If \(y=\log _e\left(\sec \left(e^{x^2}\right)\right)\), then \(\frac{d y}{d x}=\)

Consider the differential equation \(\frac{d y}{d x}+y \tan x=\sec x\), then which of the following statements are correct?
(A) It is homogeneous
(B) It has \(\sec x\) as its integrating factor
(C) Its general solution is \(y \sec x=\tan x+c\), where \(c\) is an arbitrary constant.
(D) Its degree is not defined
Choose the correct answer from the options given below :

Mean and variance of a binomial distribution are 6 and 2 respectively.
The probability of 2 successes will be

If \(\cos ^{-1} \alpha+\cos ^{-1} \beta+\cos ^{-1} \gamma=3 \pi\) then the value of \(\alpha^2 \beta^2 \gamma^2+3(\alpha+\beta+\gamma)\) is:

The straight line \(\frac{x+2}{3} = \frac{z-3}{-2}, y=2\) is

A man rows a boat such that it covers a distance upstream in twice the time taken by him to cover the same distance downstream. If the speed of stream is 5 km/h, the speed of boat in still water is:

A random variable \(X\) has the following probability distribution

x	-2	-1	0	1	2	3
p	0.1	k	0.2	2k	0.3	k

Then the mean of X is :

Given differential equation,
\(\left(1+y^2\right) d x=\left(\tan ^{-1} y-x\right) d y\)
then which of the following is/are true?
(A) Integrating factor \(=\tan ^{-1} x\)
(B) Integrating factor \(=\tan ^{-1} y\)
(C) Integrating factor \(=e^{\tan ^{-1} y}\)
(D) Degree \(=1\)
Choose the correct answer from the options given below :

Read the following paragraph carefully and answer the given questions.
Certain types of leukocytes (WBC) of our body like polymorpho-nuclear leukocytes (PMNL-neutrophils) and monocytes and natural killer (type of lymphocytes) in the blood as well as macrophages in tissues can phagocytose and destroy microbes. For determining the cause of allergy, the patient is exposed to or injected with very small doses of possible allergens, and the reactions studied. The use of drugs like anti-histamine, adrenalin and steroids quickly reduce the symptoms of allergy. Heroin, commonly called smack is chemically diacetylmorphine which is a white, odourless, bitter crystalline compound. This is obtained by acetylation of morphine. Techniques like radiography (use of X-rays), CT (computed tomography) and MRI (magnetic resonance imaging) are very useful to detect cancers of the internal organs. Different species of Plasmodium (P. vivax, P. malaria and P. falciparum) are responsible for different types of malaria. Of these, malignant malaria caused by Plasmodium falciparum is the most serious one and can even be fatal.
Which of the following drugs are commonly used to quickly reduce the symptoms of allergy?

Radiation of frequency \(2 v _0\) incident on a metal with threshold frequency \(v _0\) results in maximum kinetic energy of photoelectrons as :

Answer the question on the basis of passage given below:
Adsorption is the phenomenon of attracting and retaining the molecules of a substance on the surface of a solid resulting into a higher concentration on the surface than in the bulk. The substance adsorbed is known as adsorbate and the substance on which adsorption takes place is called adsorbent. In physisorption adsorbate is held to the adsorbent by weak van der Waals forces and in chemisorption by strong chemical bond.
In coagulation of a positive sol, the flocculating power will be in the order:
A. \(PO _4^{3-}\)
B. \(Cl ^{-}\)
C. \([ Fe ( CN )]^{3-}\)
D. \(SO _4^{2-}\)
Choose the correct answer from the options given below:

The source of electrical energy on the Apollo moon flight was:

Cosine of the acute angle between the lines \(\frac{x-3}{2}=\frac{y-2}{1}=\frac{x-5}{2}\) and \(\frac{x-1}{6}=\frac{y-3}{-3}=\frac{z+6}{2}\) is