In a triangle the sum of two sides is x and the product of the same two sides is y. If ${\mathrm{x}}^2-{\mathrm{c}}^2=\mathrm{y},$ (where c is the third side of the triangle) then the ratio of the inradius to the circumradius of the triangle is

Let $x, y$ and $z$ be positive real numbers. Suppose $x, y$ and $z$ are the lengths of the sides of a triangle opposite to its angles $X, Y$ and $Z$ respectively. If $\tan \frac{X}{2}+\tan \frac{Z}{2}=\frac{2y}{x+y+z}$, then which of the following statements is/are TRUE?

In a triangle $PQR, let \angle PQR={30}^o$ and the sides $PQ \mathrm{a}\mathrm{n}\mathrm{d} QR$ have lengths $10\sqrt{3} \mathrm{a}\mathrm{n}\mathrm{d} 10$ , respectively. Then, which of the following statement(s) is (are) TRUE?

Let \(f:\left[0, \frac{\pi}{2}\right] \rightarrow[0,1]\) be the function defined by \(f(x)=\sin ^2 x\) and let \(g:\left[0, \frac{\pi}{2}\right] \rightarrow[0, \infty)\) be the function defined by \(g(x)=\sqrt{\frac{\pi x}{2}-x^2}\).
The value of \(\frac{16}{\pi^3} \int_0^{\frac{\pi}{2}} f(x) g(x) d x\) is

Let $S$ be the reflection of a point $Q$ with respect to the plane given by
$\overrightarrow{r}=-\left(t+p\right)\hat{i}+t\hat{j}+\left(1+p\right)\hat{k}$
where $t,p$ are real parameters and $\hat{i},\hat{j},\hat{k}$ are the unit vectors along the three positive coordinate axes. If the position vectors of $Q$ and $S$ are $10\hat{i}+15\hat{j}+20\hat{k}$ and $\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}$ respectively, then which of the following is/are TRUE?

Let \(S\) denote the locus of the point of intersection of the pair of lines
\(4 x-3 y=12 \alpha\)
\(4 \alpha x+3 \alpha y=12\)
where \(\alpha\) varies over the set of non-zero real numbers. Let \(T\) be the tangent to \(S\) passing through the points \((p, 0)\) and \((0, q), q>0\), and parallel to the line \(4 x-\frac{3}{\sqrt{2}} y=0\).
Then the value of \(p q\) is

The line passing through the extremity \(A\) of the major axis and extremity \(B\) of the minor axis of the ellipse \(x^2+9 y^2=9\) meets its auxiliary circle at the point \(M\). Then, the area of the triangle with vertices at \(A, M\) and the origin \(O\) is

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

Which of the following values of $\mathrm{\alpha }$ satisfy the equation $\left|\begin{array}{ccc}{\left(1+\mathrm{\alpha }\right)}^2& {\left(1+2\mathrm{\alpha }\right)}^2& {\left(1+3\mathrm{\alpha }\right)}^2\\ {\left(2+\mathrm{\alpha }\right)}^2& {\left(2+2\mathrm{\alpha }\right)}^2& {\left(2+3\mathrm{\alpha }\right)}^2\\ {\left(3+\mathrm{\alpha }\right)}^2& {\left(3+2\mathrm{\alpha }\right)}^2& {\left(3+3\mathrm{\alpha }\right)}^2\end{array}\right|=\mathrm{ }-648\mathrm{ }\mathrm{\alpha }$ ?

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The nuclear charge \((\mathrm{Ze})\) is non-uniformly distributed within a nucleus of radius \(R\). The charge density \(\rho(r)\) [charge per unit volume] is dependent only on the radial distance \(r\) from the centre of the nucleus as shown in figure. The electric field is only along the radial direction.

Question:
The electric field at \(r=R\) is

The \(\beta\)-decay process, discovered around 1900 , is basically the decay of a neutron \((n)\). In the laboratory, a proton \((p)\) and an electron \(\left(e^{-}\right)\)are observed as the decay products of the neutron. Therefore, considering the decay of a neutron as a two-body decay process, it was predicted theoretically that the kinetic energy of the electron should be a constant. But experimentally, it was observed that the electron kinetic energy has continuous spectrum. Considering a three-body decay process, i.e.
\(n \rightarrow p+e^{-}+\bar{v}_{e}\), around 1930, Pauli explained the observed electron energy spectrum. Assuming the anti-neutrino \(\left(\bar{v}_{e}\right)\) to be massless and possessing negligible energy, and the neutron to be at rest, momentum and energy conservation principles are applied. From this calculation, the maximum kinetic energy of the electron is \(0.8 \times 10^{6} \mathrm{eV}\). The kinetic energy carried by the proton is only the recoil energy.

Question:
If the anti-neutrino had a mass of \(3 \mathrm{eV} / \mathrm{c}^{2}\) (where \(\mathrm{c}\) is the speed of light) instead of zero mass, what should be the range of the kinetic energy, \(K\), of the electron?

A bob of mass \(M\) is suspended by a massless string of length \(L\). The horizontal velocity \(v\) at position \(A\) is just sufficient to make it reach the point \(B\). The angle \(\theta\) at which the speed of the bob is half of that at \(A\), satisfies

Two coherent monochromatic point sources $S_1$ and $S_2$ of wavelength $\lambda =600nm$ are placed symmetrically on either side of the center of the circle as shown. The sources are separated by a distance $d=1.8mm.$ This arrangement produces interference fringes visible as alternate bright and dark spots on the circumference of the circle. The angular separation between two consecutive bright spots is $\mathrm{\Delta }\theta$ . Which of the following options is/are correct?