\(\int \frac{1}{(x+1)(x+2)} d x\) is equal to

If \(A\) is an invertible matrix of order 3 and the determinant of \(A\) is 9, then the determinant of \(A^{-1}\) is :

The value of the determinant \(\left|\begin{array}{ccc}a+b & b+c & c+a \\ c & a & b \\ 1 & 1 & 1\end{array}\right|\) is :

The derivative of \(\sin \left(\tan ^{-1} e^{2 x}\right)\) with respect to x is:

Let \(f(x)=x|x|\). Then \(f(x)\) is:

Interval in which the function \(f\) given by \(f(x)=\tan x-4 x, x \in\left(0, \frac{\pi}{2}\right)\) is strictly decreasing is

Two positive numbers \(x\) and \(y\) such that \(x+y=60\) and \(x y^3\) is maximum are :

Consider the differential equation \(x d y=(x+y) d x\). Which of the following are true?
(A) It is a homogenous differential equation
(B) It is a differential equation of order 2
(C) The general solution of the differential equation contains 2 arbitrary constants
(D) Integrating factor of differential equation is \(\frac{1}{x}\)
(E) Degree of the differential equation is not defined
Choose the correct answer from the options given below:

An electric source has an emf of 130 V and a terminal voltage of 110 V. If it supplies a terminal current of 20 A, its internal resistance will be

Read the passage carefully and answer
Carbohydrates are essential for both plants and animals. Some common examples of carbohydrates are glucose, fructose, sucrose, glycogen, cellulose and starch. They form a major portion of our food. Honey has been used for a long time as an instant source of energy in Ayurvedic medicine. Carbohydrates are used as storage molecules in both plants and animals. The cell wall of bacteria and plants is made of cellulose. We use cellulose in the form of wood for making furniture and as cotton fiber for clothing. They also provide raw material for textiles, paper, lacquers, and breweries. Carbohydrates are classified as Monosaccharides, Oligosaccharides and Polysaccharides on the basis of their behavior towards hydrolysis.
Monosaccharides cannot be hydrolyzed further to simpler units whereas oligosaccharides can yield 2-10 units of monosaccharides on hydrolysis. Polysaccharides, which are non-sugars yield a large number of monosaccharide units on hydrolysis.
Answer the following questions about carbohydrates:
Animal starch is popularly known as

Read the following paragraph carefully and answer the given questions.
In the year 1963 the two enzymes responsible for restricting the growth of bacteriophage in E. coli were isolated. The first restriction endonuclease - Hind II, whose functioning depended on a specific DNA nucleotide sequence was characterized five years later. It was found that Hind II always cut DNA molecules at a particular point by recognizing a specific sequence of six base pairs. This specific base sequence is known as the recognition sequence for Hind II. Besides Hind II, today we know more than 900 restriction enzymes that have been isolated from over 230 strains of bacteria, each of which recognize different recognition sequences.
When DNA from different sources is cut by the same restriction enzymes, the resultant DNA fragments have the :

Rahul invested ₹ 20000 in a mutual fund in year 2018. If the value of mutual fund increased to ₹ 32000 in year 2023, then the compound annual growth rate of his investment is : [given that \((1.6)^{\frac{1}{5}}=1.098\) ]

The function f defined by \((x)=\left\{\begin{array}{ll}a x+1, & x \leq 3 \\ b x+3, & x>3\end{array}\right.\) is continuous at x = 3 if