For some constant 'k', if the system of linear equations
\(2 x-y+3 z=1\)
\(x-2 y+z=3\)
\(k x+y-z=0\)
has a unique solution, then

The maximum value of a LPP \(z=3 x+4 y\) subject to the constraints: \(x+y \leq 6, x \geq 0, y \geq 0\) is:

If \(\cos^2 y + \sin(xy) = \lambda\), then \(\frac{dy}{dx} =\)

Which of the following terms are associated with a linear programming problem?
(A) Constraints
(B) Independent events
(C) Feasible region
(D) Objective function
Choose the correct answer from the options given below :

If the matrix \(\left[\begin{array}{ll}1 & 2 \\ 3 & x\end{array}\right]\) is a singular matrix, then the value of \(x\) is :

The integrating factor of the differential equation \(\frac{d y}{d x}+y \cot x=2 x+x^2 \cot x, x \neq 0\) is

\(\int_0^1 x e^x d x\) is equal to

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.

The reaction is an example of:

The core of a transformer is laminated, so as to

If \(\vec{a}=2 \hat{i}+m \hat{j}-n \hat{k}\) and \(\vec{b}=\hat{l} \hat{i}-3 \hat{j}+4 \hat{k}\) such that \(\vec{a}=2 \vec{b}\), then the value of \(14 l+m+n\) 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 :

Phenols are acidic in nature. Which of the following statements is incorrect about them?

A wire of length 4 m carrying current I A is bent in the form of a circle. Its magnetic moment will be :