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The correct option is C) (1111)₂

The decimal number 15 into binary will be (1111)₂

(b) 2 

f(x) = |x – 1| + |x – 3| + sin x is not differentiable at x = 1, and x = 3

Answer is (b) -1/7

f(x) = x – cos x

∴ f'(x) = d/dx (x – cos x)

= 1 – (-sin x) 

= 1 + sin x

Now, -1 ≤ sin x ≤ 1 for all x ∈ R 

∴ -1 + 1 ≤ 1 + sin x ≤ 1 for all x ∈ R 

∴ 0 ≤ f'(x) ≤ 1 for all x ∈ R 

∴ f'(x) ≥ 0 for all x ∈ R 

∴ f is increasing for all x.

Let y = 2x - 7 5x²

⇒ y ′= 2 − 10x = 0, for max/min

⇒ x = 51​

Also y′′ = −10 < 0

so maximum value of y is = y(\(\frac{1}{5}\)​) = 2⋅\(\frac{1}{5}\) ​−7 − 5 ⋅ \(\frac{1}{25}\) ​

= \(\frac{2}{5}\) − 7 − \(\frac{1}{5}\) 

​= \(\frac{1}{5}\) ​− 7 

= −\(\frac{34}{7}\)​

And minimum value of y will be −∞

min. value = 4

Since the square of any no. Is positive, the given function has no maximum value.

The minimum value exists when the quantity (5x -1)² = 0

Therefore, minimum value = 4

The functions e^-x and sin x are continuous and differentiable on their domains.

∴ f(x) = e^-x sin x is continuous on [0, π] and differentiable on (0, π).

Now, f(0) = e⁰  sin 0 = 1 × 0 = 0 

and f(π) = e^-π sin π = e^-π × 0 = 0 

∴ f(0) = f(π)

Thus, the function f satisfies all the conditions of the Rolle’s theorem.

It is given that f (x) = – (x – 3)² + 9

Here (x – 3)² ≥ 0

We can write it as

– (x – 3)² ≤ 0

By adding 9 on both sides

– (x – 3)² + 9 ≤ 0 + 9

Where f (x) ≤ 9

So the maximum value of f (x) = 9 which occurs at x = 3 which is the point of absolute maxima and minimum value does not exist.

max. value = 6

Since |x+4| is non-negative for all x belonging to R.

Therefore the least value it can have is 0 .

Hence value of function is 6

It has no minimumvalue as it can have infinitely many.

max. value = 4, min. value = 6

f(x) = sin2x+5

We know that,

-1≤sinӨ≤1

-1≤sin2x≤1

Adding 5 on both sides,

-1+5≤sin2x+5≤1+5

4≤sin2x+5≤6

Hence

max value of f(x) = 2x+5 will be 6

Min value of f(x) = 2x+5 will be 4

Tungsten is a poor conductor of heat and electricity

Tungsten was a poor conductor of heat and electricity the bulb in the circuit would not glow because there wouldn't be electricity passing through the circuit. Tungsten is also used as an electrode in gas tungsten arc welding. Electricity passes through the tungsten (negative) and moves to the work piece (positive) for most applications.

According to question,

(5i + j - 3k).(3i - 4j + 7k)

= 15 - 4 - 21 = 15 - 25 = -10

Answer is (b) -10

The general formula of magnetic field consists of the circumference of the body through which the magnetic field is passing in the denominator and in the numerator, it consists of current passing through the body and a proportionality constant.

B(2πR) = μo^I

Homopolymer : PVC, Nylon-6 

Copolymer : Terylene, Buna-S

A polymer made from only one type of repeating units is called homopolymer. in some cases the repeating unit is formed by condensation of two distinct monomers. 

Examples : Polythene, PVC. Nylon-6.

Concave mirror is used by dentists.

The dentist holds a small concave mirror in such a way that the tooth lies within its focus. A magnified image of the tooth is then seen by the dentist in the concave mirror. Since the tooth looks much bigger, it becomes easy to examine the defect in the tooth.

In heterolytic cleavage the bond breaks in such a fashion that the shared pair of electrons remains with one of the fragments.

The Banyan tree is a mythological one, and when his grandmother calls it along with the other trees to be sacred. A religious sentiment is brought out in not just the poet but the reader as well. When the tree is slaughtered on the orders of the poet's father, a mystery is revealed. First the scraggy aerial roots were brought down unleashing age old trunk that had a circumference of fifty feet. It was a tree that had witnessed ages and held all the knowledge knotted inside. When it was brought down it felt as if all the mythology was revealed to the poet. The darkness and the concealed, the enlightenment it has secreted away since so long. The reasons and the answers to all the logic all was beheld by the tree and now was slaughtered.

A banyan tree's life cycle is a unique one, it grows as an epiphyte. It may live for more than 200 years or even 400-450 years. In contrast an average human life is of 70-80 years. It is clearly visible that one banyan alone may survive generations of humans. Banyan has strangled roots knotted in its trunk that grow for centuries. It transcends human life and rolls and grows with the knowledge that it witnesses all the while. The poet has subtly juxtaposed a human life to that of a banyan. For all its mythical hugeness that makes it an enigma a human is thrown into utter bewilderment and surprise for all the knowledge that it secrets away inside its vast trunk, which grows with its aerial roots that grow to reach the ground. They resemble the lives that it has lived again and again witnessing all the human lives that descended and perished with time standing tall and erect.

There are many expressions in the poem that reveal the critical tone of the poet towards the felling of the tree: 

“Its scraggy aerial roots fell to the ground” 

“Sawing them off for seven days and the heap was huge” 

“Insects and birds began to leave the tree” 

“Fifty men with axes chopped and chopped” 

“We watched in terror and fascination this slaughter”

There are many legends surrounding the sheoga, the oudumber, the neem and most of all the banyan tree. These are considered mythologically relevant and holy in Hinduism. Old folk who are deeply religious consider it sin to cut down these trees as they are to be worshipped according to the holy scriptures. They even say that if one brings neem or peepal down by felling them they are cursed with ill fate. There are plenty of stories in our mythology that fear and plague our society with as many superstitions as possible. Thus, the poet is merely trying to convey the fears and religious beliefs of old folk like his granny.

In the climax of the poem, the poet with his family moved to Baroda. There, the poet sees banyan tree. The memory of the banyan tree in his own garden rushes to him. And now that the tree in reality is dead, for his father brought it down, the poet carries its memories in his faded dreams. The poet personifies the banyan tree by saying that this tree is the one that grows in one's dreams, i.e., in poet's dreams. There in the dreams, the tree seethes, i.e., boils. It is to be considered for why the poet has used the second term. Is it possible that the tree is in anguish, because it was brought down, and thus it boils in anger while it grows in the poet's dreams though dead in reality.

In Bhagvad Geeta, Krishna said, “Of all the trees I am Banyan Tree”. Banyan is considered to be the most pious tree in Hindu mythology. It is believed that Buddha renounced under the banyan tree in Bodh Gaya. There are many more such stories that this tree encompasses in its age old aerial roots that fall to the ground mysteriously. The poet is impressed by its huge figure as he says, “the great tree revealed its rings of two hundred years”. The physical description of the banyan carves an image of an enigma. The tree's aerial roots dangling from above to reach out to the ground. They are the proof of all the years and decades the tree has lived. It is a tough tree and it takes more that usual effort to bring it down. And when it happens, all watched it fall in terror and fascination, the slaughter of the age old banyan tree. It fees as if a raw mythology was revealed it age to the poet and his folk.

There are such words in the poem that help one understand the nature of the poet's father.

 The father was a much practical man. Probably because the whole family was moving to Baroda, the father got all the trees removed and the surrounding property demolished. He was not an emotional man but a man of actions. 

“the structures were demolished” 

“but he massacred them all” 

“My father ordered it to be removed”

Constancy is not just the theme of the poem but also it is the most important thing in any and every relationship. In the poem, there are many such phrases that express this quality. A few of the expressions are listed below. 

1. Love is not love which alters 

2. Love is not love which … bends 

3. It is an ever-fixed mark 

4. Never shaken

5. Love's not time's fool

1. The above passage has been taken from the lesson A Letter to God. The writer of this lesson is G.L. Fuentes.

2. The hailstones affected Lencho’s field badly. The corn was totally destroyed.So Lencho’s soul was filled with sadness.

Lencho needed hundred pesos to sow his field and to live until the next crop. He received only seventy pesos.

Lencho wanted from God one hundred pesos to sow his fields again and for his family to live.

Lencho, a poor farmer needed a shower badly in his fields. He was sure that it would rain as he had seen clouds in the North-East. One day big drops of rain began to fall. Lencho went out and was very happy as the raindrops amounted to coins. His fields will yield rich crops due to these raindrops. But suddenly a strong wind began to blow and along with the rain very large hailstones began to fall. The smaller boys ran out to collect hailstones which appeared like frozen pearls. For an hour the hail rained everywhere including Lencho’s corn field. The field was white as if covered with salt. The flowers were gone from the plants. The corn was totally destroyed. Lencho’s soul was filled with sadness. His single hope was-help from God. So he asked God for money.

Lencho hoped that there would be a good shower of rain for his crop.

The rain with large hailstorm had ruined Lencho’s field of ripe corn. So he was sad. His only hope was—help from God. So he asked God for money.

Lencho calls the raindrops new coins. The raindrops would enrich the harvest. He will get more money by selling it. So the raindrops have been called new coins.

Yes, Lencho is sure that it is going to rain because he has been looking at the sky since morning. The thick clouds in the north-east sky made the rain sure.

At meal time big drops of rain began to fall.

Lencho wrote in his letter to God that he badly needed one hundred pesos to sow his fields again and for his family to live until the crop. He went to town, placed a stamp on the envelope and dropped it into the mailbox at the post-office.

The postmaster collected some money from his employees. He himself gave a part of his salary. He also made several of the friends to contribute for the noble cause.

The rain turned into hailstorm and it destroyed Lencho’s crop completely.

The view could be seen from the house was beautiful. One could, in the valley, see the river and the field of ripe corn dotted with flowers. It promised a good harvest.

Lencho wrote to God to send him the rest of money as he needed it badly. But he requested God not to send the money this time through mail because the post-office employees were dishonest.

If I am13 year sold, I have gone round the sun 13 times.

Lencho was angry after opening the letter because he found only seventy pesos inside the envelope. He had demanded a hundred pesos from God. Instead of getting angry on God, he got angry on the post office employees because he thought that they must have misappropriated the money.

The postmaster read the letter.

In order to not break Lencho’s firm belief / faith in the Almighty / God, the postmaster decided to answer the letter as God himself. He collected money from his employees, friends and contributed a part of his salary to send to Lencho. He put the money in an envelope and sent it.

1. d. Lencho felt satisfied when he saw his fields covered with the curtain of rain.

2. a. The hailstones resembled silver coins.

3. b. These new silver coins turned into hailstones which destroyed all his crops and ruined his hopes.

4. a. Lencho was satisfied to see his crops.

5. a. The word 'draped' means the same as 'cover'.

Lencho was filled with sadness after the rain / hail had stopped. His corn was totally destroyed. He said to his sons that they would have no corn that year. But he was hopeful of getting help from God.

Poets are the bards celebrating the nature surrounding them. While, scientists are  the ones to harvest nature and its mechanism and mark inventions. Poets such as  Wordsworth and Keats criticise humans of exploiting nature. Whereas, scientists  on the other hand utilize the given resources of nature to create and invent.  However, it is not that there is an enmity between poets and scientists. Shelley said,  undoubtedly the promoters of utility, in this limited sense, have their appointed  office in society They make space and give time.Here we have Darwin, who enjoyed literature immensely, however, until he was thirty. He said later,' My mind seems to have become a kind of machine for grinding general laws out of large collections of facts but why this should have caused the atrophy of that part of the brain alone on which the higher tastes depend.Thus, it can be understood, while the poets celebrate the present and arrest it making it all immortal, the scientists create and invent leading us to a tomorrow, thus, marking a difference.

A In the essay patterns of creativity, S. Chandrasekhar tries to figure out the reason for the difference in the patterns of creativity among the practitioners in the arts and practitioners in the sciences. He did not answer it, rather, he made an assortment of remarks that bore the answer. He cites examples explaining how poets and scientists view each other defining the difference in their views. There are poets such as Wordsworth and Keats who are worshippers of nature, who believe that humans sabotage nature by the technological advancement. However, there are poets like Shelley, who do poetry on science. It is difficult to segregate the views and put them into water tight compartments. Darwin, for instance enjoyed literature immensely as it gave him utmost joy, but only till the age of 30. W. B. Yeats, in praise of Shelley's A Defence of Poetry, called it the profoundest essay on the foundation of poetry in the English language The author of the essay, Chandrasekhar wonders in the end that why is there no such A Defence of Science written by a scientist of equal endowment. Perhaps the answer to the question he knew already.

Yudhistira had sent Nakula to bring water. But Nakula did not return for a long time. Therefore he sent Sahadeva to see what the matter was.

There are two perspectives to every issue. While poets like Wordsworth and Keats condemn man of exploiting nature and moving towards science, Shelley is a scientific poet, who even in his poems like his Cloud. Shelley loved science and expressed it in his poetry.? It symbolised to him joy, and peace, and illumination. Charles Darwin, being such a great scientist was immensely fond of literature, especially in his youth. However, another scientist, Faraday, who was absolutely engrossed in his scientific experiments about electricity and made great invention. It is always difficult to conclude whether poets and scientists are compatible or not. There will be many such poets and scientists fond of science and poetry. While there will be many who are only concerned about their subject.