The value of \(\int e^x\left[f(x)+f^{\prime}(x)\right] d x\) is :

If \(\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=\hat{j}-\hat{k}\) and \(\vec{c}=\hat{i}+\hat{j}+\hat{k}\) such that \(\vec{r} \times \vec{b}=\vec{c} \times \vec{b}\) and \(\vec{r} \cdot \vec{a}=0\), then \(\vec{r}\) is :

The inverse of the matrix \(A=\left[\begin{array}{ll}2 & 3 \\ 1 & 2\end{array}\right]\) is:

If the minimum value of the objective function \(Z=a x+b y\) of an LPP occurs at two points \((3,5)\) and \((5,3)\), then

The solution X of the matrix equation AX = B, where A is an invertible matrix, is given by

The general solution of the differential equation \((1+y) d x-2 x d y=0\) is :

If \(A=\left[\begin{array}{cc}2 & 3 \\ -1 & 1\end{array}\right]\) and \(A^2-3 A+k I=0\), then the value of \(k\) is:

Read the passage carefully and answer the Questions
The decomposition of \(N_2O_5\ (g)\) is given as
\(N_2O_5(g) \rightarrow 2NO_2(g) + \frac{1}{2}O_2(g)\)
The above reaction is found to be of first order.
The initial concentration of \(N_2O_5\) was \(1.24 \times 10^{-2}\ M\) at \(318\ K\).
The concentration \(N_2O_5\) after \(60\ minutes\) was \(0.20 \times 10^{-2}\ M\).
The rate constant of the reaction at \(318\ K\) is ______________ (\(log\ 6.2 = 0.792\))

Read the passage carefully and answer the Questions:
Amines are classified as primary \(\left(1^{\circ}\right)\), secondary \(\left(2^{\circ}\right)\) and tertiary \(\left(3^{\circ}\right)\) depending upon the number of hydrogen atoms replaced by alkyl or aryl groups in ammonia molecules. Aliphatic amines are a stronger bases due to \(+I\) effect of an alkyl groups. Intermolecular hydrogen bonding in primary and secondary amines affects the boiling points of amines. The intermolecular association is more in primary amines than in secondary amines as there are more number of hydrogen atoms available for hydrogen bond formation. Nitro compounds are reduced to amines by passing hydrogen gas in the presence of finely divided nickel, palladium or platinum and also by reduction with metals in an acidic medium. Gabriel synthesis is used for the preparation of primary amines. Primary amines react with nitrous acid to give unstable diazonium-salts which can be used to synthesize a variety of compounds.
Identify the amine possessing least basic strength:

Identify the product 'X' in the following reaction:
\(C_2H_5COONa\ \xrightarrow[\text{Heat}]{NaOH+CaO}\ X+ Na_2CO_3\)

Which one of the given assisted reproductive technologies (ART) involves transfer of fertilized egg into the fallopian tube?

The equation of the normal to the curve \(y=2 \sin x\) at \((0,0)\) is:

A gaussian surface encloses three point charges \( q_1 = -14 \text{ nC}, q_2 = 78.85 \text{ nC} \) and \( q_3 = -56 \text{ nC} \). The electric flux for the gaussian surface due to these charges is