\(\int \frac{\cos x - \sin x}{1 + \sin 2x} dx\) is equal to:

If \(y=38.85+7.64(x-2020)\) is the equation of the straight line trend for the following data:

Year(x)	2017	2018	2019	2020	2021	2022	2023
Profit( Rs. '000) (y)	14	30	26	44	38	56	64

The trend value for the year 2021 by least square method is :

The probability distribution of a random variable X is given by

X	0	1	2
P(X)	\(1-7 a^2\)	\(\frac{1}{2} a+\frac{1}{4}\)	\(a^2\)

If \(a>0\), then P\((0< x \leq 2\)) is equal to

Let \(\tan ^{-1} y=\tan ^{-1} x+\tan ^{-1}\left(\frac{2 x}{1-x^2}\right)\). Then \(y\) is :

\(\text { If } I=\int \frac{x}{x-\sqrt{x^2-4}} d x=\alpha x^3+\beta\left(x^2-4\right)^{\frac{3}{2}}+\gamma\) where \(\gamma\) is a constant of integration, then

A box contains 2 black and 4 red balls and another box contains 4 black and 3 red balls. If a ball is drawn at random from one of the two boxes, then the probability of getting a black ball is

If \(\left[\begin{array}{cc}2 & -1 \\ 1 & 0 \\ -3 & 4\end{array}\right] A=\left[\begin{array}{ccc}-1 & -8 & -10 \\ 1 & -2 & -5 \\ 9 & 22 & 15\end{array}\right]\), then matrix \(A\) is equal to:

A short bar magnet of magnetic dipole moment \( 10^{-5} \text{ A m}^{2} \) is kept in a uniform magnetic field \( 3 \times 10^{-4} \text{ T} \) at an angle of \( 30^{\circ} \). The torque acting on it is :

The average drift speed of conduction electrons in a copper wire of cross-sectional area \( 1.0 \times 10^{-7} \text{ m}^2 \) carrying a current of 1.5 A. (Assume the density of conduction electrons to be \( 9 \times 10^{28} /\text{m}^3 \))

There are two kinds of nucleases - exonucleases and endonucleases. Exonucleases remove nucleotides from the ends of the DNA whereas, endonucleases make cuts at specific positions within the DNA. The cutting of DNA by restriction endonucleases results in the fragments of DNA. These fragments can be separated by a technique known as gel electrophoresis. Since DNA fragments are negatively charged molecules they can be separated by forcing them to move towards the anode under an electric field through a medium/matrix. Small volume cultures cannot yield appreciable quantities of products. To produce in large quantities, the development of bioreactors, where large volumes (100 - 1000 litres) of culture can be processed, was required. Thus, bioreactors can be thought of as vessels in which raw materials are biologically converted into specific products, individual enzymes, etc., using microbial plant, animal or human cells. PCR stands for Polymerase Chain Reaction. In this reaction, multiple copies of the gene (or DNA) of interest are synthesised in vitro using two sets of primers (small chemically synthesised oligonucleotides that are complementary to the regions of DNA) and the enzyme DNA polymerase.
Which one of the following enzymes remove nucleotides from the ends of the DNA?

The value of the integral \(\int \frac{e^x(1+x)}{\cos ^2\left(x e^x\right)} d x\) is equal to:

Read the passage carefully and answer the Questions
\(KMnO _4\) is prepared by the fusion of \(MnO _2\) with an alkali metal hydroxide and an oxidizing agent like \(KNO _3\) to give a dark-green manganate ion which disproportionate to give permanganate as follows.
\(\begin{array}{l}2 MnO _2+4 KOH + O _2 \rightarrow 2 K_2 MnO _4+2 H _2 O \\ 3 K_2 MnO _4+4 H ^{+} \rightarrow 2 KMnO _4+ MnO _2+2 H _2 O \end{array}\)
On heating \(K _2 MnO _4\) decomposes at 513 K to give \(K _2 MnO _4\). Permanganate ion is tetrahedral and diamagnetic
Acidified \(KMnO _4\) acts a strong oxidizing agent which oxidizes oxalic acid, ferrous ions, nitrite ion and iodides.
When one mole of \(KMnO _4\) is used for the oxidation of oxalic acid, how many moles of \(CO _2\) will be produced?

If \(X=11\) and \(Y=3\), then \(X \bmod Y=(X+a Y) \bmod Y\) holds: