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Let Vmg = velocity of man with respect to ground
Vtg = velocity of trolley with respect to ground
Vmt = velocity of man with respect to trolley = 1 m/s
Let M = mass of person = 80 kg
Let M' = the mass of trolley = 320kg
from relative motion 
Vtg = Vmg - Vmt --------------------(1)
Initially trolley and man were at rest initial momentum = 0
According to law of conservation of momentum
VmgM + VtgM' = 0
Substituting value of Vtg from equation (1) 
VmgM + (Vmg - Vmt)M' = 0 ⇒ Vmg(M + M') - VmtM'
Vmg(320 + 80) = 1(320) ⇒ Vmg = 320/400 = 0.8 m/s
Thus displacement of man with respect to observer = Vmg × time = 0.8 × 4 = 3.2 m

\(A=\begin{bmatrix}2&3\\5&7\end{bmatrix}\)

\(|A|=\begin{vmatrix}2&3\\5&7\end{vmatrix}\) = 2 x 7 - 5 x 3 = 14 - 15 = -1

∵ A_ij = (-1)^i+j M_ij

∴ A₁₁ = (-1)¹⁺¹ M₁₁ = 7,

A₁₂ = (-1)¹⁺² M₁₂ = -5,

A₂₁ = (-1)²⁺¹ M₂₁ = -3,

A₂₂ = (-1)²⁺² M₂₂ = 2

Cofactor matrix = C_A \(=\begin{bmatrix}A_{11}&A_{12}\\A_{21}&A_{22}\end{bmatrix}=\begin{bmatrix}7&-5\\-3&2\end{bmatrix}\)

Adjoint matrix of A \(=C^T_A=\begin{bmatrix}7&-5\\-3&2\end{bmatrix}^T=\begin{bmatrix}7&-3\\-5&2\end{bmatrix}\)

\(A^{-1}=\frac{adj(A)}{|A|}=\frac1{-1}\begin{bmatrix}7&-3\\-5&2\end{bmatrix}=\begin{bmatrix}-7&3\\5&-2\end{bmatrix}\)

Hence \(A^{-1}=\begin{bmatrix}-7&3\\5&-2\end{bmatrix}.\)

\(A=\begin{bmatrix}4&-2\\3&1\end{bmatrix}\)

We have A = IA

\(\begin{bmatrix}4&-2\\3&1\end{bmatrix}=\begin{bmatrix}1&0\\0&1\end{bmatrix}A\)

⇒ \(\begin{bmatrix}4&-2\\0&10\end{bmatrix}=\begin{bmatrix}1&0\\-3&4\end{bmatrix}A\) (By applying R₂ → 4R₂ - 3R₁)

⇒ \(\begin{bmatrix}1&-1/2\\0&1\end{bmatrix}=\begin{bmatrix}1/4&0\\-3/10&2/5\end{bmatrix}A\) (By applying R₁ → R₁/4 and R₂ → R₂/10)

⇒ \(\begin{bmatrix}1&0\\0&1\end{bmatrix}=\begin{bmatrix}1/10&1/5\\-3/10&2/5\end{bmatrix}A\) (By applying R₁ → R₁ + 1/2 R₂)

⇒ \(\begin{bmatrix}1&0\\0&1\end{bmatrix}=\frac1{10}\begin{bmatrix}1&2\\-3&4\end{bmatrix}A\)

We get I = A^-1A

∴ A^-1 \(=\frac1{10}\begin{bmatrix}1&2\\-3&4\end{bmatrix}.\)

Increase the power of t in every term by one and then divide the term by new power of t

As integration of 5t2-2t is =(5/3)t³-(2/2)t²

The differentiating electron i.e. last incoming electron in iron acquires 3d orbitals to give [Ar] 3d⁶, 4s² configuration with incomplete d-sub-shell, the characteristic of transition.

FeSO₂(aq) + 2NaOH(aq)  ------>Fe(OH)₂(s) + Na₂ SO₄ (aq)

(a) Rate = K[H2 O2 ][I] because second step is rate determining step.

 (b) Order = 1 + 1 = 2 

2Pb(NO₃)₂(s)+Heat ----->   2PbO(s) + 4NO₂(g) + O₂(g)

When quicklime or calcium oxide (CaO) reacts with water, slaked lime [Ca(OH)₂] is formed. During this reaction, a large amount of heat is released. So, this reaction is an exothermic reaction.
CaO(s) + H₂O(l)   ------>Ca(OH)₂(aq) +Heat
calcium oxide    water                calcium  hydroxide
  (Quicklime)                                                   (slaked lime)
                                                                

(a) This road leads to the Gandhi Maidan.

(b) I have seen this picture somewhere before.

(c) My mother cooks food.

According to Karl Marx “History is a story of the struggle between the haves and have nots”.

(a) A story was told to me by the teacher.

(b) Tools cannot be made by animals.

(c) Good men will be blessed by God.

The work of Herodotus was ‘The History of Persian Wars’.

Who Killed (kill) Caesar?

The secret journey of Prophet Mohammed from Mecca to Medina.

Karl Marx gave the economic interpretation of History.

By whom can it be done.

A story was told by mother.

Great general, dictator, reformer and the symbol of Ancient Rome.

St. Augustine wrote the book ‘The City of God’.

Shih Hu Wang Ti.

St.Augustine.

She was offered a chair.

Before the age of industrialisation the silk and cotton goods were totally dominated.The British goods were imported to India at a nominal rate.70%tax in silk and 80%tax in cotton,destroyed those industries.

(i) Silk and cotton products of India dominated the international market. India was known for its finer varieties of cotton. The Armenian and Persian merchants took these goods from Punjab to Afghanistan. Persia and Central Asia. Though most of the trade was carried through land routes, but the sea route was and Hoogly were the most important pore; which were used for trade.

(ii) A complex and complete market i Before the arrival of the outsiders, the Dade was handled by a variety of Indian merchants and bankers. The whole process of Dade basically involved three steps :
• Financing production
• Carrying or transporting goods
• Supplying goods to the exporters
Supply merchant; linked the port towns to the inland regions. They gave advances to weavers, procured the woven cloth from weaving village; and carried the supply to the ports At the port, the big shippers and export merchants had brokers, who negotiated the price. and bought goods from the supply merchants operating inland.

People wanting to see him and work in his own premises hold the author back from going out for a walk.

I usually go out with my friends.

Correct option is A) accurate

Correct option is C) How are you

Correct option is B) spirits

Correct option is E) We plant flowers/roses and tulips

Correct option is C) Sure. Here you are / Thanks

While walking I enjoy the sights along the street most.

Walking: 

Walking is an excellent form of exercise and it is good for our health. A person who walks for about three kilometres a day can maintain his physique very well. Walking gives exercise not only to the body but also the mind. As our muscles and body shake up, increasing the blood circulation, we see new things forcing our mind to think. Staying inside all the time makes one bored with life. Walking brings variety into our lives. Variety is the spice of life.

Doctors say that large numbers of people suffer from back pain due to lack of exercise. They say walking reduces back pain. When we walk, the fat accumulated in our body burns up. Then carbohydrates are burnt. Thus we clear our body from extra fat and carbohydrates.

Walking is an exercise which does not cost anything extra. You may need a good pair of walking shoes. These days there are many people who jog every morning or evening for health reasons. Jogging is walking at a greater speed, less than running. Walking is especially good for people who do sedentary jobs, which force them to sit down for long periods.

I like to go for walking with friends. When we walk, talking with friends, walking becomes doubly enjoyable. We forget the bodily exertion of walking and at the same time we exchange views on so many things. I prefer to walk early in the morning when the streets are not crowded. The air then will be fresh and without much dust. My advice to all the young boys and girls is to go to their schools on foot, if the distance is not much. Walking will make you healthy, wealthy and wise.

Two more such wall notices

i) ‘DRIVE SLOWLY’ 

ii) ‘CAUTION, SHARP CURVE’

The author does not take a vehicle to every destination. He never goes out of his way to avoid exercise. When a vehicle is essential he takes one.

I think there should be physical education in schools. I have many reasons to say so.

In Latin there is a saying ‘Mens sana in corpore sano’. In English it means ‘A sound mind in a sound body. A sickly body will not have a healthy mind. Physical education helps in the formation of a healthy and active life style. Such a life style makes the body and the mind sound.

Physical education helps in the development of leadership qualities. To be a leader one should have the capacity to organize, take initiative, guide and control his mates. He should be fearless and just. By taking part in physical education, one develops confidence, cooperation, and team spirit. Work is done in teams. Although individual excellence counts, it is the group work that makes a project successful. Any big event is the result of team work. Games are lost and won based on the team spirit. We all know that ‘United we stand, divided we fall’ and ‘Unity is strength’.

A positive attitude is very essential for success. An optimistic attitude keeps one cheerful and prepared to work hard. Perseverance is essential for any important work. Physical education helps one to build a positive attitude to life and things. When we keep our bodies in shape, our mind also gets shaped with the right attitudes. Physical education brings confidence and self-esteem. Only if we believe in ourselves, others will believe in us. Dr. Abdul Kalam said that each person is unique and he should try to remain he and no one else. We should feel that we are unique. We don’t have to imitate and copy others. Self-confidence and self-esteem grow by our being active in physical education.

Physical education also helps us to learn better. A fit body keeps the mind fit and alert. An alert mind grasps things fast and physical exercises help the brain to retain things for a long time. Physical education also instils civic virtues in us. We become more compassionate and loving. We become ready to help others. Physical education is a group activity and so group welfare, social welfare, becomes an important aspect of our training. Man is not an island. He is a social being and physical education drives into our mind this significant concept. In all schools all over the world physical education is compulsory because it has such a big roll to play in the life of an individual and also in the life of the society.

Nicholai: Welcome Ivan! Nice to see you after such a long time. 

Ivan: Niholai, you’ve changed a lot! You have become fat and old! 

Nicholai: Yes, Ivan. I’ve become fat. I spend much of my time inside the house. Ivan, you too have become old! 

Ivan: I can’t help. Time runs. Remember how young we were! Good old days! 

Nicholai: Okay, Ivan, let’s go and look around my estate. 

Ivan: How are you getting on, brother?

Nicholai: I’m doing very well. I am a landowner now. I am not the old official in the treasury. I’ve become a man of importance. 

Ivan : Good to hear that! You are happy, aren’t you?

Nicholai: I’m happy. When the peasants in my estate become sick they come to me and I treat them with soda and castor oil. On my birthday, a thanksgiving service is held in the middle of the village. I then give the peasants half a bucket of vodka. Okay, Ivan, let’s now go back to the house. It’s tea time. (They go back to the house and are having tea. There are gooseberries on the table.) 

Nicholai: Ivan, look at those gooseberries. They are from my estate. They are so nice. 

Ivan : Gooseberries. I don’t think I like them. They are too sour for me. 

Nicholai: Sour? No man, they are so delicious. I can eat a whole lot them. 

Ivan: Sorry Nicholai, I can’t eat them. You may eat. Maybe, you enjoy them so much because they grew in your estate. My idea of happiness is quite different. 

Nicholai: Well, I know. But for me they are so , delicious. 

Ivan: Enjoy them, Nicholai.

Happiness differs from person to person. That is why we say ‘One man’s meat is another man’s poison’ and ‘One man’s religion is another man’s madness’. Happiness, like beauty, is non-definable in precise terms. Still, there are some common ingredients, some characteristics that constitute a happy life. I definitely agree with the view that satisfaction, self-sufficiency and piety are some of the characteristic. Satisfaction comes when our desires are fulfilled. Desires are of various kinds. Some love adventure. Climbing the Himalayas, swimming across seas, flying into space, etc. are things some people enjoy and such things give them great satisfaction. For some satisfaction comes when they behold things of beauty. Sights, sounds and movements of nature and even people make them happy and satisfied. Standing on a hill and watching the sunset might satisfy the lovers of nature. Walking through the forest, watching the wild life might be a thrilling experience to some.

Self-sufficiency is an important characteristic of a happy life. If a person does not have the essential requirements like food, dress and shelter, he can’t be happy. He should have self-sufficiency in these matters. Depending on others for our needs is demeaning especially for an adult. Even birds and animals train their young to be self-sufficient when they grow up.

Piety is another important characteristic of a happy life. Piety does not mean mere religious piety. Piety also includes the love  and respect for our parents, elders, teachers, brethren, relatives, neighbours and people at large. We saw that piety of Mahatma Gandhi did not limit itself to some rituals and religious worship. His piety went much beyond that. A real pious man will also think of the welfare of his fellow beings. He will not do anything that brings pain and sorrow to others because he feels that all human beings are the children of the same God. “Ishwar”, “Allah” and “God” are synonyms of that Great Power that created us and that nourishes us.

There are many more characteristics of a happy life, but satisfaction, .self-sufficiency and piety seem to be most important. Once we have all these we definitely will be happy people.

The drawbacks of London are its endless noise and hustle, its smoky air, and its squalor.

The author’s excuse is ineffective in three ways: i) it is not believed; ii) it forces you to rise from your chair, go to the writing table and pretend to be writing a letter until the walk monger goes out of the room; iii) it won’t work on Sunday mornings because there is no post out till the evening.

The author’s excuse to avoid walking is to tell the people who call him for a walk that he has many letters to write.

The author says that walking for walking’s sake stops one’s brain because then a person’s power to instruct or to amuse when he is sitting on a chair or standing on a hearth-rug leaves him.

A man is not urged by his reasoning faculties to go for a walk. He is urged by something that surpasses reason, by his soul. His soul tells his body to go for a walk. The brain asks the soul where to go and for what purpose. The soul then replies that there is no definite destination or any particular mission.

have not gone out for a walk all my life. But I have been taken out for wal for t nostalgic for the old days when I had a perambulator. When I grew up I felt that one advantage of living in London was that nobody ever wanted me to come out for a walk. The endless noise and hustle of London, its smoky air and its squalor were unsuitable for walks.

When I was in the country and if there was no actual rain somebody might come any time and ask me to go out for a walk with him. People think there is something noble and virtuous in the wish to go for a walk. A person with such a desire feels that he has a right to impose his will on somebody sitting in an armchair and reading and ask him to accompany him. It is easy to say ‘No’ to an old friend. In the case of a mere acquaintance, one wants some excuse like ‘I wish I could, but …’. I always

have just one excuse: “I have some letters to write.” This excuse is unsatisfactory in three ways: i) it is not believed; ii) it forces you to rise from your chair, go to the writing table and pretend to be writing a letter until the walk monger, who does not dare to call you a liar and a hypocrite, goes out of the room; iii) it won’t work on Sunday mornings. There is no post out till this evening’, the walk monger will tell you and you will have to go with him quietly.

Walking for walking’s sake may be laudable and exemplary for those who practise it. My objection to it is that it stops the brain. Some people have told me that their brains work well when they walk along the high road or over hill and valley. Experience tells me that a person’s power to instruct or to amuse when he is sitting on a chair or standing on a hearth-rug leaves him when he takes one out for a walk. Plenty of ideas come to him when he is in a room. But

when he is out for a walk all those ideas evaporate. His encyclopaedic knowledge goes away. His imagination dries up. The man’s face becomes hard. Light goes from his fine eyes. He says that A (our host) is a thoroughly good fellow. Fifty yards further on, he adds that A is one of the best fellows he has ever met. We go for another furlong and then he says that Mrs. A is a charming woman. Then he adds that she is one of the most charming women he has ever known.

We pass an inn. He reads quickly to me, “The King’s Arms. Licensed to sell Ales and Spirits’. I foresee that during the rest of the walk he will read aloud any description that occurs. We see a milestone. He points to it with his stick and says, ‘Uxminster. 11 miles.’ We turn a sharp corner at the foot of a hill. He points at the wall and reads, ‘Drive Slowly’. Far ahead there is a small notice board. He sees it. He looks at it carefully and in due course he We pass an inn. He reads quickly to me, “The King’s Arms. Licensed to sell Ales and Spirits’. I foresee that during the rest of the walk he will read aloud any description that occurs. We see a milestone. He points to it with his stick and says, ‘Uxminster. 11 miles.’ We turn a sharp corner at the foot of a hill. He points at the wall and reads, ‘Drive Slowly’. Far ahead there is a small notice-board. He sees it. He looks at it carefully and in due course he says, Trespassers will be prosecuted.’ Poor man – he has become a mental wreck.

Luncheon at the A.S, brings his mind back. Once again he is the life and soul of the party. I think he will never go out for another walk after the bitter lesson of this morning. But an hour later I see him walking with a new companion. I watch him out of sight. I know what he is saying. He is saying that I am a rather dull man to go for a walk with. He will then say I am one of the dullest men he ever went for a walk with. Then he will start reading the inscriptions.

It is surprising how this deterioration happens in those who go for walking for walking’s sake. A man is not urged by his reasoning faculties to go for a walk. He is urged by something that surpasses reason, by his soul. His soul tells his body ‘Quick march!’ The brain then says, ‘Halt! Stand at ease!’ and sweetly asks the soul ‘To what destination and on what mission are you sending the body?’ The soul replies saying that ‘On no mission at all and to no destination’

People are always on the lookout for some Ulterior motive. The body goes out because it is a sure indication of nobility, probity and rugged grandeur of character. But the brain says it won’t be mixed up with the foolishness of the body and it would go to sleep till the walk is over. It will wake up again only when the body is safely placed inside the house.

Even if you go to some definite place, for a definite purpose, the brain wants you to take a vehicle. It does not insist on that and it will serve you well unless you are going out for a walk. While your legs are competing with each other, the brain will not do any thinking for you, but it will do any number of odd jobs for you. Even this the brain will do only if it is sure that the legs are making themselves useful and not merely fooling you about to gratify the pride of the soul.

This essay was composed in the course of a walk this morning. I do not take a vehicle to every destination. I never go out of my way to avoid exercise. I don’t despise the exercise of a walk because the sickly people are always talking about it or practising it to excess. I think in moderation it is good, physically. I will never go out for a walk until no people want me to see them or there is nothing for me to do in my own premises.

Going out for walk means you go out because you want to go out. Being taken out for a walk means you don’t go because you are keen on going, but somebody, whom you can’t refuse, forces you to go with him. If a very dear person comes and tells you to go for a walk with him, you can’t refuse.

After the walk I feel energetic and happy.