Maximum value of \(z=5 x+4 y\) subject to the constraints \(x-y \leq 0, x+y \leq 4, x \geq 0, y \geq 0\) occurs at the point \((a, b)\). Then \(4 a+5 b=\) \(\_\_\_\_\) ,

In which of the following intervals, the function \(f(x)=\frac{x}{\log x}\) is decreasing?

If \(y=\log \left[\frac{x^2}{e^2}\right]\) then value of \(\frac{d^2 y}{d x^2}\) is :

The particular solution of the differential equation \(\cos \left(\frac{d y}{d x}\right)=a,(a \in R ) ; y=2\) at \(x=0\) is given by

Let R be the feasible region for a Linear Programming Problem and let \(z=a x+b y\) be the objective function. If R is bounded, then the objective function z has :

Consider the differential equation \(x d y=(x+y) d x\). Which of the following are true?
(A) It is a homogenous differential equation
(B) It is a differential equation of order 2
(C) The general solution of the differential equation contains 2 arbitrary constants
(D) Integrating factor of differential equation is \(\frac{1}{x}\)
(E) Degree of the differential equation is not defined
Choose the correct answer from the options given below:

Match List-I with List-II (where \(c\) is an arbitrary constant)

List-I (Differential Equation)	List-II (General solution)
(A) \(\frac{d y}{d x}=\frac{y}{x}, x \neq 0\)	(I) \(y=c x, c\) is an arbitrary constant
(B) \(x d x-y d y=0 ; y \neq 0, x \neq 0\)	(II) \(x^2-y^2=c, c\) is an arbitrary constant
(C) \(\frac{\left(x^2-1\right)}{y^2+1} \frac{d y}{d x}=1\)	(III) \(2 x+3 y=c, c\) is an arbitrary constant
(D) \(2 d x+3 d y=0\)	(IV) \(\left(x^3-y^3\right)=c+3(x+y), c\) is an arbitrary constant

Choose the correct answer from the options given below :

Which of the following statements are TRUE?
(A) The variable \(t\) of \(t\)-distribution ranges from \(-\infty\) to \(\infty\).
(B) The probability curve of the \(t\)-distribution is symmetric about the line \(t=0\)
(C) The variance of the \(t\)-distribution is greater than one.
(D) As the number of degrees of freedom decreases, the \(t\)-distribution curve moves closer to the standard normal probability curve.
Choose the correct answer from the options given below :

For the following reaction, which is correct statement \(2 H ^{+}( aq )+2 e^{-} \rightarrow H _2(g)\)
under non-standard conditions.

If A and B are symmetric matrices of the same order, then which one of the following is true?

Match List-I with List-II

List-I	List-II
(A) \(f ( x )=| x |\)	(I) Not differentiable at \(x=-2\) only
(B) \(f(x)=|x+2|\)	(II) Not differentiable at \(x=0\) only
(C) \(f(x)=\left|x^2-4\right|\)	(III) Not differentiable at \(x=2\) only
(D) \(f(x)=|x-2|\)	(IV) Not differentiable at \(x=2,-2\) only

Choose the correct answer from the options given below:

Which of the following statements is correct for an enzyme?
(A) It is a biocatalyst.
(B) It can catalyze any chemical reaction.
(C) Almost all enzymes are globular proteins.
(D) They reduce the magnitude of activation energy.
Choose the correct answer from the options given below:

Read the passage carefully and give the answers to the question :
India was amongst the first countries in the world to initiate action plans and programmes at a national level to attain total reproductive health as a social goal.
These programmes called 'family planning' were initiated in 1951 and were periodically assessed over the past decades. Improved programmes covering wider reproduction-related areas are currently in operation. Creating awareness among the people about various reproduction-related aspects and providing facilities and support for building up a reproductively healthy society are the major tasks under these programmes.
Which of the following statements is incorrect with respect to Medical Termination of Pregnancy?