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Showing posts with label Class 9. Show all posts
Showing posts with label Class 9. Show all posts

Nicholas Nye - Class 9 English

Nicholas Nye

By, Walter De La Marie

Thistle and darnell and dock grew there,
And a bush, in the corner, of may,
On the orchard wall I used to sprawl
In the blazing heat of the day;

Half asleep and half awake,
While the birds went twittering by,
And nobody there my lone to share
But Nicholas Nye.

Nicholas Nye was lean and gray,
Lame of leg and old,
More than a score of donkey’s years
He had been since he was foaled;
He munched the thistles, purple and spiked,
Would sometimes stoop and sigh,
And turn his head, as if he’d said,
'Poor Nicholas Nye!'

Alone with his shadow he’d drowse in the meadow,
Lazily swinging his tail,
At break of day he used to bray,-
Not much too hearty and hale;
But a wonderful gumption was under his skin,
And a clean calm light in his eye,
And once in a while; he’d smile:-
Would Nicholas Nye.

Seem to be smiling at me, he would,
From his bush in the corner, of may,-
Bony and ownerless, widowed and worn,
Knobble-kneed, lonely and gray;
And over the grass would seem to pass
‘Neath the deep dark blue of the sky,
Something much better than words between me
And Nicholas Nye.

But dusk would come in the apple boughs,
The green of the glow-worm shine,
The birds in nest would crouch to rest,
And home I’d trudge to mine;
And there, in the moonlight, dark with dew,
Asking not wherefore nor why,
Would brood like a ghost, and as still as a post,
Old Nicholas Nye

Word meanings:

darnet    a type of grass commounly found growing in fields in Europe

dock    a plant of the buckwheat family with greenish or reddish flowers and long, broad leaves

may    the may blossom; a hawthorn plant

sprawl    slump; lounge; to lie with the arms and legs spread awkwardly

lone    lonely (state)

foal    a young horse

thistle    a prickly plant with purple flower heads surrounded by thorny leaves

gumption    courage, bravery

knoble-knee    knees that are uneven and bent inwards

    Exercises    

  1. Questions

    1. In what way does the poet feel close to the donkey?
    2. Ans: The poet feels close to the donkey as each of them give the other a company. When the speaker sprawled alone in the orchard lonely, there was only Nicholas Nye with whom he could share his loneliness. The poet could empathize his loneliness with the miserable condition of the donkey.

    3. How does the poet pass his day?
    4. Ans: The poet passes his day in the blazing heat by sprawling on the orchard wall half asleep and half awake.

    5. How does the donkey pass his day?
    6. Ans: The donkey would pass his day by munching(chewing) the thistles. Sometimes it would stoop and sigh. In evening it would bray though not full of energy and liveliness, however, there would be boldness to make a smile showing the calmness in his eye.

    7. What physical description does the poet give of the donkey?
    8. Ans: Nicholas Nye - the donkey, was thin with gray colored hair, its one leg was lame(defective) and it was old as its age was over 20 years.

    9. What characteristics does the poet see in the donkey?
      Which phrases give us clues about his character?
    10. Ans: The poet describes the donkey as lonely this clue can be derived from the phrase "Alone with his shadow he’d drowse in the meadow, and Knobble-kneed, lonely and gray;". The poet also described it as gritty, inspite of all miseries there was calmness in its eye and a smile once in a while, this trait can be derived from the phrase "Not much too hearty and hale;..., And a clean calm light in his eye, And once in a while; he’d smile:-".

    11. How are the natural aspects of the field and its surrounding described by the poet?
    12. Ans: The field is an orchard covered with thistles, darnell, dock, in one corner of the orchard was a bush with may. The orchard had walls, birds flew and twittered in the orchard.

  2. Reference to context

  3. Read these lines from the poem, then answer the questions.

    1. Half asleep and half awake,
      While the birds went twittering by,

      1. Who or what is half asleep?
      2. Ans: The poet(speaker) is half asleep.

      3. When and where is the subject half asleep?
      4. Ans: The subject is half asleep on the orchard wall.

      5. Is the subject alone?
      6. Ans: The subject was lonely in the orchard,however, there was Nicholas Nye(the donkey) as well in the orchard.

    2. But a wonderful gumption was under his skin,
      And a clear calm light in his eye,

      1. What does 'gumption' mean?
      2. Ans: Gumption means gritty (showing courage and resolve).

      3. Why is it surprising that the subject has gumption?
      4. Ans: The subject (now the donkey) has gumption(grit willed) because inspite of all his miseries there was calmness in his eyes and a smile once in a while.

      5. What is being said prior to this about the subject?
      6. Ans: Before mentioning 'gumption' the poet(speaker) tells about his physical characteristics - lean, gray haired, old, one leg lame; then the speaker tells how it would munch the thistles sometimes stooping and heaving a sigh as if to describe its miserable condition, further the speaker tells that it would be half asleep in the meadow, lazily waging its tail, in evening it would bray though not full of energy and liveliness.

The Tide Rises, the Tide Falls - Class 9 English

The Tide Rises, the Tide Falls

By, Henry Wadsworth Longfellow

The tide rises, the tide falls,
The twilight darkens, the curlew calls;
Along the sea-sands damp and brown
The traveller hastens towards the town,
    And the tide rises, the tide falls.

Darkness settles on roofs and walls,
But the sea, the sea in the darkness calls;
The little waves, with their soft, white hands,
Efface the footprints in the sands,
    And the tide rises, the tide falls.

