The sum of order and degree of the differential equation \(y=x \frac{d y}{d x}+2 \sqrt{1+\left(\frac{d y}{d x}\right)^2}\) is

If \(A=\left(\begin{array}{cc}1 & 0 \\ -1 & 7\end{array}\right), I=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)\), and \(A^2=8 A+K I\) the value of \(K\) is :

With respect to the following shaded feasible region (ABCDEFA), the maximum value of the objective function z =3x + 4y - 2 is at point(s):

The number of arbitrary constants in the particular solution of a differential equation of order 4 and degree 3 is

For the function \(f(x)=e^{-2 x}(2-x)^2\), the point of local maxima is:

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

A proton accelerated through potential difference V has de Broglie wavelength \(\lambda\). On doubling the potential, the de Broglie wavelength of the proton :

Which of the following is male sex hormone ?

"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 :"

Let \(f(x)=4 x^3-18 x^2+27 x-5, x \in \mathbb{R}\). Then which of the following statements are TRUE?
(A) \(f^{\prime \prime}(x)=24 x-36\)
(B) \(f\) has local maxima at \(x=\frac{3}{2}\) but no minima
(C) \(f\) has neither maxima nor minima
(D) \(f\) has both maxima and minima
Choose the correct answer from the options given below:

Which of the following option is correct for ammensalism?

The work function for Aluminium surface is 4.2 eV. Find the threshold wavelength for the photoelectric emission.

For the linear programming problem,
Minimize (Z) =60x + 30y
subject to : \(2 x - y \geq-5 ; 3 x + y \geq 3 ; 2 x -3 y \leq 12; x , y \geq 0\)
the optimal value of \(z\) is