The corner points of the bounded feasible region determined by the system of linear constraints are \((0,0),(5,0),(3,4)\) and \((0,5)\). Let \(Z=p x+q y\) where \(p, q>0\). Condition on \(p\) and \(q\) so that the maximum of \(Z\) occurs at both \((5,0)\) and \((3,4)\) is

Solution of \(\frac{d y}{d x}=\left(1+x^2\right)\left(1+y^2\right)\) is :

For a full wave rectifier, if the input frequency is \(5 0 ~ H z\), the output frequency will be

Let A and B be square matrices of order 3, \(\left[\left(A-A^T\right)+\left(B-B^T\right)\right]\) then det is equal to

Let \(A=\left[\begin{array}{ccc}2 & -1 & 3 \\ 1 & 2 & -1 \\ 4 & 1 & 2\end{array}\right]\). \(M_{i j}\) and \(A_{i j}\) respectively denote the minor and cofactor of an element \(a_{i j}\) of matrix \(A=\left[a_{i j}\right]\).
(A) \(M_{23}=6\)
(B) \(A_{22}=-8\)
(C) \(A_{13}=7\)
(D) \(M_{32}=-5\)
Choose the correct answer from the options given below:

For the differential equation \(x \frac{d y}{d x}+2 y=x^2 \log _e x\)
(A) Integrating factor is \(2 x\)
(B) Integrating factor is \(x^2\)
(C) General Solution is \(y=\frac{x^2}{16}\left(4 \log _e|x|-1\right)+C x^{-2}\), where \(C\) is an arbitrary constant.
(D) General Solution is \(y=\frac{x^4}{16}\left(4 \log _e|x|-1\right)+C\), where \(C\) is an arbitrary constant.
Choose the correct answer from the options given below :

If \(P(A)=\frac{3}{5}, P(B)=\frac{1}{2}\) and \(P(A \cap B)=\frac{1}{4}\), then \(P(\bar{A} \mid \bar{B})\) is

Identify the INCORRECT statement(s) in the context of phenomenon of interference.
1. All bright and dark fringes have equal width.
2. All bright fringes have the same intensity.
3. Fringe width depends on the distance of screen from the slits.
4. All dark and bright fringes are not equally spaced.
Choose the correct answer from the options given below:

If \(A\) and \(B\) are symmetric matrices of order \(3 \times 3\) then the matrix \(2 A B-B A\) is :

Light from a point source in air falls on a spherical glass surface. If \(\mu = 1.5\) and radius of curvature \(= 20\) cm. The distance of light source from the glass surface is 100 cm, at what position will the image be formed?

Arrange the following in decreasing order of their basic strength in aqueous solution
(A)C₆H₅NH₂
(B)C₂H₅NH₂
(C) (C₂H₅)₂NH
(D)NH₃

Which of the following is also known as 'molecular scissors'?

"Answer the question on basis of passage given below :
Carboxylic acids are weaker than mineral acids, but they are stronger acids than alcohols and many simple phenols (\(\text{pKa}\) is ~ 16 for ethanol and 10 for phenol). The higher acidity of carboxylic acids as compared to phenols can be understood similarly. The conjugate base of carboxylic acid, a carboxylate ion, is stabilised by two equivalent resonance structures in which the negative charge is at the more electronegative oxygen atom. The conjugate base of phenol, a phenoxide ion, has non-equivalent resonance structures in which the negative charge is at the less electronegative carbon atom. Further, the negative charge is delocalised over two electronegative oxygen atoms in carboxylate ion whereas it is less effectively delocalised over one oxygen atom and less electronegative carbon atoms in phenoxide ion. Thus, the carboxylate ion is more estabilised than phenoxide ion, so carboxylic acids are more acidic than phenols.
Effect of substituents on the acidity of carboxylic acids: substituents may affect the stabliity of the conjugate base and thus, also affect the acidity of the carboxylic acids. Electron withdrawing group increase the acidity of carboxylic acids by stabilising the conjugate base through delocalisation of the negative charge by inductive and/or resonance effects. Conversely, electron donating groups decrease the acidity by destabilising the conjugate base.
The weakest among the following acid is :"