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If y=y(x) is the solution curve of the differential equation x^{2} \mathrm{~d} y+\left(y-\frac{1}{x}\right) \mathrm{d} x=0 ; x>0$, and $y(1)=1, then y\left(\frac{1}{2}\right) is equal to:
Option: 1 3+e
Option: 2 3+e
Option: 3 3+e
Option: 4 3+e
Option: 5 3-\mathrm{e}
Option: 6 3-\mathrm{e}
Option: 7 3-\mathrm{e}
Option: 8 3-\mathrm{e}
Option: 9 \frac{3}{2}-\frac{1}{\sqrt{e}}
Option: 10 \frac{3}{2}-\frac{1}{\sqrt{e}}
Option: 11 \frac{3}{2}-\frac{1}{\sqrt{e}}
Option: 12 \frac{3}{2}-\frac{1}{\sqrt{e}}
Option: 13 3+\frac{1}{\sqrt{\mathrm{e}}}
Option: 14 3+\frac{1}{\sqrt{\mathrm{e}}}
Option: 15 3+\frac{1}{\sqrt{\mathrm{e}}}
Option: 16 3+\frac{1}{\sqrt{\mathrm{e}}}

x^{2}dy+\left ( y-\frac{1}{x} \right )dx= 0
\Rightarrow x^{2}\frac{dy}{dx}+y= \frac{1}{x}
\Rightarrow\frac{dy}{dx}+\frac{y}{x^{2}}= \frac{1}{x^{3}}
I\cdot F= e^{\int \frac{1}{x^{2}}dx}= e^{-\frac{1}{x}}
\Rightarrow y\cdot e^{-\frac{1}{x}}= \int e^{-\frac{1}{x}}\cdot \frac{1}{x^{3}}dx
Let\, \frac{-1}{x}= t\Rightarrow \frac{1}{x^{2}}dx= dt
\Rightarrow y\cdot e^{-\frac{1}{x}}= -\int e^{t}\cdot t\: dt
\Rightarrow y\cdot e^{-\frac{1}{x}}= -\left [ t\cdot e^{t}-\int e^{t} \; dt\right ]
\Rightarrow ye^{-\frac{1}{x}}= e^{-\frac{1}{x}}+\frac{1}{x}\cdot e^{-\frac{1}{x}}+C
\Rightarrow y= 1+\frac{1}{x}+C\cdot e^{\frac{1}{x}}

As\: y\left ( 1 \right )= 1
\Rightarrow 1= 2+C\cdot e\Rightarrow C= \frac{-1}{e}
\Rightarrow y= 1+\frac{1}{x}-\frac{1}{e}\cdot e^{\frac{1}{x}}
\therefore y\left ( \frac{1}{2} \right )= 1+2-\frac{1}{e}\cdot e^{2}
= 3-e

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Posted by

Kuldeep Maurya

The number of rational terms in the binomial expansion of \left(4^{\frac{1}{4}}+5^{\frac{1}{6}}\right)^{120}is ________
 

21

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Posted by

Anjali

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Study the table given below. Identify A, B, C, D, E and F in the table

Name of the 

Drug

Scientific name of

source plant

Effect on human 

organ/system
opioids 'A' 'B'
'C' Cannabis sativa 'D'
Cocaine 'E' F

 

 

 

 

 
 
 
 
 

A. opium poppy

B. Brain Cells that influence pleasure and pain,

C. marijuana

D. Brain Cells that influence pleasure, memory, thinking etc.

E. Coca plant

F. Brain cells as well as cause serious long term health problems.

 

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Posted by

Priyanka Kumari

What are recombinant DNA vaccines? Give two examples of such vaccines. Discuss their advantages.

In recombinant DNA vaccine, only one or two antigen is inserted into the DNA of a suitable host. Recombinant DNA technology enables to make vaccines on a larger scale with least chances of contamination. These are cheaper. Examples include Hepatitis B and HPV Vaccine

Cell culture is used to do large scale production. A suitable animal (ex - horse) is injected with strains of pathogen and antigens are produced in the horse's cells. Antigens are then extracted from horse's serum. However, there are many shortcomings to this method –

  • Due to the possibility of contamination, there are chances of contracting other diseases
  • The method is challenged by Animal Rights Activists.

