Remainder Theorem Proof

Theorem functions on an actual case that a polynomial is comprehensively dividable, at least one time by its factor in order to get a smaller polynomial and ‘a’ remainder of zero. This acts as one of the simplest ways to determine whether the value ‘a’ is a root of the polynomial P(x).

That is when we divide p(x) by x-a we obtain

p(x) = (x-a)·q(x) + r(x),

as we know that Dividend = (Divisor × Quotient) + Remainder

But if r(x) is simply the constant r (remember when we divide by (x-a) the remainder is a constant)…. so we obtain the following solution, i.e

p(x) = (x-a)·q(x) + r

Observe what happens when we have x equal to a:

p(a) = (a-a)·q(a) + r

p(a) = (0)·q(a) + r

p(a) = r

Hence, proved.

It is very difficult to say about the unknown pilot who helped the narrator. But probably it was the narrator himself that helped him to overcome the fear in the storm as no other plane was seen in the radar except the narrator’s Dakota plane. In that fearsome situation, he might have been hallucinating. He himself was a good pilot and brave enough who helped himself land safely

The woman in the Control Centre looked at the narrator strangely because the narrator asked him about the black aeroplane and she saw no one except the narrator’s in the sky during the storm. Even the radar showed only the narrator’s plane that night in the sky.

As the pilot entered the storm, his plane started jumping and twisting. He could not see anything outside the plane as it was black. When he looked at compass and other instruments they had stopped to function due to storm. It was a terrible and fearsome experience for him. The fuel tank was almost empty and he could not fly more than ten minutes. Then, he saw another black aeroplane by his side and the pilot of the plane signalled him to follow. It was a surprise for the narrator, as the other black plane had no light. He followed him without any choice and landed safely on the runway.

Answer: the narrator was happy to be alone high up above the sleeping country side he was flying his old Dakota aeroplane

He was dreaming of his holiday and looking forward to being with his family

The young seagull overcomes his fear due to his hunger the young seagull where forced to fly due to his hunger .

his mother tempted him to fly showing food when he start to fly he first feared and then enjoyed flying with his family 

ANSWER

Interval                Tally marks          Frequency

800−810            ∣∣∣                              3

810−820            ∣∣                               2

820−830            ∣                                1

830−840           ∣∣∣∣ ∣∣∣                        9

840−850           ∣∣∣∣                             5

850−860            ∣                                1

860−870            ∣∣∣                              3

870−880            ∣                                1

880−890            ∣                                1

Success can't be taken for granted. One has to struggle and finally to win it. The story of the young seagull is the story of overcoming hesitations, doubts and fears that stand between us and our success. Hunger makes him take the final plunge. This leads him to make his first flight and soar into space.

The young seagull was different from his brothers and little sister as they had had the courage to take the plunge from the ledge and learned to flap their wings and fly, whereas he had been too afraid to leap off the ledge. So, he was still stuck alone on that ledge whereas his brothers and sister were learning to perfect flying with the aid of their parents.

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Food items or substances that are combined to make a particular dish are called its ingredients. 

Anything which we eat to live is called food. For example: Roti ,bread ,butter ,fruits vegetables ,pulses, vegetable oil ,butter ,desi ghee ,eggs ,meat ,chicken ,fish ,egg etc.

The young seagull was afraid to fly because he felt certain that his wings would never support him to fly. He was frightened to see the sea around him. He was afraid that he would fall down into the sea. He had not eaten anything since the previous night

Primary schools were also opened in six villages to teach children. Kasturba taught the ashram rules on personal cleanliness and community sanitation. In order to improve the miserable health conditions Gandhiji got a doctor. He volunteered his services for six months.

Kasturba talked to the women to get rid of their filthy state of clothes. During his long stay at Champaran, Gandhiji taught people self-reliance and freedom from the fear of the British. Thus he paved the path for the freedom of india.

To make sure he had conquered fear, Douglas went to Lake Wentworth in New Hampshire Dived in and swan two miles across the lake to Stamp Act Island. Only once when he was in the middle of the lake, did the terror return. But he confronted it and swam on. He then swam across Warm Lake to the other shore and back .Thus, he was sure of having conquered his fear of water.

Jumbling: suhum

And: Humus

.Organisms which can make their own food from simple substances are called autotrophs

The term “autotrophic” is formed by the combination of two terms, “auto” meaning self, and “trophic” meaning nutrition. The literal meaning of this term is self-nutrition.

The autotrophic organisms contain a green coloured pigment called chlorophyll which helps in trapping energy from the sun. All green plants possess an autotrophic mode of nutrition. They prepare their own food by utilizing solar energy, water, and carbon dioxide by the process of photosynthesis. This results in the formation of glucose.

Not only humans but animals to cherish freedom. Freedom can't be bargained. In this case, a tiger may be well-fed and protected. However, the curtailment of his freedom keeps him in quiet rage'. He resents being behind the bars. He is a different animal when he is in his natural habitat, the jungle. He roams around the water hole and ambushes his favourite plump deer.

The vertical line in the Cartesian plane which determines the position of a point is called THE Y-AXIS

The horizontal line in the Cartesian plane which determines the position of a point is called THE X-AXIS

The vertical line in the Cartesian plane which determines the position of a point is called THE Y-AXIS

The horizontal line in the Cartesian plane which determines the position of a point is called THE Y-AXIS

At night, the caged tiger hears the last voices of the patrolling cars. Being locked in the cage, he looks towards the sky. He goes on staring the brilliant stars that are shining freely in the sky like his brilliant eyes.

we know ,

a³ + b³ + c³ -3abc = (a + b + c )(a² + b² + c² -ab -bc-ca)

now ,

a + b + c = 5
ab + bc + ca = 10

(a + b + c)² = a² + b² + c² +2(ab + bc+ca)
(5)² -2×10 = a² + b² + c²
a² + b² + c² =5

hence ,

a³ + b³ +c³ -3abc = ( a + b + c )(a² + b² + c² -ab- bc-ca)

=( 5)( 5 - 10) = 5 × (-5) = -25

hence proved

The tiger feels slave in the concrete cell. It feels like a prisoner. It misses its freedom in the forest where it should be lurking in the shadows; stalking its prey in the long grass near water hole, where deer come to drink water and where it lurks to pounce on them.
It can only walk a few steps in the cage. Many visitors come to see it but it does not pay any heed to them.
At night after everyone retires, it can only look at the stars from its cage.

Greece had been part of the Ottoman Empire since the fifteenth century.

• The growth of revolutionary nationalism in Europe sparked off a struggle for independence amongst the Greeks which began in 1821.

• Nationalists in Greece got support from other Greeks living in exile and also from many West Europeans who had sympathies for ancient Greek culture.

• Poets and artists lauded Greece as the cradle of European civilisation and mobilised public opinion to support its struggle against a Muslim empire.

• The English poet Lord Byron organised funds and later went to fight in the war, where he died of fever in 1824.

• Finally, the Treaty of Constantinople of 1832 recognised Greece as an independent nation.

Had he not been bound in his cage, the tiger would have been roaming around freely. He should have been growling around the houses of the human settlement at the outskirts of the jungle. There, he would open out his sharp long white teeth and paws just to terrorise the villagers.

The tiger is bound in his cage. Stripes on his skin are clearly visible. His heels look like pads an of velvet. On his pads of velvet, he walks quietly within his cage. He doesn’t seem to be happy to be in his unnatural abode. So, without showing his anger, he walks around quietly in his cage.

If a plant cell is kept in hypertonic solution, water comes out from the cell sap due to exosmosis. Water loss from the cell sap results in contraction of protoplast, which is referred to as plasmolysis. The cell is surrounded by the hypertonic solution, and the cell wall allows entry of most of the solutions. The hypertonic solution enters the cell wall freely but is not allowed to cross the cell membrane. 

Thus, the space between the cell wall and condensed protoplast is filled with hypertonic solution in a plasmolyzed cell.

• A n s w e r:
• Carbohydrates, proteins, fats, vitamins and minerals are essential components of food, these components are called nutrients.
• Carbohydrates, proteins, fats, vitamins and minerals are the components of food.