Let $G$ be a circle of radius $R>0$. Let $G_1,G_2,\dots ,G_n$ be $n$ circles of equal radius $r>0$. Suppose each of the $n$ circles $G_1,G_2,\dots ,G_n$ touches the circle $G$ externally. Also, for $i=1,2,\dots ,n-1$, the circle $G_i$ touches $G_{i+1}$ externally, and $G_n$ touches $G_1$ externally. Then, which of the following statements is/are TRUE?

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Let \(p, q\) be integers and let \(\alpha, \beta\) be the roots of the equation, \(x^{2}-x-1=0\), where \(\alpha \neq \beta\). For \(n=0,1,2, \ldots\), let \(a_{n}=p \alpha^{n}+q \beta^{n}\)
FACT: If \(a\) and \(b\) are rational numbers and \(a+b \sqrt{5}=0\), then \(a=0=b\).

Question:
If \(a_{4}=28\), then \(p+2 q=\)

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

Let $n_1<n_2<n_3<n_4<n_5$ be positive integers such that $n_1+n_2+n_3+n_4+n_5=20$ . Then the number of such distinct arrangements $\left(n_1, n_2, n_3, n_4, n_5\right)$ is _________

Let $\widehat{u}=u_1\widehat{i}+u_2\widehat{j}+u_3\widehat{k}$ be a unit vector in ${\mathbb{R}}^3$ and $\widehat{w}=\frac{1}{\sqrt{6}} \left(\widehat{i}+ \widehat{j}+2\widehat{k}\right).$ Given that there exists a vector $\overrightarrow{v}$ in ${\mathbb{R}}^3$ such that $\left|\widehat{u}\times \overrightarrow{v}\right|=1$ and $\widehat{w} . \left(\widehat{u}\times \overrightarrow{v}\right)=1.$ Which of the following statement(s) is (are) correct ?

For each positive integer $n$, let $y_n={\frac{1}{n}\left((n+1)(n+2)\mathrm{...}(n+n)\right)}^{\frac{1}{n}}$. If $\underset{n\to \infty }{\lim }{y}_n=L,$ then the value of $\left[L\right]$ (where $\left[x\right]$ is the greatest integer less than or equal to $x$) is ____

Let $C_1$ be the circle of radius $1$ with center at the origin. Let $C_2$ be the circle of radius $r$ with center at the point $A=(4,1)$, where $1<r<3$. Two distinct common tangents $PQ$ and $ST$ of $C_1$ and $C_2$ are drawn. The tangent $PQ$ touches $C_1$ at $P$ and $C_2$ at $Q$. The tangent $ST$ touches $C_1$ at $S$ and $C_2$ at $T$. Midpoints of the line segments $PQ$ and $ST$ are joined to form a line which meets the $x$-axis at a point $B$. If $AB=\sqrt{5}$, then the value of $r^2$ is

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Box \(1\) contains three cards bearing numbers \(1,2,3\); box \(2\) contains five cards bearing numbers \(1,2,3,4,5\); and box \(3\) contains seven cards bearing numbers \(1,2,3,4,5,6,7\). A card is drawn from each of the boxes. Let \(x_{i}\) be the number on the card drawn from the \(i^{t h}\) box, \(i=1,2,3\).

Question:
The probability that \(x_{1}, x_{2}, x_{3}\) are in an arithmetic progression, is

If the value of \(n(Y)+n(Z)\) is \(k^2\), then \(|k|\) is
Let \(S=\{1,2,3,4,5,6\}\) and \(X\) be the set of all relations \(R\) from \(S\) to \(S\) that satisfy both the following properties:
i. \(R\) has exactly 6 elements.
ii. For each \((a, b) \in R\), we have \(|a-b| \geq 2\).
Let \(Y=\{R \in X\) : The range of \(R\) has exactly one element \(\}\) and \(Z=\{R \in X: R\) is a function from \(S\) to \(S\}\).
Let \(n(A)\) denote the number of elements in a set \(A\).

In the following reactions, $\mathbf{P},\mathbf{Q},\mathbf{R}$ and $\mathbf{S}$ are the major products.




The correct statement about $\mathbf{P},\mathbf{Q},\mathbf{R}$ and $\mathbf{S}$ is

Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.

Column 1	Column 2	Column 3
(I) $x^2+y^2=a^2$	(i) $my=m^{2}x+a$	(P) $\left(\frac{a}{m^2},\frac{2a}{m}\right)$
(II) $x^2+a^2{y}^2=a^2$	(ii) $y=mx+a \sqrt{m^2+1}$	(Q) $\left(\frac{-ma}{\sqrt{m^2+1}},\frac{a}{\sqrt{m^2+1}}\right)$
(III) $y^2=4ax$	(iii) $y=mx+ \sqrt{a^2{m}^2-1}$	(R) $\left(\frac{-a^{2}m}{\sqrt{a^2{m}^2+1}},\frac{1}{\sqrt{a^2{m}^2+1}}\right)$
(IV) $x^2-a^2{y}^2=a^2$	(iv) $y=mx+\sqrt{a^2{m}^2+1}$	(S) $\left(\frac{-a^{2}m}{\sqrt{a^2{m}^2-1}},\frac{-1}{\sqrt{a^2{m}^2-1}}\right)$

The tangent to a suitable conic (Column 1) at $\left(\sqrt{3},\frac{1}{2}\right)$ is found to be $\sqrt{3}x+2y=4$ , then which of the following options is the only Correct combination?

A cylindrical vessel of height \(500 \mathrm{~mm}\) has an orifice (small hole) at its bottom. The orifice is initially closed and water is filled in it upto height \(H\). Now the top is completely sealed with a cap and the orifice at the bottom is opened. Some water comes out from the orifice and the water level in the vessel becomes steady with height of water column being 200 \(\mathrm{mm}\). Find the fall in height (in \(\mathrm{mm}\) ) of water level due to opening of the orifice.
[Take atmospheric pressure \(=1.0 \times 10^5 \mathrm{Nm}^{-2}\), density of water \(=1000 \mathrm{~kg} \mathrm{~m}^{-3}\) and \(g=10\) \(\mathrm{ms}^{-2}\). Neglect any effect of surface tension.]

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,