Which of the following are first order linear differential equations?
(A)\(\frac{d x}{d y}+P_1(y) x=Q_1(y): P_1(y)\) and \(Q_1(y)\) are functions of y or constant functions
(B)\(\frac{d y}{d x}+P_2(x) y=Q_2(x): P_2(x)\) and \(Q_2(x)\) are functions of a or constant functions
(C)\((x+y) \frac{d y}{d x}=x-2 y\)
(D)\(\left(1+x^2\right) \frac{d y}{d x}-2 x y=x^2+3\)
Choose the correct answer from the options given below:

The common region determined by all constraints of a Linear programming problem is called:

Let X be a random variable. Let E(X) and Var(X) denote the mean and the variance of X respectively.
Match List - I with List - II

List - I	List - II
(A) If \(\operatorname{Var}( X )=\alpha\), then \(\operatorname{Var}(2 X +3)\) is	(I) \(11 \alpha\)
(B) If \(E ( X )=\alpha\), then \(E (2 X )\) is	(II) \(6 \alpha\)
(C) If \(\operatorname{Var}( X )=\alpha\), then \(\operatorname{Var}(3 X -\alpha)+\operatorname{Var}(\sqrt{2} X +\beta)\) is	(III) \(4 \alpha\)
(D) If \(E(X) = 3\) \(\frac{5\alpha}{12}, \text{ then E(12X + } \alpha)\) is	(IV) \(2 \alpha\)

Choose the correct answer from the options given below:

The function \(f: R \rightarrow R , f(x)=x^2\) is

The value of \(\cot \left(\cos ^{-1} \frac{7}{25}\right)\) is

In a 200 m race, Rohan completes the race in 8 seconds and Vivan completes the race in 10 seconds. By how much distance Rohan beats Vivan?

If \(A\) is an invertible matrix, then which of the following statement(s) is/are TRUE?
(A) \(\left|A^{-1}\right|=|A|\)
(B) \(\left(A^{-1}\right)^{-1}=A\)
(C) \(A^{-1}=\frac{\operatorname{adj} A}{|A|}\)
(D) \(\left(A^T\right)^{-1}=\left(A^{-1}\right)^T\)
Choose the correct answer from the options given below :

Two solenoids of the same length and the same cross-sectional area have 400 turns and 250 turns, respectively. The ratio of self-inductance of the two solenoids, respectively, is

Read the following paragraph carefully and answer the given questions.
In the year 1963 the two enzymes responsible for restricting the growth of bacteriophage in E. coli were isolated. The first restriction endonuclease - Hind II, whose functioning depended on a specific DNA nucleotide sequence was characterized five years later. It was found that Hind II always cut DNA molecules at a particular point by recognizing a specific sequence of six base pairs. This specific base sequence is known as the recognition sequence for Hind II. Besides Hind II, today we know more than 900 restriction enzymes that have been isolated from over 230 strains of bacteria, each of which recognize different recognition sequences.
DNA fragments can be separated by a technique known as:

The probability that in a year of the 22nd century chosen at random, there will be 53 Sundays is:

Read the passage carefully and answer the given questions.
Recombinant DNA Technology
Asexual reproduction preserves the genetic information, while sexual reproduction permits variation. Traditional hybridization procedures used in plants and animals breeding very often lead to inclusion and multiplication of undesirable genes along with the desired genes. The techniques of genetic engineering, which include creation of recombinant DNA, use of gene cloning and gene transfer, overcome this limitation and allow us to isolate and introduce only one or a set of desirable genes without introducing genes into the target organism.
In Recombinant DNA Technology, what is the role of the "selectable marker gene"?

An infinitely long straight conductor carries a current of 150 A. At what distance from the conductor is the magnetic field caused by the current equal to \( 10^{-4} \text{ T} \)?

Solution of \(\frac{dy}{dx} = 2^{y-x}\) is: