\[
\text { Match the conics in Column I with the statements/expressions in Column II. }
\]

A pack contains n cards numbered from 1 to n. Two consecutive numbered cards are removed from the pack and the sum of the numbers on the remaining cards is 1224. If the smaller of the numbers on the removed cards is k, then $\text{k}-20=$

Let $M$ be a $3\times 3$ invertible matrix with real entries and let $I$ denote the $3\times 3$ identity matrix. $M^{-1}=adj\left(adjM\right)$, then which of the following statements is/are ALWAYS TRUE?

Let complex numbers α and $\frac{1}{\overline{\alpha }}$   lie on circles ${\left(x-x_0\right)}^2+{\left(y-y_0\right)}^2={\text{r}}^2\text{, }$  and  ${\left(x-x_0\right)}^2+{\left(y-y_0\right)}^2={4\text{r}}^2\text{, }$respectively. If $z_0=x_0+i{y}_0$ satisfies the equation $2{\left|z_0\right|}^2=r^2+2\text{,}\text{then}\left|\alpha \right|=$

Let \((x, y, z)\) be points with integer coordinates satisfying the system of homogeneous equations \(3 x-y-z=0,-3 x+z=0,-3 x+2 y+z=0\). Then, the number of such points for which \(x^2+y^2+z^2 \leq 100\) is

The ellipse \(E_{1}: \frac{x^{2}}{9}+\frac{y^{2}}{4}=1\) is inscribed in a rectangle \(R\) whose sides are parallel to the coordinate axes. Another ellipse \(E_{2}\) passing through the point \((0,4)\) circumscribes the rectangle \(R\). The eccentricity of the ellipse \(E_{2}\) is

For $\text{a}\in \text{R}$ (the set of all real numbers), $\text{a}\ne -1$, $\underset{\text{n}\to \infty }{\text{lim}}\frac{\left(1^{\text{a}}+2^{\text{a}}+\dots +{\text{n}}^{\text{a}}\right)}{{\left(\text{n}+1\right)}^{\text{a}-1}\left[\left(\text{na}+1\right)+\left(\text{na}+2\right)+\dots +\left(\text{na}+\text{n}\right)\right]}=\frac{1}{60}$ Then a =

The total number of real solutions of the equation \(\theta=\tan ^{-1}(2 \tan \theta)-\frac{1}{2} \sin ^{-1}\left(\frac{6 \tan \theta}{9+\tan ^2 \theta}\right)\) is (Here, the inverse trigonometric functions \(\sin ^{-1} x\) and \(\tan ^{-1} x\) assume values in \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) and \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\), respectively.)

The value of ${\left({\left({\log }_{2}9\right)}^2\right)}^{\frac{1}{{\log }_2\left({\log }_{2}9\right)}}\times {\left(\sqrt{7}\right)}^{\frac{1}{{\log }_{4}7}}$ is_____.

The structure of a peptide is given below.
If the absolute values of the net charge of the peptide at $\mathrm{pH}=2,\mathrm{pH}=6,$ and $\mathrm{pH}=11$ are $\left|{\mathrm{z}}_1\right|,\left|{\mathrm{z}}_2\right|,$ and $\left|{\mathrm{z}}_3\right|,$ respectively, then what is $\left|{\mathrm{z}}_1\right|+\left|{\mathrm{z}}_2\right|+\left|{\mathrm{z}}_3\right|$?

A steel wire of diameter 0.5 mm and Young's modulus \(2 \times 10^{11} Nm ^{-2}\) carries a load of mass M . The length of the wire with the load is 1.0 m . A vernier scale with 10 divisions is attached to the end of this wire. Next to the steel wire is a reference wire to which a main scale, of least count 1.0 mm , is attached. The 10 divisions of the vernier scale correspond to 9 divisions of the main scale. Initially, the zero of vernier scale coincides with the zero of main scale. If the load on the steel wire is increased by \(1.2 k g\), the vernier scale division which coincides with a main scale division is _____________ . Take \(g =10 m s^{-1}\) and \(\pi=3.2\).

In ${\mathrm{R}}^3$ , let L be a straight line passing through the origin. Suppose that all the points on L are at a constant distance from the two planes ${\mathrm{P}}_1\mathrm{ }:\mathrm{x}+2\mathrm{y}-\mathrm{z}+1=0\mathrm{ }$ and ${\mathrm{P}}_2\mathrm{ }:2\mathrm{x}-\mathrm{y}+\mathrm{z}-1=0.$ Let M be the locus of the feet of the perpendiculars drawn from the points on L to the plane ${\mathrm{P}}_1$ . Which of the following points lie (s) on M ?

Let $E$ be the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$. For any three distinct points $P,Q$ and $Q^{\text{'}}$ on $E$, let $M\left(P,Q\right)$ be the mid-point of the line segment joining $P$ and $Q$, and $M\left(P,Q^{\text{'}}\right)$ be the mid-point of the line segment joining $P$ and $Q^{\text{'}}$. Then the maximum possible value of the distance between $M\left(P,Q\right)$ and $M\left(P,Q^{\text{'}}\right)$, as $P,Q$ and $Q^{\text{'}}$ vary on $E$, is _____.