The relation \(R\) on the set \(N \times N\) defined by \((a, b) R(c, d) \Longleftrightarrow a+d=b+c\), \(\forall(a, b),(c, d) \in N \times N\), where N is the set of natural numbers, is:

A random variable y has the following probability distribution

y	1	2	3	4	5
P(y)	2k	3k	k	4k	5k

Match List-l with List-ll

List - I	List - II
(A) P(y > 2)	(I) \(\frac{2}{5}\)
(B) k	(II) \(\frac{2}{3}\)
(C) \(P(y \leq 3)\)	(II) \(\frac{8}{15}\)
(D) \(P(2 \leq y \leq 4)\)	(II) \(\frac{1}{15}\)

Choose the correct answer from the options given below

Match List 1 with List 2.

List 1 (Differential equation)	List 2 (Sum of order and degree)
(A) \(\frac{d^2 y}{d x^2}+\frac{d y}{d x}+3 y=\sin x\)	(I) 2
(B) \(\frac{d y}{d x}=\sin (x+y)\)	(II) 3
(C) \(\sqrt{1+\left(\frac{d y}{d x}\right)^2}=\frac{d^2 y}{d x^2}\)	(III) 4
(D) \(x^2\left(\frac{d^2 y}{d x^2}\right)^3+y\left(\frac{d y}{d x}\right)^4+y^3=0\)	(IV) 5

Choose the Correct answer from the options given below :

In the formula for Z -test statistic \(Z=\frac{\bar{x}-\mu_0}{\sigma / \sqrt{n}}\), mark the incorrect statement.

If \(x=t^3\) and \(y=t^2\), then \(\frac{d^2 y}{d x^2}\) is equal to :

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 :

Which of the following is NOT true?

The standard reduction potential for \(Zn^{2+}/Zn\) is \(-0.76\) V and that for \(Fe^{2+}/Fe\) is \(-0.41\) V. The galvanic cell can be correctly represented by :

Select the correct statement.

Over 95% of all existing transgenic animals are:

Read the following passage carefully and answer the given questions
The earthworm is referred to as the farmer's friend. This is because they help in the breakdown of complex organic matter as well as in loosening of soil. Similarly, decomposers break down complex organic matter into inorganic substances like carbon dioxide, water and nutrients and the process is called decomposition. Dead plant remains such as leaves, bark, flowers and dead remains of animals, including fecal matter, constitute detritus, which is the raw material for decomposition. Some of the steps in the process of decomposition are fragmentation, catabolism, leaching, humification etc.
The degradation of detritus into simpler inorganic substances by the bacterial and fungal enzymes is called as-

The integral \(\int_0^1 x(1-x)^n d x\) is equal to :

Two point charges placed a distance d apart in vacuum exert a force of magnitude F on each other. One of the two charges is doubled. To keep the magnitude of force same the separation between the charges should be changed to