A plane passes through \((1,-2,1)\) and is perpendicular to two planes \(2 x-2 y+z=0\) and \(x-y+2 z=4\), then the distance of the plane from the point \((1,2,2)\) is

One Indian and four American men and their wives are to be seated randomly around a circular table. Then, the conditional probability that the Indian man is seated adjacent to his wife given that each American man is seated adjacent to his wife, is

Let \(a, b, c, p, q\) be real numbers.
Suppose, \(\alpha, \beta\) are the roots of the equation \(x^2+2 p x+q=0\) and \(\alpha, \frac{1}{\beta}\) are the roots of the equation \(a x^2+2 b x+c=0\), where \(\beta^2 \notin\{-1,0,1\}\).
Statement \(1\left(p^2-q\right)\left(b^2-a c\right) \geq 0\).
Statement \(2 b \neq p a\) or \(c \neq q a\).

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Let \(F_{1}\left(x_{1}, 0\right)\) and \(F_{2}\left(x_{2}, 0\right)\), for \(x_{1}<0\) and \(x_{2}>0\), be the foci of the ellipse \(\frac{x^{2}}{9}+\frac{y^{2}}{8}=1 .\) Suppose a parabola having vertex at the origin and focus at \(F_{2}\) intersects the ellipse at point \(M\) in the first quadrant and at point \(N\) in the fourth quadrant.


Question:

The orthocentre of the triangle \(F_{1} M N\) is

If \(I_n=\int_{-\pi}^\pi \frac{\sin n x}{\left(1+\pi^x\right) \sin x} d x ; n=0,1,2\), \(\ldots\), then

Let the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) be defined by
\(f(x)=\frac{\sin x}{e^{\pi x}} \frac{\left(x^{2023}+2024 x+2025\right)}{\left(x^2-x+3\right)}+\frac{2}{e^{\pi x}} \frac{\left(x^{2023}+2024 x+2025\right)}{\left(x^2-x+3\right)}\).
Then the number of solutions of \(f(x)=0\) in \(\mathbb{R}\) is

Let m and n be two positive integers greater than 1. If $\underset{\alpha \to 0}{\lim }\left(\frac{e^{\mathrm{c}\mathrm{o}\mathrm{s}\left({\alpha }^n \right)}-e}{{\alpha }^m}\right)= -\left(\frac{e}{2}\right)$ then the value of $\frac{m}{n}$ is

When 100 mL of 1.0 M HCl was mixed with 100 mL of 1.0 M NaOH in an insulated beaker at constant pressure, a temperature increase of ${5.7}^{\mathrm{o}}\mathrm{C}$ was measured for the beaker and its contents (Expt.1). Because the enthalpy of neutralization of a strong acid with a strong base is a constant $(-57.0\mathrm{ }\mathrm{k}\mathrm{J}\mathrm{ }\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1})$ , this experiment could be used to measure the calorimeter constant. In a second experiment (Expt. 2), 100 mL of 2.0 M acetic acid $({\mathrm{K}}_{\mathrm{a}}=2.0\times {10}^{-5})$ was mixed with 100 mL of 1.0 M NaOH (under identical conditions to Expt. 1) where a temperature rise of ${5.6}^{\mathrm{o}}\mathrm{C}$ was measured. (Consider heat capacity of all solutions as $4.2\mathrm{ }\mathrm{J}\mathrm{ }{\mathrm{g}}^{-1}\mathrm{ }{\mathrm{K}}^{-1}$ and density of all solutions as $1.0\mathrm{ }\mathrm{g}\mathrm{ }\mathrm{m}{\mathrm{L}}^{-1}$ )
Enthalpy of dissociation $\left(\mathrm{i}\mathrm{n}\mathrm{ }\mathrm{k}\mathrm{J}\mathrm{ }\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1}\right)$ of acetic acid obtained from the Expt. 2 is

Let \(f: I R \rightarrow I R\) be defined as \(f(x)=|x|+\left|x^{2}-1\right|\). The total number of points at which \(f\) attains either a local maximum or a local minimum is

Let $f\text{ : }\left[\frac{1}{2},1\right]⟶R$ (the set of all real numbers) be a positive, non-constant and differentiable function such that $f^{\text{'}}\left(x\right)<2f\left(x\right)$ and $f\left(\frac{1}{2}\right)=1$. Then the value of $\underset{1/2}{\overset{1}{\int }}f\left(x\right)\mathit{\text{dx}}$ lies in the interval:

For 3 x 3 matrices M and N, which of the following statement(s) is (are) not correct?

Column I		Column II
$\left(\mathrm{A}\right)$	In a triangle $∆XYZ$ let $a,b$ and $\mathrm{c}$ be the lengths of the angles $X,Y$ and $\mathrm{Z}$, respectively. If $2\left(a^2-b^2\right)=c^2$ and $\lambda =\frac{\sin \left(X-Y\right)}{\mathrm{sinZ}}$then possible values of $n$ for which $\cos \left(n\pi \lambda \right)=0$ is are	$\left(\mathrm{P}\right)$	$1$
$\left(\mathrm{B}\right)$	In a triangle $\mathrm{\Delta }XYZ,$ let $a,b$ and $c$ be the lengths of the sides opposite to the angles $X,Y$ and $Z$, respectively. If $1+\cos 2x-2\cos 2y$$=2\sin x\sin y$, then possible value(s) of $\frac{a}{b}$ is (are)	$\left(\mathrm{Q}\right)$	$2$
$\left(\mathrm{C}\right)$	In ${\mathbb{R}}^2,$ let $\sqrt{3}\mathrm{ }\hat{\mathrm{i}}+\hat{\mathrm{j}},\mathrm{ }\hat{\mathrm{i}}+\sqrt{3}\mathrm{ }\hat{\mathrm{j}}$ and $\beta \hat{\mathrm{i}}+\left(1-\beta \right)\mathrm{ }\hat{\mathrm{j}}$ be the position vectors of $X,Y$ and $Z$ with respect to the origin $O$, respectively. If the distance of $Z$ from the bisector of the acute angle of $\overrightarrow{OX}$ with $\overrightarrow{OY}$ is $\frac{3}{\sqrt{2}}$, then possible value (s) of $\left|\beta \right|$ is (are)	$\left(\mathrm{R}\right)$	$3$
$\left(\mathrm{D}\right)$	Suppose that $F\left(\alpha \right)$ denotes the area of the region bounded by $x=0,\mathrm{ }x=2, y^2=4x$ and$y=\left|ax-1\right|+$$\left|\alpha x-2\right|+\alpha x,$  where $\alpha \in \left\{0, 1\right\}.$ Then the value (s) of $F\left(\alpha \right)+\frac{8}{3} \sqrt{2},$ when $\alpha =0$ and $\alpha =1$, is (are)	$\left(\mathrm{S}\right)$	$5$
		$\left(\mathrm{T}\right)$	$6$

 

A ring and a disc are initially at rest, side by side, at the top of an inclined plane which makes an angle $60°$ with the horizontal. They start to roll without slipping at the same instant of time along the shortest path. If the time difference between their reaching the ground is $\left(2-\sqrt{3}\right)/\sqrt{10} \mathrm{s}$, then the height of the top of the inclined plane (in meters) is __________. Take $g=10\mathrm{ }\mathrm{m}\mathrm{ }{\mathrm{s}}^{-2}$.