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Consider the circle \(x^2+y^2=9\) and the parabola \(y^2=8 x\). They intersect at \(P\) and \(Q\) in the first and the fourth quadrants, respectively. Tangents to the circle at \(P\) and \(Q\) intersect the \(\mathrm{X}\)-axis at \(R\) and tangents to the parabola at \(P\) and \(Q\) intersect the \(\mathrm{X}\)-axis at \(S\).Question:
The radius of the incircle of the \(\triangle P Q R\) is

A rectangular sheet of fixed perimeter with sides having their lengths in the ratio $8:15$ is converted into an open rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed squares is $100$, the resulting box has maximum volume. Then the lengths of the sides of the rectangular sheet are:

The tangent \(P T\) and the normal \(P N\) to the parabola \(y^2=4 a x\) at a point \(P\) on

Paragraph: Let $S$ be the circle in the $xy$ – plane defined by the equation $x^2 + y^2 = 4.$

Question : Let $E_1 E_2$ and $F_1 F_2$ be the chords of $S$ passing through the point $P_0 \left(\mathrm{1,1}\right)$ and parallel to the $x$ – axis and the $y$ – axis, respectively. Let $G_1 G_2$ be the chord of $S$ passing through $P_0$ and having slope $-1.$ Let the tangents to $S$ at $E_1$ and $E_2$ meet at $E_3$, the tangents to $S$ at $F_1$ and $F_2$ meet at $F_3,$ and the tangents to $S$ at $G_1$ and $G_2$ meet at $G_3$. Then, the points $E_3, F_3,$ and $G_3$ lie on the curve

Let $f: \left(0, \infty \right)\to R$ be a differentiable function such that $f^{\text{'}}\left(x\right)=2-\frac{f\left(x\right)}{x}\forall x\in (0, \infty )$ and $f\left(1\right)\ne 1$. Then,

For $x\in \mathrm{ℝ}$, let ${\tan }^{-1}\left(x\right)\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$. Then the minimum value of the function $f:\mathrm{ℝ}\to \mathrm{ℝ}$ defined by $f\left(x\right)={\int }_0^{x{\tan }^{-1}x}\frac{e^{\left(t-\cos t\right)}}{1+t^{2023}}dt$ is

For any positive integer $n$, let $S_n:(0,\infty )\to R$ be defined by
$S_n\left(x\right)=\sum _{k=1}^n{\cot }^{-1}\left(\frac{1+k(k+1)x^2}{x}\right)$
where for any $x\in R,{\cot }^{-1}\left(x\right)\in (0,\pi )$ and ${\tan }^{-1}\left(x\right)\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$. Then which of the following statements is (are) TRUE ?

A particle of mass \(m\) is under the influence of the gravitational field of a body of mass \(M(\gg m)\). The particle is moving in a circular orbit of radius \(r_0\) with time period \(T_0\) around the mass \(M\). Then, the particle is subjected to an additional central force, corresponding to the potential energy \(V_{\mathrm{c}}(r)=m \alpha / r^3\), where \(\alpha\) is a positive constant of suitable dimensions and \(r\) is the distance from the center of the orbit. If the particle moves in the same circular orbit of radius \(r_0\) in the combined gravitational potential due to \(M\) and \(V_{\mathrm{c}}(r)\), but with a new time period \(T_1\), then \(\left(T_1^2-T_0^2\right) / T_1^2\) is given by
[ \(G\) is the gravitational constant.]

In the following monobromination reaction, the number of possible chiral products is

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Let \(V_r\) denotes the sum of the first \(r\) terms of an arithmetic progression \((A P)\) whose first term is \(r\) and the common difference is \((2 r-1)\). Let \(T_r=V_{r+1}-V_r-2\) and \(Q_r=T_{r+1}-T_r\) for \(r=1,2, \ldots\)
Question:
Which one of the following is a correct statement?

Let $\mathrm{M}=\left[\begin{array}{cc}{\sin }^4\mathrm{\theta }& -1-{\sin }^2\mathrm{\theta }\\ 1+{\cos }^2\mathrm{\theta }& {\cos }^4\mathrm{\theta }\end{array}\right]=\mathrm{\alpha }\mathrm{I}+\mathrm{\beta }{\mathrm{M}}^{-1},$ where $\mathrm{\alpha }=\mathrm{\alpha }\left(\mathrm{\theta }\right)$ and $\mathrm{\beta }=\mathrm{\beta }\left(\mathrm{\theta }\right)$ are real number, and $\mathrm{I}$ is the $2\times 2$ identity matrix. If ${\mathrm{\alpha }}^{\mathrm{*}}$ is the minimum of the set { $\mathrm{\alpha }\left(\mathrm{\theta }\right):\mathrm{\theta }\in \left[0,\mathrm{ }2\mathrm{\pi }\right)\}$ and ${\mathrm{\beta }}^{\mathrm{*}}$ is the minimum of the set $\left\{\mathrm{\beta }\left(\mathrm{\theta }\right):\mathrm{\theta }\in \left[0,\mathrm{ }2\mathrm{\pi }\right)\right\},$ then the value of ${\mathrm{\alpha }}^{\mathrm{*}}+{\mathrm{\beta }}^{\mathrm{*}}$ is

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p-amino-N, N-dimethylaniline is added to a strongly acidic solution of \(X\). The resulting solution is treated with a few drops of aqueous solution of \(Y\) to yield blue colouration due to the formation of methylene blue. Treatment of the aqueous solution of \(Y\) with the reagent potassium hexacyanoferrate(II) leads to the formation of an intense blue precipitate. The precipitate dissolves an excess addition of the reagent. Similarly, treatment of the solution of \(Y\) with the solution of potassium hexacyanoferrate(III) leads to a brown colouration due to the formation of \(Z\).
Question:
The compound \(Z\), is

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One twirls a circular ring (of mass \(M\) and radius \(R\) ) near the tip of one's finger as shown in Figure 1 . In the process the finger never loses contact with the inner rim of the ring. The finger traces out the surface of a cone, shown by the dotted line. The radius of the path traced out by the point where the ring and the finger is in contact is \(r\). The finger rotates with an angular velocity \(\omega_{0}\). The rotating ring rolls without slipping on the outside of a smaller circle described by the point where the ring and the finger is in contact (Figure 2). The coefficient of friction between the ring and the finger is \(\mu\) and the acceleration due to gravity is \(g\).








Question:

The minimum value of \(\omega_{0}\) below which the ring will drop down is