If \(\frac{3+2 i \sin \theta}{1-2 i \sin \theta}\) is a real number and \(0 < \theta < 2 \pi\), then \(\theta\) is equal to

If $f:R\to R$ is defined as $f\left(x\right)={\left(2020-x^{2019}\right)}^{1/2019},\forall x\in R,$ find $(f\circ f\circ f\circ f)\left(\frac{2019}{2020}\right)$

A bag contains 6 white and 4 black balls. Two balls are drawn at random. The probability that they are of the same colour, is

If $y=\left({\log }_{\cot x}\tan x\right)\left({\log }_{\tan x}\cot x\right)+{\tan }^{-1}\left(\frac{4x}{4-x^2}\right)$, then $\frac{dy}{dx}=$

A bag consists of 3 red balls, 5 blue balls and 8 green balls. A ball is selected at random. What is the probability of not getting a blue ball?

a,b,c are three vectors such that \(|\mathbf{a}|=\mathbf{l},|\mathbf{b}|=2,|\mathbf{c}|=3\) and \(\mathbf{b}. \mathbf{c}=\mathbf{0}\). If the projection of \(\mathbf{b}\) along \(\mathbf{a}\) is equal to the projection of \(\mathbf{c}\) along \(\mathbf{a}\), then \(|2 \mathbf{a}+3 \mathbf{b}-3 \mathbf{c}|=\)

If \(|x| \lt 1\), then the number of terms in the expansion of \(\left[\frac{1}{2}\left(1.2+2.3 x+3.4 x^2+\ldots . . \infty\right)\right]^{-25}\) is

A point charge \(q\) is placed at origin. Let \(\mathbf{E}_{A^{\prime}} \mathbf{E}_B\) and \(\mathbf{E}_C\) be the electric fields at three points \(A\) \((1,2,3), B(1,1,-1)\) and \(C(2,2,2)\) respectively due to the charge \(q\). Then, the relation between them is
1. \(\mathbf{E}_A \perp \mathbf{E}_B\)
2. \(\mathbf{E}_A \| \mathbf{E}_C\)
3. \(\left|\mathbf{E}_B\right|=4\left|\mathbf{E}_C\right|\)
4. \(\left|\mathbf{E}_B\right|=8\left|\mathbf{E}_C\right|\)

Match of the following.
\(\begin{array}{ll}
\hline \text{ Column I } & \text{ Column II } \\
\hline \begin{array}{ll}
\text{A. Ratio of } \frac{\Delta Q}{\Delta U} \text{ in an isobaric} \\
\text{process}
\end{array} & 1. \frac{T_1}{\left(T_1-T_2\right)} \\
\hline \begin{array}{ll}
\text{B. Ratio of } \frac{\Delta Q}{\Delta W} \text{ in an isobaric} \\
\text{process}
\end{array} & 2. \frac{T_2}{\left(T_1-T_2\right)} \\
\hline \begin{array}{l}
\text{C. Coefficient of performance} \\
\text{of a refrigerator}
\end{array} & 3. \frac{\gamma}{\gamma-1} \\
\hline \begin{array}{l}
\text{D. Coefficient of performance} \\
\text{of a heat pump}
\end{array} & 4. \gamma \\
\hline
\end{array}\)
Codes
\(\begin{array}{llll}A & B & C & D\end{array}\)

The \(\mathrm{pH}\) of \(0.05 \mathrm{M}\) acetic acid is \(\left(K_a=2 \times 10^{-5}\right)\)

If \(A\) is a non-zero square matrix of order \(n\) with \(\operatorname{det}(I+A) \neq 0\) and \(A^3=O\), where \(I, O\) are unit and null matrices of order \(n \times n\) respectively, then \((I+A)^{-1}\) is equal to

If a diagonal of a square is along the line \(8 x-15 y=0\) and one of its vertices is \((1,2)\), then the equations of the sides of the square passing through this vertex are

The time \(T\) of oscillation of a simple pendulum of length \(L\) is governed by \(T=2 \pi \sqrt{\frac{L}{g}}\), where \(g\) is constant. The percentage by which the length be changed in order to correct an error of loss equal to 2 minutes of time per day is