Let \(A=\{1,2,3\}\) and \(R=\{(1,1),(1,3),(3,1),(2,2),(2,1),(3,3)\}\) be a relation on \(A\). Then, \(R\) is:

Let \(A\) and \(B\) be \(3 \times 3\) matrices such that \(A \neq B\). If \(A^3=B^3\) and \(A^2 B=B^2 A\), then the determinant of \(A^2+B^2\) is :

Consider the LPP: Max \(Z = 5x + 3y\) subject to \(3x + 5y ≤ 15, 5x + 2y ≤ 10 x ≥ 0, y ≥ 0\)
match List - I with List - II

List-I	List-II
(A) Objective function	(I) \(3x + 5y ≥ 15\)
(B) One constraint	(II) \(x, y ≥ 0\)
(C) Non-negative restrictions	(III) \(Z = 5x + 3y\)
(D) Point (1, 2) does not lie in the region	(IV) \(3x + 5y ≤ 15\)

Choose the correct answert from the option given below :

If the matrix \(\left[\begin{array}{ccc}-3 & x-y & 5 \\ 1 & 0 & z \\ x+y & 4 & 7\end{array}\right]\) is symmetric, then the correct option of the following, is :

When a slab of insulating material 4 mm thick is introduced between the plates of a parallel plate capacitor of separation 4 mm , it is found that the distance between the plates has to be increased by 3.2 mm to restore the capacity to its original value. The dielectric constant of the material is

The value of the integral \(\int_{-1}^1|x| d x\) is :

The co-ordinates of the point where the line\(\frac{x+3}{3}=\frac{y-1}{-1}=\frac{z-5}{-5}\) cuts the yz--plane are :

The general solution of the differential equation \(x\left(\frac{d y}{d x}\right)=y+x \tan \left(\frac{y}{x}\right)\) is

If A and B are two events such that \(P(A)=\frac{1}{2}, P(B)=\frac{1}{3}\) and \(P(A \cap B)=\frac{1}{4}\), then which of the following statements are true?
(A) A and B are independent events
(B) \(P(A \mid B)=\frac{3}{4}\)
(C) \(P\left(A^{\prime} \mid B^{\prime}\right)=\frac{5}{8}\)
(D) \(P\left(A^{\prime} \mid B\right)=\frac{1}{4}\)
Choose the correct answer from the options given below:

In the electrochemical series written in the increasing order of reduction potential
(A) The elements which are placed above the hydrogen in the electrochemical series are stronger oxidizing agents.
(B) The elements which are placed above the hydrogen in the electrochemical series are stronger reducing agents.
(C) The elements with the positive electrode potential values lie below the hydrogen in the electrochemical series.
(D) The activity of both the metals and non-metals in the electrochemical series decreases from top to bottom.
Choose the correct answer from the options given below:

Read the given passage carefully and answer the questions
Antibiotics have greatly improved our capacity to treat deadly diseases such as plague, whooping cough (kali khansi), diphtheria (gal ghotu) and leprosy (kusht rog), which used to kill millions all over the globe.
Streptokinase produced by the bacterium Streptococcus and modified by genetic engineering is used as a 'clot buster' for removing clots from the blood vessels of patients who have undergone myocardial infarction leading to heart attack. Microbes are diverse-protozoa, bacteria, fungi and microscopic animal and plant viruses, viroids and also prions that are proteinacious infectious agents. The puffed-up appearance of dough is due to the production of CO₂ gas. Similarly the dough, which is used for making bread, is fermented using baker's yeast (Saccharomyces cerevisiae). Primary sewage treatment basically involve physical removal of particles - large and small from the sewage through filtration and sedimentation. These are removed in stages; initially, floating debris is removed by sequential filtration. Then the grit (soil and small pebbles) are removed by sedimentation. All solids that settle form the primary sludge, and the supernatant forms the effluent. The effluent from the primary settling tank is taken for secondary treatment. The primary effluent is passed into large aeration tanks, where it is constantly agitated mechanically and air is pumped into it. This allows vigorous growth of useful aerobic microbes into flocs. While growing, these microbes consume the major part of the organic matter in the effluent.
Genetically modified Streptokinase is used as a 'clot buster' for removing clots from the blood vessels is produced by :

Match List I with List II.

List-I	List-II
A. aq \(N a C l\) solution with \(P t\) electrodes	I.oxygen gas evolution at the anode
B. Dilute solution of \(H _2 SO _4\) with Pt electrode	II. Metal loss at anode.
C. aq \(CuCl _2\) with Pt electrode	III. Hydrogen gas evolved at cathode
D. aq \(CuCl _2\) with Cu electrode	IV. Metal deposited at cathode.

Choose the correct answer from the options given below:

The random variable X has the following probability distribution

X	0	1	2	3
P(X)	a	a	b	b

such that E\(\left(x^2\right)\) = 2E(x), then the value of b is: