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A circle \(C\) of radius 1 is inscribed in an equilateral \(\triangle P Q R\). The points of contact of \(C\) with the sides \(P Q, Q R, R P\) are \(D, E, F\) respectively. The line \(P Q\) is given by the equation
\(\sqrt{3} x+y-6=0\) and the point \(D\) is \(\left(\frac{3 \sqrt{3}}{2}, \frac{3}{2}\right)\). Further, it is given that the origin and the centre of \(C\) are on the same side of the line \(P Q\).Question:
Equations of the sides \(Q R, R P\) are

Let $X$ be a set with exactly $5$ elements and $Y$ be a set with exactly $7$ elements. If $\alpha$ is the number of one-one functions from $X\to Y$ and $\beta$ is the number of onto functions from $Y\to X$, then the value of $\frac{1}{5!}\left(\beta -\alpha \right)$ is _____-.

There are three bags ${\mathrm{B}}_1,{\mathrm{B}}_2$ and ${\mathrm{B}}_3.$ The bag ${\mathrm{B}}_1$ contains $5$ red and $5$ green balls, ${\mathrm{B}}_2$ contains $3$ red and $5$ green balls, and ${\mathrm{B}}_3$ contains $5$ red and $3$ green balls, Bags ${\mathrm{B}}_1,{\mathrm{B}}_2$ and ${\mathrm{B}}_3$ have probabilities $\frac{3}{10},\frac{3}{10}$ and $\frac{4}{10}$ respectively of being chosen. A bag is selected at random and a ball is chosen at random from the bag. Then which of the following options is/are correct?

Equation of the plane containing the straight line \(\frac{x}{2}=\frac{y}{3}=\frac{z}{4}\) and perpendicular to the plane containing the straight lines \(\frac{x}{3}=\frac{y}{4}=\frac{z}{2}\) and \(\frac{x}{4}=\frac{y}{2}=\frac{z}{3}\) is

Let \(M\) and \(N\) be two \(3 \times 3\) non-singular skew-symmetric matrices such that \(M N=N M\). If \(P^T\) denotes the transpose of \(P\), then \(M^2 N^2\left(M^T N\right)^{-1}\left(M N^{-1}\right)^T\) is equal to

Suppose that $\overrightarrow{p}, \overrightarrow{q} and \overrightarrow{r}$ are three non - coplanar vectors in ${\mathbb{R}}^3$ . Let the components of a vector $\overrightarrow{s}$ along \(p, q\) and \(r\) be 4,3 and 5 respectively. If the components of $\left(-\overrightarrow{p}+ \overrightarrow{q}+ \overrightarrow{r}\right), \left(\overrightarrow{p}- \overrightarrow{q}+ \overrightarrow{r}\right) and \left(- \overrightarrow{p}- \overrightarrow{q}+ \overrightarrow{r}\right)$ are x, y and z, respectively, then the value of $2x+y+z$ is

Let $f\left(x\right)=\frac{\sin \pi x}{x^2}, x>0.$
Let $x_1<x_2<x_3<\dots <x_n<\dots$ be all the points of local maximum of $f$ and $y_1<y_2<y_3<\dots <y_n<\dots$ be all the points of local minimum of $f.$
Then which of the following options is/are correct?

A rod of mass $m$ and length $L,$ pivoted at one of its ends, is hanging vertically. A bullet of the same mass moving at speed $v$ strikes the rod horizontally at a distance $x$ from its pivoted end and gets embedded in it. The combined system now rotates with an angular speed $\omega$ about the pivot. The maximum angular speed ${\omega }_{\mathrm{M}}$ is achieved for $x=x_{\mathrm{M}}$. Then

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: Consider the statements:

\(P\) : There exists some \(x \in \mathrm{R}\) such that \(f(x)+2 x\)

\(=2\left(1+x^{2}\right)\)

\(Q\) : There exists some \(x \in \mathrm{R}\) such that \(2 f(x)+1\) \(=2 x(1+x)\)

Then

As shown in the figures, a uniform rod \(O O^{\prime}\) of length \(l\) is hinged at the point \(O\) and held in place vertically between two walls using two massless springs of same spring constant. The springs are connected at the midpoint and at the top-end \(\left(O^{\prime}\right)\) of the rod, as shown in Fig. 1 and the rod is made to oscillate by a small angular displacement. The frequency of oscillation of the rod is \(f_1\). On the other hand, if both the springs are connected at the midpoint of the rod, as shown in Fig. 2 and the rod is made to oscillate by a small angular displacement, then the frequency of oscillation is \(f_2\). Ignoring gravity and assuming motion only in the plane of the diagram, the value of \(\frac{f_1}{f_2}\) is:

Let $n\ge 2$ be a natural number and $f:\left[0, 1\right]\to \mathrm{ℝ}$ be the function defined by $f\left(x\right)=\left\{\begin{array}{ccc}n\left(1-2nx\right)& \mathrm{if}& 0\le x\le \frac{1}{2n}\\ 2n\left(2nx-1\right)& \mathrm{if}& \frac{1}{2n}\le x\le \frac{3}{4n}\\ 4n\left(1-nx\right)& \mathrm{if}& \frac{3}{4n}\le x\le \frac{1}{n}\\ \frac{n}{n-1}\left(nx-1\right)& \mathrm{if}& \frac{1}{n}\le x\le 1\end{array}\right.$

If $n$ is such that the area of the region bounded by the curves $x=0, x=1, y=0$ and $y=f\left(x\right)$ is $4$, then the maximum value of the function $f$ is

A block of mass $2\mathrm{M}$ is attached to a massless spring with spring-constant $\mathrm{k}.$ This block is connected to two other blocks of masses $\mathrm{M}$ and $2\mathrm{M}$ using two massless pulleys and strings. The accelerations of the blocks are ${\mathrm{a}}_1,{\mathrm{a}}_2$ and ${\mathrm{a}}_3$ as shown in figure. The system is released from rest with the spring in its unstretched state. The maximum extension of the spring is ${\mathrm{x}}_0.$ Which of the following option(s) is/are correct?
[ $\mathrm{g}$ is the acceleration due to gravity. Neglect friction]

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_________