The Pot Maker MCQ Class 9 English Chapter 2

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The Pot Maker Class 9 MCQ

Class 9 English Chapter 2 MCQ Online Test

MCQ of The Pot Maker Class 9

Extract Based Questions

1. On days when she managed to stay at home while her parents went to the fields, Sentila approached expert potters to learn the skill. They were at first amused by the little girl’s insistence as they thought that she would soon outgrow her childish passion for the craft. Sentila did not disclose her fascination with pot making at home as she had overheard a conversation between her parents one night. Her mother was complaining about Sentila’s indifference to weaving. She said, “I shall not teach her pot making as it has brought no joy to me and only a pittance for my troubles. The riverbank, where the grey and red clay required for making pots is found, is sixteen kilometres from here. I need to climb down a sheer drop to the riverbank and my back aches from carrying the heavy load uphill all the way to the village.

(i) Why did Sentila hide her interest in pot making from her mother?
(A) She was afraid of being punished
(B) Her mother disapproved of pot making
(C) She wanted to surprise her parents
(D) She did not trust her parents
Answer:
(B) Her mother disapproved of pot making

(ii) What does the word ‘pittance’ suggest in the passage?
(A) A large profit
(B) A regular income
(C) A very small amount of money
(D) A reward for hard work
Answer:
(C) A very small amount of money

(iii) Fill in the blanks with appropriate word from the extract:
Carrying the clay uphill caused Arenla’s _________ (head / back) to ache.
Answer:
back

(iv) What is the central conflict in the extract?
(A) Sentila vs. her parents
(B) Sentila vs. potters
(C) Sentila’s interest vs. expectations her mother’s
(D) Mother vs. village
Answer:
(C) Sentila’s interest vs. expectations her mother’s

(v) Choose the image which BEST describes the expert potters’ reaction to Sentila’s willingness to learn pottery:

Answer:
(B)

2. Arenla heard of her daughter’s visits but pretended to be ignorant of them. By late afternoon, Sentila had to return home quickly, so that when her mother came back from the fields, she would be there. Gradually, Sentila’s visits l became a topic of village gossip. People wondered ; why Arenla was refusing to teach the skill to her daughter. They were worried that if all pot l makers followed suit, then there would be no ; expert potters to take their place. One day, j Sentila’s father, Mesoba was summoned by the village council and asked to explain why Arenla was refusing to pass on the skill to her daughter. He replied in a humble tone, “Uncles and elder brothers, Arenla has never said that she will not teach our daughter pot making; it is only that we wanted her to grow stronger after her illness. You j ivill soon see Sentila making the best pots in the village.

(i) Read the statement and write whether it is True or False.
Arenla was aware of Sentila’s visits to the potters.
Answer:
True

(ii) Choose the correct meaning of the word ‘summoned’ as used in the passage.
(A) Invited casually
(B) Called officially
(C) Forced to leave
(D) Chanted incantations to call someone
Answer:
(B) Called officially

(iii) Why did Sentila return home quickly every afternoon?
Answer:
Sentila returned quickly so that she would be at home when her mother returned from the fields.

(iv) What had become a topic of village gossip?
Answer:
Sentila’s visits to the potters had Become topic of village gossip.

(v) How did Mesoba explain Arenla’s actions to the village council?
Answer:
He said that Arenla had not refused to teach Sentila, but they wanted her to grow stronger after her illness.

(vi) Mesoba’s tone in this extract was:
(A) Concerned and Sad
(B) Affirmative and Hopeful
(C) Modest and Reassuring
(D) Submissive and Obedient
Answer:
(C) Modest and Reassuring

(vii) “We wanted her to grow stronger after her illness.” Who is sick? How do you know?
Answer:
Sentila was the one who was sick. When Mesoba says, “we wanted her to grow stronger,” the “her” refers back to Sentila, the subject of the council’s inquiry. He uses the illness as the official reason for the delay in her training.

3. Before long, she realised that the pot was ready. She started on the next one, and like a sprinter who had suddenly found momentum, she continued making pot after pot with the same speed and dexterity that she had noticed in her mother’s hands. Finally, when she looked at her row of pots, she saw that she had made just one short of her mother’s tally. Exhausted from the labour, she decided to go into the house and join her mother for lunch. When she reached the threshold, she found her mother lying on the floor. She was not breathing. Sentila ran towards the common area in the village for help.

(i) What helped Sentila make pots quickly and skillfully?
(A) Her mother’s direct help
(B) Her natural talent alone
(C) Her careful observation and practice
(D) Instructions from villagers
Answer:
(C) Her careful observation and practice

(ii) Why did Sentila go inside the house?
(A) To rest after working
(B) To call her mother
(C) To have lunch with her father
(D) To avoid more work
Answer:
(A) To rest after working

(iii) What sudden change occurs at the end of the extract?
(A) Sentila becomes tired
(B) Sentila improves her skill
(C) Her mother falls ill
(D) Her mother is found dead
Answer:
(D) Her mother is found dead

(iv) Why is the mention of ‘threshold’ important?
(A) It shows the house structure
(B) It marks a transition from success to tragedy
(C) It describes the village setting
(D) It indicates rest
Answer:
(B) It marks a transition from success to tragedy

(v) Arrange the following events in the correct
1. Sentila finds her mother lying on the floor
2. Sentila gains momentum while making pots
3. Sentila checks her tally of pots
4. Sentila decides to go inside the house
(A) 2 → 3 → 4 → 1
(B) 2 → 4 → 3 → 1
(C) 3 → 2 → 4 → 1
(D) 2 → 3 → 1 → 4
Answer:
(A) 2 → 3 → 4 → 1

4. The villagers raced to the house and sent for Mesoba. When Arenla’s body was being carried out of the house the next morning, Sentila ran after it, shouting, “Mother, I did not wish it to happen this way; it simply came to me. Please forgive me.” Those who heard her, did not understand what she meant, except for Onula. She intuitively sensed that something momentous had happened. On her way back, Onula noticed that the door to the work shed was slightly ajar. Out of curiosity, she stepped inside and abruptly stopped in her tracks; two neat rows of newly-made pots stood side by side. She could find nothing to tell one batch from the other. She was sure thatdt was not the handiwork of one person alone. Onula stood there for a long time as if trying to absorb a new phenomenon.

(i) Which scene is being described in the extract? Answer as per the extract:
(A) A funeral procession
(B) A wedding procession
(C) A village celebration
(D) A market gathering
Answer:
(A) A funeral procession

(ii) What did Sentila’s statement suggest about how she feels?
(A) Regret
(B) Miserable
(C) Hopeless
(D) Confused
Answer:
(A) Regret

(iii) Complete the sentence using words from the passage.
Sentila says that what happened “simply” to her.”
Answer:
came

(iv) What does the presence of two identical rows of pots suggest about Sentila’s learning?
Answer:
It suggests that Sentila has achieved the same level of skill as her mother and has become a true pot maker.

(v) Which detail in the image MOST CLOSELY reflects the ‘phenomenon’ mentioned in the extract?

Answer:
(B)

Match the following terms with their correct descriptions :

(a) Deftly	(i) A small amount of money
(b) Pittance	(ii) Tiring or boring
(c) Malleable	(iii) Skillfully
(d) Tedious	(iv) Tool used for shaping pots
(e) Spatula	(v) Able to be shaped without breaking

Answer:

(a) Deftly	(iii) Deftly: Skillfully
(b) Pittance	(i) Pittance: A small amount of money
(c) Malleable	(v) Malleable: Able to be shaped without breaking
(d) Tedious	(ii) Tedious: Tiring or boring
(e) Spatula	(iv) Spatula: Tool used for shaping pots

Choose the correct Determiner to complete the sentence:

“Arenla believed that _______ (some/several/much) more money could be made from weaving than pottery.”
Answer:
Much

Identify the type of clause underlined below:
“Sentila, whose passion for pottery was deep, practised every day.”
Answer:
Adjectival / Relative Clause.

Match the Words with Their Meanings

A	B
1. Deftly	a. Very important
2. Indifference	b. Skilled and quick in movement
3. Consistency	c. Lack of interest
4. Ruin	d. Thickness or firmness
5. Malleable	e. To destroy completely
6. Momentous	f. Easily shaped

Answer:

A	B
1. Deftly	b. Skilled and quick in movement
2. Indifference	c. Lack of interest
3. Consistency	d. Thickness or firmness
4. Ruin	e. To destroy completely
5. Malleable	f. Easily shaped
6. Momentous	a. Very important

Choose suitable words from the box:
(pittance, kiln, fascinated, obliged, dexterity, tedious, gossip)

1. The villagers were _______ to preserve their traditional craft.
Answer: obliged

2. Sentila watched the potters, completely _______ by their skill.
Answer: fascinated

3. The pots were fired in the _______ carefully.
Answer: kiln

4. The work of pounding clay was long and _______ .
Answer: tedious

5. The reward for months of labour was only a _______ .
Answer: pittance

6. The news soon became a topic of village _______ .
Answer: gossip

7. Sentila worked with speed and _______ like her mother.
Answer: dexterity.

Write whether the underlined word is a noun, verb, adjective or adverb.

1. The pot emerged slowly from the lump of clay.
Answer:
slowly – Adverb

2. The villagers showed great concern.
Answer:
concern – Noun

3. Sentila worked diligently.
Answer:
diligently-Adverb

4. The clay became malleable after soaking.
Answer:
malleable-Adjective

5. Arenla pretended to be ignorant.
Answer:
pretended-Verb

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How I Taught My Grandmother to Read MCQ Class 9 English Chapter 1

Practicing Class 9 English MCQ and Class 9 English Kaveri Chapter 1 How I Taught My Grandmother to Read MCQs with Answers daily helps in time management.

How I Taught My Grandmother to Read Class 9 MCQ

Class 9 English Chapter 1 MCQ Online Test

MCQ of How I Taught My Grandmother to Read Class 9

Extract Based Questions

1. “All of us would wait eagerly for the bus, which used to come with the papers, weekly magazines and the post. At that time, Triveni was a very popular writer in the Kannada language. Her style was easy to read and very convincing.”

(i) Complete the sentence with the correct option.
The eagerness with which people waited for the bus suggests that printed magazines were _______________.
(A) a valued source of news and entertainment
(B) an unimportant part of village daily life
(C) disliked by readers in the village
(D) easily available in every household
Answer:
(A) a valued source of news and entertainment

(ii) Choose the correct option.
The description of Triveni’s writing style suggests that her stories were _______________.
(A) difficult and confusing
(B) unrelated to people’s lives
(C) simple and engaging for readers
(D) written only for scholars
Answer:
(C) simple and engaging for readers

(iii) Identify one phrase from the extract that shows the villagers’ dependence on the bus.
Answer:
“wait eagerly for the bus”

(iv) Which of the following aspects is NOT emphasised in the given extract?
(A) the popularity of Triveni as a writer
(B) the arrival of newspapers and magazines
(C) the villagers’ eagerness for reading material
(D) the emotional struggles of characters . in Triveni’s stories
Answer:
(D) the emotional struggles of characters in Triveni’s stories

(v) Which image best reflects the popularity of Triveni’s writing?

Answer:
(A)

(vi) Select the option that best expresses the meaning of the word ‘convincing’ as used in the extract. (An) [Vocabulary Building]
(A) difficult to understand
(B) easy to believe
(C) unrelated to reality
(D) meant for scholars
Answer:
(B) easy to believe

Explanation:
If a story is ‘convincing,’ it means the characters and situations feel real and believable to the reader, rather than fake or forced.

2. “Every Wednesday the magazine would come and I would read the next episode of the story to her. During that time she would forget all her work and listen with the greatest concentration. Later, she could repeat the entire text by heart. My grandmother too never went to Kashi, and she identified herself with the novel’s protagonist.”

Complete the sentence with the most appropriate option.

(i) The grandmother’s ability to repeat the story by heart shows her ____________.
(A) strong memory and learning ability
(B) lack of interest in daily work
(C) habit of ignoring responsibilities
(D) preference for listening over reading
Answer:
(A) strong memory and learning ability

(ii) Complete the following with a suitable reason.
The grandmother identified herself with the novel’s protagonist because
Answer:
both of them had never been to Kashi and shared the same desire to visit the holy place.

(iii) Identify the line from the extract that shows the grandmother remembered the story very well.
Answer:
“Later, she could repeat the entire text by heart.”

(iv) Which of the following aspects is NOT emphasised in the given extract?
(A) the narrator reading the story aloud every week
(B) the grandmother listening with deep concentration
(C) the grandmother memorising parts of the story
(D) the author describing the setting of Kashi in detail
Answer:
(D) the author describing the setting of Kashi in detail

(v) Select the option that best expresses the meaning of the word “protagonist” as used in the extract.
(A) the main character in a story
(B) the writer of the story
(C) the narrator of the story
(D) the reader of the story
Answer:
(A) the main character in a story

3. “You know when you were away, Karmaveera came as usual. I opened the magazine. I saw the picture that accompanies the story of Kashi Yatre and I could not understand anything that was written. Many times I rubbed my hands over the pages wishing they coidd understand what was written. But I knew it was not possible.”

(i) Complete the sentence with the appropriate option.
The grandmother’s act of repeatedly rubbing her hands over the pages shows her _______________.
(A) curiosity about the pictures
(B) frustration at being unable to read
(C) dislike for the magazine
(D) anger towards the narrator
Answer:
(B) frustration at being unable to read

(ii) Select the correct option to complete the statement.
The phrase “came as usual” suggests that the magazine was _______________.
(A) published once in a while
(B) read only by children
(C) borrowed occasionally
(D) part of a regular routine
Answer:
(D) part of a regular routine

(iii) Identify the line from the extract that shows the grandmother’s strong desire to understand the story.
Answer:
“Many times I rubbed my hands over the pages wishing they could understand what was written.”

(iv) Complete the following with a suitable reason.
The grandmother looked at the picture accompanying the story because
Answer:
she wanted to understand the story even though she could not read the words.

(v) Select the option that best expresses the meaning of the word ‘accompanies’ as used in the extract.
(A) replaces
(B) goes along with
(C) hides behind
(D) removes from
Answer:
(B) goes along with

4. “She bent down and touched my feet. I was surprised and taken aback. Elders never touch the feet of youngsters. We have always touched the feet of God, elders and teachers. We consider that as a mark of respect.”

(i) Complete the sentence with the correct option.
The narrator’s surprise shows that the grandmother’s action was _______________.
(A) disrespectful
(B) common
(C) unexpected
(D) meaningless
Answer:
(C) unexpected

(ii) Choose the correct option.
The grandmother touching the narrator’s feet symbolises _______________.
(A) anger towards the narrator
(B) fear of the narrator
(C) regret for the past
(D) respect for the narrator
Answer:
(D) respect for the narrator

(iii) What makes the grandmother’s action unusual in the cultural context described in the extract?
Answer:
The grandmother’s action is unusual because elders are normally respected by younger people, but here she touches the narrator’s feet as a mark of respect.

(iv) The line “We consider that as a mark of respect” suggests that touching the feet of elders is mainly associated with
(A) expressing gratitude
(B) showing respect and reverence
(C) asking for forgiveness
(D) seeking permission
Answer:
(B) showing respect and reverence

(v) Select the option that best expresses the meaning of the phrase “taken aback” as used in the extract.
(A) greatly amused
(B) suddenly surprised
(C) deeply offended
(D) quietly pleased
Answer:
(B) suddenly surprised

True or False:

1. Krishtakka had attended school but forgotten how to read.
Answer: True

2. The narrator eventually became a Computer Science teacher.
Answer: False

Vocabulary Focus (Write the meaning):

1. Ardent: ________.
Answer:
Ardent: Very keen; showing strong feelings.

2. Immensely: ________.
Answer:
Immensely: Extremely or to a great extent.

3. Taken aback: ________.
Answer:
Taken aback: Surprised by something unexpected.

Fill in the blanks with the correct form of the verbs given in brackets.

One evening, the sky A. ________ (grow) dark suddenly, and strong winds B. ________ (begin) to blow. By the time the villagers C. ________ (understand) the danger, heavy rain D. ________ (flood) the fields. Some people E. ________ (move) their cattle to safer places before the river F. ________ (overflow). When the rescue team G. ________ (arrive), many families H. ________ (take) shelter in the school building. Although the storm I. ________ (cause) great damage, the villagers J. ________ (show) courage and unity.
Answer:
A. grew
B. began
C. understood
D. had flooded
E. had moved
F. overflowed
G. arrived
H. had taken
I. caused
J. showed.

Fill in the blanks with the simple past and past perfect tense forms of the verbs given in brackets.

A. When the rain ________ (stop), the children ________ already ________ (finish) their homework.
Answer:
stopped, had already finished

B. After the doctor ________ (examine) the patient, he ________ (prescribe) the medicine.
Answer:
had examined, prescribed.

C. Before the train ________ (depart), the passengers ________ (board) their compartments.
Answer:
departed, had boarded (board).

D. By the time Meera ________ (realise) her mistake, she ________ (submit) the wrong file.
Answer:
realised, had submitted

E. When the teacher ________ (enter) the classroom, the students ________ (arrange) their desks neatly.
Answer:
entered, had arranged.

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Shaping of the Earth’s Surface Class 9 Worksheet with Answers SST Chapter 2

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Shaping of the Earth’s Surface Worksheet Class 9

Class 9 Shaping of the Earth’s Surface Worksheet

Worksheet On Shaping of the Earth’s Surface Class 9

Multiple Choice Questions

Question 1.
The theory of Plate Tectonics, which explains the slow movement of the Earth’s crustal plates over the mantle, was formulated by which of the following scientists?
(a) Alfred Wegener
(b) W.J. Morgan
(c) Harry Hess
(d) Arthur Holmes
Answer:
(b) W.J. Morgan

Question 2.
Consider the following case and choose the correct option.
A student claims that tectonic plates are fixed and do not move, and that the Earth’s surface remains unchanged over time. Based on the concept of lithosphere and asthenosphere, what is the correct understanding?
(a) Yes, because the lithosphere is completely rigid and does not allow any movement.
(b) No, because tectonic plates are massive slabs of rock that move slowly over the semi-molten asthenosphere,
(c) Yes, because only the mantle moves, not the crust or plates.
(d) No, because the asthenosphere is solid and prevents plate movement.
Answer:
(b) No, because tectonic plates are massive slabs of rock that move slowly over the semi-molten asthenosphere,

Question 3.
Which one of the following pairs is correctly matched?
(a) Continental crust – 5-7 km thick
(b) Oceanic crust – 30-40 km thick
(c) Oceanic crust – 5-7 km thick
(d) Continental crust – 100 km thick
Answer:
(c) Oceanic crust – 5-7 km thick

Question 4.
Find the odd one out from the following.
(a) Inner Core
(b) Asthenosphere
(c) Outer Core
(d) Lithosphere
Answer:
(b) Asthenosphere

Question 5.
Identify the term with the help of the following information.
• It is the rigid outer layer of the Earth.
• It consists of the crust and the uppermost mantle.
• It is broken into tectonic plates.

(a) Asthenosphere
(b) Lithosphere
(c) Mantle
(d) Outer Core
Answer:
(b) Lithosphere

Question 6.
Which of the following plate boundaries is characterised by plates sliding horizontally past each other without creating or destroying the crust, such as along the San Andreas Fault?
(a) Convergent Boundary
(b) Divergent Boundary
(c) Transform Boundary
(d) Subduction Zone
Answer:
(c) Transform Boundary

Question 7.
The primary engine driving the continuous lateral movement of massive tectonic plates across the globe is Competency.
(a) The gravitational pull of the moon acting on tectonic boundaries
(b) Convection currents in the mantle powered by interior radioactive and thermal energy!
(c) Frictional heating generated by the spinning of the inner metal core
(d) Centrifugal force generated by the Earth’s diurnal rotation
Answer:
(b) Convection currents in the mantle powered by interior radioactive and thermal energy!

Question 8.
The mud volcano at Baratang Island in the Andaman and Nicobar Islands is a unique natural feature where mud, instead of lava, erupts due to underground gases and pressure. Over time, nearby rocks are broken down into smaller pieces by changes in temperature, water, and other natural processes without being transported from their original place.

Which of the following processes is taking place in the rocks?
(a) Weathering, because the rocks are broken into smaller pieces without being moved.
(b) Erosion, because the broken rocks remain at the same place.
(c) Deposition, because sediments are being laid down at the same location.
(d) Volcanism, because all rock breaking is caused by volcanic eruptions.
Answer:
(a) Weathering, because the rocks are broken into smaller pieces without being moved.

Question 9.
Arrange the following stages in the correct sequence and choose the correct option.
I. Transportation of sediments
II. Weathering of rocks
III. Deposition of sediments
IV. Erosion by natural agents

Options
(a) II, IV, I, III
(b) IV, II, I, III
(C) II, I, IV, III
(d) III, II, IV, I
Answer:
(a) II, IV, I, III

Question 10.
Identify the appropriate option to fill in the empty boxes.

Types of Erosion

Water Erosion	?	?
Caused by rivers, rain, or ocean waves.	Common in dry and sandy areas.	Where moving ice scrapes and carries rocks.

(a) Wind erosion, Glacial erosion
(b) Glacial erosion, Wind erosion
(c) Weathering, Wind erosion
(d) Deposition, Glacial erosion
Answer:
(a) Wind erosion, Glacial erosion

Question 11.
Consider the following statements regarding weathering and choose the correct option.
(i) Weathering breaks down rocks on the Earth’s surface into smaller fragments.
(ii) Weathering requires the physical transportation of broken materials to distant locations by wind or water.
(a) Only (i) is correct.
(b) Only (ii) is correct.
(c) Both (i) and (ii) are correct.
(d) Neither (i) nor (ii) is correct.
Answer:
(a) Only (i) is correct.

Question 12.
Earthen embankments built along contour lines of a hillside to slow surface run-off, reduce soil erosion, and increase water infiltration are called.
(a) Contours
(b) Bunds
(c) Terraces
(d) Deltas
Answer:
(b) Bunds

Question 13.
The ancient irrigation dam called Grand Anicut (Kallanai) in Tamil Nadu represents a historic human adaptation to which type of running water feature?
(a) Glacial Moraine
(b) Coastal Wave Cliff
(c) River Meander
(d) Limestone Sinkhole
Answer:
(c) River Meander

Question 14.
Choose the correct option to fill in the blank.

Options
(a) Delta
(b) Coastal Erosion
(c) Glacier
(d) Sand dune
Answer:
(b) Coastal Erosion

Question 15.
Read the given infographic carefully and choose the correct option that identifies the landform described.

Options
(a) Plateau II
(b) Beach
(c) Valley
(d) Gorge
Answer:
(b) Beach

Question 16.
If two distinct glaciers merge in a high-altitude mountain valley, which deposit is most likely to form down the central axis of the combined glacial flow?
(a) Lateral Moraine
(b) Medial (Mid) Moraine
(c) Terminal Moraine
(d) Yardang Ridge
Answer:
(b) Medial (Mid) Moraine

Question 17.
An agricultural community in a dry, sandy desert observes that the wind is systematically eroding fine sand grains, leaving behind a hard, flat surface consisting solely of interlocking gravels and pebbles. This newly exposed geologic feature is known as a
(a) Deflation Hollow
(b) Yardang
(c) Desert Pavement
(d) Star Dune
Answer:
(c) Desert Pavement

Question 18.
In a limestone region experiencing high rainfall, water mixes with carbon dioxide to form a weak acid. This acidic water leaks down joint cracks, dissolving the rock over thousands of years. If a large underground cavern is formed, and its roof suddenly collapses due to gravity, which land form is created directly at the surface?
(a) Stalagmite
(b) Sinkhole (Doline)
(c) Wave-cut Platform
(d) Oxbow Lake
Answer:
(b) Sinkhole (Doline)

Assertion-Reason Questions

Directions (Question Nos. 19-25)In the following questions, a statement of Assertion (A) is followed by a statement of Reason (R), Choose the correct option from the choices given below.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true, but R is not the correct explanation of A.
(c) A is true, but R is false.
(d) A is false, but R is true.

Question 1.
Assertion (A) Fold mountains like the Himalayas are formed at convergent plate boundaries.
Reason (R) When two continental plates collide, the immense compressive force buckles and folds the rock layers upwards.
Answer:
(a) Both A and R are true and R is the correct explanation of A.

Question 2.
Assertion (A) The region surrounding the Pacific Ocean is highly vulnerable to devastating volcanic eruptions and earthquakes.
Reason (R) The “Ring of Fire” corresponds to active subduction zones and tectonic plate boundaries along the Pacific Plate margins.
Answer:
(a) Both A and R are true and R is the correct explanation of A.

Question 3.
Assertion (A) In his ancient treatise Brihatsamhita, the scholar Varahamihira dedicated an entire section to predicting earthquakes (bhukampa).
Reason (R) Varahamihira possessed modern seismographs and GPS equipment to track active fault slip lines across India.
Answer:
(c) A is true, but R is false.

Question 4.
Assertion (A) River deltas are typically densely populated and serve as thriving economic and agricultural hubs.
Reason (R) The continuous deposition of nutrient-rich alluvial soils at the river mouth makes deltas exceptionally fertile and highly productive for crops like rice and jute.
Answer:
(a) Both A and R are true and R is the correct explanation of A.

Question 5.
Assertion (A) Sea stacks are isolated vertical pillars of rock standing in the ocean near steep coastlines.
Reason (R) They are formed when sea waves systematically deposit layers of wind-blown desert sand along wave-cut platforms.
Answer:
(a) Both A and R are true and R is the correct explanation of A.

Question 6.
Assertion (A) Deforestation significantly increases the hazard of massive landslides on steep mountain slopes during heavy rainstorms. Reason (R) Tree roots act as natural binders that physically anchor the soil and stabilise slope structures against gravity.
Answer:
(a) Both A and R are true and R is the correct explanation of A.

Question 7.
Assertion (A) Glacial Lake Outburst Floods (GLOFs) present a major hazard to hydropower projects and human settlements located downstream.
Reason (R) Rising global temperatures accelerate glacial melting, increasing the water volume in lakes dammed by fragile, loose moraines until the dams burst.
Answer:
(a) Both A and R are true and R is the correct explanation of A.

Match the Columns

Question 1.
Match Column 1 with Column II.

Column I (Tectonic Process)	Column II (Geological Feature / Example)
A. Convergent Boundary Collision	1. Mid-Atlantic Ridge
B. Divergent Boundary Seafloor Spreading	2. San Andreas Fault
C. Transform Slip	3. The Himalaya
D. Plate Boundaries	4. Edges where tectonic plates meet

Options
(a) A – 3, B – 2, C – 1, D – 4
(b) A – 2, B – 3, C – 4, D – 1
(c) A – 1, B – 4, C – 2, D – 3
(d) A – 3, B – 1, C – 2, D – 4
Answer:

Column I (Tectonic Process)	Column II (Geological Feature / Example)
A. Convergent Boundary Collision	3. The Himalaya
B. Divergent Boundary Seafloor Spreading	1. Mid-Atlantic Ridge
C. Transform Slip	2. San Andreas Fault
D. Plate Boundaries	4. Edges where tectonic plates meet

Question 2.
Match Column I with Column II.

Column I (System)	Column II (Description of Technique)
A. Zabo System	1. Digging trenches along equal elevation lines to slow rainwater infiltration
B. Contour Trenching	2. Integrated hill farming in Nagaland using earthen bunds and check dams
C. Bunding	3. Earthen embankments built along contour lines to reduce soil erosion

Options
(a) A – 2, B – 1, C – 3
(b) A – 3, B – 2, C – 1
(c) A – 1, B – 3, C – 2
(d) A – 1, B – 2, C – 3
Answer:

Column I (System)	Column II (Description of Technique)
A. Zabo System	2. Integrated hill farming in Nagaland using earthen bunds and check dams
B. Contour Trenching	1. Digging trenches along equal elevation lines to slow rainwater infiltration
C. Bunding	3. Earthen embankments built along contour lines to reduce soil erosion

Question 3.
Match Column I with Column II.

Column I (Course of the River)	Column II (Primary Characteristic Landforms)
A. Upper Course	1. Meanders, Oxbow Lakes, Floodplains
B. Middle Course	2. Fan-shaped Alluvial Deltas, Levees
C. Lower Course	3. Steep V-shaped Valleys, Waterfalls, Rapids
D. River Mouth	4. Deposition of sediments into a sea or lake

Options
(a) A – 3, B – 1, C – 2, D – 4
(b) A – 2, B – 3, C – 4, D – 1
(c) A – 1, B – 2, C – 3, D – 4
(d) A – 4, B – 1, C – 2, D – 3
Answer:

Column I (Course of the River)	Column II (Primary Characteristic Landforms)
A. Upper Course	3. Steep V-shaped Valleys, Waterfalls, Rapids
B. Middle Course	1. Meanders, Oxbow Lakes, Floodplains
C. Lower Course	2. Fan-shaped Alluvial Deltas, Levees
D. River Mouth	4. Deposition of sediments into a sea or lake

Question 4
Match Column I with Column II.

Column I (Coastal Feature)	Column II (Erosional or Depositional Origin)
A. Sea Stack	1. Erosional hollow formed at the base of a sea cliff by wave action
B. Beach	2. Opening formed when waves erode through a headland
C. Sea Cave	3. Depositional feature consisting of sand or pebbles along the shoreline
D. Sea Arch	4. Erosional feature formed when a sea arch collapses

Options
(a) A – 4, B – 3, C – 1, D – 2
(b) A – 2, B -1, C – 4, D – 3
(c) A – 1, B – 2, C – 3, D – 4
(d) A – 3, B – 4, C – 2, D -1
Answer:

Column I (Coastal Feature)	Column II (Erosional or Depositional Origin)
A. Sea Stack	4. Erosional feature formed when a sea arch collapses
B. Beach	3. Depositional feature consisting of sand or pebbles along the shoreline
C. Sea Cave	1. Erosional hollow formed at the base of a sea cliff by wave action
D. Sea Arch	2. Opening formed when waves erode through a headland

Question 5.
Match Column I with Column II.

Column I (Desert Feature)	Column II (Formative Process/Shape)
A. Barchan Dune	1. Depression formed by wind deflation that reaches the water table
B. Yardang	2. Crescent-shaped sand mound pointing in the direction of wind flow
C. Oasis	3. Streamlined, narrow rock ridge carved by wind erosion
D. Desert Pavement	4. Surface covered with pebbles and gravels left after fine particles are removed by wind

Options
(a) A – 2, B – 3, C – 1, D – 4
(b) A – 3, B – 4, C – 2, D – 1
(c) A – 1, B – 2, C – 4, D – 3
(d) A – 4, B – 1, C – 3, D – 2
Answer:

Column I (Desert Feature)	Column II (Formative Process/Shape)
A. Barchan Dune	2. Crescent-shaped sand mound pointing in the direction of wind flow
B. Yardang	3. Streamlined, narrow rock ridge carved by wind erosion
C. Oasis	1. Depression formed by wind deflation that reaches the water table
D. Desert Pavement	4. Surface covered with pebbles and gravels left after fine particles are removed by wind

Picture-Based Questions

Question 1.
Based on the schematic diagram of the Earth’s interior, what is the approximate combined thickness of the lithosphere and the asthenosphere, and which of these layers contains the crust?

(a) 300 km; the asthenosphere contains the crust.
(b) 300 km; the lithosphere contains the crust.
(c) 2100 km; the lithosphere contains the crust.
(d) 100 km; both layers contain the crust.
Answer:
(b) 300 km; the lithosphere contains the crust.

Question 2.
Refer to the heat transport mechanism inside the Earth’s layers in the given figure. Which layer accounts for the highest individual percentage of total interior heat flow, and what is its primary heat transport mechanism shown by the circular arrows?

(a) Crust (24%); Conduction
(b) Core (22%); Radiation
(c) Upper Mantle (22%); Advection
(d) Lower Mantle (32%); Convection
Answer:
(d) Lower Mantle (32%); Convection

Question 3.
Study the given figure carefully and answer the following questions.

(a) Name the landform formed at the base of a waterfall.
Answer:
The landform formed at the base of a waterfall is a Plunge Pool.

(b) Explain how this landform is formed.
Answer:
It is formed due to the continuous erosion caused by the falling water at the base of the waterfall.

Very Short Answer Questions

Question 1.
Have you ever seen pictures of mountains or volcanoes? Can you think of how the slow movement of tectonic plates helps in their formation?
Answer:
The slow movement of tectonic plates causes them to collide, separate or slide past one another. These movements lead to the formation of mountains when plates collide and volcanoes when magma rises through cracks in the Earth’s crust.

Question 2.
“The lithosphere is broken into tectonic plates that move continuously.” Explain how the asthenosphere helps in the movement of these plates and why this movement is important for shaping the Earth’s surface.
Answer:
The asthenosphere is a hot, mobile layer of partially molten rock beneath the lithosphere. It allows tectonic plates to move slowly. Their movement forms mountains, volcanoes and earthquakes, continuously reshaping the Earth’s surface over long periods.

