The function \(f(x)=x^x\) has a critical point at :

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

If \(I=\int \frac{x^4+x^2+1}{x^2-x+1} d x=a x+\beta x^2+\gamma x^3+\delta\), where \(\delta\) is the constant of integration, then \((a+2 \beta+3 \gamma)\) equals

Which of the following can be an objective function for a Linear Programming problems in two variables x and y?

If the function \(f(x)=x^2\) is one-to-one and onto, then the domain D and range R of \(f(x)\) are respectively :

If \(y=\log \left(x+\sqrt{x^2+a^2}\right)\) then \(\frac{d y}{d x}\) =

Let a pair of dice be thrown and the random variable X be the sum of numbers on the two dice. Then, Match List I with List II

LIST - I	LIST - II
A. \(P(X=2)\)	I. \(4 / 36\)
B. \(P(X=4)\)	II. \(5 / 36\)
C. \(P(X=5)\)	III. \(1 / 36\)
D. \(P(X=6)\)	Iv. \(3 / 36\)

Choose the correct answer from the options given below

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.
Which one of the following has minimum osmotic pressure at 298 K ?

The variance of the number of heads in two tosses of a coin is :

Arrange the metal ions of the following compounds in increasing order of oxidation number:
(A) \(TiCl _3\)
(B) \(CrF _6\)
(C) ZnO
(D) \(Mn _2 O _7\)
Choose the correct answer from the options given below:

The random variable X has a probability distribution \(P(X=r)=\left\{\begin{array}{ll}r k, & \text { if } r \leq 2, \\ (r-1) k, & \text { if } 2 < r \leq 4, \text { where } r \in N \cup\{0\} \text { and } k \in R . \\ 0, & \text { otherwise },\end{array}\right.\), where \(n\) is the set of natural numbers
(A) \(k=\frac{1}{9}\)
(B) \(P(2 \leq X \leq 3)=\frac{1}{2}\)
(C) \(P(X=4)=\frac{1}{3}\)
(D) \(P(X>1)=\frac{7}{8}\)
Choose the correct answer from the options given below:

Which of the following is NOT correct in the case of the output obtained from a full wave rectifier?

A machine costing ₹ \(36000\) has an effective life of 5 years with scrap value of ₹ \( 5000\) following a linear method of depreciation. Which are correct?
(A) The value of the machine after 1 year is ₹ \(31000\)
(B) The value of the machine after 2 years is ₹ \(23600\)
(C) The value of the machine after 3 years is ₹ \(18400\)
(D) The value of the machine after 4 years is ₹ \(11200\)
Choose the correct answer from the options given below :