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The circle \(x^2+y^2-8 x=0\) and hyperbola \(\frac{x^2}{9}-\frac{y^2}{4}=1\) intersect at the points \(A\) and \(B\).Question:
Equation of the circle with \(A B\) as its diameter is

Let \(\mathbf{a}, \mathbf{b}, \mathbf{c}\) be unit vectors such that \(\mathbf{a}+\mathbf{b}+\mathbf{c}=\mathbf{0}\). Which one of the following is correct?

Let $f:\left[a, b\right]\to \left[1, \infty \right)$ be a continuous function and let $g :R\to R$ be defined as
$g\left(x\right)= \left\{\begin{array}{ccc}0& if& x<a,\\ \underset{a}{\overset{x}{\int }}f\left(t\right)dt& if& a \le x \le b,\\ \underset{a}{\overset{b}{\int }}f\left(t\right)dt& if& x>b\end{array}\right.$ . Then

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Tangents are drawn from the point \(P(3,4)\) to the ellipse \(\frac{x^2}{9}+\frac{y^2}{4}=1\) touching the ellipse at points \(A\) and \(B\).Question:
The coordinates of \(A\) and \(B\) are

Let \(g(x)=\frac{(x-1)^n}{\log \cos ^m(x-1)} ; 0 < x < 2, m\) and \(n\) are integers, \(m \neq 0, n>0\) and let \(p\) be the left hand derivative of \(|x-1|\) at \(x=1\). If \(\lim _{x \rightarrow 1^{+}} g(x)=p\), then

If \(\alpha=\int_{\frac{1}{2}}^2 \frac{\tan ^{-1} x}{2 x^2-3 x+2} d x\)
then the value of \(\sqrt{7} \tan \left(\frac{2 \alpha \sqrt{7}}{\pi}\right)\) is ____________.
(Here, the inverse trigonometric function \(\tan ^{-1} x\) assumes values in \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\).)

Let $\alpha ,\beta \in R$ be such that $\underset{x\to 0}{\lim }\frac{x^2\sin \left(\beta x\right)}{\alpha x-\sin x}=1.$ Then $6\left(\alpha +\beta \right)$ equals

A light source, which emits two wavelengths ${\mathrm{\lambda }}_1=400\mathrm{ }\mathrm{n}\mathrm{m}$ and ${\mathrm{\lambda }}_2=600\mathrm{ }\mathrm{n}\mathrm{m}$ , is used in a Young's double slit experiment. If recorded fringe widths for ${\mathrm{\lambda }}_1$ and ${\mathrm{\lambda }}_2$ are ${\mathrm{\beta }}_1$ and ${\mathrm{\beta }}_2$ and the number of fringes for them within a distance y on one side of the central maximum are ${\mathrm{m}}_1$ and ${\mathrm{m}}_2$ , respectively, then

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?

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Redox reaction play a pivotal role in chemistry and biology. The values of standard redox potential \(\left(E^{\circ}\right)\) of two half-cell reactions decide which way the reaction is expected to proceed. A simple example is a Daniel cell in which zinc goes into solution and copper gets deposited. Given below are a set of half-cell reactions (acidic medium) along with their \(E^{\circ}(V\) with respect to normal hydrogen electrode) values. Using this data obtain the correct explanations to Questions 14-19.
\(\mathrm{I}_2+2 e^{-} \longrightarrow 2 \mathrm{I}^{-} E^{\circ}=0.54 \)
\( \mathrm{Cl}_2+2 e^{-} \longrightarrow 2 \mathrm{Cl}^{-} E^{\circ}=1.36 \)
\( \mathrm{Mn}^{3+}+e^{-} \longrightarrow \mathrm{Mn}^{2+} E^{\circ}=1.50 \)
\( \mathrm{Fe}^{3+}+e^{-} \longrightarrow \mathrm{Fe}^{2+} E^{\circ}=0.77 \)
\( \mathrm{O}_2+4 \mathrm{H}^{+}+4 e^{-} \longrightarrow 2 \mathrm{H}_2 \mathrm{O} E^{\circ}=1.23\)
Question :
Sodium fusion extract, obtained from aniline, on treatment with iron (II) sulphate and \(\mathrm{H}_2 \mathrm{SO}_4\) in presence of air gives a prussian blue precipitate. The blue colour is due to the formation of

Statement I The total translational kinetic energy of all the molecules of a given mass of an ideal gas is \(1.5\) times the product of its pressure and its volume.
Statement II The molecules of a gas collide with each other and the velocities of the molecules change due to the collision.

Match List I with List II and select the correct answer using the code given below the lists :
 

	List - I		List - II
A.	${\left(\frac{1}{{\text{y}}^2}\left(\frac{\text{cos}\left({\text{tan}}^{-1}\text{y}\right)+\text{y sin}\left({\text{tan}}^{-1}\text{y}\right)}{\text{cot}\left({\text{sin}}^{-1}\text{y}\right)+\text{tan}\left({\text{sin}}^{-1}\text{y}\right)}\right)+{\text{y}}^4\right)}^{1/2}takes\; value$	P.	$\frac{1}{2}\sqrt{\frac{5}{3}}$
B.	​If cosx + cosy + cosz = 0 = sinx + siny + sinz then possible value of cos $\frac{\text{x}-\text{y}}{2}is$	Q.	$\sqrt{2}$
C.	If cos $\left(\frac{\pi }{4}-\text{x}\right)$ cos2x + sinx sin 2x secx = cosx sin2x secx + cos $\left(\frac{\pi }{4}+\text{x}\right)$ cos2x then possible value of secx is	R.	$\frac{1}{2}$
D.	If cot $\left({\text{sin}}^{-1}\sqrt{1-{\text{x}}^2}\right)=\text{sin}\left({\text{tan}}^{-1}\left(\text{x}\sqrt{6}\right)\right)\text{,}x\ne 0\text{,}$  then possible value of x is	S.	1

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_____.