Integrating factor of \(\frac{dy}{dx} + y \cot x = 2x + x^2 \cot x\) (\(x \neq 0\)) is:

The supply function of a commodity is \(P=x^3+2 x+18\). When 4 units of the commodity are sold, then producer surplus is:

Let \(A\) be any square matrix of order 3 and \(B=\left[\begin{array}{ccc}0 & -4 & 2 \\ 4 & 0 & 3 \\ -2 & -3 & 0\end{array}\right]\).Then the matrix \(A B A^T\) is a:

If A is an invertible matrix of order 2 ; then \(\operatorname{det}\left(A^{-1}\right)\) is equal to:

The curve \(y = f ( x )\) is normal probability curve, then which of the following statements are correct?
(A) mean, median and mode of the distribution coincide.
(B) the area bounded by the curve \(y = f ( x )\) and \(x\)-axis is one unit.
(C) The curve is symmetrical about the line \(x =\mu\), where \(\mu\) is the mean.
(D) y -axis is an asymptote to the curve.
Choose the correct answer from the options given below :

If \(\int_0^a 3 x^2 d x=8\), then the value of \(a\) is:

For the objective function \(Z = 3x + 5y\) subject to constraints \(x + 3y ≥ 3, x + y ≥ 2, x ≥ 0, y ≥ 0\):

Consider the LPP, \(\operatorname{Min} Z=2 x-3 y\) subject to the conditions
\(x+y \leq 3\),
\(x \leq 3\),
\(x \geq 0, y \geq 0\)
then the minimum value of the objective function is at the point.

Match List - I with List - II

List - I (Laws/properties)	List - II (Statements)
(A) Colligative properties	(I) for a solution of volatile liquids, the partial vapour pressure of each component of the solution is directly proportional to its mole fraction present in the solution.
(B) Henry's law	(II) the total pressure over the solution phase in the container will be the sum of the partial pressures of all the components of the solution.
(C) Raoult's law	(III) depends on the number of solute particles, irrespective of their nature.
(D) Dalton's law of partial pressures	(IV) at a constant temperature, the solubility of a gas in a liquid is directly proportional to the partial pressure of the gas present above the surface of a liquid or solution.

Choose the correct answer from the options given below:

Read the passage carefully and answer the questions
The speed at which a chemical reaction takes place is called the rate of reaction. The rate of reaction depends on various factors like concentration of the reactants, temperature, etc. The relation between the rate of reaction and the concentration of reacting species is represented by the equation r = k[A]^x[B]^y, where x and y are the order of the reaction with respect to the reactants A and B, respectively. The overall order of the reaction is x + y. The rate of reaction can also be increased by the use of a catalyst which provides an alternate pathway of lower activation energy. It increases the rate of forward and backward reaction to an equal extent. It does not alter the Gibbs energy of the reaction.
If the rate of reaction becomes twenty-seven times upon increasing the concentration of reactant by three times, the order of this reaction is

Eight charged water droplets, each of radius 1 mm and carrying a charge \( 5 \times 10^{-10} \text{ C} \), coalesce to form a single drop. The potential of the bigger drop is.

Which one of the given sacred groves is not found in Madhya Pradesh?

The molality of solution obtained by dissolving 1.5 g of ethanoic acid \(CH _3 COOH\) in \(2 5 ~ g\) of benzene is :