Recombinant DNA technology can address these issues as yeast is required to produce the vaccine, and large scale production can be done at a lower cost.

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infoexpert21

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What is the Pulse Polio Programme of Government of India? What is OPV? Why is it that India is yet to eradicate Polio?

Pulse Polio: Launched in 1995-96, it is a mass immunisation programme aimed at eradicating the poliomyelitis virus from India. This programme has been undertaken with active cooperation from government agencies, NGOs, UNICEF, and CDC.

Key Objectives of Pulse Polio:

  • Not a single child should be missed.
  • High level of surveillance.
  • Reporting of any new case within 14 days.

To cover all the children below age 5, Polio booths are established throughout the country and volunteers are hired to administer the polio vaccine.

OPV (Oral Polio Vaccine): OPV is an attenuated vaccine, which is produced at a sub-physiological temperature by the passage of the virus through non-human cells. This produces spontaneous mutation in the viral genome. A single vial of OPV usually contains 10-20 doses.

India was declared a Polio free nation on 27th March 2014 as there were no new cases in the previous three years.

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infoexpert21

Drugs like LSD, barbiturates, amphetamines, etc., are used as medicines to help patients with mental illness. However, excessive doses and abusive usage are harmful. Enumerate the major adverse effects of such drugs in humans.

Adverse Effects of LSD: Most common side-effect of LSD is depression. Other side-effects include Schizophrenia or sever uterine contractions (in female) 

Adverse Effects of Barbiturates: Side-effects of Barbiturates include -

  •             Tolerance - Person needs higher doses as time passes
  •             Dependence - Person cannot live without this drug.

Adverse Effects of Amphetamines: Mental depression and bouts of hallucinations are common side effects.

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infoexpert21

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What are the methods of cancer detection? Describe the common approaches for the treatment of cancer.

Following are the methods of cancer detection:

Biopsy: A small fragment of the concerning tissue is extracted and then goes through histopathological study.

Blood Test and Bone Marrow Test: Increased levels of cell count can be gauged from this test.

Imagery Technique: Internal organs can be diagnosed by X-ray, CT scan and MRI.

Apart from the above mentioned techniques, antibodies against cancer-specific antigens and molecular biotechnology are also used in certain cases.

Common approaches of treatment involve surgery, radiotherapy, and immunotherapy:

  1. Surgery: It can be used to remove the tumour.
  2. Radiotherapy: Radioactive rays can be utilised to destroy cancerous cells, with minimal impact to normal cells
  3. Immunotherapy: Chemotherapeutic drugs are used to destroy cancer cells. It involves the use of Interferon alfa, which helps the body's immune system to respond to cancer.

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Posted by

infoexpert21

In your locality, if a person is addicted to alcohol, what kind of behavioural changes do you observe in that person? Suggest measures to overcome the problem.

A person addicted to alcohol may avoid socialising, may become easily irritable and suffers from spells of rage and depression. Some people even resort to petty crimes if they are unable to afford the money to buy alcohol.

To address the situation:

  • A qualified counsellor should be pressed into service.
  • People closest to them should spend more time with the person
  • The person must get help to reduce or totally withdraw from any alcohol consumption

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infoexpert21

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 Why do some adolescents start taking drugs? How can this be avoided?

Many adolescents who start taking drugs do so because of their curiosity and adventure-seeking nature. Moreover, under peer pressure, many adolescents end up taking drugs to experiment, and continued use leads to addiction.

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Posted by

infoexpert21

Compare the lifestyle of people living in urban areas with those of rural areas and briefly describe how lifestyle affects their health.

People in Urban Areas

People in Rural Areas

Most of the people do jobs which involve sitting for long hours.

Most of people do jobs which involve manual work

Have a sedentary lifestyle

Have an active lifestyle

Can easily avail processed food

Primarily eat unprocessed and homemade

Environmental pollution is high

Environmental pollution is low

High risk of Diabetes and cardiovascular

Low risk of Diabetes and cardiovascular

Can receive quality medical attention

May not receive quality medical attention

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infoexpert21

JEE Main high-scoring chapters and topics

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