Question 3.
Name two major tectonic plates of the world that meet along active plate boundaries and briefly explain them.
Answer:
Two major tectonic plates that meet along active plate boundaries are the Pacific Plate and the Eurasian Plate. The Pacific Plate is the largest oceanic plate, while the Eurasian Plate covers Europe and much of Asia. Their interaction along active boundaries causes earthquakes, volcanic eruptions and the formation of mountains.

Question 4.
What physical phenomenon is observed when two continental plates collide along a convergent boundary?
Answer:
When two continental plates collide at a convergent boundary, the Earth’s crust folds and rises to form fold mountains. This process creates high mountain ranges, such as the Himalayas, through continuous plate movement.

Question 5.
Observe a nearby riverbank, roadside, hill slope or open ground. Can you identify whether weathering or erosion is taking place? Give a reason for your answer.
Answer:
If I observe that soil and small stones have been washed away from a riverbank after heavy rainfall, it is an example of erosion because the weathered materials are carried from one place to another by flowing water.

Question 6.
Distinguish between physical weathering and chemical weathering.
Answer:
The differences between physical weathering and chemical weathering are

Physical Weathering	Chemical Weathering
Breaks rocks into smaller pieces without changing their chemical composition.	Changes the chemical composition of rocks by forming new substances.
Caused by temperature changes, frost or wind.	Caused by reactions with water, air or acids.

Question 7.
A farmer builds small earthen embankments around his field to prevent rainwater from washing away the fertile soil. Name this soil conservation practice and explain how it helps conserve soil.
Answer:
The soil conservation practice is called bunding. It involves constructing earthen embankments along contour lines to slow down the flow of rainwater, reduce surface run-off, and prevent soil erosion. It also helps increase water infiltration and retains soil moisture, thereby conserving fertile topsoil.

Question 8.
Relate the different stages of a river (upper and lower course) with the landforms formed in each stage.
Answer:
The different stages of a river are
• Upper course Forms V-shaped valleys, waterfalls and rapids due to steep slopes and strong erosion.
• Lower course Forms deltas, levees and alluvial fans due to heavy deposition of sediments near the river’s mouth.

Question 9.
What is a sea stack, and how is it related to a sea arch?
Answer:
A sea stack is an isolated vertical pillar of rock formed along the coast by wave erosion. It is directly related to a sea arch because when a sea arch is continuously eroded and collapses due to wave action, the remaining separated rock mass stands as a sea stack.

Question 10.
Define Karst Topography. What rock type is necessary for its formation?
Answer:
Karst topography is a landscape formed by the dissolution of soluble rocks through groundwater action. It is characterised by features such as caves and sinkholes. Limestone is the essential rock type required for its formation.

Short Answer Questions

Question 1.
Discuss how convection currents in the mantle act as the main mechanism for plate tectonic movements.
Answer:
Convection currents in the mantle are produced by heat from the Earth’s core. Hot, semi-molten material rises, while cooler material sinks back into the mantle. This continuous circular movement creates convection currents that push and pull the tectonic plates resting on the lithosphere.

As a result, the plates move slowly in different directions. Their movement leads to the formation of mountains, volcanoes, earthquakes and ocean basins, making convection currents the main driving force behind plate tectonics.

Question 2.
Explain the physical process of subduction that occurs when an oceanic plate collides with a continental plate.
Answer:
Subduction is the process in which a denser oceanic plate sinks beneath a continental plate when the two plates collide at a convergent boundary. Since the oceanic plate is heavier, it is forced downward into the mantle beneath the continental plate.

As it descends, the intense heat and pressure cause part of the plate to melt and form magma. This magma may rise to the Earth’s surface and create volcanoes.

The collision also forms deep ocean trenches and generates earthquakes. Thus, subduction is an important geological process that continuously reshapes the Earth’s surface by forming new landforms and releasing internal energy.

Question 3.
Contrast the main differences between weathering and erosion. Use a table to present your comparison.
Answer:
The main differences between weathering and erosion are as follows

Basis of Comparison	Weathering	Erosion
Meaning	The process of breaking rocks into smaller pieces.	The process of removing and transporting weathered materials.
Movement of Material	Rocks break down but remain at the same place.	Rock fragments are carried away to another place.
Agents Involved	Physical, chemical, and biological processes.	Running water, wind, glaciers, sea waves and groundwater.

Question 4.
Describe the “Zabo System” of water and soil conservation practiced in Nagaland, highlighting its relevance to modern ecological sustainability.
Answer:
The Zabo System is a traditional method of water and soil conservation practiced in Nagaland that makes efficient use of natural resources. In this system, rainwater flowing from forested hill slopes is collected and stored in ponds before being used for irrigation, livestock and household purposes. It also reduces soil erosion, improves soil fertility and helps recharge groundwater. The Zabo System promotes sustainable agriculture by conserving water and protecting the environment.

It remains highly relevant today because it supports ecological sustainability through the wise and eco-friendly management of natural resources.

Question 5.
“Landforms have played a major role in shaping the history of human civilisations.” Substantiate this statement with examples of how rivers and mountains influenced Indian history.
Answer:
Landforms have played a major role in shaping the history of human civilisations by influencing settlement, agriculture, trade and defence. Fertile plains of the Indus and Ganga rivers supported agriculture and led to the growth of early civilisations and towns. Rivers also provided water, transport, and trade routes. The Flimalayas acted as a natural barrier, protecting India from many invasions and influencing its climate. At the same time, mountain passes such as the Khyber Pass allowed trade and cultural exchanges with neighbouring regions. Thus, rivers and mountains significantly shaped India’s history, economy and culture.

Question 6.
Explain the formation of a waterfall, detailing how hard and soft rock layers facilitate its vertical drop.
Answer:
A waterfall is formed when a river flows over alternating layers of hard and soft rocks. The softer rock is eroded more quickly by the flowing water, while the harder rock resists erosion. As the softer rock is worn away, it creates a steep vertical drop, leaving the hard rock projecting over the edge. Eventually, the unsupported hard rock collapses, and the waterfall gradually retreats upstream. This continuous process of erosion and collapse shapes the waterfall and the surrounding landscape over time.

Question 7.
Choose any two agents of erosion (running water, sea waves, glaciers, wind or groundwater) and find out the following.
(i) Which landforms are created by each agent?
(ii) Are the landforms formed mainly by erosion, deposition, or both?
(iii) How are these landforms useful to people (such as for tourism, agriculture, transport or fishing)?
Answer:
Two agents of erosion: running water and sea waves

Agent of Erosion	Landforms Created	Main Process	Usefulness to People
Running water	V-shaped valleys, waterfalls, floodplains, deltas	Both erosion and deposition	Floodplains and deltas support agriculture and settlements.
Sea waves	Cliffs, sea caves, sea arches, sea stacks, wave-cut platforms	Mainly erosion	Coastal landforms attract tourists and some coasts support fishing and harbours.

