The shortest distance between the following lines :
\(\vec{r}=(\hat{i}+\hat{j}-\hat{k})+s(2 \hat{i}+\hat{j}+\hat{k})\)
\(\vec{r}=(\hat{i}+\hat{j}+2 \hat{k})+t(4 \hat{i}+2 \hat{j}+2 \hat{k})\), where \(s\) and \(t\) are scalars, is :

The maximum value of \(z=2 x+3 y\) subject to the constraints \(3 x-3 y \geq 0,2 x+2 y \leq 12, x \geq 0, y \geq 0\) occurs at the point:

If an asset costs Rs. 50,000 with an estimated useful life of 6 years and a scrap value of Rs. 5000. Then by using a linear depreciation method, the annual depreciation of the asset will be:

If the matrix \(M=\left[\begin{array}{ccc}0 & -1 & 3 \alpha \\ 1 & \beta & -5 \\ -6 & 5 & 0\end{array}\right]\) is skew-symmetric, then

If a moneylender charges interest at the rate of 10 rupees per 100 rupees per half year, payable in advance, then the effective rate of interest per annum is :

If a set \(P\) contains 5 elements and the set \(Q\) contains 8 elements, then the number of one-one functions from \(P\) to \(Q\) is:

The value of the integral \(\int \frac{d x}{e^x+e^{-x}}\) is equal to:

The compound which will not respond to Fehling's test is

In an AC generator, a coil of 20 turns each with area \( 2 \times 10^{-2} \text{ m}^2 \) is rotated at 50 revolutions per sec in a uniform magnetic field of 2.5 T. The maximum induced emf produced is:

A motorbike costing Rs. 1, 25, 000 has a scrap value of Rs. 25, 000.
If the annual depreciation charge is Rs. 12, 500, then the useful life of the bike is (by using linear method):

Light of frequency \( 8.27 \times 10^{14} \text{ Hz} \) is incident on a metal surface. Electrons with maximum speed \( 6.6 \times 10^5 \text{ m/s} \) are ejected. The threshold frequency for photoemission of electrons will be (given \( h = 6.6 \times 10^{-34} \text{ Js} \), and mass of electron = \( m_e = 9 \times 10^{-31} \text{ kg} \)) :

Read the passage carefully and answer the questions
The osmotic pressure is one of the four colligative properties which depends on the number of particles of solute in the solution, irrespective of their nature. The osmotic pressure is equal to the product of concentration, gas constant and temperature. It is the most commonly used property for the determination of the molar mass of biomolecules. In many situations, the molecules get dissociated or associated when they are dissolved in a solvent, thereby changing the number of molecules in the solution. As a result, an abnormal molecular mass is obtained. The extent of dissociation or association is given by van't Hoff factor (i), which is the ratio of normal molar mass to abnormal molar mass. It is related with the degree of association or ionization. The introduction of van't Hoff factor in the osmotic pressure equation modifies the equation.
How is the osmotic pressure of a concentrated solution related to the osmotic pressure of a dilute solution of the same substance at a given temperature?

In the (i) absence of electric field, and in the (ii) presence of electric field, the paths of electrons between successive collisions with the positive ions of the metal, are