Let $a$ and $b$ be positive real numbers. Suppose $\overrightarrow{PQ}=a\hat{i}+b\hat{j}$ and $\overrightarrow{PS}=a\hat{i}-b\hat{j}$ are adjacent sides of a parallelogram $PQRS.$ Let $\overrightarrow{u}$ and $\overrightarrow{v}$ be the projection vectors of $\overrightarrow{w}=\hat{i}+\hat{j}$ along $\overrightarrow{PQ}$ and $\overrightarrow{PS}$, respectively. If $\left|\overrightarrow{u}\right|+\left|\overrightarrow{v}\right|=\left|\overrightarrow{w}\right|$ and if the area of the parallelogram $PQRS$ is $8,$ then which of the following statements is/are TRUE?

A factory has a total of three manufacturing units, \(M_1, M_2\), and \(M_3\), which produce bulbs independent of each other. The units \(M_1, M_2\), and \(M_3\) produce bulbs in the proportions of 2:2:1, respectively. It is known that \(20 \%\) of the bulbs produced in the factory are defective. It is also known that, of all the bulbs produced by \(M_1, 15 \%\) are defective. Suppose that, if a randomly chosen bulb produced in the factory is found to be defective, the probability that it was produced by \(M_2\) is \(\frac{2}{5}\).
If a bulb is chosen randomly from the bulbs produced by \(M_3\), then the probability that it is defective is ____.

An ellipse intersects the hyperbola \(2 x^2-2 y^2=1\) orthogonally. The eccentricity of the ellipse is reciprocal to that of the hyperbola. If the axes of the ellipse are along the coordinate axes, then

Tangent is drawn at any point \(P\) of a curve which passes through \((1,1)\) cutting \(X\)-axis and \(Y\)-axis at \(A\) and \(B\), respectively. If \(A P: B P=3: 1\), then

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

Let $b$ be a nonzero real number. Suppose $f:\mathrm{ℝ}\to \mathrm{ℝ}$ is a differentiable function such that $f\left(0\right)=1$. If the derivative $f^{\text{'}}$ of $f$ satisfies the equation $f^{\text{'}}\left(x\right)=\frac{f\left(x\right)}{b^2+x^2}$ for all $x\in \mathrm{ℝ},$ then which of the following statements is/are TRUE?

Let $b_i>1$ for $i=1, 2,\dots .,101.$ Suppose ${\log }_e{b}_1,{\log }_e{b}_2,\dots ..,{\log }_e{b}_{101}$ are in Arithmetic Progression (A.P.) with the common difference ${\log }_{e}2.$ Suppose $a_1, a_2,\dots .,a_{101}$ are in A.P. such that $a_1=b_1$ and $a_{51}=b_{51}.$ If $t=b_1+b_2+\dots +b_{51}$ and $s=a_1+a_2+\dots +a_{51}$ then

A charged particle is introduced at the origin $x=0, y=0, z=0$ with a given initial velocity $\overrightarrow{v}$ . A uniform electric field $\overrightarrow{E}$ and a uniform magnetic field $\overrightarrow{B}$ exist everywhere. The velocity $\overrightarrow{v}$ , electric field $\overrightarrow{E}$ and a uniform magnetic field $\overrightarrow{B}$ are given in the columns below

Column 1	Column 2	Column 3
1. Electron with $\overrightarrow{v}=2\frac{E_0}{B_0}\widehat{x}$	(i) $\overrightarrow{E}=E_0\widehat{z}$	(P) $\overrightarrow{B}=-B_0\widehat{x}$
2. Electron with $\overrightarrow{v}=2\frac{E_0}{B_0}\widehat{y}$	(ii) $\overrightarrow{E}=-E_0\widehat{y}$	(Q) $\overrightarrow{B}=B_0\widehat{x}$
3.Proton with $\overrightarrow{v}$ =0	(iii) $\overrightarrow{E}={-E}_0\widehat{x}$	(R) $\overrightarrow{B}=-B_0\widehat{y}$
4. Proton with $\overrightarrow{v}=2\frac{E_0}{B_0}\widehat{x}$	(iv) $\overrightarrow{E}=E_0\widehat{x}$	(S) $\overrightarrow{B}=-B_0\widehat{z}$

In which case will the particle describe a helical path with axis along positive z direction?

The total number of distinct naturally occurring amino acids obtained by complete acidic hydrolysis of the peptide shown below is

The number of solutions of the pair of equations \(2 \sin ^2 \theta-\cos 2 \theta=0\) and \(2 \cos ^2 \theta-3 \sin \theta=0\) in the interval \([0,2 \pi]\) is

A moving coil galvanometer has 50 turns and each turn has an area $2\times {10}^{-4} {\mathrm{m}}^2$ . The magnetic field produced by the magnet inside the galvanometer is $0.02 \mathrm{T}$ . The torsional constant of the suspension wire is ${10}^{-4} \mathrm{N}\mathrm{ }\mathrm{m}\mathrm{ }{\mathrm{rad}}^{-1}$ . When a current flows through the galvanometer, a full scale deflection occurs if the coil rotates by $0.2 \mathrm{rad}$ . The resistance of the coil of the galvanometer is $50 \mathrm{\Omega }$ . This galvanometer is to be converted into an ammeter capable of measuring current in the range $0-1.0 \mathrm{A}$ . For this purpose, a shunt resistance is to be added in parallel to the galvanometer. The value of this shunt resistance, in ohms, is __________.

In the Newman projection for 2,2-dimethylbutane

\(X\) and \(Y\) can respectively be

There are two vernier calipers both of which have $1\mathrm{cm}$ divided into $10$ equal divisions on the main scale. The Vernier scale of one of the calipers $\left(C_1\right)$ has $10$ equal divisions that correspond to $9$ main scale divisions. The Vernier scale of the other caliper $\left(C_2\right)$ has $10$ equal divisions that correspond to $11$ main scale divisions. The readings of the two calipers are shown in the figure. The measured values (in $\mathrm{cm}$) by calipers $C_1$ and $C_2$ respectively, are