If \(f(x) = \begin{vmatrix} 0 & x-a & x-b \\ x+a & 0 & x-c \\ x+b & x+c & 0 \end{vmatrix}\), then \(f(0)\) is:

Domain of function \(f(x)=\cos ^{-1} \sqrt{2 x-1}\) is:

The solution of the differentiable equation \(2 x \frac{d y}{d x}+y=14 x^3, x>0\), is:

Match List-I with List-II
If \(P(A)=\frac{3}{7}, P(B)=\frac{4}{7}\) and \(P(A \cup B)=\frac{5}{7}\)

List-I	List-II
(A) \(P(A \cap B)\)	(I) \(\frac{2}{3}\)
(B) \(P(A \mid B)\)	(II) \(\frac{5}{7}\)
(C) \(P(B \mid A)\)	(III) \(\frac{1}{2}\)
(D) \(P\left(A^{\prime} \cup B^{\prime}\right)\)	(IV) \(\frac{2}{7}\)

Choose the correct answer from the options given below :

A \(95 \%\) confidence interval for a population mean was reported to be 152 to 160 , if sample standard deviation \(\sigma=15\), then sample size used in this study is (Given \(Z_{0.025}=1.96\) )

The area of triangle with vertices \(P, Q, R\) is given by (where \(\overrightarrow{A B}=\) position vector of point \(B-\) position vector of point \(A\) )

The second order derivative of \(x^3\) log x is:

If a matrix \(A=\left[\begin{array}{cc}5 & -8 \\ -3 & 5\end{array}\right]\) then which of the following is / are TRUE?
(A) |A| = 1
(B) A is a singular matrix.
(C) \(-2 A=\left[\begin{array}{cc}10 & -16 \\ -6 & 10\end{array}\right]\)
(D) \(A I=\left[\begin{array}{ll}5 & 0 \\ 0 & 5\end{array}\right] I\) is an identity matrix of order 2.
Choose the correct answer from the options given below:

The speed of propagation of light wave in a dispersive medium does NOT depend on the :

Read the passage carefully and answer the questions
Biomolecules are the organic compounds present as an essential constituents of living organisms in different cells. molecules are generally termed as macromolecules, which are carbohydrates, enzymes, nucleic acids, amino acids and proteins The etc. Carbohydrates are classified as sugars and non-sugars include sucrose, glucose, fructose, starch and cellulose. Enzymes are coordinated with various chemical reactions in nucleic acid. Nucleic acids are found in living organisms made up of proteins and atural polymers. Two types that are generally found in biological systems are DNA and RNA nucleic acid. Amino acids bifunctional groups of carboxyl and amino groups. Proteins are a class of biologically important compounds. They are crucial are to virtually all processes in living systems. Some of them are hormones, which serve as chemical messengers that coordinate certain biochemical activities.
The zwitterionic form of amino acid is

A thick long straight wire of radius 'a' is carrying steady current I. The current I is uniformly distributed across the cross-section. The ratio of magnetic fields at a distance a/2 and 3a/2 from the central axis of the wire is

The angle of minimum deviation if the refracting angle of a prism is \(2θ\), and the refractive index of the material of the prism is \(\cot θ\) is:

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.
What is the value of the van't Hoff factor for\(NaCl , K _2 SO _4\) and KCl,respectively?