Let \(a_{n}\) denote the number of all \(n\)-digit positive integers formed by the digits 0,1 or both such that no consecutive digits in them are 0 . Let \(b_{n}=\) the number of such \(n\)-digit integers ending with digit 1 and \(c_{n}=\) the number of such \(n\)-digit integers ending with digit 0 .

Question:
Which of the following is correct?

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

Let \(\vec{w}=\hat{i}+\hat{j}-2 \hat{k}\), and \(\overrightarrow{\mathrm{u}}\) and \(\overrightarrow{\mathrm{v}}\) be two vectors such that \(\vec{u} \times \vec{v}=\vec{w}\) and \(\vec{v} \times \vec{w}=\vec{u}\). Let \(\alpha, \beta, \gamma\) and \(t\) be real numbers such that \(\vec{u}=\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k},-t \alpha+\beta+\gamma=0, \alpha-t \beta+\gamma=0\), and \(\alpha+\beta-t \gamma=0\).
Match each entry in List-I to the correct entry in List-II and choose the correct option.

	LIST - I		LIST - II
(P)	\(|\vec{v}|^2\) is equal to	(1)	0
(Q)	If \(\alpha=\sqrt{3}\),then \(\gamma^2\) is equal to	(2)	1
(R)	If \(\alpha=\sqrt{3}\),then \((\beta+\gamma)^2\) is equal to	(3)	2
(S)	If \(\alpha=\sqrt{2}\),then \(t+3\) is equal to	(4)	3
		(5)	5

Let \(L=\lim _{x \rightarrow 0} \frac{a-\sqrt{a^2-x^2}-\frac{x^2}{4}}{x^4}\), \(a>0\). If \(L\) is finite, then

The value of ${\left({\left({\log }_{2}9\right)}^2\right)}^{\frac{1}{{\log }_2\left({\log }_{2}9\right)}}\times {\left(\sqrt{7}\right)}^{\frac{1}{{\log }_{4}7}}$ is_____.

Paragraph:
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 $w=\frac{\sqrt{3}+\text{i}}{2}$ and $\text{P}=\left\{w^{\text{n}}\text{: }\text{n}=\text{1, 2, 3,}\dots \right\}$.Further ${\text{H}}_1=\left\{\text{z}\in \text{C}\text{ : }\text{Re z}>\frac{1}{2}\right\}$ and ${\text{H}}_2=\left\{\text{z}\in \text{C}\text{ : }\text{Re z}<\frac{-1}{2}\right\}$, where C is the set of all complex numbers. If ${\text{z}}_1\in \text{P}\cap {\text{H}}_1\text{, }{\text{z}}_2\in \text{P}\cap {\text{H}}_2$ and O represents the origin, then $\angle {\text{z}}_1\text{O}{\text{z}}_2=$

The set of values of \(\theta\) satisfying the inequation \(2 \sin ^2 \theta-5 \sin \theta+2>0\), where \(0 < \theta < 2 \pi\), is

A conducting wire of parabolic shape, initially $\mathrm{y}={\mathrm{x}}^2,$ is moving with velocity $\overrightarrow{\mathrm{V}}={\mathrm{V}}_0\widehat{\mathrm{i}}$ in a non-uniform magnetic field $\overrightarrow{\mathrm{B}}={\mathrm{B}}_0\left(1+{\left(\frac{\mathrm{y}}{\mathrm{L}}\right)}^{\mathrm{\beta }}\right)\widehat{\mathrm{k}},$ as shown in figure. If ${\mathrm{V}}_0,{\mathrm{B}}_0,\mathrm{L}$ and $\mathrm{\beta }$ are positive constants and $\mathrm{\Delta }\mathrm{\varphi }$ is the potential difference developed between the ends of the wire, then the correct statement(s) is/are:

Paragraph : In electromagnetic theory, the electric and magnetic phenomena are related to each other. Therefore, the dimensions of electric and magnetic quantities must also be related to each other. In the questions below, $\left[E\right] \mathrm{a}\mathrm{n}\mathrm{d} \left[B\right]$ stand for dimensions of electric and magnetic fields respectively, while $\left[{\epsilon }_0\right]$ and $\left[{\mu }_0\right]$ stand for dimensions of the permittivity and permeability of free space respectively. $\left[L\right]\mathrm{a}\mathrm{n}\mathrm{d} \left[T\right]$ are dimensions of length and time respectively. All the quantities are given in $SI$ units

Question : The relation between $\left[{ϵ}_0\right] \mathrm{a}\mathrm{n}\mathrm{d} \left[{\mu }_0\right]$ is

A small particle of mass $\mathrm{m}$ moving inside a heavy, hollow and straight tube along the tube axis, undergoes elastic collision at two ends. The tube has no friction and it is closed at one end by a flat surface while the other end is fitted with a heavy movable flat piston as shown in figure. When the distance of the piston from closed end is $\mathrm{L}={\mathrm{L}}_0$ the particle speed is $\mathrm{v}={\mathrm{v}}_0.$ The piston is moved inward at a very low speed $\mathrm{V}$ such that $\mathrm{V}<<\frac{\mathrm{d}\mathrm{L}}{\mathrm{L}}{\mathrm{v}}_0,$ where $\mathrm{d}\mathrm{L}$ is the infinitesimal displacement of the piston. Which of the following statement(s) is/are correct?

The reaction of $4$-methyloct-$1$-ene $(\mathrm{P},2.52 \mathrm{g})$ with $\mathrm{HBr}$ in the presence of ${\left({\mathrm{C}}_6{\mathrm{H}}_5\mathrm{CO}\right)}_2{\mathrm{O}}_2$ gives two isomeric bromides in a $9: 1$ ratio, with a combined yield of $50\%$. Of these, the entire amount of the primary alkyl bromide was reacted with an appropriate amount of diethylamine followed by treatment with aq. ${\mathrm{K}}_2{\mathrm{CO}}_3$ to give a non-ionic product $\mathrm{S}$ in $100\%$ Yield. The mass (in $\mathrm{mg}$ ) of $\mathrm{S}$ obtained is
[Use molar mass (in ${\mathrm{gmol}}^{-1}$ ): $\mathrm{H}=1,\mathrm{C}=12, \mathrm{N}=14,\mathrm{Br}=80$ ]

A series \(R\) - \(C\) circuit is connected to \(A C\) voltage source. Consider two cases; \((A)\) when \(C\) is without a dielectric medium and \((B)\) when \(C\) is filled with dielectric of constant 4. The current \(I_R\) through the resistor and voltage \(V_C\) across the capacitor are compared in the two cases. Which of the following is/are true?