Class 9

AP State Syllabus AP Board 9th Class Maths Solutions Chapter 4 Lines and Angles InText Questions and Answers.

AP State Syllabus 9th Class Maths Solutions 4th Lesson Lines and Angles InText Questions

Question
Observe your surroundings carefully and write any three situations of your daily life where you can observe lines and angles. [Page No. 71]
Solution:
The edges of a blackboard; the edges of a ruler/scale and the edges of a table gives the idea of lines and their corners give the idea of angles.

Question
Draw the pictures in your notebook and collect some pictures. [Page No. 71]
Solution:

Think, Discuss and Write

What is the difference between inter-secting lines and concurrent lines ? [Page No. 74]
Solution:
If the number of lines meeting at a point are only two then they are said to be intersecting lines whereas if the lines are three or more than three meeting at a point then they are said to be concurrent lines.

Do These

Question 1.
Write the complementary, supplementary and conjugate angles for the following angles, a) 45° b) 75° c) 215° d) 30° e) 60° f) 90° g) 180° [Page No. 76]
Solution:

Question 2.
Which pairs of the following angles become complementary or supplementary. [Page No. 76]

Solution:
The angles formed by the figures (i) and (ii) are complementary.
The angles formed by the figures (ii) and (iii) are supplementary.

Try This

Question 1.
Find the pairs of adjacent and non- adjacent angles in the given figures. [Page No. 77]

Solution:
In figure (i) ∠1 and ∠2 are pair of adjacent angles.
In figure (ii), no adjacent angles.
In figure (iii) (∠1, ∠2), (∠2, ∠3) are pairs of adjacent angles.
In figure (iv) ∠1 and Z2 are adjacent angles.

ii) List the adjacent angles in the given figure. [Page No. 77]

Solution:
In the given figure
(∠1, ∠2), (∠3, ∠4), (∠4,∠5) and (∠3, ∠5) are the pairs of adjacent angles.

Think Discuss and Write

Question
Linear pair of angles are always supplementary. But supplementary angles need not form a linear pair. Why ? [Page No. 77]
Solution:
A pair of supplementary angles need not necessarily a linear pair because they may exists in separate figures.

Activity

Measure the angles in the following figure and complete the table.
[Page No. 78]

Solution:

Measure the four angles 1,2,3,4 in each of the above figure and complete the table:
[Page No. 79]

Solution:

Do This

Question 1.
Classify the given angles as pairs of complementary, linear pair, vertically opposite and adjacent angles. [Page No. 80]

Solution:
The pairs of angles a and b in the fig.(i) are linear pair of angles.
The pairs of angles a and b in the fig.(ii) are adjacent angles.
The pairs of angles a and b in the fig.(iii) are complementary angles*
The pairs of angles a and b in the fig.(iv) are vertically opposite angles.

Question 2.
Find the measure of angle ‘a’ in each figure. Give reasons in each case. [Page No. 81]
i)

a = 180° – 50° = 130°
(linear pair of angles)

ii)

a = 43°
( ∵ vertically opposite angles)

iii)

a = 360° – (209° + 96°)
= 360° – 305° = 55°
(∵ complete angle = 360°)

iv)

a = 90° – 63° = 27°
(pair of complementary angles )

Do These

Question 1.
Find the measure of each angle indi-cated in each figure where / and m are parallel lines intersected by a transversal n. [Page No. 87]

Solution:
x = 110° (alternate exterior angles)

Solution:
y = 84° (alternate interior angles)

Solution:
z = 180° – 100° = 80° .
(interior angles on the same side of transversal)

Solution:
s° = 53° (pair of corresponding angles)

Question 2.
Solve for x and give reasons. [Page No. 88]

11x + 2 = 75°
11x = 75 – 2 = 73
∴ x = \(\frac{73}{11}\)
(∴ pair of corresponding angles are equal).

Solution:
8x – 4 = 60°
8x = 60 + 4 = 64
∴ x = \(\frac{64}{8}\) = 8
(∴ alternate interior angles are equal)

Solution:
(14x- 1)°.= (12x + 17)°
14x – 12x = 17 + 1
2x = 18
x = \(\frac{18}{2}\) = 9
(∴ alternate exterior angles are equal)

Solution:
13x- 5 = 17x + 5
13x- 17x = 5 + 5
– 4x = 10
x = \(\frac{10}{-4}=\frac{-5}{2}\)
(∴ Pair of, corresponding angles are equal).

Activity

Question 1.
Take a scale and a ‘set sqaure’. Arrange the set sqare on the scale as shown in figure. Along the slant edge of set sqare draw a line with the pencil. Now slide your set square along its horizontal edge and again draw a line. We observe that the lines are parallel. Why are they parallel ? Think and discuss with your friends. [Page No. 88]
Solution:

A. Student Activity (For Reference .)
All slant lines are parallel making an angle of 60° with the horizontal line. In this figure horizontal line is transversal to the slant line and corresponding angles are equal.

Do This

Question
Draw a line \(\overline{\mathbf{A D}}\) and mark points B and C on it. At B and C, construct ∠ABQ and ∠BCS equal to each other as shown. Produce QB and SC on the other side of AD to form two lines PQ and RS.

Draw common perpendiculars EF and GH for the two lines PQ and RS. Measure the lengths of EF and GH. What do you observe ? What can you conclude from that ? Recall that if the perpendicular distance between two lines is the same, then they are parallel lines. [Page No. 89]


Solution:
As ∠ABQ =∠BCS and they lie on the same line AD we can say that BQ // CS. Now EF and GH are the perpendicular distances between two parallel lines PQ and R, we say EF = GH.

Try This

Question i)
Find the measure of the question marked angle in the given figure. [Page No. 90]

Solution:
? = 70°
[ ∵ from the figure, these two angles are exterior angles on the same side of the transversal]

ii) Find the angles which are equal to ∠P.
Solution:
∠P = ∠Q = ∠R = 110°
(corresponding angles)

Activity

Question
Draw and cut out a large triangle as shown in the figure.
Number the angles and tear them off.
Place the three angles adjacent to each other to form one angle as shown below. [Page No. 97]

1. Identify angle formed by the three adjacent angles ? What is its mea-sure ?
2. Write about the sum of the measures of the angles of a triangle. Now let us prove this statement
using the axioms; and theorems related to parallel lines.
Solution:
Student Activity.

Think, Discuss and Write

Question
If the sides of a triangle are produced in order, what will be the sum of exterion angles formed ? [Page No. 99]
Solution:
Let ΔABC and the sides of the triangle is formed by exterior angles.

∠3 = ∠B + ∠C
∠1 + ∠2 + Z∠3 = 2[∠A + ∠B + ∠C]
= 2 x 180° = 360°
∴ Sum of the exterior angles are 360°.

AP Board 9th Class Maths Solutions

AP State Syllabus AP Board 9th Class English Textbook Solutions Chapter 4A What is Man without the Beasts? Textbook Questions and Answers.

AP State Syllabus 9th Class English Solutions Chapter 4A What is Man without the Beasts?

9th Class English Chapter 4A What is Man without the Beasts? Textbook Questions and Answers

Look at the picture and answer the questions that follow.

Question 1.
What do you think the picture is about?
Answer:
The picture shows the dead body of an elephant. It also shows a fallen tree. It presents man’s cruelty to animals and trees. It shows man’s selfishness as well as senselessness. It reminds us of the need to protect plant life and animal life.

Question 2.
What can you do to save nature?
Answer:
We can do a lot to save nature. First of all, not harming nature itself is a great service. Then, we can go on planting trees at various places like school grounds, road sides. We can promote animal life. We can spread awareness among people the need to and methods of saving nature. Preventing pollution, avoiding excessive use of resources, limiting deforestation and encouraging afforestation, promoting animal life are some of the steps we can initiate.

Comprehension

Answer the following questions.

Question 1.
Why does Chief Seattle say that the Earth is sacred to his people?
Answer:
Chief Seattle says that the Earth is sacred to them. It is the creation of god. It is ash of our ancestors. It supports life. It has warmth. It is holy and pious. It is precious too. That is why people love it. They respect it. They worship it too.

Question 2.
The speaker says, ‘I am a savage’. Who do you think is a savage, the Red Indian or the White? Why?
Answer:
Chief Seattle belongs to a primitive Red Indian tribe. They live in the lap of nature. Civilization has not touched them. Hence he humbly calls himself a savage. But at heart, he is good, cultured, and concerned. The White is civilized. But at heart, he is the real savage.

Question 3.
Why does the Chief say ‘The destiny is a mystery to us’?
Answer:
Chief Seattle says that the destiny is a mystery to them. They know how to live with nature. They don’t know destroying nature. They don’t understand why the White want to control them. They fail to understand why animals and trees are being killed and felled. Hence he says it is a mystery.

Question 4.
“What is man without the beasts? If all the beasts were gone, man would die from a great loneliness of the spirit.” Why?
Answer:
God has a scheme. Every living being has its own important role in nature. All animals and even plants help one another in various ways. This fact is all the more important in the lives of tribes. Life without beasts is unimaginable for them. Man without beasts becomes lonely in spirit.

Question 5.
Is man the sole owner of the Earth? Pick out sentences from the speech to justify your answer.
Answer:
Man is not at all the owner of the earth. Earth does not belong to man. Man belongs to Earth. We are a part of the Earth. All animals and trees share the same breath. The Earth is God’s creation, not the man’s property.

Question 6.
Why does the speaker say that if we spit on the ground, we spit on ourselves?
Answer:
The writer says that the earth is the mother of many lives. If we harm mother earth, we are harming ourselves. If something bad happens to the earth, it is bad to us. If men spit upon the ground, they spit upon themselves.

Question 7.
“All belong to the same family.” The speaker says this to mean
(a) all animais belong to one family.
(b) all animals and plants belong to the same family.
(c) everything on the earth belongs to one family.
Answer:
(c) everything on the earth belongs to one family.

Question 8.
In his speech, Chief Seattle asks the audience a number of questions. He also creates vivid pictures in the minds of the audience. What are the other features of the speech? List them.
Answer:
Chief Seattle makes use of many literary features. He creates many images. Repetition of some powerful expressions is frequent. Metaphors like “the river of blood of ancestors”, “the land of ashes of forefathers” are very piercing.

Vocabulary

Read the following expression.
The warmth of the land
The word warmth here suggests love, care, affection, intimacy, etc.

I. Which of the following words can be used to work out new combinations with the word warmth?

e.g. the warmth of friendship
Answer:
the warmth of love ; the warmth of relationship; the warmth of freedom.

Why do you think certain combinations are not possible?
Are there any other expressions of this kind in the reading material?
Answer:
Yes. There are other expressions of this kind in the reading material. They are : the freshness of air; the sparkle of water; the body heat of the pony; the blood of our ancestors; the ashes of our grandfathers

Work out new combinations and use them in your own sentences.
Answer:
a) the pain of the suppressed
If only every one feels the pain of the suppressed, the planet will be a better place to live on.

b) the mood of the listeners
Some speakers do not at all take into consideration the mood of the listeners and go on speaking.

c) the depth of their feeling
Can you measure the depth of their feeling?

In the speech you can see a few words suggesting movement. For example, in the sentence ‘The sap which courses through the trees carries the memories of the red man’ the word ‘course’ means ‘flow’ or ‘move rapidly’.

II. Pick out from the speech other words that denote movement. If necessary, refer to a dictionary.

Answer:
The other words used in the speech denoting “movement” are: moves; carry; go; passing; walks.

Grammar

The Verb Phrase (a part of the predicate)

• A sentence is a group of words with complete sense.
• A sentence has two parts : 1) The Subject and 2) The Predicate
• The part of the sentence about which something is said is the subject of that sentence.
• The part of the sentence that says something about the subject is the predicate.
  
• The subject is either a noun or a noun equivalant.
• The predicate has a verb and some other elements.
• The predicate begins with a verb.
• The predicate may have only the verb or the verb along with other elements.

Note the following examples.

• You must have noticed that the verb begins the predicate.
• The verb may be a single word. Then it must be the main verb.
• The ‘verb’ may be a group of words with one or more helping verbs. Such groups of verbs (helping verb + main verb) are called the VERB PHRASES.

Read the following sentences taken from the text.
1. We are a part of the Earth.
2. You must teach your children that the ground beneath their feet is the ashes of our grandfathers.

1. What is the relationship of the above underlined parts with the first part of the sentences?
Answer:
You must have noticed that the underlined parts are the PREDICATES and the other parts are the SUBJECTS.

2. What type of word do you see at the beginning of the underlined part?
Answer:
The first word in the predicate is the VERB.

3. Which is the most important word in it?
Answer:
The verb is the most important word in the predicate.

I. List all the verb phrases in the following passage. Identify the Main verb.

Look at the river. It has very little water in it. Once it was flowing well. Now it is dying. Can’t you hear the cries of the dying river? The water in it is polluted. Do you get its stink? You cannot drink it. You cannot give this water even to animals. Animals will not go near it. We must make the river live forever. We must make the river our own companion.
Answer:
List of Verb Phrases

look (main verb)
has (main verb)
was flowing (flowing – main verb)
is dying (dying – main verb)
can’t hear (hear – main verb)
is (main verb)
do get (get – main verb)
cannot drink (drink – main verb)
cannot give (give – main verb)
will not go (go – main verb)
must make (make – main verb)

II. Analyze the verb phrases you have already identified and list the Main verb and the Auxiliary verbs in each of them.

III. If Clauses

Read the following sentences.
1. If you leave the door of the cage open, the bird will fly out. .
2. If Mother Teressa were alive today, how would people receive her?
3. If I had played well, I would have won the game.

• What is the meaning of each sentence?
• What are the two parts in each sentence?
• What is the role of ‘If’?

Types of ‘If Clauses’

1. Sentence (1) means there is a possibility of the bird flying out when we leave the door of the cage open.
2. In sentence (2) the speaker is not thinking about a real situation. He is only imagining the future happening ( because Mother Teressa is not alive today.).
There is no possibility of future happening. (People receiving her.)
3. In sentence (3) the speaker talks about something that might have happened in the past, but it didn’t.

Let us identify the ‘verb and tense’ elements in each of these sentences both in the ‘If- clause’ and in the ‘Main clause’.

Complete the following sentences using possible main clauses.
1. If I were asked to stay at home and do only household chores, ……………..
2. If I had attended the function, …………….. .
3. If Raju comes to school every day, …………….. .
4. If Lalit had helped him do his homework, …………….. .
5. If I were the class teacher of class-IX, …………….. .
Answer:

1. I would not accept.
2. my friend would have felt very happy.
3. he will shine in studies.
4. he would have escaped imposition.
5. I would try my best to improve the students’ performance.

Writing

I. Read the last part of the speech of Chief Seattle once again.
Answer:
Are all the sentences of the same length? Obviously, not. The speech contains long sentences (But in your… over the red man), short ones (Where is the thicket?) and even fragments (Gone). Why do writers vary the length of the sentences? Again,why do they use different types of structures (sometimes statements, questions,exclamations, fragments, etc.)?

Good writers vary their sentences to make them interesting.
Sentence length can influence the mood of the piece. If you’re concerned that your writing is either too choppy or too flowery, review it with an eye toward sentence length.

• Is it varied?
• Does it fit the mood you’re trying to convey?

Do you heavily favour short, simple sentences, or does the piece contain too many paragraph-long sentences?

To make your texts more interesting, you should use sentences of varying lengths with a variety of structures.

Read the following text.

Dogs are our great companions, aren’t they? Any idea when the friendship between man and dogs began? Maybe, thousands of years back. Who knows! What is important is that they are man’s best friends for obvious reasons. Dogs, as such, need to be walked. Walks keep their owners healthy; and the owners can talk to their dogs. They never tell secrets. Dogs don’t care what their owners observe while walking or what they watch on TV. As long as dogs are taken care of, they are happy.
What are the features of this write-up? In this the variety in sentences has been obtained in a number of ways:
Using different sentence types (statements, questions, exclamations, etc.)
Using elements such as tags, connectives (and, as such, etc.)

Read the following write-up.

You must have heard about the Hussain Sagar Lake. It is in Hyderabad. It is one of the largest man-made lakes. Hyderabad and Secunderabad are twin cities of the State. The lake connects these cities. It was originally constructed to supply drinking water. Now it is not used as a drinking water source. People say, “This is sad. There is plenty of water. Nobody can drink it.” The lake faces a few threats. The main threat is encroachment by both private and public agencies. The lake also faces the problem of pollution. One of the locals said: ‘Oh, sometimes it stinks horribly.” This is due to the continuous discharge of domestic wastes and industrial chemicals. Hence it is our duty to save the Hussain Sagar Lake.

I. Rewrite the sentences according to the direction given in the brackets.

1. The lake connects these cities. (Change into a question).
2. It is in Hyderabad. It is one of the largest man-made lakes. (Combine the two sentences using ‘and’.)
3. Hyderabad and Secunderabad are twin cities of the State. The lake connects these cities. (Begin the sentence with ‘The lake … ‘and connect the two sentences.)
4. It was originally constructed to supply drinking water. Now it is not used as a drinking water source. ( Connect using ‘though’)
5. “This is sad,… ?” (Complete the sentence with a question tag.)
6. “Oh, sometimes it stinks horribly.”(Rewrite the sentence beginning with ‘How )
Answer:

1. Does the lake connect these cities?
2. It is in Hyderabad and one of the largest man-made lakes.
3. The lake connects Hyderabad and Secunderabad which are twin cities of the state.
4. Though it was originally constructed to supply drinking water, now it is not used as a drinking water source.
5. This is sad, isn’t this (it)?
6. How horribly it stinks sometimes!

Read the speech made by Chief Seattle once again and reflect on the following features.
1) The beginning and ending of the speech
2) The arguments and the building up of arguments in a sequential manner
3) The emotive and argumentative language used
4) The variety of sentences used
5) Mental images created
Answer:

1. The speech begins with a question that has no valid answer. And the speech ends with a paradox packed with sarcasm.
2. The arguments are developed gradually and in a logical and powerful sequence.
3. The language used is filled with emotions and rhetoric.
4. A wide variety of sentences add force and interest to the argument.
5. A good number of mental images employed in the speech serve a very valuable purpose.

II. Prepare a speech that you would like to make on ‘the World Environment Day’. You can make use of the following hints :

1) The threats to animal and plant life
2) The pollution of air, water and earth and the consequences
3) The need to preserve our environment for ourselves and for the future generations
Answer:
Good evening everyone. We are all here on the occasion of the ‘World Environment
Day’. It is definitely not a happy gathering. Are we not aware of the most serious problem we face today ? Yes, it is fast polluting environment. Who, then, is responsible ? It is we, the makind. And the sufferers ? Plants and animals too ! And they suffer for none of their faults.

Was pollution a problem centuries ago too ? No, it wasn’t. Why, then, now it poses a serious threat ? Is man today not as intelligent as his forefathers? No, it is not the question of intelligence. It isthe question of wisdom. In good olden days man was wise and selfless. Now man has become clever and selfish.

Mother Earth can satisfy every one’s needs. But man wants his greed to be satisfied. The result? The very existence of LIFE on the earth becomes doubtful in a few hundred years. Why?

Man’s selfishness results in senseless falling of trees. Industrialisation leads to pouring out of poison into air, water, and the earth. Automobiles add to air and noise pollution. Excessive use of pesticides and fertilisers leave food items, ground water, and even the earth full of harmful chemicals. The list, in fact, is too long to be completed now.

What, then can we do now? Should we silently suffer? Can we remain as mere spectators? Does Mother Earth excuse us if we don’t respond? Can we forgive ourselves if we don’t react? No. Let us do our bit. Let us plant a tree each on every important event. Let us stop using automobiles at least once a week. Let us promote awareness among villagers once a month. Let us ask the public representatives to make effective laws to prevent pollution. Let us pledge to become strong soldiers in the war against pollution. Let us promise to bring peace into the minds of every living being. Let us strive to see Mother Earth have green smile all over her surface! Say Green is Great!

Jai Green Earth ! Bye Dear friends!

Study Skills

I. Study the following table, which shows the ranking of districts in Andhra Pradesh in terms of industrial pollution intensity.

Answer the following questions:

1. Which district tops the list in pollution intensity?
Answer:
Krishna district tops the list in pollution intensity.

2. Which region of Andhra Pradesh has the least pollution intensity?
Answer:
In terms of regions, it is the Rayalaseema region that has the least pollution intensity.

3. What percentage of land is affected by industrial pollution in Guntur?
Answer:
46.87% of land is affected in Guntur.

4. What percentage of land is not affected by industrial pollution in Krishna district?
Answer:
42.37% of land in Krishna district is not affected.

5. What are the comparisons that you can make related to the two regions in terms of pollution intensity?
Answer:
Rayalaseema is the least affected Area. In Andhra region too, Krishna, the Godavari districts, Visakhapatnam, Sreekakulam district have high intensity of pollution.

II. Write an analytical report on the pollution intensity in Andhra Pradesh.
Answer:
The Pollution Intensity figures for Andhra Pradesh are rather alarming. Krishna, East Godavari too join together in alerting us to awake. What comforts us a little bit is the position of pollution in districts like Kadapa and Ananthapur. But that doesn’t leave any scope for sleep or rest. The picture presents a grave problem. We are all to work for a solution. That too as early as possible. The quicker we work out the solution, the better for us!

Listening

Practise listening carefully. Then you will be able to speak.
Listen to the ‘Earth Song’ by Michael Jackson and answer the following questions.

EARTH SONG

What about sunrise
What about rain
What about all the things
That you said we were to gain …
What about killing fields
Is there a time
What about all the things
That you said was yours and mine…
Did you ever stop to notice
All the blood we’ve shed before
Did you ever stop to notice
The crying Earth the weeping shores?
What have we done to the world
Look what we’ve done
What about all the peace
That you pledge your only son…
What about flowering fields
Is there a time
What about all the dreams
That you said was yours and mine…
Did you ever stop to notice
All the children dead from war
Did you ever stop to notice
The crying Earth the weeping shores
– Michael Jackson

1. How does Michael Jackson respond to man’s attack on Nature?
Answer:
Man has been attacking Nature mercilessly. Michael Jackson is moved. He is concerned. He asks everyone to think. He asks everyone to notice the crying earth. He appeals for peace, protection, preservation. He uses a series of questions to convince the reader.

2. What is the mood of the song? Indicate by putting a tick(✓).
a) sadness ( )
b) anger ( )
c) anguish ( )
d) joy ( )
Answer:
c) anguish (✓)

Oral Activity

Discuss the following in groups.

1. Why do writers often write about these things (things like earth)?
2. In what ways does nature influence man?
3. Can it shape one’s personality?
Answer:
“Mother earth motivates writers in many ways. Poets find Mother earth as an endless source of inspiration. Earth’s colour, shape, textures, life it supports and many more things have formed themes of a lot many works.”

“But these days the major theme has been “how man abuses Mother earth”. People have become not only selfish but also senseless. Their cruelty to earth is going to cost a heavy price to man.”

“Nature is man’s best benefactor. It gives us rich food, fresh water, and clean air. It provides us with chances to play. It entertains us. It educates us. It fills our minds with great ideas. It sets models of behaviour.”

“Nature helps one develop one’s personality well. It creates plenty of opportunities to learn, to correct, to improve, to perform and to progress. Hence great persons, poets, philosophers appreciate Nature’s glory and acknowledge its service. They appeal for the preservation of nature for posterity.”

Literary Terms

Literary elements :
The essential techniques used in literature (e.g., characterization, setting, plot, theme).

Literary devices :
Tools used by the author to enliven and provide voice to the writing (e.g., dialogue, alliteration).

Metaphor:
The comparison of two, unlike things in which no words of comparison (like or as)are used (e.g., That new kid in class is really a squirrel.).

Narrative :
A story, actual or fictional, expressed orally or in writing.

Personification :
An object or abstract idea given human qualities or human form (e.g., Flowers danced about the lawn.).

Satire :
A literary tone used to ridicule or make fun of human vice or weakness.

Simile :
A comparison of two, unlike things in which a word of comparison (like or as) is used(e.g., She eats like a bird.).

Metaphor vs. Simile :
A metaphor is direct relationship where one thing is another (e.g. “Juliet is the sun”). A simile, on the other hand, is indirect and usually only likened to be similar to something else. Similes usually use “like” or “as” (e.g. “Your eyes are like the ocean”).

What Is Man Without The Beasts? Summary in English

Chief Seattle’s speech “What is Man Without the Beasts” is a heart-rending, thought provoking, and environment-protecting piece. These words flowed directly out of Seattle’s heart when the white chief proposed to buy the land in which the Red Indians were living. Seattle says that the idea of buying is strange. We are a part of the earth. The earth is not a part of us. We cannot own the warmth of the land, the sparkle of the water, the freshness of the air or the vastness of the sky. Then, how can we buy or sell them ? The land is the ash of our grandfathers. The river is the blood of our brothers. The air that we breathe binds all the living beings together. The plants and animals around us are all our brothers and sisters. The land, the water, and the air are sacred to us. They are precious to us. We respect them. We worship them. They protect us. They sustain us. Without beasts, we become lonely in spirit. If we kill them, we will die later. If we harm the earth, it harms us. The white man’s words of buying and offering make no sense to me. What we know is to live in the lap of nature. What we teach our children is to respect every living being. We know only living not surviving.

What Is Man Without The Beasts? Glossary

beasts (n – plural) : wild animals

resonate (v) : resound, echo

gospel (n) : the Bible; final and pious saying

sparkle (n) : shine; brightness

sacred (adj) : holy; pious

pine (n) : a kind of tree

sap (n) : the liquid in plants carrying food to various parts

courses (verb- simple present) : passes; moves ; flow

perfumed (adj) : sweet scented

crests (n – plural) : tops of hills ; rocks

pony (n) : a type of a small horse

ancestors (n – plural) : forefathers

ghostly (adj) : of a ghost

quench (v) : satisfy; fulfil

canoes (n – plural) : light, narrow boats

numb (adj) : insensitive (Note that the letter ‘b’ is silent).

stench (n) : strong, unpleasant smell

savage(n) : an uncivilized person

rotting (adj; v+ing) : decaying, decomposing

prairie (n) : a piece of land covered mainly with grass (with not many trees)

the smoking iron horse (phrase) : the train (Notice the creativity of the innocent person with a kind heart and wise mind.)

befalls (v – simple present tense) : happen to (particularly something bad)

web (n) : a complicated pattern of things

strand (n) : a single piece/line of thread

compassion (n) : a strong feeling of symathy

heap (v) : put a lot

contempt (n) : lack of respect; not treating someone with due respect

perishing (n – v+ing) : destruction

dominion (n) : authority; control

slaughtered (v – past tense) : killed

blotted (v-past tense) : covered

thicket (n) : a group of bushes

AP State Syllabus AP Board 9th Class Maths Solutions Chapter 13 Geometrical Constructions InText Questions and Answers.

AP State Syllabus 9th Class Maths Solutions 13th Lesson Geometrical Constructions InText Questions

Try These

Question
Observe the sides, angles and diagonals of quadrilateral BEFD. Name the figures given below and write properties of figures. [Page No. 283]

Solution:
In fig. (1)
\(\overrightarrow{\mathrm{BF}}\) is the bisector of ∠B and ∠F.
In quad BEFD
BE = BD = DF = EF
It is a rhombus

In fig. (2)
BD = BE
FD = FE
∴ BEFD is a kite.
BF is bisector of ∠B and ∠F.

Try This

Question
Draw a circle, identify a point on it. Cut arcs on the circle with the length of the radius in succession. How many parts can the circle be divided into ? Give reasons. [Page No. 284]
Solution:

Let P be the centre of the circle.
A is any point on its circumference.
It can be divided into 2π parts
∴ \(\frac{\text { Circumference }}{\text { Radius }}=\frac{2 \pi r}{r}=2 \pi\)

Think, Discuss and Write

Question
Can you construct a triangle ABC with BC = 6 cm, ∠B = 60° and AB + AC = 5 cm? If , not, give reasons. (Page No. 286)
Solution:
We can’t construct a triangle with measures ∠B = 60°; BC = 6 cm and AB + AC = 5 cm.
∵ AB + AC < BC
Sum of any two sides of a triangle must be greater than the third side.

Think, Discuss and Write

Question
Can you construct the triangle ABC with the same measures by changing the base angle ∠C instead of ∠B ? Draw a rough sketch and construct it.
BC = 4.2cm. ∠C = 30°, AB – AC = 1.6 cm (Page No. 287)

Solution:

Steps of construction:

1. Construct ΔBCD where BC = 4.2 cm and ∠C = 30° and AC – AB = 1.6 cm.
2. Draw perpendicular bisector of BD which meets \(\overline{\mathrm{CD}}\) produced at A.
3. Join B, D.
4. ΔABC is the required triangle.

Try These

Can you draw the triangle with the same measurements as shown in the figure in alternate way ? (Page No. 289)
[Measurements : ∠B = 6Q°, ∠C = 45° and AB + BC + CA =11 cm]
[Hint: Take ∠YXL = 60°/2 = 30° and ∠XYM = 45°/2 = 22 \(\frac { 1 }{ 2 }\) ]

Solution:
→ Draw XY = 11 cm [AB + BC + CA = 11 cm]
Construct ∠YXP = 30° at X \(\left[\frac{B}{2}=\frac{60^{\circ}}{2}=30^{\circ}\right]\)
Construct ∠XYQ = 22\(\frac { 1 }{ 2 }\) at Y \(\left[\frac{\mathrm{C}}{2}=\frac{45^{\circ}}{2}=22 \frac{1}{2}^{\circ}\right]\)
→ \(\overrightarrow{\mathrm{XP}}\) and \(\overrightarrow{\mathrm{YQ}}\) meet at A.
→ At A, draw \(\overrightarrow{\mathrm{AB}}\) such that ∠XAB = 30° where B is a point on XY.
→ Also draw \(\overrightarrow{\mathrm{AC}}\) such that ∠YAC = 22\(\frac{1}{2}^{\circ}\) where C is a point on XY.
→ Δ ABC is the required triangle.

Try These

Question
What happens if the angle in the. circle segment is right angle ? What kind of segment do you obtain ? Draw the figure and give reason. [Page No. 290]
Solution:
If the angle in the circle segment is right angle i.e., 90°, then the angle subtended by it at the centre is 2 x 90° = 180°
Thus the line segment becomes the diameter and the circle segment becomes the semi-circle.

AP State Syllabus AP Board 9th Class English Textbook Solutions Chapter 3C Homework Textbook Questions and Answers.

AP State Syllabus 9th Class English Solutions Chapter 3C Homework

9th Class English Chapter 3C Homework Textbook Questions and Answers

Answer the following questions.

Question 1.
What is the main purpose of giving homework to children?
Answer:
Strengthening the child’s learning is the main purpose of homework.

Question 2.
How can excessive homework decrease students’ interest in studies?
Answer:
Excessive homework makes students dull in their studies. Heavy work makes them feel tired and bored. It decreases their interest in studies. Researches have proved this point beyond doubt.

Question 3.
Why do many teachers defend large amounts of homework?
Answer:
Many teachers defend heavy homework. They say it is necessary in the highly competitive atmosphere of these days.

Question 5.
How does homework take time away from important elements of daily life?
Answer:
Heavy homework takes the entire time of children. They don’t find time to play, exercise, develop social and family bonds, have fun and sleep for sufficient time.

Question 6.
Is the writer in favour of or against homework? Justify your reasons.
Answer:
The writer is against the heavy homework. He welcomes limited, interesting, enjoyable and productive homework. He lists the harmful effects of excessive homework.

Project Work

Conduct a survey to find out the kind of reading material the children like to be included in the textbook which makes them more interesting.

Ask them which of the following they enjoy most and complete the table :

Work in groups and consolidate the data you have collected.
Collect the data in the given format.

Prepare a pie chart with the collected data.
Analyse the pie chart and name the materials they enjoyed reading.

Homework Summary in English

The essay is an excellent analysis of the advantages and the otherwise of’homework.’ It uses a number of examples, statistics, arguments and critical surveys. Both the sides of the coin are presented in a balanced way.
Homework is mainly to strengthen the pupil’s learning. It should by rule be limited, interesting, enjoyable, purposeful and productive. In recent times, homework has, on the other hand, become heavy, boring and counter productive. Competition and baseless thinking on the part of some teachers and parents have added to the burden of homework.
Excessive homework is very harmful in more than one way. It eats into the child’s free time. Children miss their social life. Family relations do not develop in the desired manner. Children don’t find time to play and exercise. Even academically also, they become dull. Thus, burdensome homework injures children physically, mentally, emotionally, socially and morally. Hence ‘less is more’ should be the rule. Interesting, enjoyable homework in a limited quantity is advisable and welcome.

Homework Glossary

reinforce (v) : strengthen

counter-productive (adj) : harmful; producing negative results

curriculum (n) : syllabus

obesity (n) : over weight; a condition in which a person’s body is heavier than what is desirable, healthy

low self-esteem (n) : lower confidence levels

depression (n) : a state of total hopelessness

precious (adj) : very valuable

resort to (v) : adopt; follow

assignments (n-plural) : tasks to be completed; homework

AP State Syllabus AP Board 9th Class English Textbook Solutions Chapter 3B Not Just a Teacher, but a Friend Textbook Questions and Answers.

AP State Syllabus 9th Class English Solutions Chapter 3B Not Just a Teacher, but a Friend

9th Class English Chapter 3B Not Just a Teacher, but a Friend Textbook Questions and Answers

I. Answer the following questions.

Question 1.
Why does the student consider the teacher his friend?
Answer:
The student is sad. He has no one to share his sorrow. He doesn’t know where to go. Then, he meets his teacher. The teacher’s company makes him comfortable. Hence the boy consideres the teacher his friend.

Question 2.
How does the teacher take the student out of his gloom?
Answer:
The teacher listens to the boy. He offers the boy comfort. He provides the boy with the needed courage. He brings out the boy’s real soul. He finds time to see if the boy is fine. In the company of this teacher-friend, the boy is all smiles.

Question 3.
Why does the student fail to estimate the teacher correctly?
Answer:
The boy moves to the higher class. He is worried that he would miss his teacher-friend. As a young boy, he fails to understand that great teacher’s concern for him.

Question 4.
How does the poet glorify the relationship between the teacher and the student?
Answer:
The poet lays bare the boy’s heart. That brings out the dizzy heights of the teacher’s personality. The boy is sad. The teacher makes him glad. The boy feels lonely. The teacher makes him lively. The relationship between them is that of love and affection. They become a model teacher student pair. The bond between them makes everyone fond of them.

Not Just a Teacher, but a Friend Summary in English

The poem is a moving account of a noble teacher being a kind friend of a student with no one to depend on. The boy was sad. He had not a soul to share his sorrow with. He had nowhere to go. Then he saw this teacher. He trusted the teacher. He poured out his feelings. The teacher extended his healing hand. The boy felt relieved. Whenever the boy felt unhappy, he would go to his teacher and found comfort in his company.

When the boy moved to a higher grade, he was worried that he would miss his favourite teacher. But he was wrong. The relationship only grew strong. The teacher would find time to greet the boy. The boy would smile and feel happy. Hence the boy feels that the teacher is also his friend.

Not Just a Teacher, but a Friend Glossary

trust (v) : believe in ; have faith in

true soul (phrase) : the real self : the hidden power

next grade (phrase) : higher class

fade (v) : become dull; pale

fate (v) : destiny; future

AP State Syllabus AP Board 9th Class Maths Solutions Chapter 13 Geometrical Constructions Ex 13.1 Textbook Questions and Answers.

AP State Syllabus 9th Class Maths Solutions 13th Lesson Geometrical Constructions Exercise 13.1

Question 1.
Construct the following angles at the initial point of a given ray and justify the construction.
a) 90°
Solution:

• Let AB be the given ray.
• Produce BA to D.
• Taking A as centre draw a semi circle with some radius.
• With X and Y as Center draw two intersecting arcs of same radius.

Or

|

• Let \(\overrightarrow{\mathrm{AB}}\) be the given ray.
• With A as centre draw an arc of any radius.
• Mark off two equal arcs from X as shown in the figure with the same radius taken as before.
• Bisect the second segment.
• Join the point of intersection of above arcs, with A.
• ∠BAC is the required right angle.
• Join the point of intersection ‘C’ and ‘A’.
• ∠BAC = 90°

In ΔAXY; ∠YAX = 60° and
in ΔAYC ∠YAC = 30° ∠BAC = 90°

b) 45°
Solution:
Steps:

• Construct 90° with the given ray AB.
• Bisect it from ∠BAD = 45°

Or

Steps:

• Construct ∠BAC = 60°
• Bisect ∠BAC = ∠DAC = 30°
• Bisect ∠DAC such that ∠DAE = ∠FAC = 15°
• ∠BAE=45°

ΔAXZ is equilateral
and ∠YAZ = 15°
∴∠XAY = 45°

Question 2.
Construct the following angles using ruler and compass and verify by measuring them by a protractor.
a) 30°
Solution:

• Construct ∠ABY = 60°
• Bisect ∠ABY = 60°
• Such that ∠ABC = ∠CBY = 30°

b) 22\(\frac{1}{2}^{\circ}\)
Solution:

• Construct ∠ABD = 90°.
• Bisect ∠ABD such that ∠ABC = ∠CBD = 45°
• Bisect ∠ABC such that
  ∠ABE = ∠EBC = 22\(\frac{1}{2}^{\circ}\)

c) 15°
Solution:

Steps of construction : ,

• Construct ∠BAE = 60°
• Bisect ∠BAE such that ∠BAC = ∠CAE = 30°
• Bisect ∠BAC such that ∠BAF = ∠FAC = 15°

d) 75°
Solution:

Steps of construction :

• Construct ∠BAC = 60°
• Construct ∠CAD = 60°
• Bisect ∠CAD such that ∠BAE = 90°
• Bisect ∠CAE such that ∠BAF = 75°

e) 105°
Solution:

Steps of construction:

• Construct ∠ABC = 90°
• Construct ∠CBE = 30°
• Bisect ∠CBE such that the angle formed ∠ABD = 105°

f) 135°
Solution:

Steps of construction:

• Construct ∠ABC = 120°
• Construct ∠CBD = 30°
• Bisect ∠CBD such that the angle formed ∠ABE = 135°

Question 3.
Construct an equilateral triangle, given its side of length of 4.5 cm and justify the constraction.
Solution:
A.

• Draw a line segment AB = 4.5 cm.
• With B and A as centres draw two arcs of radius 4.5 cm meeting at C.
• Join C to A and B.
• ΔABC is the required triangle.

Justification:
In ΔABC
AB = ∠C ⇒ ∠C = ∠B
Also AB = BC ⇒ ∠C = ∠A
Hence ∠A = ∠B = ∠C
But ∠A + ∠B + ∠C = 180°
∴ ∠A = ∠B = ∠C = \(\frac{180^{\circ}}{3}\) = 60°

Question 4.
Construct an isosceles triangle, given its base and base angle and justify the construction. [Hint: You can take any measure of side and angle]
Solution:
A.

Steps:

• Draw a line segment AB of any given length.
• Construct ∠BAX and ∠ABY at A and B such that ∠A = ∠B.
• \(\overrightarrow{\mathrm{AX}}\) and \(\overrightarrow{\mathrm{BY}}\) will intersect at C.
• ΔABC is the required triangle.

Justification:
Drop a perpendicular CM to AB from C.
Now in ΔAMC and ΔBMC
∠AMC = ∠BMC [Right angle]
∠A = ∠B [Construction]
CM = CM (Common)
∴ ΔAMC ≅ ΔBMC
⇒ AC = BC [CPCT]

AP State Syllabus AP Board 9th Class English Textbook Solutions Chapter 3A Swami is Expelled from School Textbook Questions and Answers.

AP State Syllabus 9th Class English Solutions Chapter 3A Swami is Expelled from School

9th Class English Chapter 3A Swami is Expelled from School Textbook Questions and Answers

Look at the picture and answer the questions that follow.

Question 1.
Why do you think the teacher is punishing that student?
Do you approve of this action of the teacher? Give at least one reason for your opinion.
Answer:
I think the teacher is punishing the student because he might cause nuisance. He might not finish his homework. He might be indisciplined in the class. I usually don’t approve this kind of action of the teacher because corporal punishment should not be initiated against children. It is common with the school children behaving improperly at their tender age. They don’t know the importance of education and how to behave. Teachers need great patience to change them.

Question 2.
Can you suggest a few steps to correct the students causing nuisance?
Answer:
Various different kinds of steps are at hand to correct erring students. Some of them are:

1. explaining to them in the way they appreciate what is wrong with them and how they can be good.
2. making them imagine that they are at the receiving end.
3. making them realise the value of the fruits of being good.
4. providing them with examples of bad behaviour ruining careers and lives.
5. engaging them in productive activities of their choice.
6. repeatedly attempting to understand their feelings, sentiments, and view points.
7. giving them opportunities to express themselves, to lead, to organise, to suggest programmes and activities.
8. ignoring their misdeeds and allowing and encouraging them to correct themselves.
9. informing their parents about their positive qualities.
10. encouraging them to participate in social service activities.

Comprehension

I. Answer the following questions.

Question 1.
Why do you think the headmaster entered the class with a flushed face and a hard ominous look?
Answer:
Swami and his friends had not attended the classes the previous day. Moreover they had broken the panes of ventilators of the headmaster’s room. The school peon saw that. And he Informed the headmaster. That was why the headmaster was very angry.

Question 2.
Why did the headmaster send for the peon?
Answer:
The headmaster sent for the peon. He wanted to show to the students the proof of their mischief. The peon was the witness to their wrong deeds.

Question 3.
“I don’t care for your dirty school.” Why did Swami mutter so?
Answer:
The headmaster went on beating Swami. Swami complained of severe pain. That Invited four more hits. Swami could no longer bear the pain. He was in a desperate situation, That made him bold. So he took his books and walked out of the class. He said he least cared for that dirty school,

Question 4.
Do you justify the headmaster’s behaviour? If not, state your reasons.
Answer:
No, I don’t justify the headmaster’s behaviour. Corporal punishment is not the correct way to correct the wrong doing. And continuous corporal punishment is the worst part of it. That only provokes revolt not remorse.

Question 5.
If you were in Swami’s place, how would you feel?
Answer:
If I were in Swami’s place, I would feel the same way as Swami did. But I doubt I would have broken the panes. And I am afraid I could have muttered as Swami did.

II. Here are some utterances from the story. Complete the table.

Answer:

III. Work in Groups

Give reasons for Swami’s decision to leave the school:
1. ________________
2. ________________
3. ________________
Answer:

1. Swami was unable to bear the pain of the cane any more.
2. The headmaster had any how announced his dismissal.
3. More than the physical pain, the insult was unbearable.
4. In the desperation, Swami couldn’t think of any other alternative.
5. Swami must have thought of joining another school.

Vocabulary

I. Pick out words from the story which are synonyms of the following words.

Answer:

II. Look at the following phrasal verbs taken from the text.

1. keep away
2. look around
3. look at
4. bring down
5. cut off

These phrases are verbs followed by prepositions or adverbial particles. You may understand that they are phrasal verbs.

You will notice that the following phrasal verbs ‘keep away’, ‘bring down’ and ‘cut off’ can be split as shown below.
Examples:

1. Keep the files away.
2. Bring the patient down.
3. Cut it off.

The other two phrasal verbs cannot be split.
Use the following phrasal verbs in your own sentences and decide whether you can split them as shown in the above examples.

Answer:

III. Refer to a dictionary and pick out the phrasal verbs that begin with the following verbs and write down sentences using them.

Grammar

I. Read the following imaginary conversation between Swami and the headmaster.

Headmaster : Why didn’t you come to school yesterday?
Swami : Sir, my mother was suffering from fever.
Headmaster : I don’t believe your words. You always say something or the other to escape from school.
Swami : I’m speaking the truth, sir.
Headmaster : Well, I’ll come to your house tomorrow and talk to your parents.

In the indirect speech, the above conversation can be written like this.

The headmaster asked Swami why he had not come to school the day before. Swami replied respectfully that his mother had been suffering from fever. The headmaster retorted that he did not believe his words and added that he always said something or the other to escape from school. Then Swami replied respectfully that he was speaking the truth. Then the headmaster told him that he would go to his house the next day and talk to his parents.

As you can see, while converting the direct speech into indirect speech, the words in the bold are added to express the feelings, emotions, attitudes of the speaker, and the sequences of the actions.

Read the imaginary conversation between Swami and his father:

Swami’s Father : My dear Swami, why are you looking so dull? Why haven’t you gone to school today?
Swami : Daddy, I don’t like the school. The headmaster beats me every day.
Swami’s Father : Why does your headmaster beat you every day without any reason? I’m sure you must be causing a lot of nuisance in school.
Swami : No, Dad. The headmaster beats all my friends in the same way.
Swami’s Father : OK. What do you want to do now? Don’t you go to school and continue your studies?
Swami : No, Dad. I’ll join some other school.

Now change the conversation into indirect speech.
Answer:
Swami’s father lovingly addressed his son and asked him why he was looking very dull. He further enquired why he had not gone to school that day. Swami replied with pain that he did not like the school and complained that the headmaster beat him everyday. Fatherdisbelievingly asked why the headmaster beat him every day without any reason. He added that he was sure Swami must be causing a lot of nuisance in school. Swami emphatically denied that and added that the headmaster beat all his friends in the same way. Father agreeingly asked what he wanted to do then. He further asked if he would not go to school and continue his studies. Swami replied that he would join some other school.

II. Noun Clause
Look at the sentences taken from the story.

1. One student said that he had an attack of a headache.
This sentence has two clauses.
a) One student said (Principal clause)
b) that he had an attack of a headache. (Subordinate clause)

The Subordinate clause is the object of the verb ‘said’. It is a noun clause. The noun clause can also appear in the subject position as can be seen in the following sentence.

2. What you say is not clear to me.
If we replace the underlined part with ‘it’ in the above sentence, the sentence structure will be complete.

Note: Sometimes the word ‘that'(conjunction) can be left out in spoken English.
e.g. He felt that punishment was not enough, (that- adjective)
Now read the following passage carefully and identify the noun clauses.
Replace the underlined words ‘that’, ‘so’, and ‘it’ with suitable noun clauses.

The headmaster entered the class furiously and said that he wanted to know the reason for the absence of some students in the class the day before. One student said that he had suffered from a severe headache. The headmaster said, “I don’t believe that”. The second said that somebody stopped him from coming to school. The headmaster said, “I don’t think so”. The third said that he too had suffered from a bad headache. On hearing that the headmaster shouted in anger. The fourth said that he had suffered from a terrible toothache. The headmaster said, “I don’t believe it”. The fifth said, “My grandmother died suddenly”. The headmaster retorted that he would ascertain it. He said, “I will come to your house tomorrow to know the fact.”
Answer:
A) Noun Clauses

1. that he wanted to know the reason for the absence of some students in the class the day before.
2. that he had suffered from a severe headache.
3. that somebody stopped him from coming to school.
4. that he too had suffered from a bad headache.
5. that he had suffered from a terrible toothache.
6. that he would ascertain it.

B) Replacing ’that1 ‘so1 and ‘it’ with suitable noun clauses.

1. “I don’t believe that.” = “I don’t believethat you have suffered from a severe headache”.
2. “I don’t thinkso” = “I don’t thinkthat somebody stopped you from coming to school”.
3. “I don’t believe it.” = “I don’t believethat you have suffered from a terrible toothache.”

III. Editing

Read the following passage and edit (correct) the underlined parts.

Swami went home and says that the headmasterbeats him severely. The parents asked that why the headmaster had beaten him. Swami said that the headmaster beats him yesterday. Swami’s father said why the headmaster has beaten him without any cause. Swami replied to his father that the headmasterbeats him every day. Swami’s mother told to Swami to attend the classes regularly.
Answer:
Swami went home and said that the headmaster had beaten him severely. The par¬ents asked why the headmaster had beaten him. Swami said that the headmaster had beaten him the day before/the previous day. Swami’s father asked Swami why the head- master had beaten him without any cause. Swami repted that the headmaster beats him every day. Swami’s mother told Swami to attend the classes regularly.

Writing

I. What do you think Swami might have thought after he had left his school? Write down his reflections in a paragraph.
Answer:
Swami controlled his tears with a great effort. He came out of the school muttering “I don’t care for your dirty school.” As he came out he must have thought thus, “What a wretched school! How cruelty has taken the shape of this headmaster? Don’t they understand our feelings? Weren’t they too children at one time? Didn’t they cause any nuisance when they were children? Yes, I did break the panes in a moment of childish behaviour. Have I not been honest in not denying the charge? Is it not enough to scold us in the whole class? He went on insulting us, beating us, and threatening us with dismissals. Is that the way to correct us? Shouldn’t they give us a chance to set ourselves right? I wish I weren’t a student of this horrible school. Thank God! At least now, I gathered enough courage to come out of the school. I can join some other school. I think any other school will definitely be better than this bloody school. Anyhow, I too should and will be more careful about my behaviour and friends. I should think of my studies and my parents too. Yes, in one way, the incident is a lesson for me. I should make utmost use of it!

II. Construction of a Narrative

Look at the concluding part of the story.
He restrained the tears that were threatening to rush out, jumped down, and, grasp¬ing his books, rushed out, muttering, ‘I don’t care for your dirty school.’
Now imagine what happens to Swami after going away from school. Write a narrative which should include dialogues, sensory perceptions etc.
You may include things such as the following.
1) Swami rushed out from the class.
2) His parents asked him what happened at school.
3) His mother looked at the scars on his shoulders.
4) His father wanted him to go to school.
5) Swami did not like to go to school.
Answer:
Swami rushed out of the school saying, “I don’t care for your dirty school.” As he walked homewards, the tears that had been controlled till then found their way out. And they came out in floods. Swami didn’t mind them. His mind was swarmed with different kinds of thoughts. His heart ached with insult. His body burned with pain. His mind was hot with a heavy flow of thoughts. He did not notice passing persons. He didn’t even take care to walk to the left of the road. In fact, his legs took him along the regularly used path. At last, somehow he reached home with school bag on his back. The face with the marks of tears, the shoulders with the prints of the cane, and the heart, most importantly, with the load of insult and suffering!

Swami’s mother was moved on seeing Swami’s pathetic state. Her eyes turned wet as she saw the red scars on Swami’s shoulders. With choked throat, she asked Swami what had happened. Even the anxious father eagerly looked forward to listening to Swami. His parents’ anxiety touched Swami. He tried to be as cool as possible. He started narrating the incident at school very briefly.

“Our headmaster beats me everyday. For one reason or the other, he scolds me, and that too in a foul way. He repeatedly warns me with dire consequences.” As he continued, it became difficult for him to control his tears. Still, he tried to suppress the sobs and paused for a while. Father took that opportunity to ask, “Why does your headmaster beat you alone everyday without any reason ?”

That question hit Swami’s heart like an arrow. Yet he restrained his feelings and said in an emotion filled voice, “Dad, he beats everyone, not me alone; that is his nature, his hobby, his practice. You can check it with any of our school students. The only difference is on the days of visits by officers or village elders.”

“Then,” interrupted mother with concern, “What shall we do ?” “I shall join some other school,” prompt was the reply from Swami lest Father should announce a different decision.

Noticing the resolute voice of Swami, even father wanted Swami’s words to prevail. Mother was too willing. She sighed in relief as there was no opposition from father. Swami started dreaming of his new school!

Study Skills

I. Read the data given in the table and answer the questions that follow.
Reasons for Dropouts among children Aged 5-14 Years – 1997-98

Source: Ministry of Human Resource Development (MHRD)
Answer the following questions:
1. What is the major reason for dropouts in rural and urban areas?
2. What percentage of female children are dropped out as their parents not being interested in their children’s studies in rural areas?
3. Which is the less significant reason for dropouts?
4. Which of the following statements are true with reference to the data given in the above table? Tick (S) the true statements.
a. The dropout-rate due to child not being interested in studies is more among the urban children than that of the rural children.
b. Participation in other economic activities is high among urban female children when compared with rural female children.
c. If we create interest in studies among the children, the literacy rate will increase in our country.
Answer:

1. The major reason for dropouts among children both in rural and urban areas is the child being not interested in studies.
2. 9.2 percent of girls are dropouts in rural areas because of lack of interest on the part of parents.
3. Working for wages/salaries is the least significant reason for dropouts.
4. a – ✗ b – ✓ c – ✓

II. Write a report on dropouts using the information given in the above table.

You may begin the report like this.
This report is based on the data provided by MHRDfor the year 1997-98for children who dropped out in the age group of 5-14 years…
Include the following.
1. The financial reasons for dropouts
2. Personal reasons for the dropouts
3. Whether the dropout rate is more among girls/boys
4. Whether the dropout rate is more among rural/urban
Answer:
This is a report on dropout rates among children aged between 5 -14. The information comes from the survey conducted by the Ministery of Human Resource Development (MHRD) for the year 1997-1998. Both the rural and urban areas (separately) were taken into consideration and the data categorises male and female children separately. Dropout rate caused by lack of interest on the part of the child tops the list with 37.2% in rural areas and 34.7% in urban areas. Working for wages is the least significant reason with just 2.5% in villages and 3.6% in towns. Dropout rate among girls is more than that of the boys by 10% in rural areas and by about 6% in urban areas. Economic, social, personal, and other reasons contribute for this undesirable state of affairs.

Listening

Practise listening carefully. Then you will be able to speak.
Listen to the 2 speeches and answer the questions that follow.

Speeches!
Speech 1
Good evening to all the people present here today. It’s a pleasure to start this occasion by welcoming everybody. I welcome our headmaster to preside over the function. Now I would like to invite the honourable chief guest, our M.L.A to come onto the dais. I welcome my teachers, my fellow – friends, and schoolmates. Today we all have gathered here for the Annual Day celebration. I welcome you all and I hope you enjoy the programme.
Now I request our headmaster to start the function.
Thank you.

Speech 2 :
I have a great pleasure in welcoming our principal to preside over the Children’s Day programme that we have today. It is a great privilege for me to invite our chief guest, the D.E.O. of our district, who has kindly consented to be the chief guest for the day, I also welcome other distinguished guests, who also have consented to be with us today despite their tight schedule. I cordially welcome my colleagues and non-teaching staff too. In today’s celebration our main heroes are our students. So I welcome them and their parents too along with all others.

I hope you enjoy every aspect of this event and request the president to begin the proceedings.
Thank you.
Now answer the following questions:
1. Who is the speaker of speech 1?
Answer:
A student is the speaker of speech 1.

2. What is the occasion mentioned in the first speech?
Answer:
The occasion is Annual Day celebration. (1st speech)

3. What is the occasion mentioned in the second speech?
Answer:
The occasion is Children’s Day programme. (2nd speech)

4. Who is the chief guest mentioned in the second speech?
Answer:
The D.E.O of the district is the chief guest. (2nd speech)

5. Who is addressing the gathering in the second speech?
Answer:
A teacher is addressing the gathering. (2nd speech)

Oral Activity

Imagine that you are the School Pupils’ Leader (SPL). Compere on the Republic Day celebrations in your school.
You may include the following in your speech :
1) Welcome address
2) Inviting the guests onto the dais
3) Importance of the occasion
4) Request to continue the proceedings
Answer:
A brilliant morning to every soul gracing the grand occasion of our Republic Day celebrations. A noble national festival and our patriotic feelings have brought us all together here. On this memorable event, I extend my warm welcome to you all and the greetings of the day too. Now I invite our honourable headmaster to come over to the dais to preside over the proceedings. I feel it my privilege and pleasure to invite our distinguished guest of the day, our beloved collector on to the dais. Let us all extend our thunderous applause as our collector comes to grace the occasion. It is my pleasure to invite on to the dais our dear teachers who have won our hearts with their excellent teaching skills and touching attitude.

Now my dear fellow students, parents, and guests let us cherish each moment of the programmes it gradually unfolds, majestically moves ahead, and colourfully culminates. May I now request the president of the programme to lead the proceedings.
Thank you one and all.

Swami is Expelled from School Summary in English

Rasipuram Krishnaswamy Narayan’s novel Swami and Friends is an interesting story of Swami. The present part deals with one incident in Swami’s school life. One day Swami and his friends did not attend classes. Moreover, they broke the window panes of the Headmaster’s room. The peon saw that act. The following day the headmaster entered Swami’s class with a cane and a furious face. He started scolding them. He also demanded every one to explain to him why they had been absent the previous day. When they cooked up some stories, he asked for proof. He used the cane, asked them to stand up on their benches, announced suspension, etc. When it was the turn of Swami, he had no stories to tell the headmaster. His silence added insult to the headmaster’s injury. He went on beating Swami on the shoulders, the back with his cane. Swami’s complaints of pain made the headmaster more violent. Not able to bear the torture any more, Swami left the class

Swami is Expelled from School Glossary

ominous (adj) : indicating the happening of bad

but (preposition) (here) : except

eminence (n) : fame, respect

loafer(n) : a person who wastes time without working

ascertain (v) : find to be true or not

pause(n) : a short stop ; gap

knuckles (n-plural-the first ‘k’ silent : joints in fingers
(Note : The ‘k’ at the beginning of a word followed by ‘n’ is silent, e.g. : know, knife, knight)

resolutely (adv) : with a strong decision

gazing (v-ing) : looking

intently (adv) : with all attention

acute (adj) : intense

rap (n) : a sharp hit

gaol(n) : jail, prison

discreet (adj) : careful; tactful

stammered (v-past tense) : spoke with difficulty ; repeating sounds

thrash (v) : hit, beat

sinister (adj) : evil, dangerous

grunted (v-past tense) : made short, low sounds in the throat to show Irritation

ventilators (n-plural) : window like arrangements close to the ceiling

zest (n) : enthusiasm

whack (n) : a hit

brigand (n) : a member of a group of criminals

defiant (adj) : refusing to obey

deny (v) : to refuse to admit

charge (n) : accusation

ejaculated (v-past tense) : shouted suddenly

staring (v+ing) : looking

idiot (n) : a stupid person ; a fool

desperation (n) : hopeless condition

restrained (v – past tense) : controlled

muttering (v + ing) : saying in a low voice

AP State Syllabus AP Board 9th Class Maths Solutions Chapter 1 Real Numbers InText Questions and Answers.

AP State Syllabus 9th Class Maths Solutions 1st Lesson Real Numbers InText Questions

Do This

Question 1.
Represent \(\frac{-3}{4}\) on a number line. (Page No.3)
Solution:

Step – 1: Divide each unit into four equal parts to the right and left side of zero on the number line.
Step – 2 : Take 3 parts after zero on its left side.
Step -3 : It represents \(\frac{-3}{4}\)

Question 2.
Write 0, 7, 10, -4 in p/q form (Page No.2)
Solution:

Question 3.
Guess my number : Your friend chooses an integer between 0 and 100. You have to find out that number by asking questions, but your friend can only answer ‘Yes’ or ‘No’. What strategy would you use ? (Page No.3)
Solution:
Let my friend choosen 73; then my questions may be like this.

Q : Does it lie in the first 50 numbers ?
A: No .

Q : Does it lie between 50 and 60 ?
A: No

Q : Does it lie between 60 and 70 ?
A: No

Q : Does it lie between 70 and 80 ?
A: Yes
[then my guess would be “it is a number from 70 to 80]

Q : Is it an even number ?
A: No
[my guess : it should be one of the numbers 71, 73, 75, 77 and 79]

Q : Is it a prime number ?
A: Yes
[my guess : it may be 71, 73 or 79]

Q : Is it less than 75 ?
A: Yes
[my guess : it may be either 71 or 73]

Q : Is it less than 72 ?
A: No

Then the number is 73.
Therefore the strategy is
→ Use number properties such as even, odd, composite or prime to determine the number.

Do This

Question (i)
Find five rational numbers between 2 and 3 by mean method. (Page No.4)
Solution:
We know that \(\frac{a+b}{2}\) is a rational
between any two numbers a and b.
Let a = 2 and b = 3

Question (ii)
Find 10 rational numbers between \(\frac{-3}{11}\) and \(\frac{8}{11}\) (Page No.4)
Solution:
\(\frac{-3}{11}<\left[\frac{-2}{11}, \frac{-1}{11}, \frac{0}{11}, \frac{1}{11}, \frac{2}{11}, \frac{3}{11}, \frac{4}{11}, \frac{5}{11}, \frac{6}{11}, \frac{7}{11}\right]<\frac{8}{11}\)

Do This

Question
Find the decimal form of
(i) \(\frac{1}{17}\) (Page No.5)
Solution:

\(\frac{1}{19}\) = 0.052631

(ii) \(\frac{1}{19}\)(Page No.5)
Solution:

Try These

Question 1.
Find the decimal values of the following. (Page No.6)
Solution:
i) \(\frac{1}{2}\) = 0.5
ii) \(\frac{1}{2^{2}}=\frac{1}{4}\) = 0.25
iii) \(\frac{1}{5}\) = 0.2
iv) \(\frac{1}{5 \times 2}=\frac{1}{10}\) = 0.1
v) \(\frac{3}{10}\) = 0.3
vi) \(\frac{27}{25}\)

vii) \(\frac{1}{3}\)
Solution:

viii) \(\frac{7}{6}\)
Solution:

ix) \(\frac{5}{12}\)
Solution:

x) \(\frac{1}{7}\)
Solution:

Think, Discuss and Write

Question 1.
Kruthi said √2 can be written as \(\frac{\sqrt{2}}{1}\) which is in form. So √2 is a rational number. Do you agree with her argument ? (Page No.10)
Solution:
No.
Writing √2 = \(\frac{\sqrt{2}}{1}\) is not in the \(\frac { p }{ q }\) form.
Since p and q are integers and √2 is not an integer.

Try These

Question 1.
Find the value of √3 upto six decimals.(Page No.10)
Solution:

Step 1 :
Write 3 as 3.00 00 00 00 00 00 00
Step – 2 :
Group the zeros in pairs (i.e.) make periods.
Step – 3 :
Find the square root using long division method.
∴ √3 = 1.732050

Try These

Question 1.
Locate √5 and -√5 on the number line [Hint 5 = 2² + 1²](Page No.12)
Solution:
Step – 1 :
At zero draw a rectangle of length 2 units and breadth 1 unit.

Step – 2 :

Step – 3 :
Using compass draw an arc of radius OB with centre ‘O’ which cuts the num-ber line at D and D¹.

Step – 4 :
D represents √5 and D¹ represents – √5 on the number line.

Activity

Question
Constructing the ‘Square root spiral’. (Page No. 15)
Solution:
Take a large sheet of paper and construct the ‘Square root spiral’ in the following manner.
Step – 1 : Start with point ‘O’ and draw a line segment \(\overline{\mathrm{OP}}\) of 1 unit length.
Step – 2 : Draw a line segment \(\overline{\mathrm{PQ}}\) perpendicular to \(\overline{\mathrm{OP}}\) of unit length (where OP = PQ = 1) (see Fig.)
Step – 3 : Join 0, Q. (OQ = √2 )
Step – 4 : Draw a line segment OR of unit length perpendicular to \(\overline{\mathrm{OQ}}\)
Step – 5 : Join O, R. (OR = √3 )
Step – 6 : Draw a line segment RS of unit length perpendicular to \(\overline{\mathrm{OR}}\).
Step – 7 : Continue in this manner for some more number of steps, you will create a
beautiful spiral made of line segment \(\overline{\mathrm{PQ}}, \overline{\mathrm{QR}}, \overline{\mathrm{RS}}, \overline{\mathrm{ST}}, \overline{\mathrm{TU}}\) …………etc. Note that the line segments \(\overline{\mathrm{OQ}}, \overline{\mathrm{OR}}, \overrightarrow{\mathrm{OS}}, \overline{\mathrm{OT}}, \overline{\mathrm{OU}}\) …… etc., denote the lengths √2,√3, √4, √5, √6 respectively.

Do This

Question 1.
Find rationalising factors of the denominators of (Page No. 20)
i) \(\frac{1}{2 \sqrt{3}}\)
Solution:
\(\frac{1}{2 \sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}}=\frac{\sqrt{3}}{2 \times 3}=\frac{\sqrt{3}}{6}\)
∴ The R.F is √3

ii) \(\frac{1}{\sqrt{5}}\)
Solution:
\(\frac{3}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}}=\frac{3 \sqrt{5}}{5}\)
∴ The R.F is √5

iii) \(\frac{1}{\sqrt{8}}\)
Solution:

Do This;

Question
Simplify : (Page No. 23)

i) (16) ^1/2
Solution:
(16)^1/2 = (4 x 4)^1/2 = (42)^1/2 = 4^2/2 = 4

ii) (128)^1/7
Solution:
(128)^1/7 =(2 X 2 X 2 X 2 X 2 X 2 X 2)^1/7 = (2⁷)^1/7 = 2

iii) (343)^1/5
Solution:
(343)^1/5= (3 x 3 x 3 x 3 x 3)^1/5 = (3⁵)^1/5= 3

Question 1.
Write the following surds in exponen-tial form (Page No. 24)
i) √2
Solution:
√2 = 2^1/2

ii) 3√9
Solution:
\(\sqrt[3]{9}=\sqrt[3]{3 \times 3}=\sqrt[3]{3^{2}}=3^{\frac{2}{3}}\)

iii) \(\sqrt[5]{20}\)
Solution:
\(\begin{aligned}
\sqrt[5]{20}=\sqrt[5]{2 \times 2 \times 5} &=\sqrt[5]{2^{2} \times 5} \\
&=2^{\frac{2}{5}} \times 5^{\frac{1}{5}}
\end{aligned}\)

iv) 17√19
Solution:
\(\sqrt[17]{19}=19^{\frac{1}{17}}\)

Question 2.
Write the surds in radical form. (Page No. 24)
i) 5^1/7 = \(\sqrt[7]{5}\)
ii) 17^1/6 = \(\sqrt[6]{17}\)
iii) 5^2/5 = \(\sqrt[5]{5^{2}}=\sqrt[5]{5 \times 5}=\sqrt[5]{25}\)
iv) 142^1/2 = \(\sqrt{142}\)

AP State Syllabus AP Board 9th Class Maths Solutions Chapter 12 Circles Ex 12.5 Textbook Questions and Answers.

AP State Syllabus 9th Class Maths Solutions 12th Lesson Circles Exercise 12.5

Question 1.
Find the values of x and y in the figures given below.
i)

Solution:
From the figure x = y [ ∵ angles opp. to opp. to equal sides]
But x + y + 30° = 180°
∴ x + y = 180° – 30° = 150°
⇒ x + y = \(\frac { 150° }{ 2 }\) = 75°

ii)

Solution:
From the figure x° + 110° = 180°
[ ∵ Opp. angles of a cyclic quad, are supplementary]
y + 85°= 180°
∴ x= 180° – 110°; y = 180° – 85°
x = 70°; y = 95°

iii)

Solution:
From the figure x = 90° [angle in a semi-circle]
∴ y = 90° – 50° [∵ angle sum property]
= 40°

Question 2.
Given that the vertices A, B, C of a quadrilateral ABCD lie on a circle. Also ∠A + ∠C = 180°, then prove that the vertex D also lie on the same circle.
Solution:
Given : ∠A + ∠C = 180°
∴ ∠B + ∠D = 360° – 180°
[ ∵ sum of the four angles of a quad. is 360 ].
Now in □ABCD, sum of the pairs of opp. angles is 180°.
∴ □ABCD must be a cyclic quadrilateral, i.e., D also lie on the same circle on which the vertices A, B and C lie. Hence proved.

Question 3.
Prove that a cyclic rhombus is a square.
Solution:

Let □ABCD be a cyclic rhombus,
i.e., AB = BC = CD = DA and
∠A + ∠C = ∠B + ∠D = 180°
But a rhombus is basically a parallelo-gram.

Question 4.
For each of the following, draw a circle and inscribe the figure given. If a poly¬gon of the given type can’t be in-scribed, write not possible.
a) Rectangle
b) Trapezium .
c) Obtuse triangle
d) Non-rectangular parallelogram
e) Acute isosceles triangle
f) A quadrilateral PQRS with PR as diameter.
Solution:

AP State Syllabus AP Board 9th Class Maths Solutions Chapter 15 Proofs in Mathematics Ex 15.4 Textbook Questions and Answers.

AP State Syllabus 9th Class Maths Solutions 15th Lesson Proofs in Mathematics Exercise 15.4

Question 1.
State which of the following are mathematical statements and which are not ? Give reason.
i) She has blue eyes.
Solution:
This is not a mathematical statement. no mathematics is involved in it.

ii) x + 7 = 18
Solution:
This is not a statement, as its truthness cant be determined.

iii) Today is not Sunday.
Solution:
This is not a statement. This is an am biguous open sentence.

iv) For each counting number x, x + 0 = x.
Solution:
This is a mathematical statement.

v) What tune is it?
Solution:
This is not a riathematical statement.

Question 2.
Find counter examples to disprove the following statements.
i) Every rectangle is a square.
Solution:

A rectangle and square are equiangular i.e., all the four angles are right angles. This doesn’t mean that they have equal sides.

ii) For any integers x and y,
\(\sqrt{x^{2}+y^{2}}\) = x + y
Solution:
Let x = 3; y = 8
\(\sqrt{x^{2}+y^{2}}=\sqrt{3^{2}+8^{2}}\)
= \(\sqrt{9+64}=\sqrt{73}\)

x + y = 3 + 8 = 11
Here, √73 ≠ 11
i.e., \(\sqrt{x^{2}+y^{2}}\) ≠ x + y

iii) If n is a whole number then 2n² +11 is a prime.
Solution:
If n = 11 then 2n²+ 11 = 2 (11)² + 11
= 11 (2 × 11 + 1) = 11 × (22 + 1)
= 11 × 23 is not a prime.

iv) Two triangles are congruent if all their corresponding angles are equal.
Solution:

If the corresponding angles are equal then the triangles are only similar.

v) A quadrilateral with all sides are equal is a square.
Solution:
A rhombus is not a square, but all its sides are equal.

Question 3.
Prove that the sum of two odd numbers is even.
Solution:

Question 4.
Prove that the product of two even numbers is an even number.
Solution:

Question 5.
Prove that if x is odd, then x² is also odd.
Solution:
Let x be an odd number.
Then x = 2m + 1
(general form of ah odd number) Squaring on both sides,
x² = (2m + 1)²
= 4m² + 4m +1
= 2 (2m² + 2m) + 1
= 2K + 1 where K = 2m²+ 2
Hence x² is also odd.

Question 6.
Examine why they work ?
Choose a number. Double it. Add nine. Add your original number. Divide by three. Add four. Subtract your original number. Your result is seven.
Solution:
Choose a number = x say
Double it = 2x
Add nine = 2x + 9
Add your original number
= 2x + 9 + x = 3x + 9
Divide by 3 = (3x + 9) ÷ 3
= \(\frac{3 x}{3}+\frac{9}{3}\) = x + 3
Add 4 ⇒ x + 3 + 4 = x + 7
Subtract your original number =
x + 7 – x = 7
Your result is 7 – True.

ii) Write down any three digit number (for example, 425). Make a six digit number by repeating these digits in the same order 425425. Your new number is divisible by 7, 11 and 13.
Solution:
Let a three digit number be xyz.
Repeat the digit = xyzxyz
= xyz × (1001)
= xyz × (7 × 11 × 13)
Hence the given conjecture is true.

AP State Syllabus AP Board 9th Class Maths Solutions Chapter 15 Proofs in Mathematics Ex 15.3 Textbook Questions and Answers.

AP State Syllabus 9th Class Maths Solutions 15th Lesson Proofs in Mathematics Exercise 15.3

Question 1.
Take any three consecutive odd numbers and find their product, for example 1 × 3 × 5 = 15;
3 × 5 × 7 = 105: 5 × 7 × 9 = ……………
ii) Take any three consecutive even numbers and add them, say,
2 + 4 + 6 = 12; 4 + 6 + 8 = 18:
6 + 8 + 10 = 24; 8 + 10 + 12 = 30 ….
so on. Is there any pattern you can guess in these sums ? What can von conjecture about them ?
Solution:
i) 1 × 3 × 5 = 15
3 × 5× 7 = 105
5 × 7 × 9 = 315
7 × 9 × 11 = 693

1. The product of any three consecutive odd numbers is odd.
2. The product of any three consecutive odd numbers is divisible by ’3′.
3. 2 + 4 + 6 = 12; 4 + 6 + 8= 18;
   6 + 8 + 10 = 24; 8 + 10 + 12 = 30
4. The sum of any three consecutive even numbers is even.
5. The sum of any three consecutive even numbers is divisible by 6
6. The sum of any three consecutive even numbers is a multiple of 6.

Question 2.
Go back to Pascal’s triangle.
Line-1: 1=11°
Line-2: 11 = 11¹
Line-3 : 121 = 11²
Make a conjecture about line – 4 and line – 5.

Does your conjecture hold ? Does your conjecture hold for line – 6 too ?
Solution:
Line-4 : 1331 = 11³
Line-5 : 14641 = 11⁴
Line – 6 : 11⁵
∴ Line – n = 11^n-1
Yes the conjecture holds good for line – 6 too.

Question 3.
Look at the following pattern.
i) 28 = 2² × 7¹;
Total number of factors
(2+ 1)(1 + 1) = 3 × 2 = 6
28 is divisible by 6 factors i.e.,
1, 2, 4, 7, 14, 28.
ii) 30 = 2¹ × 3¹ × 5¹, Total number of .
factors (1 + 1) (1 + 1) (1 + 1) = 2 × 2 × 2 = 8
30 Is divisible by 8 factors i.e., 1, 2, 3, 5, 6, 10, 15 and 30
Find the pattern.
[Hint : Product of every prime base exponent +1)
Solution:
24 = 2³ × 3¹
24 has (3+1) (1 + 1) = 4 × 2 = 8 factors
[1, 2, 3, 4, 6, 8, 12 and 24]
36 = 2² × 3²
Number of factors = (2 + 1) (2 + 1)
3 × 3 = 9 [ 1, 2, 3. 4, 6, 9, 12, 18 and 36]
If N = a^p. b^q . c^r…….. where N is a natural number.
a. b, c … are primes and p, q, r ……. are positive integers then, the number of factors of N =(p- 1)(q+ 1)(r + 1)

Question 4.
Look at the following pattern :
1² = 1
11² = 121
111² = 12321;
1111² = 1234321
11111² = 123454321
Make a conjecture about each of the following
111111² =
1111111² =
Check if your conjecture is true.
Solution:
111111² = 12345654321
1111111² = 1234567654321
(111………. n-times)² = (123 … (n- 1) n (n – 1) (n – 2) 1)
The conjecture is true.

Question 5.
List five axioms (postulates) used in text book.
Solution:

1. Things which are equal to the same things are equal to one another.
2. If equals are added to equals, the sums are equal.
3. If equals are subtracted from equals, the differences are equal.
4. When a pair of parallel lines are in-tersected by a transversal, the pairs of corresponding angles are equal.
5. There passes infinitely many lines through a given point.

Question 6.
In a polynomial p(x) = x² + x + 41 put different values of x and find p(x). Can you conclude after putting different values of x that p(x) is prime for all. Is ‘x’ an element of N ? Put x = 41 in p(x). Now what do you find ?
Solution:
p(x) = x² + x + 41
p(0) = 0² + 0 + 41 = 41 – is a prime
p(1) = 1² + 1 + 41 = 43 – is a prime
p(2) = 2² + 2 + 41 = 47 – is a prime
p(3) = 3² + 3 + 41 = 53 – is a prime
p(41) = 41² + 41 + 41
= 41(41 + 1 + 1)
= 41 x 43 is not a prime.
∴ p(x) = x²+ x + 41 is not a prime for all x.
∴ The conjecture “p(x) = x² + x + 41 is a prime” is false.

AP State Syllabus AP Board 9th Class Maths Solutions Chapter 15 Proofs in Mathematics Ex 15.2 Textbook Questions and Answers.

AP State Syllabus 9th Class Maths Solutions 15th Lesson Proofs in Mathematics Exercise 15.2

Question 1.
Use deductive reasoning to answer the following.

i) Human beings are mortal. Jeevan is a human being. Based on these two statements, what can’ you conclude about Jeevan?
Solution:
From the above statements we can deduce that Jeevan is mortal as it is given that all humans are mortal and Jeevan is a human.

ii) All Telugu people are Indians. X is an Indian. Can you conclude that X belongs to Telugu people?
Solution:
No. X may belong to any other language like Tamil, Kannada, Malayali…. etc.

iii) Martians have red tongues. Gulag is a Martian. Based on these two statements, what can you conclude about Gulag?
Solution:
Gulag had red, tongue.

iv) What is the fallacy in the Raju’s reasoning in the cartoon below

Solution:
All smarts need not be a President. There could be some other persons who are smart too.

Question 2.
Once again you are given four cards. Each card has a number printed on one side and a letter on the other side. Which are the only two cards you need to turn .over to check whether the following rule holds ?
“If a card has a consonant on one side, then it has an odd number on the other side”.

Solution:
You need to turn over B and 8 only. If B has an even number then the rule has broken. Similarly if 8 has a consonant on the other side then also the rule has been broken.

Question 3.
Think of this puzzle. What do you need to find a chosen number from this square ? Four of the clues below are true but do nothing to help in finding the number. Four of the clues are necessary for finding it.
Here are eight clues to use:
a) The number is greater than 9.
b) The number is not a multiple of 10.
c) The number is a multiple of 7.
d) The number is odd.
e) The number is not a multiple of 11.
f) The number is less than 200.
g) Its ones digit is larger than its tens digit.
h) Its tens digit is odd.
What is the number ?

Solution: