Let \((x, y)\) be any point on the parabola \(y^2=4 x\). Let \(P\) be the point that divides the line segment from \((0,0)\) to \((x, y)\) in the ratio \(1: 3\). Then, the locus of \(P\) is

Let \(z=x+i y\) be a complex number, where \(x\) and \(y\) are integers. Then, the area of the rectangle whose vertices are the roots of the equation \(z \bar{z}^3+\bar{z} z^3=350\) 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 \(p(x)\) be a real polynomial of least degree which has a local maximum at \(x=1\) and a local minimum at \(x=3\). If \(p(1)=6\) and \(p(3)=2\), then \(p^{\prime}(0)\) is

Two adjacent sides of a parallelogram \(A B C D\) are given by \(\overrightarrow{\mathbf{A B}}=2 \hat{\mathbf{i}}+10 \hat{\mathbf{j}}+11 \hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{A D}}=-\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}\). The side \(A D\) is rotated by an acute angle \(\alpha\) in the plane of the parallelogram so that \(A D\) becomes \(A D^{\prime}\). If \(A D^{\prime}\) makes a right angle with the side \(A B\), then the cosine of the angle \(\alpha\) is given by

Let $f :\left(0, \infty \right)\to R$ be given by $f\left(x\right)= \underset{\frac{1}{x}}{\overset{x}{\int }}{e}^{-\left(t+\frac{1}{t}\right)}\frac{dt}{t},$ then

If \(\vec{a}, \vec{b}\) and \(\vec{c}\) are unit vectors satisfying \(|\vec{a}-\vec{b}|^{2}+|\vec{b}-\vec{c}|^{2}+|\vec{c}-\vec{a}|^{2}=9\), then \(|2 \vec{a}+5 \vec{b}+5 \vec{c}|\) is

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Reimer-Tiemann reaction introduces an aldehyde group, on to the aromatic ring of phenol, ortho to the hydroxyl group. This reaction involves electrophilic aromatic substitution. This is a general method for the synthesis of substituted salicylaldehydes as depicted below.

Question :
The electrophile in this reaction is

The total number of ways in which 5 balls of different colours can be distributed among 3 persons so that each person gets at least one ball is

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Two trains \(A\) and \(B\) are moving with speeds \(20 \mathrm{~m} / \mathrm{s}\) and \(30 \mathrm{~m} / \mathrm{s}\), respectively in the same direction on the same straight track, with \(B\) ahead of \(A\). The engines are at the front ends. The engine of train \(A\) blows a long whistle.
Assume that the sound of the whistle is composed of components varying in frequency from \(f_1=800 \mathrm{~Hz}\) to \(f_2=1120 \mathrm{~Hz}\), as shown in the figure. The spread in the frequency (highest frequency - lowest frequency) is thus \(320 \mathrm{~Hz}\). The speed of sound in still air is \(340 \mathrm{~m} / \mathrm{s}\).
Question:
The spread of frequency as observed by the passengers in train \(B\) is

Let ${\tan }^{-1}\left(x\right)\in \left(-\frac{\pi }{2}, \frac{\pi }{2}\right)$, for $x\in \mathrm{ℝ}$. Then the number of real solutions of the equation $\sqrt{1+\cos \left(2x\right)}=\sqrt{2}{\tan }^{-1}\left(\tan x\right)$ in the set $\left(-\frac{3\pi }{2}, -\frac{\pi }{2}\right)\cup \left(-\frac{\pi }{2}, \frac{\pi }{2}\right)\cup \left(\frac{\pi }{2}, \frac{3\pi }{2}\right)$ is equal to

Three vectors $\overrightarrow{P},\overrightarrow{Q}$ and $\overrightarrow{R}$ are shown in the figure. Let, $S$ be any point on the vector $\overrightarrow{R}$ . The distance between the points $P$ and $S$ is $b\left|\overrightarrow{R}\right|$. The general relation among vectors $\overrightarrow{P},\overrightarrow{Q}$ and $\overrightarrow{S}$ are

Let \(S\) denote the locus of the point of intersection of the pair of lines
\(4 x-3 y=12 \alpha\)
\(4 \alpha x+3 \alpha y=12\)
where \(\alpha\) varies over the set of non-zero real numbers. Let \(T\) be the tangent to \(S\) passing through the points \((p, 0)\) and \((0, q), q>0\), and parallel to the line \(4 x-\frac{3}{\sqrt{2}} y=0\).
Then the value of \(p q\) is