The value of \(\int_{0}^{\frac{\pi}{2}} \log \left( \frac{5 + 4 \sin x}{5 + 4 \cos x} \right) dx\) is:

Match List - I with List - II

List - I	List - II
(A) \(\int \frac{d x}{a^2-x^2}\)	(I) \(\frac{1}{2 a} \log \left|\frac{x-a}{x+a}\right|+C\)
(B) \(\int \frac{d x}{x^2-a^2}\)	(II) \(\frac{1}{2 a} \log \left|\frac{a+x}{a-x}\right|+C\)
(C) \(\int \frac{d x}{x^2+a^2}\)	(III) \(\sin ^{-1}\left(\frac{x}{a}\right)+C\)
(D) \(\int \frac{d x}{\sqrt{a^2-x^2}}\)	(IV) \(\frac{1}{a} \tan ^{-1} \frac{x}{a}+C\)

Choose the correct answer from the options given below:

Let \(P\) and \(Q\) be any two invertible matrices of the same order. Then Match List-I with List-II

List - I (Matrix)	List - II (Equivalent matrix)
(A) \((P Q)^{-1}\)	(I) \(Q^{-1} P\)
(B) \(\left(P^{-1} Q\right)^{-1}\)	(II) \(Q P^{-1}\)
(C) \(\left(P Q^{-1}\right)^{-1}\)	(III) \(Q^{-1} P^{-1}\)
(D) \(\left(P^{-1} Q^{-1}\right)^{-1}\)	(IV) \(Q P\)

Choose the Correct answer from the options given below :

For the matrix \(A=\left[\begin{array}{ccc}2 & -1 & -1 \\ 0 & 2 & 3 \\ 1 & -2 & 1\end{array}\right]\), which of the following statements are correct?
(A) The order of the matrix is 3 × 3
(B) |A| = 21
(C) |adj A| = 225
(D) A is skew symmetric matrix
Choose the correct answer from the options given below:

The projection of the vector \(5 \hat{i}+\hat{j}-3 \hat{k}\) on the vector \(\hat{i}+2 \hat{j}-3 \hat{k}\) is

On simplifying \(\cot \left(2 \tan ^{-1} x\right)\) where \(|x|<1\), we get:

A metal wire under constant potential difference, when heated, shows that the drift velocity of the electron :

A boat takes 1 hour 30 minutes less to travel 36 km downstream than to travel the same distance upstream.
If the speed of the boat in still water is 10 km/hr, then the speed of the stream is :

For independent events \( A_{1} , A_{2}, A_{3} , \ldots, A_{n}\) , if \(P(A_{i}\) \(= \frac{1}{i + 1}, \; i = 1, 2, 3 \ldots, n\), then the probability that none of the events occur is :

Identify the role of deoxyribonucleoside triphosphates in DNA replication.
A. Initiate replication.
B. Act as substrate.
C. Provide energy for polymerisation.
D. Activate ligases.
Choose the correct answer from the options given below:

If A is a square matrix of order \(3 \times 3\) and \(\left|A^T\right|=-3\), then \(\left|A A^T\right|\) is equal to

The number of all possible matrices of order \(2 \times 2\) with each entry ' \(a\) ', ' \(b\) ' or ' \(c\) ' is :

Read the following paragraph and answer the given questions:
DRUGS AND ALCOHOL ABUSE
The drugs, commonly abused are opioids, cannabinoids, and coca alkaloids. Opioids are the drugs, which bind to specific opioid receptors present in our central nervous system and gastrointestinal tract. Heroin, commonly called smack is chemically diacetylmorphine which is a white, odourless, bitter crystalline compound. Cannabinoids are a group of chemicals, which interact with cannabinoid receptors present principally in the brain. Natural cannabinoids are obtained from the inflorescences of the plant Cannabis sativa. The flower tops, leaves and the resin of cannabis plant are used in various combinations to produce marijuana, hashish, charas and ganja. Generally taken by inhalation and oral ingestion, these are known for their effects on cardiovascular system of the body. It interferes with the transport of the neuro-transmitter dopamine. Cocaine, commonly called coke or crack is usually snorted. It has a potent stimulating action on central nervous system, producing a sense of euphoria and increased energy. Excessive dosage of cocaine causes hallucinations. Other well-known plants with hallucinogenic properties are Atropa belladona and Datura. These days cannabinoids are also being abused by some sportspersons. Drugs like barbiturates, amphetamines, benzodiazepines, and other similar drugs, that are normally used as medicines to help patients cope with mental illnesses like depression and insomnia, are often abused. Morphine is a very effective sedative and painkiller, and is very useful in patients who have undergone surgery. Several plants, fruits and seeds having hallucinogenic properties have been used for hundreds of years in folk-medicine, religious ceremonies and rituals all over the globe.
'Coca alkaloids' are obtained from the plant-