The centres of two circles \(C_1\) and \(C_2\) each of unit radius are at a distance of 6 units from each other. Let \(P\) be the mid-point of the line segment joining the centres of \(C_1\) and \(C_2\) and \(C\) be a circle touching circles \(C_1\) and \(C_2\) externally. If a common tangents to \(C_1\) and \(C\) passing through \(P\) is also a common tangent to \(C_2\) and \(C\), then the radius of the circle \(C\) is

Let the straight line \(\mathrm{y}=2 \mathrm{x}\) touch a circle with center \((0, \alpha), \alpha>0\), and radius \(\mathrm{r}\) at a point \(\mathrm{A}_1\). Let \(\mathrm{B}_1\) be the point on the circle such that the line segment \(A_1 B_1\) is a diameter of the circle. Let \(\alpha+r=5+\sqrt{5}\).
Match each entry in List-I to the correct entry in List-II.

The correct option is

The vector(s) which is/are coplanar with vectors \(\hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}}\) and \(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}}\), are perpendicular to the vector \(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}\) is/are

Let \(\omega\) be a complex cube root of unity with \(\omega \neq 1\). A fair die is thrown three times. If \(r_1, r_2\) and \(r_3\) are the numbers obtained on the die, then the probability that \(\omega^{r_1}+\omega^{r_2}+\omega^{r_3}=0\) is

Tangents are drawn to the hyperbola \(\frac{x^{2}}{9}-\frac{y^{2}}{4}=1\), parallel to the straight line \(2 x-y=1\). The points of contact of the tangents on the hyperbola are

The equation of a plane passing through the line of intersection of the planes \(x+2 y+3 z=2\) and \(x-y+z=3\)

and at a distance \(\frac{2}{\sqrt{3}}\) from the point \((3,1,-1)\) is

Let $f :\mathbb{R}\to \mathbb{R}\mathbb{ }$ be a continuous odd function, which vanishes exactly at one point and $f\left(1\right)=\frac{1}{2}.$ Suppose that $F\left(x\right)= \underset{-1}{\overset{x}{\int }}f\left(t\right)dt$ for all $x\in \left[-1, 2\right]$ and $G\left(x\right)= \underset{-1}{\overset{x}{\int }}t\left|f\left(f\left(t\right)\right)\right|dt$ for all $x\in \left[-1, 2\right].$ If $\underset{x\to 1}{\lim }\frac{F\left(x\right)}{G\left(x\right)}=\frac{1}{14},$ then the value of $f\left(\frac{1}{2}\right)$ is

Let \(A B C D\) be a quadrilateral with area 18 , with side \(A B\) parallel to the side \(C D\) and \(A B=2 C D\). Let \(A D\) be perpendicular to \(A B\) and \(C D\). If a circle is drawn inside the quadrilateral \(A B C D\) touching all the sides, then its radius is

Two small particles of equal masses start moving in opposite directions from a point \(A\) in a horizontal circular orbit. Their tangential velocities are \(v\) and \(2 v\) respectively, as shown in the figure. Between collisions, the particles move with constant speeds. After making how many elastic collisions, other than that at \(A\), these two particles will again reach the point \(A\) ?

Paragraph:

Let \(a, r, s, t\) be nonzero real numbers. Let \(P\left(a t^{2}, 2 a t\right), Q, R\left(a r^{2}, 2 a r\right)\) and \(S\left(a s^{2}, 2 a s\right)\) be distinct points on the parabola \(y^{2}=4 a x\). Suppose that \(P Q\) is the focal chord and lines \(Q R\) and \(P K\) are parallel, where \(K\) is the point \((2 a, 0)\).


Question:

If \(s t=1\), then the tangent at \(P\) and the normal at \(S\) to the parabola meet at a point whose ordinate is

The surface of copper gets tarnished by the formation of copper oxide. $N_2$ gas was passed to prevent the oxide formation during heating of copper at 1250 K. However, the $N_2$ gas contains 1 mole % of water vapour as impurity. The water vapour oxidises copper as per the reaction given below:

$2Cu\left(s\right)+ H_{2}O\left(g\right)\to C{u}_{2}O\left(s\right) + H_2\left(g\right)$

$P_{H_2}$ Is the minimum partial pressure of $H_2$ (in bar) needed to prevent the oxidation at 1250 K. The value of $\ln \left(p{H}_2\right)$ is ____.

(Given: total pressure \(=1\) bar, R (universal gas constant) \(=8 J K ^{-1} mol^{-1}, \ln (10) =2.3 \cdot Cu (s)\) and \(Cu _2 O (s)\) are mutually immiscible.

At \(1250 K: 2 Cu ( s )+\frac{1}{2} O _2(g) \rightarrow Cu _2 O ( s ) ;\) \(\Delta G^{\ominus}=-78,000 J mol ^{-1}\)

\(H _2(g)+\frac{1}{2} O _2(g) \rightarrow H _2 O ( g ) ; \quad \Delta^{\ominus}=\)\(-1,78,000 J mol ^{-1} ; G\) is the Gibbs energy)/mi> \(\Theta=\) \(-78,000 J mol -1\)

\(H _2(g)+\frac{1}{2} O _2(g) \rightarrow H _2 O ( g ) ; \quad\)\(\Delta^{\ominus}=-1,78,000 J mol ^{-1} ; G\) is the Gibbs energy)

The figure below shows the variation of specific heat capacity ($C$) of a solid as a function of temperature ($T$). The temperature is increased continuously from $0$ to $500 \mathrm{K}$ at a constant rate. Ignoring any volume change, the following statement (s) is (are) correct to a reasonable approximation.

Let \(A_n=\left(\frac{3}{4}\right)-\left(\frac{3}{4}\right)^2+\left(\frac{3}{4}\right)^3+\ldots+(-1)^{n-1}\)\(\left(\frac{3}{4}\right)^n\) and \(B_n=1-A_n\). Find a least odd natural number \(n_0\), so that \(B_n>A_n, \forall n \geq n_0\).