Let $\mathrm{f}\mathrm{ }:\mathbb{R}\to \mathbb{R}\mathrm{ }$ be a function defined by $\mathrm{f}\left(\mathrm{x}\right)=\mathrm{ }\left\{\begin{array}{cc}\left[\mathrm{x}\right],& \mathrm{x}\mathrm{ }\le 2\\ 0,& \mathrm{x}>2\end{array}\right.$ , where $\left[\mathrm{x}\right]$ is the greatest integer less than or equal to $\mathrm{x}$ . If $\mathrm{I}=\mathrm{ }\underset{-1}{\overset{2}{\int }}\frac{\mathrm{x}\mathrm{f}\left({\mathrm{x}}^2\right)}{2+\mathrm{f}(\mathrm{x}+1)}\mathrm{ }\mathrm{d}\mathrm{x},$ then the value of $(4\mathrm{I}-1)$ is

From a point $P\left(\lambda , \lambda , \lambda \right),$ perpendiculars PQ and PR are drawn respectively on the lines $y=x, z=1$ and $y= -x, z= -1$ . If P is such that $\angle QPR$ is a right angle, then the possible value (s) of $\lambda$ is (are)

Let $P_1: 2x + y - z = 3 \mathrm{a}\mathrm{n}\mathrm{d} P_2: x + 2y + z = 2$ be two planes. Then, which of the following statement(s) is (are) TRUE?

The total number of local maxima and local minima of the function \(f(x)=\left\{\begin{array}{lc}(2+x)^3 ; & -3 < x \leq-1 \\ x^{\frac{2}{3}} ; & -1 < x < 2\end{array}\right.\) is

Coefficient of ${\mathrm{x}}^{11}$ in the expansion of ${\left(1+{\mathrm{x}}^2\right)}^4\mathrm{ }{\left(1+{\mathrm{x}}^3\right)}^7\mathrm{ }{\left(1+{\mathrm{x}}^4\right)}^{12}$ is

The value of
\(\sec ^{-1}\left(\frac{1}{4} \sum_{k=0}^{10} \sec (\frac{7 \pi}{12}+\frac{k \pi}{2}\right) \sec (\frac{7 \pi}{12}\) \(+\frac{(k+1) \pi}{2}))\) in the interval \(\left[-\frac{\pi}{4}, \frac{3 \pi}{4}\right]\) equals

Let $\mathrm{y}\left(\mathrm{x}\right)$ be a solution of the differential equation $\left(1+{\mathrm{e}}^{\mathrm{x}}\right)\mathrm{ }{\mathrm{y}}^{\mathrm{\prime }}+\mathrm{y}{\mathrm{e}}^{\mathrm{x}}=1.$ If $\mathrm{y}\left(0\right)=2$ , then which of the following statements is (are) true?

Let \(\gamma \in \mathbb{R}\) be such that the lines \(\mathrm{L}_1: \frac{x+11}{1}=\frac{\mathrm{y}+21}{2}=\frac{\mathrm{z}+29}{3}\) and \(L_2: \frac{x+16}{3}=\frac{y+11}{2}=\frac{z+4}{\gamma}\) intersect. Let \(\mathrm{R}_1\) be the point of intersection of \(\mathrm{L}_1\) and \(\mathrm{L}_2\). Let \(\mathrm{O}=(0,0,0)\), and \(\hat{\mathrm{n}}\) denote a unit normal vector to the plane containing both the lines \(\mathrm{L}_1\) and \(\mathrm{L}_2\).
Match each entry in List-I to the correct entry in List-II.

The correct option is

Paragraph:
When a metal rod \(M\) is dipped into an aqueous colourless concentrated solution of compound \(\mathrm{N}\), the solution turns light blue. Addition of aqueous \(\mathrm{NaCl}\) to the blue solution gives a white precipitate \(\mathrm{O}\). Addition of aqueous \(\mathrm{NH}_3\) dissolves \(\mathrm{O}\) and gives an intense blue solution.
Question:
The compound \(N\) is

The correct statement(s) regarding,
(i) $\mathrm{H}\mathrm{C}\mathrm{l}\mathrm{O}$
(ii) $\mathrm{H}\mathrm{C}\mathrm{l}{\mathrm{O}}_2$
(iii) $\mathrm{H}\mathrm{C}\mathrm{l}{\mathrm{O}}_3$ and
(iv) $\mathrm{H}\mathrm{C}\mathrm{l}{\mathrm{O}}_4$ is /are

A ball moves over a fixed track as shown in the figure. From \(A\) to \(B\) the ball rolls without slipping. If surface \(B C\) is frictionless and \(K_A, K_B\) and \(K_C\) are kinetic energies of the ball at \(A, B\) and \(C\) respectively, then

Consider the kinetic data given in the following table for the reaction $\mathrm{A}+\mathrm{B}+\mathrm{C}\to$ Product.

Ex peri ment No.	\([\text A]\) \((\text{mol}\) \(\text{dm} ^{-3})\)	\([\text B]\) \((\text{mol}\) \(\text{dm} ^{-3})\)	\([\text C]\) \((\text{mol}\) \(\text{dm} ^{-3})\)	Rate of reaction \((\text{mol}\) \(\text{dm} ^{-3}\) \(\text s^{-1})\)
1	0.2	0.1	0.1	\(6.0 \times\) \(10^{-5}\)
2	0.2	0.2	0.1	\(6.0 \times \) \(10^{-5}\)
3	0.2	0.1	0.2	\(1.2 \times \) \(10^{-4}\)
4	0.3	0.1	0.1	\(9.0 \times \) \(10^{-5}\)

The rate of the reaction for $\left[\mathrm{A}\right]=0.15\mathrm{ }\mathrm{m}\mathrm{o}\mathrm{l}\mathrm{ }\mathrm{d}{\mathrm{m}}^{-3},\mathrm{ }\left[\mathrm{B}\right]=0.25\mathrm{ }\mathrm{m}\mathrm{o}\mathrm{l}\mathrm{ }\mathrm{d}{\mathrm{m}}^{-3}$ and $\left[\mathrm{C}\right]=0.15\mathrm{ }\mathrm{m}\mathrm{o}\mathrm{l}\mathrm{ }\mathrm{d}{\mathrm{m}}^{-3}$ is found to be $\mathrm{Y}\times {10}^{-5}\mathrm{ }\mathrm{m}\mathrm{o}\mathrm{l}\mathrm{ }\mathrm{d}{\mathrm{m}}^{-3}{\mathrm{s}}^{-1}.$ The value of $\mathrm{Y}$ is __________

Paragraph:
The reaction of \(\mathrm{K}_{3}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]\) with freshly prepared \(\mathrm{FeSO}_{4}\) solution produces a dark blue precipitate called Turnbull's blue. Reaction of \(\mathrm{K}_{4}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]\) with the \(\mathrm{FeSO}_{4}\) solution in complete absence of air produces a white precipitate \(\mathbf{X}\), which turns blue in air. Mixing the \(\mathrm{FeSO}_{4}\) solution with \(\mathrm{NaNO}_{3}\), followed by a slow addition of concentrated \(\mathrm{H}_{2} \mathrm{SO}_{4}\) through the side of the test tube produces a brown ring.
Question:
Precipitate \(\mathbf{X}\) is