Question 8.
Explain how moraines are formed. Mention any two types of moraines with their positions relative to a glacier.
Answer:
Moraines are formed when a glacier melts and deposits the rocks, soil and other debris (till) it has carried. These deposited materials accumulate to form different types of moraines.
Two types of moraines with their positions relative to a glacier are
(i) Lateral moraine Forms along the sides of a glacier.
(ii) Terminal moraine Forms at the front (snout/end) of a glacier, marking its farthest advance.

Question 9.
Explain the development and physical structure of a Yardang. How does it differ from a Dune?
Answer:
A yardang is a long, narrow ridge formed by the continuous erosion of soft rocks by strong winds in desert regions. Strong winds erode the softer rocks more rapidly than the harder rocks through abrasion and deflation, leaving behind long, streamlined ridges called yardangs.

These ridges are aligned parallel to the prevailing wind direction and have steep sides. In contrast, a dune is formed by the deposition of sand carried by the wind. Thus, a yardang is an erosional landform made of rock, whereas a dune is a depositional landform made of loose sand.

Question 10.
While exploring a limestone cave, a tourist observed pointed formations hanging from the roof and others rising from the floor. Explain how these formations are created by the action of underground water.
Answer:
The pointed formations hanging from the roof and rising from the floor of a limestone cave are formed by the depositional action of underground water. Rainwater absorbs carbon dioxide and forms a weak carbonic acid, which dissolves limestone as it passes through cracks in the rocks. When the water enters the cave and drips from the roof, it loses carbon dioxide and deposits calcium carbonate, forming stalactites. The dripping water continues to deposit calcium carbonate on the cave floor, forming stalagmites. Over time, these formations may join together to form a pillar or column.

Long Answer Questions

Question 1..
Describe the physical layered structure of the Earth’s interior. Explain the thickness, state of matter (solid/liquid), chemical composition and unique geological functions.
Answer:
The Earth’s interior is divided into four main layers: the crust, mantle, outer core and inner core, each have the following distinct characteristics and functions.

Layer	Thickness	State of Matter	Chemical Composition	Geological Function
Crust	30-40 km under continents; 5-7 km under oceans	Solid	Silicate rocks	Forms the Earths outer surface and supports continents and oceans.
Mantle	About 2,900 km	Mostly solid with semi-molten upper part	Silicate rocks rich in magnesium and iron	Convection currents in the mantle drive the movement of tectonic plates.
Outer Core	About 2,200 km	Liquid	Iron and nickel	Generates the Earths magnetic field through the movement of molten metals.
Inner Core	About 1,250 km	Solid	Iron and nickel	The hottest and densest layer that provides heat to the Earths interior and helps maintain the magnetic field.

Question 2.
Provide a detailed, comparative analysis of the three primary types of tectonic plate boundaries (Convergent, Divergent and Transform). For each boundary, explain the relative direction of plate movement, the crustal consequences (whether crust is created, destroyed or conserved), and at least one prominent geographic feature associated with it.
Answer:
The differences between three primary types of tectonic plate boundaries are convergent, divergent and transform boundaries are

Plate Boundary	Direction of Plate Movement	Crustal Consequence	Major Geographic Feature
Convergent Boundary	Two plates move towards each other.	Crust is destroyed when one plate subducts beneath another or is folded to form mountains.	Himalayas, Deep Ocean Trenches
Divergent Boundary	Two plates move away from each other.	New crust is created as magma rises and solidifies.	Mid-Atlantic Ridge
Transform Boundary	Two plates slide past each other horizontally.	Crust is neither created nor destroyed.	San Andreas Fault

Question 3.
“Traditional Indian water and soil conservation systems showcase a deep ecological understanding of the local landscape.” Elaborate on this statement by analysing the techniques used during the Sindhu-Sarasvati civilisation and the guidelines found in Kautilya’s Arthashastra.
Answer:
The techniques used during the Sindhu-Sarasvati civilisation and the guidelines found in Kautilya’s Arthashastra are

1. The Sindhu-Sarasvati Civilisation used techniques such as contouring, bunding, terracing, dams and canals to conserve water and prevent soil erosion.
2. These methods stored rainwater, improved irrigation and supported agriculture.
3. Kautilya’s Arthashastra provided guidelines for assessing land and classifying it according to its fertility and productivity.
4. It also emphasised proper land management and the efficient use of water resources.
5. These traditional practices conserved natural resources, protected the environment and promoted sustainable agriculture.

Question 4.
If you are planning a trek in a snow-covered mountain region, what precautions should you take to reduce the risk of avalanches?
Answer:
If I were planning a trek in a snow-covered mountain region, I would take the following precautions to reduce the risk of avalanches.

1. Check weather forecasts and avalanche warnings before starting the trek and avoid travelling during or after heavy snowfall.
2. Stay on marked trails and avoid walking on steep snow-covered slopes where avalanches are more likely to occur.
3. Follow the instructions of local authorities and experienced guides and never ignore warning signs.
4. Carry essential safety equipment, such as an avalanche beacon, shovel, probe, first-aid kit and communication device, for emergencies.
5. Avoid activities that may disturb unstable snow, such as trekking in restricted areas or making loud vibrations and always trek in a group so that help is available in case of an emergency.

Case-Based Questions

Question 1.
Read the case given below and answer the following questions.

The Ring of Fire and the 2001 Gujarat Earthquake
Plate tectonics plays a major role in shaping the Earth’s surface. The movement of plates leads to the formation of mountains, valleys, ocean basins, volcanoes and earthquakes. It also explains the distribution of continents and oceans across the Earth. Most earthquakes and volcanoes occur along plate boundaries, especially around the Pacific Ocean, an area known as the Ring of Fire.

The theory of plate tectonics helps us understand how the Earth’s surface is constantly changing. It explains the origin of many landforms and natural disasters. This theory is very important for identification of earthquake and volcano prone regions and managing disasters arising from them. India has experienced some major earthquakes in the past, resulting in thousands of deaths. A large earthquake in a densely populated country like India can cause severe damage to life and environment.

(i) Why is the theory of plate tectonics useful for disaster management?
Answer:
The theory of plate tectonics is useful for disaster management because it helps identify earthquake- and
volcano-prone regions, enabling better planning and preparedness.

(ii) Why are regions around the Pacific Ocean more vulnerable to earthquakes and volcanoes?
Answer:
Regions around the Pacific Ocean are more vulnerable to earthquakes and volcanoes because they are located along active plate boundaries where tectonic plates frequently interact.

