Let $l_1,l_2...l_{100}$ be consecutive terms of an arithmetic progression with common difference $d_1$, and let $w_1,w_2$, $\dots ,w_{100}$ be consecutive terms of another arithmetic progression with common difference $d_2$, where $d_1{d}_2=10$. For each $i=1,2,3..........100$, let $R_i$ be a rectangle with length $l_i$, width $w_i$ and area $A_i$.If $A_{51}-A_{50}=1000$, then the value of $A_{100}-A_{90}$ is _______.

Let \(\int(x)=\left\{\begin{array}{r}x^{2}\left|\cos \frac{\pi}{x}\right|, \quad x \neq 0 \\ 0, \quad x=0\end{array}, x \in R\right.\) then \(\int\) is

If the value of \(n(Y)+n(Z)\) is \(k^2\), then \(|k|\) is
Let \(S=\{1,2,3,4,5,6\}\) and \(X\) be the set of all relations \(R\) from \(S\) to \(S\) that satisfy both the following properties:
i. \(R\) has exactly 6 elements.
ii. For each \((a, b) \in R\), we have \(|a-b| \geq 2\).
Let \(Y=\{R \in X\) : The range of \(R\) has exactly one element \(\}\) and \(Z=\{R \in X: R\) is a function from \(S\) to \(S\}\).
Let \(n(A)\) denote the number of elements in a set \(A\).

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\text { Match the statements given in Column I with the values given in Column II. }
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One Indian and four American men and their wives are to be seated randomly around a circular table. Then, the conditional probability that the Indian man is seated adjacent to his wife given that each American man is seated adjacent to his wife, is

Let \(S=\left\{A=\left(\begin{array}{lll}0 & 1 & c \\ 1 & a & d \\ 1 & b & e\end{array}\right): a, b, c, d, e \in\{0,1\}\right.\) and \(\left.|A| \in\{-1,1\}\right\}\), where \(|A|\) denotes the determinant of \(A\). Then the number of elements in \(S\) is

Three lines are given by $\overrightarrow{\mathrm{r}}=\mathrm{\lambda }\hat{\mathrm{i}}, \lambda \in \mathrm{R}$
$\overrightarrow{\mathrm{r}}=\mathrm{\mu }\left(\widehat{\mathrm{i}}+\widehat{\mathrm{j}}\right),\mathrm{\mu }\in \mathrm{R}$ and
$\overrightarrow{\mathrm{r}}=\mathrm{\nu }\left(\widehat{\mathrm{i}}+\widehat{\mathrm{j}}+\widehat{\mathrm{k}}\right),\mathrm{\nu }\in \mathrm{R}.$
Let the lines cut the plane $\mathrm{x}+\mathrm{y}+\mathrm{z}=1$ at the points $\mathrm{A},\mathrm{ }\mathrm{B}$ and $\mathrm{C}$ respectively. If the area of the triangle $\mathrm{A}\mathrm{B}\mathrm{C}$ is $\mathrm{\Delta }$ then the value of ${\left(6\mathrm{\Delta }\right)}^2$ equals_________

Match the orbital overlap figures shown in List-I with the description given in List-II and select the correct
answer using the code given below the lists.

	List – I		List - II
A.		P	$\mathrm{p}-\mathrm{d}\mathrm{}\mathrm{\pi }$ antibonding
B.		Q	$\mathrm{d}-\mathrm{d}\mathrm{}\mathrm{\sigma }$ bonding
C.		R	$\mathrm{p}-\mathrm{d}\mathrm{}\mathrm{\pi }$ bonding
D.		S	$\mathrm{d}-\mathrm{d}\mathrm{}\mathrm{\sigma }$ antibonding

The carboxyl functional group \((-\mathrm{COOH})\) is present in

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Consider an evacuated cylindrical chamber of height \(h\) having rigid conducting plates at the ends and an insulating curved surface as shown in the figure. A number of spherical balls made of a light weight and soft material and coated with a conducting material are placed on the bottom plate. The balls have a radius \(r \ll h\). Now a high voltage source \((HV)\) is connected across the conducting plates such that the bottom plate is at \(+V_{0}\) and the top plate at \(-V_{0} .\) Due to their conducting surface, the balls will get charged, will become equipotential with the plate and are repelled by it. The balls will eventually collide with the top plate, where the coefficient of restitution can be taken to be zero due to the soft nature of the material of the balls. The electric field in the chamber can be considered to be that of a parallel plate capacitor. Assume that there are no collisions between the balls and the interaction between them is negligible. (Ignore gravity)



Question:

The average current in the steady state registered by the ammeter in the circuit will be

Benzene and naphthalene form an ideal solution at room temperature. For this process, the true statement(s) is (are)

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Let \(V_r\) denotes the sum of the first \(r\) terms of an arithmetic progression \((A P)\) whose first term is \(r\) and the common difference is \((2 r-1)\). Let \(T_r=V_{r+1}-V_r-2\) and \(Q_r=T_{r+1}-T_r\) for \(r=1,2, \ldots\)
Question:
\(T_r\) is always

Correct option(s) for the following sequence of reactions is(are)