Let $\mathrm{A}=\{2,3,5,7,9\}$. Let R be the relation on A defined by x Ry if and only if $2 \mathrm{x} \leq 3 \mathrm{y}$. Let $\ell$ be the number of elements in R , and m be the minimum number of elements required to be added in R to make it a symmetric relation. Then $\ell+\mathrm{m}$ is equal to:

If $\int_{0}^{1} 4 \cot ^{-1}\left(1-2 \mathrm{x}+4 \mathrm{x}^{2}\right) \mathrm{dx}=\mathrm{a} \tan ^{-1}(2)-\operatorname{blog}_{\mathrm{e}}(5)$, where , then is equal to .

A spherical body of radius $r$ and density $\sigma$ falls freely through a viscous liquid having density $\rho$ and viscosity and attains a terminal velocity . Estimated maximum error in the quantity is : (Ignore errors associated with and g , gravitational acceleration)

Let the line $L_{1}$ be parallel to the vector $-3 \hat{i}+2 \hat{j}+4 \hat{k}$ and pass through the point and the line be parallel to the vector and pass through the point . If the line is parallel to the vector and intersects the lines and at the points C and D , respectively, then is equal to :

As shown in the diagram, when the incident ray is parallel to base of the prism, the emergent ray grazes along the second surface.$\\$ $\\$If refractive index of the material of prism is $\sqrt{2}$, the angle of prism is.

The correct order of the rate of the reaction for the following reaction with respect to nucleophiles is: $\\ \mathrm{CH}_{3} \mathrm{Br}+\mathrm{Nu}^{\Theta} \longrightarrow \mathrm{CH}_{3} \mathrm{Nu}+\mathrm{Br}^{\Theta}$

If\\ $\left(\frac{1}{{ }^{15} \mathrm{C}_{0}}+\frac{1}{{ }^{15} \mathrm{C}_{1}}\right)\left(\frac{1}{{ }^{15} \mathrm{C}_{1}}+\frac{1}{{ }^{15} \mathrm{C}_{2}}\right) \ldots\left(\frac{1}{{ }^{15} \mathrm{C}_{12}}+\frac{1}{{ }^{15} \mathrm{C}_{13}}\right) =\frac{\alpha^{13}}{{ }^{14} \mathrm{C}_{0}{ }^{14} \mathrm{C}_{1} \ldots{ }^{14} \mathrm{C}_{12}}$, then is equal to

Match List-I with List-II.

List-I	List-II
Pair of Compounds	Type of Isomers
A. 2-Methylpropene and but-1-ene	I. Stereoisomers
B. Cis-but-2-ene and trans-but-2-ene	II. Position isomers
C. 2-Butanol and diethyl ether	III. Chain isomers
D. But-1-ene and but-2-ene	IV. Functional group isomers

Given below are two statements : Statement-I: Compound (X), shown below , dissolves in $\mathrm{NaHCO}_{3}$ solution and has two chiral Statement - II : Compound (Y), shown below, has two carbons with hybridization, one carbon with and one carbon with sp hybridization In the light of the above statements, choose the correct answer from the options given below :

The first and second ionization constants of $\mathrm{H}_{2} \mathrm{X}$ are $2.5 \times 10^{-8}$ and respectively. The concentration of in solution is . (Nearest Integer)

Given below are two statements: Statement I: Crystal Field Stabilization Energy (CFSE) of $\left[\mathrm{Cr}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{2+}$ is greater than that of Statement II: Potassium ferricyanide has a greater spin-only magnetic moment than sodium Ferrocyanide. In the light of the above statements, choose the correct answer from the options given below:

The charge stored by the capacitor C in the given circuit in the steady state is $\_\_\_\_$ $\mu \mathrm{C}.\\$

A random variable $X$ takes values $0,1,2,3$ with probabilities $\frac{2 \mathrm{a}+1}{30}, \frac{8 \mathrm{a}-1}{30}, \frac{4 \mathrm{a}+1}{30}$, b respectively, where . Let and respectively be the mean and standard deviation of such that Then is equal to :

Let $\alpha$ and $\beta$ be the roots of equation $x^{2}+2 a x+(3 a+10) =0$ such that . Then the set of all possible values of is :

An infinitely long straight wire carrying current I is bent in a planer shape as shown in the diagram. The radius of the circular part is r . The magnetic field at the centre O of the circular loop is :$\\$

Consider the following data: $\\ \Delta_{\mathrm{f}} \mathrm{H}^{\Theta}($ methane, g Enthalpy of sublimation of graphite Dissociation enthalpy of The bond enthalpy of bond is given by:

Let $\mathrm{y}^{2}=12 \mathrm{x}$ be the parabola with its vertex at O . Let P be a point on the parabola and A be a point on the x -axis such that $\angle \mathrm{OPA}=90^{\circ}$. Then the locus of the centroid of such triangles OPA is :

The largest $\mathrm{n} \in \mathrm{N}$, for which $7^{\mathrm{n}}$ divides 101!, is :

Consider the following spectral lines for atomic hydrogen: $\\$A. First line of Paschen series $\\$B. Second line of Balmer series $\\$C. Third line of Paschen series $\\$D. Fourth line of Bracket sereisThe correct arrangement of the above lines in ascending order of energy is:

The energy of an electron in an orbit of the Bohr's atom is $-0.04 \mathrm{E}_{0} \mathrm{eV}$ where $\mathrm{E}_{0}$ is the ground state energy. If L is the angular momentum of the electron in this orbit and h is the Planck's constant, then is :