Let \(z=\cos \theta+i \sin \theta\). Then, the value of \(\sum_{m=1}^{15} \operatorname{Im}\left(z^{2 m-1}\right)\) at \(\theta=2^{\circ}\) is

Let $f:\left(0, 1\right)\to \mathrm{ℝ}$ be the function defined as $f\left(x\right)=\sqrt{n}$ if $x\in [\frac{1}{n+1}, \frac{1}{n})$ where $n\in \mathrm{\mathbb{N}}$. Let $g:\left(0, 1\right)\to \mathrm{ℝ}$ be a function such that ${\int }_{x^2}^x\sqrt{\frac{1-t}{t}}dt<g\left(x\right)<2\sqrt{x}$ for all $x\in \left(0, 1\right)$. Then $\underset{x\to 0}{\lim }f\left(x\right) g\left(x\right)$

A computer producing factory has only two plants $T_1$ and $T_2$. Plant $T_1$ produces $20\%$ and plant $T_2$ produces $80\%$ of the total computers produced. $7\%$ of computers produced in the factory turn out to be defective. It is known that,  $P$(computer turns out to be defective given that it is produced in plant $T_1$) $=10P$(computer turns out to be defective given that it is produced in plant $T_2$). Where, $P\left(E\right)$ denotes the probability of an event $E.$ A computer produced in the factory is randomly selected and it does not turn out to be defective. Then the probability that it is produced in plant $T_2$ is

An engineer is required to visit a factory for exactly four days during the first $15$ days of every month and it is mandatory that no two visits take place on consecutive days. Then the number of all possible ways in which such visits to the factory can be made by the engineer during $1-15$ June $2021$ 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?

Consider the lines \(L_{1}\) and \(L_{2}\) defined by
\[
L_{1}: x \sqrt{2}+y-1=0 \text { and } L_{2}: x \sqrt{2}-y+1=0
\]
For a fixed constant \(\lambda\), let \(C\) be the locus of a point \(P\) such that the product of the distance of \(P\) from \(L_{1}\) and the distance of \(P\) from \(L_{2}\) is \(\lambda^{2}\). The line \(y=2 x+1\) meets \(C\) at two points \(R\) and \(S\), where the distance between \(R\) and \(S\) is \(\sqrt{270}\).
Let the perpendicular bisector of \(R S\) meet \(C\) at two distinct points \(R^{\prime}\) and \(S^{\prime}\). Let \(D\) be the square of the distance between \(R^{\prime}\) and \(S^{\prime}\).
The value of $D$ is

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 common tangent of the two circles is

In a bimolecular reaction, the steric factor $\mathrm{P}$ was experimentally determined to be $4.5$. The correct option(s) among the following is(are)

Match the statements in Column I with the properties Column II.

The sum of the spin only magnetic moment values (in B.M.) of \(\left[\mathrm{Mn}(\mathrm{Br})_6\right]^{3-}\) and \(\left[\mathrm{Mn}(\mathrm{CN})_6\right]^{3-}\) is _____.

Statement 1 For every chemical reaction at equilibrium, standard Gibbs energy of reaction is zero.
Statement 2 At constant temperature and pressure, chemical reactions are spontaneous in the direction of decreasing Gibbs energy.

A solid sphere of radius \(R\) has a charge \(Q\) distributed in its volume with a charge density \(\rho=k r^a\), where \(k\) and \(a\) are constants and \(r\) is the distance from its centre. If the electric field at \(r=\frac{R}{2}\) is \(\frac{1}{8}\) times that at \(r=R\), find the value of \(a\).

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 in \(A\) is