Let the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) be defined by
\(f(x)=\frac{\sin x}{e^{\pi x}} \frac{\left(x^{2023}+2024 x+2025\right)}{\left(x^2-x+3\right)}+\frac{2}{e^{\pi x}} \frac{\left(x^{2023}+2024 x+2025\right)}{\left(x^2-x+3\right)}\).
Then the number of solutions of \(f(x)=0\) in \(\mathbb{R}\) is

If the line $x=\alpha$ divides the area of the region $R=\{\left(x,y\right)\in R^2:x^3\le y\le x, 0\le x\le 1\}$ into two equal parts, then

Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.

Column 1	Column 2	Column 3
(I) $x^2+y^2=a^2$	(i) $my=m^{2}x+a$	(P) $\left(\frac{a}{m^2},\frac{2a}{m}\right)$
(II) $x^2+a^2{y}^2=a^2$	(ii) $y=mx+a \sqrt{m^2+1}$	(Q) $\left(\frac{-ma}{\sqrt{m^2+1}},\frac{a}{\sqrt{m^2+1}}\right)$
(III) $y^2=4ax$	(iii) $y=mx+ \sqrt{a^2{m}^2-1}$	(R) $\left(\frac{-a^{2}m}{\sqrt{a^2{m}^2+1}},\frac{1}{\sqrt{a^2{m}^2+1}}\right)$
(IV) $x^2-a^2{y}^2=a^2$	(iv) $y=mx+\sqrt{a^2{m}^2+1}$	(S) $\left(\frac{-a^{2}m}{\sqrt{a^2{m}^2-1}},\frac{-1}{\sqrt{a^2{m}^2-1}}\right)$

The tangent to a suitable conic (Column 1) at $\left(\sqrt{3},\frac{1}{2}\right)$ is found to be $\sqrt{3}x+2y=4$ , then which of the following options is the only Correct combination?

Paragraph:
Let \(A, B, C\) be three sets of complex numbers as defined below.
\(A=\{z ; \operatorname{lm} z \geq 1\} ; B=\{z:|z-2-i|=3\} ;\) \(C=\{z: \operatorname{Re}((1-i) z)=\sqrt{2}\}\).
Question:
Let \(z\) be any point in \(A \cap B \cap C\). The \(|z+1-i|^2+|z-5-i|^2\) lies between

Lines \(L_1: y-x=0\) and \(L_2: 2 x+y=0\) intersect the line \(L_3: y+2=0\) at \(P\) and \(Q\), respectively. The bisector of the acute angle between \(L_1\) and \(L_2\) intersects \(L_3\) at \(R\).
Statement I The ratio \(P R: R Q\) equals \(2 \sqrt{2}: \sqrt{5}\).
Statement II In any triangle, bisector of an angle divides the triangle into two similar triangles.

Let $f_1: \mathbb{R}\to \mathbb{R}, f_2\left(-\frac{\pi }{2},\frac{\pi }{2} \right)\to \mathbb{R}, f_3:\left(-1, e^{\frac{\pi }{2}}-2\right)\to \mathbb{R}$ and $f_4: \mathrm{ℝ}\to \mathrm{ℝ}$ be functions defined by
(i) $f_1\left(x\right)=\sin \left(\sqrt{1-e^{-x^2}}\right)$
(ii) $f_2\left(x\right)=\left\{\begin{array}{ll}\frac{\left|\sin x\right|}{{\tan }^{-1}\left(x\right)}& , x\ne 0\\ 1& , x=0\end{array}\right.$ , where the inverse trigonometric function ${\tan }^{-1}x$ assumes values in $\left(-\frac{\pi }{2},\frac{\pi }{2}\right)$
(iii) $f_3\left(x\right)=\left[\sin \left({\log }_{\text{e}}\left(x+2\right)\right)\right]$ , where, for $t\in R, \left[t\right]$ denotes the greatest integer less than or equal to $t$ ,
(iv) $f_4\left(x\right)=\left\{\begin{array}{cc}{x}^2\sin \left(\frac{1}{x}\right) & , x\ne 0\\ 0 & , x=0\end{array}\right.$

