A spherical body of radius $r$ and density $\sigma$ falls freely through a viscous liquid having density $\rho$ and viscosity $\eta$ and attains a terminal velocity $\mathrm{v}_{0}$. Estimated maximum error in the quantity $\eta$ is : (Ignore errors associated with $\sigma, \rho$ and g , gravitational acceleration)

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.

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 :

Let [ • denote the greatest integer function and $f(\mathrm{x})=\lim _{\mathrm{n} \rightarrow \infty} \frac{1}{\mathrm{n}^{3}} \sum_{\mathrm{k}=1}^{\mathrm{n}}\left[\frac{\mathrm{k}^{2}}{3^{\mathrm{x}}}\right]$. Then is equal to

For a triangle $A B C$, let $\vec{p}=\overrightarrow{B C}, \vec{q}=\overrightarrow{C A}$ and . If and , where is the angle between and , then is equal to:

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

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 :$\\$

If the system of equations $\\3 \mathrm{x}+\mathrm{y}+4 \mathrm{z}=3\\$ has no solution, then the value of is equal to:

Consider the above sequence of reactions. The number of bromine atom(s) in the final product (P) will be

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:

For the matrices $\mathrm{A}=\left[\begin{array}{ll}3 & -4 \\ 1 & -1\end{array}\right]$ and , if , then among the following which one is true?

The osmotic pressure of a living cell in 12 atm at 300 K . The strength of sodium chloride solution that is isotonic with the living cell at this temperature is $\_\_\_\_$ $\mathrm{g} \mathrm{L}^{-1}$. (Nearest integer) Given : Assume complete dissociation of NaCl (Given : Molar mass of Na and Cl are 23 and 35.5 respectively.)

Let $\mathrm{A}=\{2,3,5,7,9\}$. Let R be the relation on A defined by x Ry if and only if . Let 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 is equal to:

Let one end of a focal chord of the parabola $y^{2}=16 x$ be (16, 16). If $\mathrm{P}(\alpha, \beta)$ divides this focal chord internally in the ratio , then the minimum value of is equal to :

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 :

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:

A large drum having radius R is spinning around its axis with angular velocity $\omega$, as shown in figure. The minimum value of $\omega$ so that a body of mass M remains stuck to the inner wall of the drum, taking the coefficient of friction between the drum surface and mass M is $\mu$, is :

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 :

Two known resistance of $\mathrm{R} \Omega$ and $2 \mathrm{R} \Omega$ and and one unknown resistance $\mathrm{X} \Omega$ are connected in a circuit as shown in the figure. If the equivalent resistance between points A and B in the circuit is , then the value of X is

Decomposition of A is a first order reaction at $\mathrm{T}(\mathrm{K})$ and is given by $\mathrm{A}(\mathrm{g}) \rightarrow \mathrm{B}(\mathrm{g})+\mathrm{C}(\mathrm{g})$. In a closed 1 L vessel, 1 bar is allowed to decompose at . After 100 minutes, the total pressure was 1.5 bar . What is the rate constant (in ) of the reaction ? (