The integral \(I=\int \frac{e^x}{1-e^{2 x}} d x\) is equal to :

The value of \(-\operatorname{cosec}^2\left(\cot ^{-1} y\right)+\sec ^2\left(\tan ^{-1} x\right)\) is equal to :

A dietician wishes to mix two types of food in such a way that the vitamin contents of the mixture contains atleast 8 units of Vitamin A and 10 units of Vitamin C. Food 'I' contains 2 units/kg of Vitamin A and 1 unit/kg of Vitamin C while food 'II' contains 1 unit/kg of Vitamin A and 2 units/kg of Vitamin C. It costs ₹5 per kg to purchase food I and ₹7 per kg to purchase food II. If 'x' kg of food I is mixed with 'y' kg of food II, then the constraint pertaining to vitamin C contents will be:

If \(f: R \rightarrow R\) is defined by f(x) = sin x + x then f(f(x)) is:

The domain of the function \(y=\sin ^{-1}(x-1)+\cos ^{-1} \sqrt{x-1}\) is:

The order and degree of differential equation \(\sqrt[3]{2 y+\frac{d y}{d x}}=\left(x+\frac{d^3 y}{d x^3}\right)\) respectively are :

Consider the LPP: Max \(Z = 5x + 3y\) subject to \(3x + 5y ≤ 15, 5x + 2y ≤ 10 x ≥ 0, y ≥ 0\)
match List - I with List - II

List-I	List-II
(A) Objective function	(I) \(3x + 5y ≥ 15\)
(B) One constraint	(II) \(x, y ≥ 0\)
(C) Non-negative restrictions	(III) \(Z = 5x + 3y\)
(D) Point (1, 2) does not lie in the region	(IV) \(3x + 5y ≤ 15\)

Choose the correct answert from the option given below :

Which one of the following is not the autosome-linked recessive Mendelian disorder?

Match List I with List II.

LIST I (Name of the organism)	LIST II (Chromosome number in a meiocyte)
A. Rat	I. \(38\)
B. Dog	II. \(42\)
C. Cat	III. \(1260\)
D. Ophioglossum	IV. \(78\)

Choose the correct answer from the options given below:

In a parallel plate capacitor of capacitance C, a metal sheet is inserted between the plates parallel to them. If the thickness of the sheet is one third of the separation between the plates, the capacitance of the capacitor will be

Read the following passage carefully and answer the given questions
The earthworm is referred to as the farmer's friend. This is because they help in the breakdown of complex organic matter as well as in loosening of soil. Similarly, decomposers break down complex organic matter into inorganic substances like carbon dioxide, water and nutrients and the process is called decomposition. Dead plant remains such as leaves, bark, flowers and dead remains of animals, including fecal matter, constitute detritus, which is the raw material for decomposition. Some of the steps in the process of decomposition are fragmentation, catabolism, leaching, humification etc.
The process of decomposition largely requires -

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:

Match List-I with List-II

List-I (Immunity barriers)	List-II (Examples)
(A) Physical barriers	(I) Interferons
(B) Cellular barriers	(II) Acid in the stomach
(C) Cytokine barriers	(III) Skin
(D) Physiological barriers	(IV) WBCS

 Choose the correct answer from the options given below: