The number of distinct solutions of the equation $\frac{5}{4}{\cos }^2\mathrm{ }2\mathrm{x}+{\cos }^4\mathrm{x}+{\sin }^4\mathrm{x}+{\cos }^6\mathrm{x}+{\sin }^6\mathrm{x}=2$ in the interval $[0,\mathrm{ }2\mathrm{\pi }]$ is

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 _________

If \(\lim _{x \rightarrow \infty}\left(\frac{x^{2}+x+1}{x+1}-a x-b\right)=4\), then

Match the statements in Column I with the properties Column II.

Let $O$ be the origin and let $PQR$ be an arbitrary triangle. The point$S$ is such that $\overrightarrow{OP}.\overrightarrow{OQ}+\overrightarrow{OR}.\overrightarrow{OS}=\overrightarrow{OR}.\overrightarrow{OP}+\overrightarrow{OQ}.\overrightarrow{OS}=\overrightarrow{OQ}.\overrightarrow{OR}+\overrightarrow{OP}.\overrightarrow{OS}$ then triangle $PQR$ has $S$ as its

Let $a, b, x$ and $y$ be real numbers such that $a-b=1$ and $y\ne 0$ . If the complex number $z=x+iy$ satisfies $Im\left(\frac{az+b}{z+1}\right)=y,$ then which of the following is (are) possible value (s) of $x?$

Consider three planes
\[
P_1: x-y+z=1, P_2: x+y-z=-1
\]
and \(P_3: x-3 y+3 z=2\)
Let \(L_1, L_2\) and \(L_3\) be the lines of intersection of the planes \(P_2\) and \(P_3, P_3\) and \(P_1, P_1\) and \(P_2\), respectively.
Statement 1 Atleast two of the lines \(L_1, L_2\) and \(L_3\) are non-parallel.
Statement 2 The three planes do not have a common point.

Consider the Bohr's model of a one-electron atom where the electron moves around the nucleus. In the following List-I contains some quantities for the $n^{\mathrm{t}\mathrm{h}}$ orbit of the atom and List-II contains options showing how they depend on $n$ .
 

List - I	List - II
(I) Radius of the $n^{\mathrm{t}\mathrm{h}}$ orbit	$\left(\mathrm{P}\right)$ $\propto n^{-2}$
(II) Angular momentum of the electron in the $n^{th}$ orbit	$\left(\mathrm{Q}\right)$ $\propto n^{-1}$
(III) Kinetic energy of the electron in the $n^{\mathrm{t}\mathrm{h}}$ orbit	$\left(\mathrm{R}\right)$ $\propto n^0$
(IV) Potential energy of the electron in the $n^{\mathrm{t}\mathrm{h}}$ orbit	$\left(\mathrm{S}\right)$ $\propto n^1$
	$\left(\mathrm{T}\right)$ $\propto n^2$
	$\left(\mathrm{U}\right)$ $\propto n^{1/2}$

Which of the following options has the correct combination considering List-I and List-II?

Let $P=\left[\begin{array}{ccc}1& 0& 0\\ 4& 1& 0\\ 16& 4& 1\end{array}\right]$ and $I$ be the identity matrix of order $3$. If $Q=\left[q_{ij}\right]$ is a matrix such that $P^{50}-Q=I,$ then the value of $\frac{q_{31}+q_{32}}{q_{21}}$ is equal to

A cylindrical furnace has height $\left(H\right)$ and diameter $\left(D\right)$ both $1\mathrm{m}$. It is maintained at temperature $360\mathrm{K}$. The air getsheated inside the furnace at constant pressure$P_a$and its temperature becomes $T=360\mathrm{K}$. The hot air with density $\rho$rises up a vertical chimney of diameter $d=0.1\mathrm{m}$ and height $h=9\mathrm{m}$ above the furnace and exits the chimney (seethe figure). As a result, atmospheric air of density${\rho }_a=1.2\mathrm{kg}{\mathrm{m}}^{-3}$,pressure $P_a$and temperature $T_a=300\mathrm{K}$ entersthe furnace. Assume air as an ideal gas, neglect the variations in$\rho$and $T$ inside the chimney and the furnace. Also ignore the viscous effects.
[Given: The acceleration due to gravity$g=10\mathrm{m}{\mathrm{s}}^{-2}$and$\pi =3.14$]


When the chimney is closed using a cap at the top, a pressure difference $∆P$develops between the top and the bottom surfaces of the cap. If the changes in the temperature and density of the hot air, due to the stoppage of air flow, are negligible then the value of $∆P$is _____ $\mathrm{N}{\mathrm{m}}^{-2}$.

An aqueous solution of X is added slowly to an aqueous solution of Y as shown in List I. The variation in conductivity of these reactions is given in List II. Match List I with List II and select the correct using answer the code given below the lists :

List - I	List - II
a. \(\underset{\quad\quad~~ X }{( C _2 H _5)_3 N}+\underset{ Y }{CH _3 COOH}\)	p. Conductivity decreases and then increases
b. \(\underset{\quad\quad~~ X }{KI (0.1 M )}+\underset{ Y }{AgNO _3(0.01 M )}\)	q. Conductivity decreases and then does not change much
c. \(\underset{\text X }{CH _3 COOH}+\underset{\text Y }{KOH}\)	r. Conductivity increases and then does not change much
d. \(\underset{\text X }{NaOH}+\underset{\text Y }{HI}\)	s. Conductivity does not change much and then increases

Choose the reaction(s) from the following options, for which the standard enthalpy of reaction is equal to the standard enthalpy of formation.

A mixture of two salts is used to prepare a solution

The correct option (s) for the salt mixture is(are)