LIST-I	LIST-II
A. the function $f_1$ is	P. NOT continuous at $x=0$
B. The function $f_2$ is	Q. continuous at $x = 0$ and NOT differentiable at $x=0$
C. The function $f_3$ is	R. differentiable at $x=0$ and its derivative is NOT continuous at $x=0$
D. The function $f_4$ is	S. differentiable at $x=0$ and its derivative is continuous at $x=0$

The correct option is :

Paragraph:
Let \(M=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x^{2}+y^{2} \leq r^{2}\right\}\),
where \(r>0 .\) Consider the geometric progression \(a_{n}=\frac{1}{2^{n-1}}, n=1,2,3, \ldots .\) Let \(S_{0}=0\) and, for \(n \geq 1\), let \(S_{n}\) denote the sum of the first \(n\) terms of this progression. For \(n \geq 1\), let \(C_{n}\) denote the circle with center \(\left(S_{n-1}, 0\right)\) and radius \(a_{n}\), and \(D_{n}\) denote the circle with center \(\left(S_{n-1}, S_{n-1}\right)\) and radius \(a_{n}\).

Question:
Consider \(M\) with \(r=\frac{\left(2^{199}-1\right) \sqrt{2}}{2^{198}} .\) The number of all those circles \(D_{n}\) that are inside \(M\) is

Molar volume \(\left(V_m\right)\) of a van der Waals gas can be calculated by expressing the van der Waals equation as a cubic equation with \(V_m\) as the variable. The ratio (in \(\mathrm{mol} \mathrm{dm}^{-3}\) ) of the coefficient of \(V_m^2\) to the coefficient of \(V_m\) for a gas having van der Waals constants \(a=6.0 \mathrm{dm}^6 \mathrm{~atm} \mathrm{~mol}^{-2}\) and \(b=0.060 \mathrm{dm}^3 \mathrm{~mol}^{-1}\) at 300 K and 300 atm is _____ .
Use: Universal gas constant \((R)=0.082 \mathrm{dm}^3 \mathrm{~atm} \mathrm{~mol}{ }^{-1} \mathrm{~K}^{-1}\)

A point charge $q$ of mass $m$ is suspended vertically by a string of length $l.$ A point dipole of dipole moment $\overrightarrow{p}$ is now brought towards $q$ from infinity so that the charge moves away. The final equilibrium position of the system including the direction of the dipole, the angles and distances is shown in the figure below. If the work done in bringing the dipole to this position is $N\times \left(mgh\right)$, where $g$ is the acceleration due to gravity, then the value of $N$ is ____________ (Note that for three coplanar forces keeping a point mass in equilibrium, $\frac{\mathrm{F}}{\sin \theta }$ is the same for all forces, where $F$ is any one of the forces and $\theta$ is the angle between the other two forces)

Which one of the following statement is WRONG in the context of X-rays generated from X-ray tube?

Paragraph:
An aqueous solution of a mixture of two inorganic salts, when treated with dilute \(H C\) ?, gave a precipitate \((\boldsymbol{P})\) and a filtrate \((\boldsymbol{Q})\). The precipitate \(\boldsymbol{P}\) was found to dissolve in hot water. The filtrate \((\boldsymbol{Q})\) remained unchanged, when treated with \(\mathrm{H}_{2} \mathrm{~S}\) in a dilute mineral acid medium. However, it gave a precipitate \((\boldsymbol{R})\) with \(H_{2} S\) in an ammoniacal medium. The precipitate \(\boldsymbol{R}\) gave a coloured solution \((\boldsymbol{S})\), when treated with \(\mathrm{H}_{2} \mathrm{O}_{2}\) in an aqueous \(\mathrm{NaOH}\) medium.
Question:
The coloured solution \(\boldsymbol{S}\) contains

Match the following columns

Let $a, b, x$ and $y$ be real numbers such that $a-b=1$ and $y\ne 0$ . If the complex number $z=x+iy$ satisfies $Im\left(\frac{az+b}{z+1}\right)=y,$ then which of the following is (are) possible value (s) of $x?$