(\frac{-3}{5}\right)=12\) and \(g\left(\frac{-1}{3}\right)=2\), then the value of \(\left(f\left(\frac{1}{4}\right)+g(-2)-8\right) g(0)\) is . ________.

Let S be the set of all non - zero real numbers $\alpha$ such that the quadratic equation $\alpha x^2-x+\alpha =0$ has two distinct real roots $x_1 and x_2$ satisfying the inequality $\left|x_1-x_2\right|<1.$ Which of the following intervals is(are) a subset(s) of S?

The maximum value of the expression \(\frac{1}{\sin ^2 \theta+3 \sin \theta \cos \theta+5 \cos ^2 \theta}\) is

Let $f:\mathrm{ℝ}\to \mathrm{ℝ}$ and $g:\mathrm{ℝ}\to \mathrm{ℝ}$ be functions satisfying $f\left(x+y\right)=f\left(x\right)+f\left(y\right)+f\left(x\right)f\left(y\right)\text{ and }f\left(x\right)=xg\left(x\right)$ for all $x,y\in \mathrm{ℝ}.$ If $\underset{x\to 0}{\lim }g\left(x\right)=1,$ then which of the following statements is/are TRUE?

For a polynomial $g\left(x\right)$ with real coefficients, let $m_g$ denote the number of distinct real roots of $g\left(x\right)$. Suppose $S$ is the set of polynomials with real coefficients defined by $S=\left\{{\left(x^2-1\right)}^2\left(a_0+a_{1}x+a_2{x}^2+a_3{x}^3\right) :a_0, a_1, a_2, a_3\in R\right\}$. For a polynomial $f$, let $f\text{'}$ and $f^{″}$ denote its first and second order derivatives respectively. Then the minimum possible value of $\left(m_{f^{\prime }}+m_{f^{″}}\right)$, where $f\in S$, is ____

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Let \(O\) be the origin, and \(\overrightarrow{O X}, \overrightarrow{O Y}, \overrightarrow{O Z}\) be three unit vectors in the directions of the sides \(\overrightarrow{Q R}, \overrightarrow{R P}\), \(\overrightarrow{P Q}\), respectively, of a triangle \(P Q R\).


Question:

If the triangle \(P Q R\) varies, then the minimum value of

\(\cos (P+Q)+\cos (Q+R)+\cos (R+P)\) is

The sum of the number of lone pairs of electrons on each central atom in the following species is \(\left[ TeBr _6\right]^{2-},\left[ Br F _2\right]^{+}, S N F_3\) and \(\left[ XeF _3\right]^{-}\)[Atomic number : \(N=7, F=9, S =16, B r=35,\) \(T e=52, X e=54]\)

In the Newman projection for 2,2-dimethylbutane

\(X\) and \(Y\) can respectively be

Match the statements given in Column I with the intervals/union of intervals given in Column II.

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The noble gases have closed-shell electronic configuration and are monoatomic gases under normal conditions. The low boiling points of the lighter noble gases are due to weak dispersion forces between the atoms and the absence of other interatomic interactions.
The direct reaction of xenon with fluorine leads to a series of compounds with oxidation numbers \(+2,+4\) and \(+6 . \mathrm{XeF}_4\) reacts violently with water to give \(\mathrm{XeO}_3\). The compounds of xenon exhibit rich stereochemistry and their geometries can be deduced considering the total number of electron pairs in the valence shell.
Question:
Argon is used in are welding because of its

The \(x-t\) graph of a particle undergoing simple harmonic motion is shown below. The acceleration of the particle at \(t=\frac{4}{3} \mathrm{~s}\) is

A planet of radius $\mathrm{R}=\frac{1}{10}\times$ (radius of Earth) has the same mass density as Earth. Scientists dig a well of depth $\frac{\mathrm{R}}{5}$ on it and lower a wire of the same length and of linear mass density ${10}^{-3}\mathrm{ }\mathrm{k}\mathrm{g}{\mathrm{m}}^{-1}$ into it. If the wire is not touching anywhere, the force applied at the top of the wire by a person holding it in place is (take the radius of Earth $=6\times {10}^6\mathrm{ }\mathrm{m}$ and the acceleration due to gravity of Earth is $10\mathrm{ }\mathrm{m}{\mathrm{s}}^{-2}$ )