\(\int \frac{\sin 2 x d x}{\sqrt{9-\cos ^4 x}}\) equals

In a game, a man wins ₹ \( 8\) for getting a number greater than 3 and loses ₹ 3 otherwise, when a fair die is thrown.
The man decided to throw a die 4 times but to quit as and when he gets a number greater than 3.
If \(X\) denotes the amount which the man wins or loses, then which of the following are correct?
(A) All the possible values of \(X\) are \(8,5,2\) and -1 .
(B) The probability distribution of \(X\) is :

X	8	5	2	-1	-12
P(X)	1/2	1/4	1/8	1/16	1/16

(C) The mean value of \(X\) is \(\frac{75}{16}\).
(D) The variance of \(X\) is \(\frac{6615}{256}\).
Choose the correct answer from the options given below :

For real numbers x and y, the relation R is defined as R = {(x, y) : x + y - \(\pi\) is an irrational number}, then the relation R is:

Pipe A can fill a tank in 24 hours and pipe B in 32 hours. If both the pipes are opened in an empty tank at the same time, then the time taken to fill the tank is:

Determine the value of k for which the function \(f(x)=\left\{\begin{array}{ll}\frac{x^2-9}{x-3}, & x \neq 3 \\ k, & x=3\end{array}\right.\) is continuous at x = 3

Match List I with List II

LIST I	LIST II
A. If \(A^{-1}=\left[\begin{array}{ll}3 & 4 \\ 5 & 6\end{array}\right]\) then \(A\) is	I. \(\left[\begin{array}{ll}5 & 2 \\ 7 & 3\end{array}\right]\)
B. If \(A=\left[\begin{array}{cc}2 & -1 \\ 3 & 1\end{array}\right]\) then \(A A^T\) is	II. \(\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]\)
C. If \(A^{-1}=\left[\begin{array}{ll}5 & 7 \\ 2 & 3\end{array}\right]\) then \(\left(A^T\right)^{-1}\) is	III. \(\left[\begin{array}{cc}-3 & 2 \\ \frac{5}{2} & -\frac{3}{2}\end{array}\right]\)
D. If \(A=\left[a_{i j}\right]_{2 \times 2}\) where \(a_{i j}=(i-j)^2\), then \(A\) is	IV. \(\left[\begin{array}{cc}5 & 5 \\ 5 & 10\end{array}\right]\)

Pipe A can fill the tank 3 times faster than pipe B. If both pipes A and B running together can fill the tank in 15 minutes, then the time taken by pipe B alone to fill the tank is :

Read the following passage carefully and answer the given questions.
Bioactive molecule, cyclosporin A, that is used as an immunosuppressive agent in organ-transplant patients, is produced by the fungus Trichoderma polysporum. Statins produced by the yeast Monascus purpureus have been commercialised as blood-cholesterol lowering agents. It acts by competitively inhibiting the enzyme responsible for synthesis of cholesterol. Baculoviruses are pathogens that attack insects and other arthropods. The majority of baculoviruses used as biological control agents are in the genus Nucleopolyhedrovirus. These viruses are excellent candidates for species-specific, narrow spectrum insecticidal applications. They have been shown to have no negative impacts on plants, mammals, birds, fish or even on non-target insects. BOD refers to the amount of the oxygen that would be consumed if all the organic matter in one liter of water were oxidised by bacteria. The sewage water is treated till the BOD is reduced. Cheese, is one of the oldest food items in which microbes were used. Different varieties of cheese are known by their characteristic texture, flavour and taste. For example, the large holes in ‘Swiss cheese’ are due to production of a large amount of CO₂ by a bacterium named Propionibacterium sharmanii.
Bioactive molecule, cyclosporin A, that is used as an immunosuppressive agent in organ-transplant patients, is produced by the fungus

Read the Passage carefully and answer the the questions
The DNA 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. The separated DNA fragments can be visualised only after staining the DNA with a compound known as ethidium bromide followed by exposure to UV radiation. The repeated amplification of DNA is achieved by the use of a thermostable DNA polymerase (isolated from a bacterium, Thermus aquaticus). After completion of the biosynthetic stage, the product has to be subjected through a series of processes before it is ready for marketing as a finished product. The downstream processing and quality control testing vary from product to product. The convention for naming restriction enzymes is the first letter of the name comes from the genus and the second two letters come from the species of the prokaryotic cell from which they were isolated.
In agarose gel electrophoresis, the DNA fragments can be visualised only after staining the DNA with a compound known as

The unit's digit of \(2^{2025}\) is

How much calcium will be obtained when 4 Faradays of electricity is passed through molten CaCl2?

Match List-l with List-II:

List-I (Products)	List-II (Organisms)
(A) Statin	(I) Streptococcus
(B) Clot buster	(II) Trichoderma
(C) Swiss cheese	(III) Monascus
(D) Cyclosporin-A	(IV) Propionibacterium

Choose the correct answer from the options given below:

Which of the following are not involved in intra uterine devices?
(A) Lippes Loop
(B) LNG-20
(C) Saheli
(D) Implants
Choose the correct answer from the options given below: