Unit vectors perpendicular to the plane of vectors \(\vec{a}=\hat{i}-\hat{j}-3 \hat{k}, \vec{b}=\hat{i}+3 \hat{j}-\hat{k}\) are:

The general solution of the differential equation \(e^x d y+\left(y e^x+2 x\right) d x=0\) is

The marginal cost of production of \(X\) units of a commodity is \(56+\frac{3}{2} x\).
It is known that fixed costs are Rs.115.
Then the total cost of producing 50 units is :-

For the function \(f(x)=x^{1 / x}, x>0\), which of the following are correct?
(A) \(x=0\) is the only point where extremum may occur.
(B) The given function is maximum at \(x=e\).
(C) The function has no extreme value for \(x>0\).
(D) The maximum value of the function \(f(x)\) is \(e^{1 / e}\)
Choose the correct answer from the options given below :

Area bounded by \(y=|x-5|\) and the \(x\)-axis between \(x=2\) and \(x=4\) is:

Match List I with List II

LIST I	LIST II
A. \(\lim_{x \to 0} \frac{(1 - \cos 2x) \sin 5x}{x^2 \sin 3x}\)	I. 18
B. \(\lim_{x \to \infty} \frac{(3x-5)(2x-7)}{(4x-9)(5x-3)}\)	II. \(\frac{10}{3}\)
C. \(\lim_{x \to 0} \frac{2 \sin^2 3x}{x^2}\)	III. \(\frac{3}{4}\)
D. \(\lim_{x \to \frac{\pi}{4}} \frac{1 - \cos^3 x}{2 \cot x - \cot^3 x}\)	IV. \(\frac{3}{10}\)

Choose the correct answer from the options given below:

If A and B are symmetric matrices of the same order, then which of the following are true?
(A) AB - BA is a skew symmetric matrix
(B) AB is a symmetric matrix
(C) AB is a scalar matrix
(D) AB + BA is a symmetric matrix
Choose the correct answer from the options given below:

Choose the statements that are valid for the molecularity of a reaction
(A) Molecularity is equal to the number of reacting species involved in the overall reaction
(B) Decomposition of ammonium nitrite is an example of unimolecular reaction
(C) Reactions with the molecularity three are very rare
(D) Molecularity can only be determined experimentally
Choose the correct answer from the options given below:

When a charged particle moves in a uniform magnetic field, its kinetic energy:

The value of \(c \in[0, \pi]\) for which the function \(f(x)=e^x \sin x\) satisfies Rolle's theorem is:

The number of free electrons per 100 mm of ordinary copper wire is 2 \(\times 10^{21}\). The average drift speed of electrons is 0.25 mm/s. The amount of current flowing would be : (e = 1.6 \(\times 10^{-19}\) C)

Read the following paragraph and answer the given questions.
Cancer detection is based on biopsy and histopathological studies. 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. Do you know why mother's milk is considered very essential for the new-born infant? The yellowish fluid colostrum secreted by mother during the initial days of lactation has abundant antibodies (IgA) to protect the infant. 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. The exaggerated response of the immune system to certain antigens present in the environment is called allergy. The use of drugs like anti-histamine, adrenalin and steroids quickly reduce the symptoms of allergy. 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.
Which one of the following is not used for detection of cancer?

The van't Hoff Factor, \(i\) for complete dissociation of (i) NaCl , (ii) \(MgSO _4\) and (iii) \(K _2 SO _4\) are