The circle passing through the point \((-1,0)\) and touching the \(Y\)-axis at \((0,2)\), also passes through the point

Let \(a_{n}\) denote the number of all \(n\)-digit positive integers formed by the digits 0,1 or both such that no consecutive digits in them are 0 . Let \(b_{n}=\) the number of such \(n\)-digit integers ending with digit 1 and \(c_{n}=\) the number of such \(n\)-digit integers ending with digit 0 .

Question:
The value of \(b_{6}\) is

For $a\in R, \left|a\right|>1,$ let $\underset{n\to \infty }{\lim }\left(\frac{1+\sqrt[3]{2}+\dots +\sqrt[3]{n}}{n^{7/3}\left(\frac{1}{{\left(an+1\right)}^2}+\frac{1}{{\left(an+2\right)}^2}+\dots +\frac{1}{{\left(an+n\right)}^2}\right)}\right)=54.$ Then the possible value(s) of $a$ is/are:

The maximum value of the function \(f(x)=2 x^3-15 x^2+36 x-48\) on the set \(A=\left\{x \mid x^2+20 \leq 9 x\right\}\) 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) $\left(\frac{-a^{2}m}{\sqrt{a^2{m}^2-1}},\frac{-1}{\sqrt{a^2{m}^2-1}}\right)$

For $a=\sqrt{2}$ , if a tangent is drawn to a suitable conic (Column 1) at the point of contact $\left(-1, 1\right)$ , then which of the following options is the only Correct combination for obtaining its equation?

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A fair die is tossed repeatedly until a six is obtained. Let \(X\) denotes the number of tosses required.Question:
The probability that \(X=3\) equals

Let $a, b\in \mathbb{R}$ and $f\mathbb{ }:\mathbb{ }\mathbb{R}\to \mathbb{R}$ be defined by $f\left(x\right)=a\cos \left(\left|x^3-x\right|\right)+b\left|x\right|\sin \left(\left|x^3+x\right|\right)$ . Then $f$ is -

The mass \(M\) shown in the figure oscillates in simple harmonic motion with amplitude \(A\). The amplitude of the point \(P\) is

In a study about a pandemic, data of $900$ persons was collected. It was found that $190$ persons had symptom of fever, $220$ persons had symptom of cough, $220$ persons had symptom of breathing problem, $330$ persons had symptom of fever or cough or both, $350$ persons had symptom of cough or breathing problem or both, $340$ persons had symptom of fever or breathing problem or both, $30$ persons had all three symptoms (fever, cough and breathing problem). If a person is chosen randomly from these $900$ persons, then the probability that the person has at most one symptom is ______.

Using the data provided, calculate the multiple bond energy \(\left(\mathrm{kJ} \mathrm{mol}^{-1}\right)\) of a \(\mathrm{C} \equiv \mathrm{C}\) bond in \(\mathrm{C}_{2} \mathrm{H}_{2}\). That energy is (take the bond energy of a \(\mathrm{C}-\mathrm{H}\) bond as \(350 \mathrm{~kJ} \mathrm{~mol}^{-1}\) ) \(2 \mathrm{C}(\mathrm{s})+\mathrm{H}_{2}(\mathrm{~g}) \longrightarrow \mathrm{HC}=\mathrm{CH}(\mathrm{g}) ; \Delta \mathrm{H}=\) \(225 \mathrm{~kJ} \mathrm{~mol}^{-1}\) \(2 \mathrm{C}(\mathrm{s}) \longrightarrow 2 \mathrm{C}(\mathrm{g}) ; \Delta \mathrm{H}=1410 \mathrm{~kJ} \mathrm{~mol}^{-1}\) \(\mathrm{H}_{2}(\mathrm{~g}) \longrightarrow 2 \mathrm{H}(\mathrm{g}) ; \Delta \mathrm{H}=330 \mathrm{~kJ} \mathrm{~mol}^{-1}\)

Four point charges, each of \(+q\), are rigidly fixed at the four corners of a square planar soap film of side \(a\). The surface tension of the soap film is \(\gamma\). The system of charges and planar film are in equilibrium, and \(a=k\left[\frac{q^2}{\gamma}\right]^{1 / N}\), where \(k\) is a constant. Then, \(N\) is

The correct stability order for the following species is

Two identical uniform discs roll without slipping on two different surfaces AB and CD (see figure) starting at A and C with linear speeds \(v_1\) and \(v_2\), respectively, and always remain in contact with the surfaces. If they reach B and D with the same linear speed and \(v_1=3 m / s\), then \(v_2\) in \(m / s\) is \(\left( g =10 \frac{m}{ s ^2}\right)\)