The morning breaks; the steeds in their stalls
Stamp and neigh, as the hostler calls;
The day returns, but nevermore
Returns the traveller to the shore,
    And the tide rises, the tide falls.

Word meanings:

twilight    the light from the sky when the sun is below the horizon

curlew    a large wading bird

hastens    walking hurriedly

efface    to erase a mark from a surface

hostler    person employed to care for horses



    Exercises    

  1. Questions

    1. What time of day is it in each stanza?
    2. Ans: In the first stanza it is evening, in the second stanza it is night and in the third stanza it is morning.

    3. Where is the traveller going?
    4. Ans: The traveller is going back to the town.

    5. Which verb describes the traveller's movement; what idea does it create?
    6. Ans. The traveller's movement is described by "hastens", it means that the traveller is waking fast.

    7. What human attributes does the sea have, and what does it do with them?
    8. Ans. The tides of sea rises and falls and so does such resemble human attributes - ups and downs, happiness and sadness of life. As the sea waves clear the traces of footsteps on the sand and moves on so does human; the obstacles, hindrances in one's life are overcome and we keep moving on.

    9. Which parts of the poem seem old fashioned to you?
    10. Ans: The repetition of the lines "And the tide rises, the tide falls" seem old fashioned as one is already aware of this eternal and never-ending process.

    11. Which parts of the final stanza could symbolize the start of a working day?
    12. Ans: "The morning breaks";"the steeds in their stalls stamp and neigh, as the hostler calls"; "Returns the traveller to the shore" symbolizes the start of a working day.

    13. In what way are the cycles of time and tide constant (and keep going)?
    14. Ans. The tide rise and the tide fall can be co-related with life: birth and death; the rising of the tide symbolizes birth and the falling of the tide symbolizes death. This phase is eternal and we cannot evade it.

    15. What is the significance of the different times of the day mentioned in the poem?
    16. Ans. The significance of the different times of the day mentioned in the poem conjugate with the passing of time in a human life cycle; the first stanza starts at evening which signifies old age; the second stanza describes night which signifies death; the third stanza describes morning which signifies birth.

    17. What is the tone of the speaker in the poem?
    18. Ans. The tone in the poem starts with a sad, subdued note and then gradually shifting into a peaceful, calm, relaxed tone towards the end.



  2. Reference to context

  3. Read these lines from the poem, then answer the questions.
    1. Along the sea-sands damp and brown

      1. Which living things are on or near the sea-sands?
      2. Ans: The "curlew" and the "traveller".

      3. What are they doing?
      4. Ans: The curlew is calling, and the traveller is hastening towards town.

      5. What happens to all living things?
      6. Ans: All living things face death.

      7. Why do you think the poet uses the word sea-sands rather than the beach?
      8. Ans: The poet uses the word sea-sands rather than the beach to sync with the words "damp and brown", if the poet would have used beach then it would have meant a pebbly or sandy shore. Also sea-sand is a better comparative than beach to comprehend the line "Efface the footprints in the sands".

    2. ...but nevermore
      Returns the traveller to the shore,

      1. What does return to the shore?
      2. Ans: The traveller returns to the shore.

      3. Will anyone know that the traveller was there?Why/why not?
      4. Ans: This is somewhat vague as the poet doesn't specify details, he leaves us in a trail of mysteriousness.

      5. What do you think has happened to the traveller?
      6. Ans: The poet's withholding of information is key to the poem's meaning. The speaker narrated about one's life journey which is unknowable, inevitable and final. A presumption can be derived from the line "The traveller hastens towards the town," that the traveller must have reached an old age.

    3. Darkness settles on roofs and walls,
      But the sea, the sea in the darkness calls;

      1. What time of day is being referred to?
      2. Ans: The time of day being referred to is night time.

      3. Why does the sea appear to call?
      4. Ans: The sea cannot call but the poet personifies it to life and states death is calling.

      5. Give an antonym of the word "darkness".
      6. Ans: The antonym of the word "darkness" in accordance of the poem is "morning" or "hope".

    4. But the sea, the sea in the darkness calls;

      1. Who calls in the darkness?
      2. Ans: The sea calls in the darkness.

      3. What are the things being done at this point?
      4. Ans: The things that are being done at this point are : the waves of the sea erases the footprints in the sands, and the tide rises and the tide falls.

      5. Why is the word "sea" repeated in the poem?
      6. Ans: The word "sea" is repeated in the poem to personify "life" as the tides keep rising and falling and so does birth and death happens in a life-cycle.

    5. The day returns, but nevermore
      Returns the traveller to the shore,

      1. Why does the traveller not return to the shore?
      2. Ans: The traveller does not return because he must have died.

      3. What is meant by "nevermore"?
      4. Ans. "Nevemore" means never again.

      5. What time of day is mentioned here?
      6. Ans: Morning time is referred here.

Important Algebra Formulas and Identities with Step by Step

Important Algebra Formulas and Identities

Formula - Formula is a mathematical expression or rule.
Identity - An identity is an equation that is true for all the values of the variables.

Formulas

  1. (a + b)2 = a2 + b2 + 2ab
  2. (a - b)2 = a2 + b2 - 2ab
  3. a2 - b2 = (a+b)(a-b)
  4. a2 + b2 = (a+b)2 -2ab
  5. (a+b+c)2 = a 2 + b 2 + c 2 + 2ab + 2bc + 2ac
  6. (a-b-c)2 = a 2 + b 2 + c 2 - 2ab + 2bc - 2ac
  7. (a + b)3 = a3 + b3 + 3a2b + 3ab2
                    = a3 + b3 + 3ab(a+b)
  8. (a - b)3 = a3 - b3 - 3a2b + 3ab2
                    = a3 - b3 - 3ab(a-b)
  9. a3 + b3 = (a+b)(a2 - ab + b2)
  10. a3 - b3 = (a-b)(a2 + ab + b2)

Deducing the Important Formulas

1. a2 = a * a


2. (a + b)2 = (a + b)(a + b)                  [ Back to Formulas ]

                = a2 + ab + ab + b2

                = a2 + 2ab + b2


3. (a – b)2 = (a – b)(a – b)                  [ Back to Formulas ]

               = a2 - ab - ab + b2

               = a2 - 2ab + b2


4 (i). a2 + b2 = (a + b)2 – 2ab                     [ Back to Formulas ]

[As (a + b)2  = a2 + 2ab + b2 and to get a2 + b2  we only need to add – 2ab]

 (ii). a2 + b2 = (a - b)2 + 2ab  

[As (a - b)2  = a2 - 2ab + b2 and to get a2 + b2  we only need to add + 2ab]


5 (i). a2 - b2 = (a - b)2 + 2ab – 2b2                  [ Back to Formulas ]

[As (a - b)2  = a2 - 2ab + b2 and  to get a2 - b2 we only need to add  + 2ab and – 2b2]  = (a - b)2 + 2b (a – b)

          = (a – b) [ (a – b) + 2b]

          = (a – b)(a + b)

 (ii). a2 - b2 = (a + b)2 - 2ab – 2b2

[As (a + b)2  = a2 + 2ab + b2 and  to get a2 - b2 we only need to add  - 2ab and – 2b2]

          = (a + b)2  - 2b (a + b)

          = (a + b) [ (a + b) - 2b]

          = (a + b)(a – b)


6.(a + b + c)2 = (a + b + c)(a + b + c)                  [ Back to Formulas ]

                   = a(a + b + c) + b(a + b + c) + c(a + b + c)

                   = a2 + ab + ac + ab + b2 + bc + ac + bc + c2

                   = a2 + b2 + c2 + ab + ab + ac + ac + bc + bc

                   = a2 + b2 + c2 + 2ab + 2ac + 2bc


7.(a - b - c)2 = (a - b - c)(a - b - c)                  [ Back to Formulas ]

                   = a(a - b - c) - b(a - b - c) - c(a - b - c)

                   = a2 - ab - ac - ab + b2 + bc - ac + bc + c2

                   = a2 + b2 + c2 - ab - ab - ac - ac + bc + bc

                   = a2 + b2 + c2 - 2ab - 2ac + 2bc


Prepositions - Class 7, Class 9 - Grammar

Preposition - A preposition is a word placed before a noun or a noun-equivalent to show its relation to some other word in the sentence.
Phrasal Prepositions - These are phrases working as prepositions.

Example - by means of, on account of etc.

Some Important Prepositions


Since & For

Since is used before a point of time, while for is used before a period of time.

Example - I did not see you for a long time.
My cousin brother has been here since Monday last.

Since and From

Both since and from are used before a point of time but since is preceeded by a verb in the perfect tense, while from can be used with any tense.

Example - Albert has started rowing from (or since) Monday last.
Albert started rowing from yesterday. (not since)
Albert started rowing from today. (not since)
Albert will start rowing from tomorrow. not since

Before, by & within

Before and by are used with a point of time, while within is used with a period of time. Before means any time within specified limit of time and by means not after the specified limit of time.

Example - You must come back by 8pm. (not after 8pm)
You must come back before 8pm. (any time before 8pm)
Albert came back within an hour. (not before)

In & within

In means at the end of while within means before the end of.

Example - The movie will end within an hour. (before the hour is passed)
The movie will end in an hour. (at the end of the hour)

In & into

In refers to position already inside anything and into refers to a movement towards the inside of anything.

Example - I am in the garden.
I went into the garden.

In & at

In refers to a much wider space or time than at.

Example - Lady Hydari Park is at Shillong in Meghalaya.

In & after

In is used about the future time, while after is used about the past.

Example - I will come in a few minutes.
I have left after an hour.

On & at

On is used before a particular date or day and at before a particular hour.

Example - I shall come on Friday at 9 o'clock.

Between & among

Between is used about two persons or things, while among is used for referring to more than two persons or things.

Example - The money was divided between the two workers.
She is the most beautiful among all the girls in her class.

Beside & besides

Beside means by the side of while besides means in addition to.

Example - Besides Mayank, his friends also sat beside me.

By & with

By is used with the doer or agent while with is used before the object with which a person does a thing.

Example - The online test was not done by me.
The online test was done with the help of my friend.

Except & excepting

Except means without while excepting means without excluding.

Example : Everyone came to the party except Albert. (Albert did not come)
Everyone not excepting Albert came to the party. (Albert came)

To

To is used to refer direction or destination.

Example - After the lockdown, students need to go to school.

Tenses - Class 7, Class 9 - Grammar

Tenses - Class 7, Class 9 Grammar

Tense - The tense is the change of form in a verb to express the time of an action. There are three principal tenses.

  • The Present Tense describes an action in the present time.
  • The Past Tense describes an action in the past time
  • The Future Tense describes an action in the future time
  • Each of these principal tenses has four forms : Indefinite, Continuous, Perfect, Perfect Continuous.
  • Tenses Present Past Future
    Indefinite Rule Subject + V1 form Subject + V2 form Subject + will/shall + V1 form
    Use Universal truth, habitual action Action in the past Action that will happen in the future
    Continuous Rule Subject + is/am/are + V (ing) form Subject + was/were + V (ing) form Subject + will be/shall be + V (ing) form
    Use Action going on at present Action that was goint on at past Action as going on at some future time
    Perfect Rule Subject + has/have V3 form Subject + had + V3 form Subject + will have/shall have + V3 form
    Use Action just finished Action complete before another past action Action that will be completed in future
    Perfect Continuous Rule Subject + has been + V (ing) form + since/for Subject + had been + V (ing) form + since/for Subject + will have been/shall have been + V (ing) form + since/for
    Use Action going on and is not finished Action had been going on in the past No longer in practical use

    Force - Class 9 Science

    Force - Class 9 Science

    Important Points

    Force - Force generally denotes push or pull. Force can :
    i. produce motion
    ii. stop motion
    iii. change the direction of motion
    iv. change the dimension in a body
    The SI unit of Force is N

    Newton's Laws of Motion

  • Newton's First Law of Motion - A body will continue in its state of rest or uniform motion in a straight line, unless compelled by some unbalance applied force to change its state of rest or uniform motion.
  • Newton's Second Law of Motion - The rate of change of momentum of a body is directly proportional to the applied unbalanced force and takes place in the same direction in which the force is applied.
  • Newton's Third Law of Motion - To every action there is an equal and opposite reaction.
  • Momentum - The force(impact) which a body possesses due to the combined effect of mass and velocity is called momentum. Momentum is denoted by p, where p = m*v where m-mass, v-velocity
    The SI unit of momentum is kgms-1
  • Mass - Mass is the measure of an object's inertia, more the mass more the inertia. It is a property of material , so it does not change with place.
    The SI unit of mass is kg
  • Law of Conservation of Momentum - In a given system, the sum total of momentum is a constant quantity, provided no external force acts on the body or
    In a given system, when two or more bodies interact and no external force acts on them, the total momentum of all bodies is conserved.
  • Inertia - Mass of a body is the measure of its inertia, i.e., the more the mass of a body, the more is its inertia.
  • Law of Inertia - The tendency of a body to continue in its state of rest or uniform motion in a straight line, unless some external unbalanced force is applied.
  • Inertia of rest - The tendency of a body at rest to remain at rest unless acted by an external force is called inertia of rest.
  • Inertia of motion - The tendency of a body in motion to remain in uniform motion in a straight line unless acted by an external force is called inertia of motion.

  • Questions and Answers

    • Give two examples from everyday life where the Newton's third law of motion comes into place.
    • Ans. Example 1 - Rubber Ball rebounding: When we hit a rubber ball on the ground with some force(action), the ground reacts in opposite direction with an equal force and hence the rubber ball rebounds.
      Example 2. Hurting of hand when we hammer a nail on a wodden plank - When we hit a nail with hammer on a wooden plank we apply a force then the nail reacts back with equal force on the hammer. As the hammer is held firmly in our hand, therefore, we feel hurt in our hand.

    • Why does a boatman push the river bank backward with a long bamboo pole, on launching his boat in water? Explain.
    • Ans. It is based on Newton's third law of motion. When the boatman standing in the boat, pushes the river bank backward with a long bamboo pole, the surface of the bank reacts back and pushes the pole in the forward direction. As the pole is in the hands of the boatman, standing in the boat, the whole system moves in the forward direction.

    • Why is it difficult to walk on marshy land? Explain.
    • Ans. It is based on Newton's third law of motion. When we push the marshy soil with our feet the soil yields. Thus, the reaction of the marshy soil is not as much as the action done on it. This makes it difficult to walk on marshy land.

    • Why does a boatman push water backward with the oars, while rowing a boat? Explain.
    • Ans. It is based on Newton's third law of motion. When a boatman pushes water backward with the oars then water also exerts a reaction force which enables the boat to move forward.

    • Why does a boatman tie his boat to a pillar, before allowing the passengers to step on the river bank? Explain.
    • Ans. When the passengers start disembarking, they push the floor of the boat backward with the feet. Now the boat is in water, and the water yields under the impact of this force. Thus, the boat starts sliding backward. To avoid the boat from sliding backward into water, the boatman ties his boat with a rope to a pillar before allowing passengers to disembark.

    • Why do the birds flap their wings downward while taking a flight? Explain.
    • Ans. It is based on Newton's third law of motion. When the birds flap their wings downward they do action on the wind. The wind in turn reacts back and pushes the wings and hence the birds fly in the upward direction.

    • Do the action and reaction act on the same body or different bodies? How are the action and reaction related to each other in :
      (a)magnitude and (b)direction? Do they act simultaneously or not?
    • Ans. Action and reaction act on two different bodies. Action and reaction are equal in magnitude but they act in opposite directions so there is simultaneous action and reaction.

    • Why does a gun recoil backward when fired? Explain.
    • Ans. When the bullet under the impact of exploding gun powder moves out of the barrel from gun with a certain momentum, the gun, in order to conserve momentum, moves with the same momentum (momentum equal to that of bullet) in backward direction. So the net momentum is again zero. Thus, the gun recoils backward.

    • Why does an inflated baloon rise up vertically for some distance, when punctured from below? Explain.
    • Ans. Initially, the balloon and the air in it, are in a state of rest and hence, have zero momentum. However, when the balloon is punctured from below, the air rushes out of it in downward direction with a certain momentum. Thus, in order to conserve momentum the balloon rises up vertically with the same momentum.

    • Why do the pieces of cracker fall in all directions when it is burst? Explain.
    • Ans. When a cracker is not burst the momentum of the cracker is zero. When it is being burst the pieces move in all directions with certain momentum. However, if we add up the momentum of various pieces, the sum total of momentum will be zero.

    • Why is it easier to stop a tennis ball than a cricket ball, moving at the same speed? Explain.
    • Ans. It is easier to stop a tennis ball than a cricket ball, moving at the same speed because the momentum of the cricket ball will be more compared to the tennis ball as the latter's mass is higher than the former's mass hence force required to stop the tennis ball will be less compared to the cricket ball.

    How to write a Notice (Notice Writing)

    How to write a Notice, Class 9

    Notice - A notice can be described as a written or printed information. It is written to provide information about an activity or an event.

    An example of Notice

    Borkhola Boy's School, Cachar

    NOTICE

    Celebration of Yoga Day

    30th May, 2021

    Students are informed that “Yoga Day” will be observed on 21st June, 2021 in the school premises. Every student is requested to bring a “Yoga Mat”. For more information contact the undersigned.


    Malay Das
    Student Secretary


    Key points to remember while writing a notice

    • Formal/Polite language
    • Avoiding quote, writing in passive voice
    • Avoiding first person, second person
    • Should be short/compact/precise in minimum words possible
    • There should be a difference between date of issue of notice and date of event
    • The content must include complete information we should try to incorporate the 7 W's (who, what, where, why, when, in what way and for whom) while writing the notice.
    • The notice should be presented within a box

    Active and Passive Voice

    Active and Passive voice - Class 7, Class 9

  • Active voice - The form of the verb which shows the importance of its subject(doer of the action) is called the active voice.

  • Passive voice - The form of the verb which shows the importance of its object(receiver of the action) rather than the subject, it is called the passive voice.


  • General Rules to express a passive voice:

    1. The Passive voice is formed by using appropriate forms of the verb be with the third form of the main verb.
    2. The object takes the place of the subject.
    3. The subject is either left out or mentioned at the end as a mere agent of the action done.
    4. The verb does not change its tense, but its form only.
    5. The preposition by is mostly used to show the subject as an agent of action. Sometimes other prepositions like to, with, at and in etc. are used in place of by.
    6. Prepositions inseperable attached with the verb in the active voice are not dropped while using by.
    7. There is no passive voice of the verbs used in Future Continuous tense and the three (Present/Past/Future)Perfect Continuous forms.

    Rules :Chaning various forms of verbs from active voice to passive voice

  • Present Indifinite Tense - Passive voice is formed by using is, am, or are before the third form of the verb.
  • Present Continuous Tense - Passive voice is formed by using is or am or are + being + third form of verb.
  • Present Perfect Tense - Passive voice is formed by using has or have + been + third form of verb.
  • Past Indefinite Tense - Passive voice is formed by using was or were + third form of verb.
  • Past Continuous Tense - Passive voice is formed by using was + were + being + third form of the verb.
  • Past Perfect Tense - Passive voice is formed by using had + been + third form of verb.
  • Future Indefinite Tense - Passive voice is formed by using will/shall + be + third form of verb.
  • Future Perfect Tense - Passive voice is formed by using will/shall + have been + third form of verb.
  • Motion, Class 9 Science

    Motion - Class 9 Science

    Motion - Motion is the movement of any object from one point to another with respect to observer. An object is said to be in motion when it changes its position with time.

    Rest - A body is said to be at rest if it does not change its position with respect to its surroundings with time.

    Different Types of Motion

  • Linear Motion - Linear motion is a motion in which a body moves in a straight line. Example - Marchpast by soldiers, A car moving on a straight road.
  • Circular Motion - Circular motion is a motion in which a body moves along in a curved path. Example - Motion of satellites around planets, A car moving in a traffic roundabout
  • Rotational Motion - Rotational motion is a motion in which a body rotates about a fixed axis. Example - Motion of wheels around vehicles, Spinning of a top
  • Vibratory Motion - Vibratory motion is a motion in which a body moves in a to and fro direction. Example - Motion of a simple pendulum, Motion described by the string of a violin when
  • Rest and Motion are relative terms - Rest and motion are considered as relative terms because they both depend on observer's frame of reference.

    Uniform motion - When an object covers equal distance in equal intervals of time.

    Non-uniform motion - When an object covers unequal distance in equal intervals of time.

    Circular motion - Circular motion is the movement of an object along the circumference of a circle or rotation along a circular path.

    Uniform circular motion - Uniform circular motion can be described as the motion of an object in a circle at a constant speed.

    Non-uniform circular motion - Non-uniform circular motion denotes a change in the speed of a particle moving along a circular path.

    Direction of motion at any point in a circular path - As an object moves in a circle, it is constantly changing its direction. At all instances, the object is moving tangent to the circle.

    Equation of speed in a uniform circular motion, if the radius of the circle is r and time taken is t: v = circumference/time or v = 2πr/t, Note - Here v is scalar and not vector.

    Circumference of Circle = 2πr (r is the radius of the circle)
    Difference between Circular motion and linear motion
    Circular motion Linear motion
    In circular motion, the speed is constant but the direction of the object changes continuously, hence it is accelerated In linear motion, the speed and direction of the object is fixed and so it is not accelerated
    Motion of earth around the sun A car moving on a straight road

    Magnitude - Magnitude is the size or extent of a physical quantity.

  • Scalar quantity - Scalar quantity are only expressed only in magnitude. Example - Time, Distance, Mass, Temperature, Length, Area, Volume etc.
  • Vector quantity - Vector quantity are expressed in magnitude as well as the direction of the object. Eg. Velocity, Displacement, Weight, Force, Acceleration etc.
  • Difference between scalar and vector quantity
    Scalar Quantity Vector Quantity
    They are expressed in magnitude only They are expressed in magnitude and direction
    They can be added by simple arithmetic means They cannot be added by simple arithmetic means
    They cannot be easily plotted on graph They can be plotted on graph
    They are one dimensional They are multidimensional
    Area, Pressure, Density, Temperature, Mass Momentum, Force, Acceleration, Displacement, Velocity

    Distance - It is the actual length of the path travelled by a moving body irrespective of the direction in which the body moves.

    Displacement - The shortest distance of a moving body from the point of reference(initial position of body).

    Difference between Distance and Displacement
    Distance Displacement
    It is the actual length of path travelled by a moving body It is the shortest distance between the initial and final position of the body
    It has only magnitude(scalar quantity) It has both magnitude and direction(vector quantity)
    It is always positive It may be positive, negative or zero
    SI unit of distance is m SI unit of displacement is m

    Speed - Speed is the defined as the distance covered in unit time. Speed = Distance/Time

    Average Speed - Average speed is the total distance travelled in a particular interval of time or we can say average speed is the total distance travelled in total time interval. Average speed = Total Distance/Total time

    Uniform speed - When the distance travelled by a body is equal in equal interval of time.

    Non-uniform speed - When the distance travelled by a body is unequal in an equal interval of time.

    Velocity - Velocity is the displacement of the object in unit time. Velocity is speed with direction. Example - Car moving at 40km/h is its speed whereas the car moving at 40km/h westward is the velocity

    Difference between speed and velocity
    Speed Velocity
    Speed is the distance covered in unit time. Speed = Distance/Time Velocity is the displacement covered in unit time. Velocity = Displacement/Time
    Speed is a scalar quantity Velocity is a vector quantity
    Speed is rate of change of distance Velocity is rate of change of displacement
    Speed can never be zero Velocity can be positive, negative, zero
    SI unit of speed is m/s SI unit of velocity is m/s

    Acceleration(a) - Rate of change of velocity, a = change of velocity/time or, a = (v - u)/t, here v - final velocity, u - initial velocitySI unit of acceleration is m/s2

    Uniform acceleration - Uniform acceleration is the acceleration in which the velocity of a body changes at a constant rate in a given interval of time.Example - Motion of a ball dropped from a height.

    Non-uniform acceleration - Non-uniform acceleration is the acceleration in which the the velocity of a body does not change at a constant rate in a given interval of time.Example - A car travelling 60 km in 1 hour and 70 km in 2nd hour

    Equations of motion

  • v = u + at
  • s = ut + 1/2 at2
  • v2 = u2 + 2as
  • Polynomials - Class 9 Maths

    Polynomials - Class 9 Maths

    Constant - The symbol which has a fixed numerical value is called a constant

    Variable - The symbol which assumes different values in different situations is called a variable

    Coefficient - A numerical or constant quantity before the variable

    Operator - The symbol of a mathematical operation

    Exponent - A number or a value or letter written above and to the right of a mathematical expression

    Algebraic Expression - The combination of constants and variables by the use of some or all operations as x2+5y-7

    Terms - The various parts of an algebraic expression seperated by operators are called terms of the algebraic expression

    Polynomial - An algebraic expression is called a polynomial if the variable involved have only non-negative integral powers as 7x3-5x2+6x-9

    Degree of a Polynomial

    (i)When the polynomial is in one variable - In this case, the highest power of the variable is called the degree of the polynomial.
      Example - 7x2+x is a polynomial in x whose degree is 2
      7y3+y is a polynomial in y whose degree is 3
     (ii)When the polynomial is in two variable - In this case, the sum of the powers of the variables in each term is taken up and the highest sum, so obtained, is called the degree of the polynomial.
      Example - 5x2y3-5x4y3-3x2y2, Note carefully for the first expression the sum of the powers is 5, for the second expression the sum of the powers is 7 and for the third it is 4, as 7 is the highest sum so the degree of x and y is 7
      For multiple variables, we follow the above step.
    

    Types of Polynomials

    • Zero Polynomial - A polynomial which consists of 0 only is called a zero polynomial. The degree of a zero polynomial is not defined.
    • Constant Polynomial - A polynomial which consists of only one constant term is called a constant polynomial. The degree of a constant polynomial is zero. Each real number is a constant polynomial. Example 5, -7/8, -3 are constant polynomials
    • Linear Polynomial - A polynomial of degree one is called a linear polynomial. Example 5x+7 is a linear polynomial in x, 5x + 7y + 6 is a linear polynomial in x and y
    • Quadratic Polynomial - A polynomial of degree two is called a quadratic polynomial. Example 5y2+7 is a quadratic polynomialin y, xy + x + 2 is a quadratic polynomial in x and y
    • Cubic Polynomial - A polynomial of degree three is called a cubic polynomial. Example 5x3+2x2+6 is a cubic polynomial in x, 5x2y+4xy2+5 is a cubic polynomial in x and y
    • Biquadratic Polynomial - A polynomial of degree four is called a biquadratic polynomial. Example 5x4+8 is a biquadratic polynomial in x, 5x3y+6xy3+7 is a biquadratic polynomial in x and y

    Number of terms in a Polynomial

    • Monomial - A polynomial which has only one non-zero term. Example 5, -7
    • Binomial - A polynomial which has only two non-zero terms. Example 5 + 7x, x - 7y, 3x2y+2xy has
    • Trinomial - A polynomial which has only three non-zero terms. Example 5+7x+x2, xy+yz+zx

    Exercise 2(a)

    Exercise 2(b)

    Co-ordinate Geometry - Class 9 Maths

    Co-ordinate Geometry - Class 9 Maths

    Heron's formula - Class 9 Maths

    Heron's formula - Class 9 Maths

    Heron's formula or Hero's formula, named after Hero of Alexandria, gives the area of a triangle when the length of all three sides are known.

    Some important points that needs to be known

    • Perimeter of triangle whose sides are a,b,c is a+b+c
    • Semi-perimeter of triangle whose sides are a,b,c is (a+b+c)/2
    • Pythogora's theorem is H 2=P 2+B 2, here H is hypotenuse, P is perpendicular and B is base
    • Area of right angled triangle is 1/2*base*altitude
    • Equilateral triangle has all the three sides equal
    • Isosceles triangle has two sides equal

    According to Heron's formula if we have the three sides of triangle then we can find the area by the below formula :

    Area of triangle = √ s(s-a)(s-b)(s-c), here s is the semi-perimeter

    By Heron's method finding area of equilateral triangle
      Three sides of equilateral triangle are equal
      Now s = (a+a+a)/2 = 3a/2
      Putting the value of s in √ s(s-a)(s-a)(s-a) we will get
      √ 3a/2 (3a/2-a)(3a/2-a)(3a/2-a) = √ 3a/2 * a/2 * a/2 * a/2
      = √ 3 /4 a 2
    

    There are other ways by which we can find the area of triangle if we have the three sides

    • Area = √ 4a 2b 2-(a 2+b 2-c 2)/4
    • Area = √ 4b 2c 2-(b 2+c 2-a 2)/4
    • Area = √ 4a 2c 2-(a 2+c 2-b 2)/4

    Trick to remember the above 3 formulas if we take the first two sides then in bracket the third side will be subtracted

    We will do a few problems for understanding -

    Exercise

    Q1.Find the area of triangle, the lengths of whose sides are 18cm, 24cm and 30cm

    Solution -
    Given lengths are 18cm, 24cm and 30cm
    s = (18 + 24 + 30)cm/2
      = 36cm
    Area of triangle = √ s(s-a)(s-b)(s-c)
    Putting the values in the above formula we get
    Area of triangle = √ 36(36-18)(36-24)(36-30)
                   = √ 36 * 18 * 12 * 6
                   = 216 cm 2
    

    Q2.Find the are of triangle whose sides are 120cm, 150cm and 200cm.

    Solution -
    Given lengths are 120cm, 150cm and 200cm
    s = (120 + 150 + 200)cm/2
      = 235cm
    Area of triangle = √ s(s-a)(s-b)(s-c)
    Putting the values in the above formula we get
    Area of triangle = √ 235(235-120)(235-150)(235-200)
                   = √ 235 * 85 * 115 * 35
                   = 8966.57 cm 2 [Find the square root using long division method]
    

    Q3.The base of a right-angled triangle measures 48cm and its hypotenuse measures 50cm. Find the area of the triangle.

    Solution -
    Given base is 48 cm and hypotenuse is 50
    Area of right angled triangle is 1/2*base*altitude
    But we do not know the altitude(perpendicular/height), to find the altitude we will use Pythogora's theorem - H 2=P 2+B 2
    P 2 = H 2 - B 2
    Putting the values we get - P = √ 50 2 - 48 2
    Trick - As 48 2 may take some time we can use the formula a 2-b 2 = (a+b)(a-b)
    Now P = √ (98)(2)
    or P = √ 7 2*2 2
    or P = 14cm
    Now putting the values in the Area formula for right angled triangle we have
    Area = 1/2 * 48 * 14
         = 336 cm 2
    

    Q4.Find the area of triangle whose sides are 91cm, 96 cm and 105 cm in length. Find the height corresponding to this greater side.​

    Solution -
    Given lengths are 91cm, 96cm, 105cm
    s = (91 + 96 + 105)/2 = 146
    Area of triangle = √ s(s-a)(s-b)(s-c)
    Putting the values in the above formula we get
    Area of triangle = √ 146(146-91)(146-96)(146-105)
                   = √ 146 * 55 * 50 * 41
                   = √ 2 * 73 * 5 * 11 * 2 * 5 * 5 * 41
                   = 10 √ 164615
                   = 10 * 405.728 [Find the square root using long division method]
                   = 4057.28 cm 2
    Now, height corresponding to the greater side can be found if consider that in a right angled triangle resting on the hypotenuse, the base is the longest side with the altitude on the base being short
    Area of right angled triangle is 1/2*base*altitude
    Putting the values we get 4057.28 = 1/2*105*H or H = 77.28 cm
    

    Q5. Find the area of an isosceles triangle each of whose equal sides measures 13 cm and whose base measures 20 cm.

    Solution -
    For an isosceles triangle we know two sides are equal.
    Given lengths are 13cm, 13cm and 20cm
    s = (13 + 13 + 20)cm/2
      = 23cm
    Area of triangle = √ s(s-a)(s-b)(s-c)
    Putting the values in the above formula we get
    Area of triangle = √ 23(23-13)(23-13)(23-20)
                   = √ 23 * 10 * 10 * 3
                   = 10 √ 23*3 [Find the square root using long division method]
                   = 83.07 cm 2
    

    Q6.An isosceles triangle has perimeter 30 cm and each of the equal sides is 12 cm. Find the area of the triangle.

    Solution -
    For an isosceles triangle we know two sides are equal so we have two sides as 12cm, 12cm
    Given perimeter = 30cm
    Perimeter of a triangle = a + b + c (where a,b and c are the sides)
    Let the third side be x, now to find x we put the values as
    30 = 12 + 12 + x
    or x = 6
    As perimeter = 30 cm so semi-perimeter(s) = 30/2 cm = 15 cm
    Area of triangle = √ s(s-a)(s-b)(s-c)
    Putting the values in the above formula we get
    Area of triangle = √ 15(15-12)(15-12)(15-6)
                   = √ 15 * 3 * 3 * 9
                   = 9 √ 15 [Find the square root using long division method]
                   = 34.83 cm2

    Q7.The base of an isosceles triangle measures 80cm and its area is 360sq.cm. Find the perimeter of the triangle (in cm).

    Q8 (i).If the area of an equilateral triangle is 36 √ 3 cm 2, find its perimeter.

    Solution -
    Area of equilateral triangle is √ 3/4 a 2
    Putting the values we get
    36 √ 3 cm 2 = √ 3/4 a 2
    or a 2 = 36 * 4
    or a = 12 cm
    Perimeter of a triangle is a + b + c, so the perimeter of the equilateral triangle is (12 + 12 + 12)cm = 36 cm

    Q8 (ii). If the area of an equilateral triangle is 8 √ 3 cm 2 find its height

    Solution -
    Area of equilateral triangle is √ 3/4 a 2
    Putting the values we get
    8  √ 3 cm 2 = √ 3/4 a 2
    or a 2 = 32
    or a = √ 32
    or a = 4 √ 2
    As, the altitude(height) bisects the opposite side in perpendicular
    Now, area of right angled triangle is 1/2 * base * height
    Substituting the values we get
    8 √ 3 = 1/2 * 8 * height
    or height = 2 √ 6 cm
    

    Q9.The sides of a triangle are in the ratio 3:5:7 and its perimeter is 300m. find its area

    Solution -
    Let the coefficient of the ratios be x
    Now, perimeter of a triangle is a + b + c
    Putting the values we get
    300 = 3x + 5x + 7x
    or x = 300/15 = 20
    Now the sides are 3*20 cm, 5*20 cm, 7*20 cm
    or the sides are 60 cm, 100 cm, 140 cm
    As perimeter = 300 cm, then semi perimeter will be 300cm/2 = 150 cm
    By Heron's formula the area of triangle is √ s(s-a)(s-b)(s-c)
    Putting the values we get
    √ 150(150-60)(150-100)(150-140)
    or √ 150*90*50*10
    or √ (10*3*5)*(10*9)*(10*5)*(10)
    or √ 104 * 32 * 52 *3
    or 102 * 3 * 5 √ 3
    or 1500 √ 3 cm2
    

    Q10.The height of an equilateral triangle measures 9cm. Find its area, take √ 3 = 1.732

    Solution -
    
    Height of an equilateral triangle = √ 3 a/2 Given height = 9, so a = 18/√ 3 Now, area of equilateral triangle is √ 3 /4 a 2 Putting the value of a we get Area = √ 3 / 4 * (18/√ 3) 2 or Area = 3 √ 3 * 9 = 3 * 1.732 * 9 = 46.764 cm 2

    Quadrilaterals

    Quadrilateral – A quadrilateral can be defined as a closed two dimensional figure which has four sides.

    Diagonal – A diagonal is a straight line which joins the two opposite corners of a quadrilateral.

    Types of Quadrilaterals

  • Rectangle – A rectangle is a quadrilateral whose opposite sides are equal and parallel and all the angles are 90 degrees.
  • Square – A square is a quadrilateral all of whose sides are equal,the opposite sides are parallel to each other and each angle measures 90 degrees.
  • Rhombus – A rhombus is quadrilaterals whose four sides are equal and opposite sides are parallel to each other. Also the opposite angles are equal and diagonals bisect each other at 90 degrees
  • Parallelogram – A parallelogram is quadrilateral whose opposite sides are parallel to each other also the opposite angles are equal.
  • Trapezium or Trapezoid – A trapezium is a quadrilateral with one pair of sides parallel.
  • Area and perimeter of quadrilaterals
    Quadrilateral Type Area Perimeter
    Rectangle l*b [l-length, b-breadth] 2(l+b) [l-length, b-breadth]
    Square s2 [s-Side] 4*s [s-Side]
    Rhombus 1/2*d1*d2 [d-Diagonal] 4*s [s-Side]
    Parallelogram b*h [b-Base, h-Height] 2(l+b) [l-length, b-breadth]
    Trapezoid (b1*b2)/2*h [b1-Base1,b2-Base2, h-Height ] s1+s2+s3+s4 [s-Side]