In a $∆XYZ$ let $x,y,z$,  be the lengths of sides opposite to the angles, $X, Y, Z$, respectively and $2s=x+y+z.$ If $\frac{s-x}{4}=\frac{s-y}{3}=\frac{s-z}{2}$ and area of incircle of the triangle XYZ is $\frac{8\pi }{3},$ then

Let $f:R\to R$ be defined by $f\left(x\right)=\frac{x^2-3x-6}{x^2+2x+4}$ Then which of the following statements is(are) TRUE?

If $2x-y+1=0$ is a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{16}=1$ , then which of the following CANNOT be sides of a right angled triangle?

In the following \([x]\) denotes the greatest integer less than or equal to \(x\). Match the functions in Column I with the properties Column II.

Let $f\mathbb{ }:\mathbb{ }\mathbb{R}\to \mathbb{R}$ be a differentiable function such that $f\left(0\right)=0, f\left(\frac{\pi }{2}\right)=3$ and $f^{\prime }\left(0\right)=1.$ If $g\left(x\right)={\int }_x^{\frac{\mathrm{\pi }}{2}} \left[f^{\prime }\left(t\right)cosect-\cot t cosect f\left(t\right)\right]dt$ , for $x\in \left(0,\frac{\pi }{2}\right]$ then $\underset{x\to 0}{\lim }g\left(x\right)=$

Let \(\omega \neq 1\) be a cube root of unity and \(S\) be the set of all non-singular matrices of the form \(\left[\begin{array}{ccc}1 & a & b \\ \omega & 1 & c \\ \omega^2 & \omega & 1\end{array}\right]\), where each of \(a, b\) and \(c\) is either \(\omega\) or \(\omega^2\). Then, the number of distinct matrices in the set \(S\) is

A pack contains n cards numbered from 1 to n. Two consecutive numbered cards are removed from the pack and the sum of the numbers on the remaining cards is 1224. If the smaller of the numbers on the removed cards is k, then $\text{k}-20=$

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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:
Among the following, identify the correct statement

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 _________

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The figure shows a circular loop of radius \(a\) with two long parallel wires (numbered \(1\) and \(2\) ) all in the plane of the paper. The distance of each wire from the centre of the loop is \(d\). The loop and the wires are carrying the same current \(I\). The current in the loop is in the counterclockwise direction if seen from above.



Question:

When \(d \approx a\) but wires are not touching the loop, it is found that the net magnetic field on the axis of the loop is zero at a height \(h\) above the loop. In that case

A piece of ice (heat capacity \(=2100 \mathrm{~J} \mathrm{~kg}^{-1}{ }^{\circ} \mathrm{C}^{-1}\) and latent heat \(=3.36 \times 10^5 \mathrm{~J} \mathrm{~kg}^{-1}\) ) of mass \(\mathrm{m}\) gram is at \(-5^{\circ} \mathrm{C}\) at atmospheric pressure. It is given \(420 \mathrm{~J}\) of heat so that the ice starts melting. Finally when the ice-water mixture is in equilibrium, it is found that \(1 \mathrm{~g}\) of ice has melted. Assuming there is no other heat exchange in the process, the value of \(m\) is

Which of the following statement(s) is (are) correct about the spectrum of hydrogen atom?

\(\frac{d^2 x}{d y^2}\) equals