The Value of \(\int_1^3|2 x-1| d x\) equal to

The common region determined by all constraints of a Linear programming problem is called:

If a random variable \(X\) has the following probability distribution :

X	0	1	2	3	4
P(X)	k	2k	3k	\(k^2\)	\(6 k^2\)

then Match List-I with List-II

List-I	List-II
(A) k	(I) 3/7
(B) \(P(x<2)\)	(II) \(6 / 49\)
(C) \(P(X>3)\)	(III) \(1 / 7\)
(D) \(P(2 \leq X \leq 3)\)	(IV) 22/49

Choose the correct answer from the options given below :

If A and B are square matrices of order n × n, then \((A-B)^2\) is equal to:
(A) \(A^2-A B-B A+B^2\)
(B) \(A^2-2 A B+B^2\)
(C) \(A^2-2 B A+B A\)
(D) \(A^2-B^2\)
(E) \(A^2+2 A B+B^2\)

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:

Match List I with List II

LIST I	LIST II
A. \(\frac{d y}{d x}=e^{x-y}+x^2 e^{-y}\)	I. Linear Differential Equation of type \(\frac{d y}{d x}+P y=Q\)
B. \(\frac{d y}{d x}-2 y=\cos 3 x\)	II. Homogeneous Differential Equation
C. \(x^2 \frac{d y}{d x}=x^2-2 y^2+x y\)	III. Linear Differential Equation of type \(\frac{d x}{d y}+P x=Q\)
D. \(\left(x+2 y^3\right) \frac{d y}{d x}=y\)	IV. Variable separable

Choose the correct answer from the options given below:

A T.V. panel costing Rs 18000 has a useful life of 12 years. If the annual depreciation is Rs 1000, then its scrap value by linear method is :

Identify the statements which are true for both RNA and DNA.
(A) Two nucleotides are linked through 3'-5' phosphodiester linkage.
(B) Nitrogenous bases are linked to the sugar moiety.
(C) Additional -OH is present at 2'-position in the sugar.
(D) 5-methyl uracil is present.
Choose the correct answer from the options given below :

The value of \(\tan ^{-1} 5+\tan ^{-1} 3-\cot ^{-1} \frac{4}{7}\) is :

If \(\frac{1}{|x|-3} \leq \frac{1}{2}\), then the value of \(x\) is:

Read the passage carefully and answer the questions
Ethers with the general formula \(C _n H _{2 n+2} O\) and functional group C-O-C may be divided as symmetrical and unsymmetrical.
Lower symmetrical ether may be obtained by the dehydration of alcohols at low temperatures in the presence of protic acids,
such as \(H _2 SO _4\) and \(H _3 PO _4\) whereas at high temperature intramolecular dehydration takes place.
Formation of ether from primary alcohols is also accompanied by the formation of small amount of alkenes.
Due to the steric hinderance, secondary and tertiary alcohols undergo elimination reaction rather than substitution reaction yielding alkene as major product. Ethers can also be obtained by heating alkyl halides with sodium or potassium alkoxides.
Substituted Ethers (secondary & tertiary) can also be prepared using this method.
If primary halides are replaced with secondary and tertiary halides, alkenes are the major products. The compound with the lowest boiling point is

Particular solution of the differential equation \(\log \left(\frac{d y}{d x}\right)=x+y\) given that when x = 0 y = 0 is:

Read the passage carefully and answer the Questions
Carbonyl compounds such as aldehydes, ketones, and carboxylic acids play an important role in the biochemical process of plants and animal kingdom.
Many aldehydes and ketones find applications in dye, perfumery and other industries.
The carbonyl carbon atom is \(s p^2\)-hybridized forming three sigma and one pi-bond with oxygen.
C-O bonds are polar bond due to high electronegativity of oxygen than carbon.
Hence the carbonyl carbon behaves as an electrophilic center and oxygen as a nucleophilic centre.
Primary and secondary alcohols yield aldehydes and ketones respectively upon oxidation with variety of oxidizing agents.
The boiling points of aldehydes and ketones differ from hydrocarbons and ethers.
Aldehydes and ketones undergo nucleophilic addition reaction due to presence of electrophilic carbonyl carbon.
Nucleophile and a proton add across the carbon oxygen double bond.
Identify the product Y in the following reaction
\(\text{CH}_3\text{-CH = CH – CH}_2\text{OH}\) \(\xrightarrow{\text{Pyridinium Chlorochromate}}\) \(\rightarrow\text{Y}\)