The number of arbitrary constants in the particular solution of a differential equation of order 4 and degree 3 is

The probability distribution of number of heads in two tosses of a coin is:

According to Bohr's Model :
(A) The radius of the orbiting electron is directly proportional to \({'n'. } \)
(B) The speedof the orbiting electron is directly proportional to \(\frac{1}{n} \)
(C) The magnitude of the total energy of the orbiting electron is directly proportional to \(\frac{1}{n^2} \)
(D) The radius of the orbiting electron is directly proportional to \( =n^2 \)
Correct options :

The integrating factor of the differential equation \(x \frac{d y}{d x}-y=\sin x\) is :

The curve \(x=y^2\) and \(x y=k\) cut orthogonally, then \(k^2\) is equal to :

\(\int e^x\left(\tan ^{-1} x+\frac{1}{1+x^2}\right) d x\) is equal to :

If a line makes angles \(\alpha, \beta\) and \(\gamma\) with the positive directions of coordinate axes respectively, then \(\cos 2 \alpha+\cos 2 \beta+\cos 2 \gamma\) is equal to

The half life of a first order reaction having rate constant, \(k=1.7 \times 10^{-5} s^{-1}\) is

The area of the region bounded by \(x=y^2\) and the line \(x=4\) in square units is :

The correct expression for osmotic pressure of the solution involving dissociation of the solute is

A system consists of two protons and one electron. They are separated by a distance of 1 \(\text{\AA}\) from each other. If the potential energy at infinity is taken to be zero, the potential energy of the system is

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.
Malignant malaria is caused by -

The area of the parallelogram determined by the vectors \(\hat{i}+2 \hat{j}+3 \hat{k}\) and \(3 \hat{i}-2 \hat{j}+\hat{k}\) is