The solution of the differential equation \(\log _e\left(\frac{d y}{d x}\right)=3 x+4 y\) is given by:

A measure of an average yearly growth of an investment over a certain time period when returns are reinvested is

If \(A=(2,3), B=(-1,0), C=(4,6)\) then the area of the parallelogram \(A B C D\) is:

A boat takes 1 hour 30 minutes less to travel 36 km downstream than to travel the same distance upstream.
If the speed of the boat in still water is 10 km/hr, then the speed of the stream is :

The sum of the order and degree of the differential equation having \(y=\left(\sin ^{-1} x\right)^2+A \cos ^{-1} x+B\) where A and B are arbitrary constants, as its general solution is:

Let \(\vec{a}=\hat{i}+\hat{j}, \vec{b}=\hat{i}-\hat{j}\) and \(\vec{c}=\hat{i}+\hat{j}+\hat{k}\). If \(\hat{m}\) is a unit vector perpendicular to both \(\vec{a}\) and \(\vec{b}\), then \(|\vec{c} \cdot \hat{m}|\) is equal to :

\(\int_0^1 x e^x d x\) is equal to

Bacillus thuringiensis is widely used these days as:

In a series LCR circuit, \( C = 80 \, \mu\text{F} \), \( L = 5.0 \, \text{H} \) and \( R = 40 \, \Omega \) are connected to a variable frequency source of 240 V. The angular frequency of the source which drives the circuit at resonance will be:

Given below are two statements :
Statement (I) : In a ground wave communication system, low frequency signal (< 2 MHz) can be transmitted.
Statement (II) : If frequency is high then it will induce electrical signal on the surface of the earth and so attenuation takes place.
In the light of the above statements, choose the most appropriate answer from the options given below :

The properties which depend on the number of solute particles irrespective of their nature relative to the total number of particles in the solution are called colligative properties. These are relative lowering of vapour pressure of the solvent, depression of freezing point of the solvent, elevation of boiling point of the solvent and osmotic pressure of the solution. All of these properties help in calculating the observed molar mass of the solute.
Determination of osmotic pressure is considered best for the determination of molecular mass of the solute.
A 1 litre solution of glucose containing 36 g of it has an osmotic pressure of 4.98 bar at 300 K .
At the same temperature, the osmotic pressure of the solution is 1.52 bar.
The concentration of the solution at 1.52 bar is :

Arrange the following nucleophiles in the decreasing order of their attacking strength :
(A) \(C _6 H _5 COO ^{-}\)
(B) \(CH _3 CH _2 CH _2 COO ^{-}\)
(C) \(CH _3 CH _2 CH _2 CH _2 S^{-}\)
(D) \(CH _3 CH _2 CH _2 CH _2 O ^{-}\)
Choose the correct answer from the options given below :

If A is a square matrix such that \(A^2=A\), then \((I+A)^3-7 A\) is equal to: