If the matrix \(\begin{bmatrix} 3 & a+b & 4 \\ 2 & 0 & -1 \\ a-b & c & 5 \end{bmatrix}\) is symmetric, then the value of a, b and c is given by:

The inverse of the function \(f: R \rightarrow R\) given by \(f(x)=2 x+7\) is :

The function \(f(x)=x^3, x \in R\), has:

The number of arbitrary constants in the general and particular solution of a differential equation of order 4 are respectively :

Let \(A=\left[a_{i j}\right]\) be a square matrix of order 2 with elements either 0 or 1 . Then the difference between the possible number of singular and non-singular matrices is

Match List-I with List-II

List-I	List-II
(A) The equation of the line passing through the points \((-1,0,2)\) and \((3,4,6)\) is	(I) \(\vec{r}=(-\hat{i}+2 \hat{k})+\lambda(4 \hat{i}+4 \hat{j}+4 \hat{k})\)
(B) The vector equation of the line \(\frac{x-1}{2}=\frac{y+1}{3}=\frac{4-z}{5}\) is	(II) \(\vec{r}=(\hat{i}+3 \hat{j}+2 \hat{k})+\lambda(2 \hat{i}+\hat{j}+3 \hat{k})\)
(C) The equation of the line passing through \((1,3,2)\) and parallel to \(\frac{x+1}{2}=\frac{y-1}{1}=\frac{z+1}{3}\) is	(III) \(\vec{r}=(-\hat{i}+2 \hat{j}+3 \hat{k})+\lambda(2 \hat{i}+\hat{j}+4 \hat{k})\)
(D) The equation of the line along \(2 \hat{i}+\hat{j}+4 \hat{k}\) and passing through \((-1,2,3)\) is	(IV) \(\vec{r}=(\hat{i}-\hat{j}+4 \hat{k})+\lambda(2 \hat{i}+3 \hat{j}-5 \hat{k})\)

where \(\lambda\) is an arbitrary constant.
Choose the correct answer from the options given below:

The value of the integral \(I=\int_{-1}^2|x| d x\) is :

A random variable X has the following probability distribution

X	2	3	4	5
P(X)	5/k	7/k	9/k	11/k

Then the value of \(\frac{k}{4}\) is

A nucleoside comprises:

Match List-I with List-II

List-I ( Matrix A )	List-II ( Determinant of adj A )
(A) \(\left[\begin{array}{cc}2 & 1 \\0 & -1\end{array}\right]\)	(I) 6
(B) \(\left[\begin{array}{cc}0 & 1 \\4 & -1\end{array}\right]\)	(II) 5
(C) \(\left[\begin{array}{cc}1 & 2 \\-3 & -1\end{array}\right]\)	(III) \(-4\)
(D) \(\left[\begin{array}{cc}4 & -2 \\3 & 0\end{array}\right]\)	(IV) \(-2\)

Choose the correct answer from the options given below :

Read the passage carefully and answer the questions
The resistance ( \(R\) ) of an object is given by the expression \(R=\rho \frac{l}{A} .\)
The inverse of \(R\) is called conductance and the inverse of \(\rho\) is called conductivity or specific conductance represented by \(\kappa\).
The conductivity depends upon the nature of the material (insulator, semiconductor or conductor) and the temperature of the substance.
Similarly, the conductance due to the ions present in the solution (ionic conductance) depends upon the nature of the electrolyte, size of ions, the nature of the solvent and the temperature.
Molar conductivity ( \(\Lambda_m\) ) of a solution at a given concentration is the conductance of volume \(V\) containing one mole of electrolyte kept between two electrodes of area \(A\) and unit distance apart.
It is given by the expression \(\Lambda_m=\kappa V .\)
Molar conductivity increases with decrease in concentration and when concentration approaches zero, it is called limiting molar conductivity.
The decrease is linear for a strong electrolyte. However, for a weak electrolyte the change is gradual at higher concentrations and sharp near zero concentration.
Using limiting molar conductivity values of strong electrolytes we can calculate the limiting molar conductivity of weak electrolytes using Kohlrausch law of independent migration of ions.
From molar conductivity and limiting molar conductivity we can determine the degree of dissociation \(\left(\frac{\Lambda_m}{\Lambda_m^0}\right)\) and the ionization constant of a weak acid.
Select the correct statement.

If \(x=a\left(\cos t+\log \tan \frac{t}{2}\right), y=a \sin t\), then value of \(\frac{d y}{d x}\) at \(t=\pi / 4\) is

Electrolysis of molten \(NaCl\) gives :