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 $x\in R$ and let $P=\left[\begin{array}{ccc}1& 1& 1\\ 0& 2& 2\\ 0& 0& 3\end{array}\right], Q=\left[\begin{array}{ccc}2& x& x\\ 0& 4& 0\\ x& x& 6\end{array}\right]$ and $R=PQ{P}^{-1}.$
Then which of the following options is/are correct?

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

Let the set of all relations \(R\) on the set \(\{a, b, c, d, e, f\}\), such that \(R\) is reflexive and symmetric, and \(R\) contains exactly 10 elements, be denoted by \(S\). Then the number of elements in \(S\) is ______

The following integral $\underset{\frac{\mathrm{\pi }}{4}}{\overset{\frac{\mathrm{\pi }}{2}}{\int }}{\left(2\mathrm{ }\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{ }\mathrm{x}\right)}^{17}\mathrm{ }\mathrm{d}\mathrm{x}$ is equal to

Let \(P\left(x_1, y_1\right)\) and \(Q\left(x_2, y_2\right)\) be two distinct points on the ellipse
\(\frac{x^2}{9}+\frac{y^2}{4}=1\)
such that \(y_1>0\), and \(y_2>0\). Let C denote the circle \(x^2+y^2=9\), and \(M\) be the point \((3,0)\).
Suppose the line \(x=x_1\) intersects \(C\) at \(R\), and the line \(x=x_2\) intersects \(C\) at \(S\), such that the \(y\)-coordinates of \(R\) and \(S\) are positive. Let \(\angle R O M=\frac{\pi}{6}\) and \(\angle S O M=\frac{\pi}{3}\), where \(O\) denotes the origin \((0,0)\). Let \(|X Y|\) denote the length of the line segment \(X Y\).
Then which of the following statements is (are) TRUE?

If \(n(X)={ }^m C_6\), then the value of \(m\) is

Paragraph:
The reaction of \(\mathrm{K}_{3}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]\) with freshly prepared \(\mathrm{FeSO}_{4}\) solution produces a dark blue precipitate called Turnbull's blue. Reaction of \(\mathrm{K}_{4}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]\) with the \(\mathrm{FeSO}_{4}\) solution in complete absence of air produces a white precipitate \(\mathbf{X}\), which turns blue in air. Mixing the \(\mathrm{FeSO}_{4}\) solution with \(\mathrm{NaNO}_{3}\), followed by a slow addition of concentrated \(\mathrm{H}_{2} \mathrm{SO}_{4}\) through the side of the test tube produces a brown ring.
Question:
Among the following, the brown ring is due to the formation of

A resistance of \(2 \Omega\) is connected across one gap of a meter-bridge (the length of the wire is \(100 \mathrm{~cm}\) ) and an unknown resistance, greater than \(2 \Omega\), is connected across the other gap. When these resistances are interchanged, the balance point shifts by \(20 \mathrm{~cm}\). Neglecting any corrections, the unknown resistance is

Which of the following statement(s) is(are) true?

A particle of mass M and positive charge Q, moving with a constant velocity ${\overrightarrow{\text{u}}}_1=4\hat{\text{i}}{\text{ms}}^{-1}$, enters a region of uniform static magnetic field normal to the x-y plane. The region of the magnetic field extends from x = 0 to x = L for all values of y. After passing through this region, the particle emerges on the other side after 10 milliseconds with a velocity ${\overrightarrow{\text{u}}}_2=2\left(\sqrt{3}\hat{\text{i}}+\hat{\text{j}}\right){\text{ms}}^{-1}$. The correct statement (s) is (are)

Consider a star of mass \(m_2 \mathrm{~kg}\) revolving in a circular orbit around another star of mass \(m_1 \mathrm{~kg}\) with \(m_1 \gg m_2\). The heavier star slowly acquires mass from the lighter star at a constant rate of \(\gamma \mathrm{kg} / \mathrm{s}\). In this transfer process, there is no other loss of mass. If the separation between the centers of the stars is \(r\), then its relative rate of change \(\frac{1}{r} \frac{\mathrm{~d} r}{\mathrm{~d} t}\left(\right.\) in s \(\left.^{-1}\right)\) is given by:
[Note: The original question was awarded bonus marks in the JEE Advanced 2025 exam. We have changed the options to make the question valid and solvable.]

Not considering the electronic spin, the degeneracy of the second excited state (n = $3$) of H atom is $9$, while the degeneracy of the second excited state of ${\mathrm{H}}^{-}$ is :