Let the function $f:R\to R$ be defined by $f\left(x\right)=x^3-x^2+\left(x-1\right)\sin x$ and let $g:R\to R$ be an arbitrary function. Let $fg:R\to R$ be the product function defined by $\left(fg\right)\left(x\right)=f\left(x\right)g\left(x\right)$. Then which of the following statements is/are TRUE?

Consider the following lists:

	List-$\mathrm{I}$		List-$\mathrm{II}$
$\left(\mathrm{I}\right)$	$\left\{x\in \left[-\frac{2\pi }{3},\frac{2\pi }{3}\right]:\cos x+\sin x=1\right\}$	$\left(\mathrm{P}\right)$	has two elements
$\left(\mathrm{II}\right)$	$\left\{x\in \left[-\frac{5\pi }{18},\frac{5\pi }{18}\right]:\sqrt{3}\tan 3x=1\right\}$	$\left(\mathrm{Q}\right)$	has three elements
$\left(\mathrm{III}\right)$	$\left\{x\in \left[-\frac{6\pi }{5},\frac{6\pi }{5}\right]:2\cos \left(2x\right)=\sqrt{3}\right\}$	$\left(\mathrm{R}\right)$	has four elements
$\left(\mathrm{IV}\right)$	$\left\{x\in \left[-\frac{7\pi }{4},\frac{7\pi }{4}\right]:\sin x-\cos x=1\right\}$	$\left(\mathrm{S}\right)$	has five elements
		$\left(\mathrm{T}\right)$	has six elements

The correct option is:

Let $f\mathbb{ }:\mathbb{R}\to \left(0, \infty \right)$ and $g\mathbb{ }:\mathbb{R}\to \mathbb{R}$ be twice differentiable functions such that $f^{″}$ and $g^{″}$ are continuous functions on $\mathbb{R}$ . Suppose $f^{\prime }\left(2\right)=g\left(2\right)=0, f^{″}\left(2\right)\ne 0$ and $g^{\prime }\left(2\right) \ne 0.\underset{x\to 2}{\lim }\frac{f\left(x\right) g\left(x\right)}{f^{\prime } \left(x\right) g^{\prime }\left(x\right)}=1,$ then

Let $\mathrm{X}\mathrm{ }\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{ }\mathrm{Y}$ be two arbitrary, $3\mathrm{ }\times 3$ , non - zero, skew - symmetric matrices and Z be an arbitrary $3\mathrm{ }\times 3$ , non - zero, symmetric matrix. Then which of the following matrices is (are) skew symmetric ?

A tangent \(P T\) is drawn to the circle \(x^{2}+y^{2}=4\) at the point \(P(\sqrt{3}, 1)\). A straight line \(L\), perpendicular to \(P T\) is a tangent to the circle \((x-3)^{2}+y^{2}-1\).

Question: A possible equation of \(L\) is

If $f\left(x\right)=\left|\begin{array}{ccc}\cos 2x& \cos 2x& \sin 2x\\ -\cos x& \cos x& -\sin x\\ \sin x& \sin x& \cos x\end{array}\right|,$ then

Let \(\alpha\) and \(\beta\) be the roots of \(x^2-6 x-2=0\) with \(\alpha>\beta\). If \(a_n=\alpha^n-\beta^n\) for \(n \geq 1\), then the value of \(\frac{a_{10}-2 a_8}{2 a_9}\) is

On decreasing the $\mathrm{pH}$ from $7$ to $2$, the solubility of a sparingly soluble salt $\left(\mathrm{MX}\right)$ of a weak acid $\left(\mathrm{HX}\right)$ increased from ${10}^{-4} \mathrm{mol} {\mathrm{L}}^{-1}$ to ${10}^{-3} \mathrm{mol} {\mathrm{L}}^{-1}$. The ${\mathrm{pK}}_{\mathrm{a}}$ of $\mathrm{HX}$ is

Consider the matrix, \(P=\left(\begin{array}{lll}2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3\end{array}\right)\)
Let the transpose of a matrix \(X\) be denoted by \(\mathrm{X}^T\). Then the number of \(3 \times 3\) invertible matrices Q with integer entries, such that \(Q^{-1}=Q^T\) and \(P Q=Q P\) is

Let \(a_{1}, a_{2}, a_{3}, \ldots . .\) be in harmonic progression with \(a_{1}=5\) and \(a_{20}=25\). The least positive integer \(n\) for which \(a_{n} < 0\) is

A thin uniform rod of length \(L\) and certain mass is kept on a frictionless horizontal table with a massless string of length \(L\) fixed to one end (top view is shown in the figure). The other end of the string is pivoted to a point \(\mathrm{O}\). If a horizontal impulse \(P\) is imparted to the rod at a distance \(x=L / n\) from the mid-point of the rod (see figure), then the rod and string revolve together around the point \(\mathrm{O}\), with the rod remaining aligned with the string. In such a case, the value of \(n\) is _______ .

Paragraph:
In hexagonal system of crystals, a frequently encountered arrangement of atoms is described as a hexagonal prism. Here, the top and bottom of the cell are regular hexagons and three atoms are sandwiched in between them. A space-filling model of this structure, called hexagonal close-packed (HCP), is constituted of a sphere on a flat surface surrounded in the same plane by six identical spheres as closely as possible. Three spheres are then placed over the first layer so that they touch each other and represent the second layer. Each one of the three spheres touches three spheres of the bottom layer. Finally, the second layer is covered with a third layer that is identical to the bottom layer in relative position. Assume radius of every sphere to be ' \(r\) '.
Question:
The empty space in this \(\mathrm{HCP}\) unit cell is

Let \(f(x)=(1-x)^{2} \sin ^{2} x+x^{2}\) for all \(x \in I R\) and let \(g(x)=\int_{1}^{x}\left(\frac{2(t-1)}{t+1}-\ln t\right) f(t) d t\) for all \(x \in(1, \infty)\).

Question: Which of the following is true?