(iii) Explain any two ways in which the theory of plate tectonics helps us understand the Earth’s surface,
Answer:
The theory of plate tectonics helps us understand the Earth’s surface in the following ways

• It explains the formation of major landforms such as mountains, valleys, ocean basins, and volcanoes.
• It helps identify regions that are more likely to experience earthquakes and volcanic eruptions, supporting disaster preparedness and risk reduction.

Question 2.
Read the case given below and answer the following questions.
Traditional Indian Water and Soil Conservation Systems
The Sindhu-Sarasvati civilisation employed sophisticated techniques, including contouring, bunding, terracing, dams and canals for water management, Multiple Sanskrit texts document these practices, including the Vedas, Krishi-Parasara, Kautilya’s Arthashastra and specialised treatises like Vrkshayurveda. The Arthashastra contains detailed guidelines on land assessment and categorisation based on fertility and productivity.

The Zabo system in Nagaland represents an integrated farming approach using earthen bunds on hillslopes for soil and water conservation, Check dams were constructed across small streams to reduce water velocity, prevent soil erosion and allow sediment deposition. These structures were designed to slow down water flow and enhance groundwater recharge.

(i) Why did the Sindhu-Sarasvati civilisation use techniques such as contouring, bunding and terracing?
Answer:
The Sindhu-Sarasvati civilisation used techniques such as contouring, bunding and terracing to conserve water,
reduce soil erosion, and improve agricultural productivity.

(ii) How do check dams in the Zabo system help conserve water?
Answer:
Check dams in the Zabo system help conserve water by slowing down the flow of water, reducing soil erosion and increasing groundwater recharge.

(iii) Explain any two ways in which traditional Indian water and soil conservation systems promote sustainable use of natural resources.
Answer:
Traditional Indian water and soil conservation systems promote the sustainable use of natural resources in the following ways

• They conserve water by storing rainwater and reducing surface run-off.
• They prevent soil erosion and improve soil fertility, supporting sustainable agriculture and environmental conservation.

Map-Based Questions

Question 1.
On the outline map of world, locate and label the following.

(i) Himalayas
(ii) Baratang Island (Mud Volcano)
(iii) Sundarbans Delta
(iv) Nagaland
Answer:

Question 2.
On the outline map of the world, identify and shade the following.

(i) Pacific Ring of Fire
(ii) Pacific Ocean
(iii) Mid-Atlantic Ridge
(iv) San Andreas Fault
Answer:

Think and Answer

Question 1.
Two regions receive the same amount of rainfall. One has thick vegetation, while the other is almost barren. Which region is likely to experience more soil erosion? Give two reasons for your answer.
Answer:
The almost barren region is likely to experience more soil erosion because
• There are no plant roots to hold the soil together.
• Rainwater flows rapidly over the exposed land and carries away the topsoil.

Question 2.
An agricultural land developer plans to clear all natural forest cover on a series of steep hillsides to plant commercial crops, As an environmental consultant, draft an objection report explaining the geomorphic risks of this plan, including weathering, rain run-off, mass wasting and sedimentation in local rivers.
Answer:
Objection Report
I strongly oppose the removal of natural forest cover from the steep hillsides.

Reasons

1. Without trees, the soil becomes loose and is more easily broken down by weathering.
2. Heavy rain will increase surface run-off, washing away fertile topsoil.
3. The unstable slopes will become more prone to landslides and other forms of mass wasting.
4. The eroded soil will be deposited in nearby rivers, causing sedimentation, reducing river depth and increasing the risk of floods.

Question 3.
Document a case of a glacier or glacial lake outburst flood (GLOF) that occurred in India or any other country. Highlight its causes and effects on people, settlements and the environment.
Answer:
Chamoli Glacial Lake Outburst Flood (Uttarakhand, India), 2021

1. In February 2021, a massive rock and ice avalanche in the Chamoli district of Uttarakhand triggered a sudden flood in the Rishiganga and Dhauliganga rivers. The disaster is believed to have been caused by the collapse of a large mass of rock and glacier ice, leading to a surge of water and debris downstream.

Effects on people, settlements and the environment

1. Many people lost their lives, and several others went missing.
2. Houses, roads and bridges were washed away, disrupting transport and communication.
3. Hydroelectric power projects suffered severe damage.
4. River ecosystems were disturbed due to large amounts of debris and sediment.
5. Local communities faced loss of livelihoods and displacement.

This event highlighted the importance of glacier monitoring, early warning systems and disaster preparedness in Himalayan regions.

Question 4.
Imagine you are a geologist examining a rocky coastline. You observe a series of sea caves, a fully intact sea arch, and an isolated sea stack nearby. Reconstruct the step-by-step geological timeline of this coastline, explaining how wave erosion will continue to alter these features over the next 1000 years.
Answer:
The wave erosion will continue to alter these features over the next 1000 years in the following ways

1. Waves first erode cracks in the sea cliff to form sea caves.
2. Continued erosion enlarges the caves until they meet, forming a sea arch.
3. Over time, the roof of the arch collapses due to continuous wave action, leaving behind a sea stack.
4. In the future, waves will keep eroding the sea stack until it is reduced to a stump or disappears below sea level.

Question 5.
Form groups of 4-6 students. Identify and compare any two natural hazards (such as dust storms, avalanches, landslides, earthquakes or floods) using credible sources such as government websites, newspapers or disaster management reports. Present your findings in a chart or poster using bullet points, symbols, diagrams or colour coding.

Parameter	Natural Hazard 1	Natural Hazard 2	Source
Causes
Regions where it commonly occurs
Effects on people and the environment
Precautionary measures
Government/community response

Answer:

Parameter	Dust Storm	Avalanche	Source
Causes	Strong winds lift loose, dry soil and sand into the air due to drought, low rainfall and sparse vegetation.	Heavy snowfall, unstable snow layers, rising temperatures, earthquakes or human activities trigger the sudden movement of snow.	NCERT; India Meteorological Department (IMD); National Disaster Management Authority (NDMA)
Regions where it commonly occurs	Desert and semi-arid regions such as the Thar Desert in Rajasthan, Gujarat and parts of Haryana.	Himalayan regions such as Jammu & Kashmir, Himachal Pradesh, Uttarakhand and Sikkim.	NCERT; IMD; NDMA
Effects on people and the environment	Reduces visibility, damages crops, affects transport and causes breathing problems.	Causes loss of life, damages roads, buildings and forests, and blocks transport routes.	NCERT; NDMA
Precautionary measures	Stay indoors, wear masks and protective eyewear, and avoid travelling during severe dust storms.	Follow avalanche warnings, avoid steep snow-covered slopes, carry safety equipment and follow trained guides.	NDMA; IMD
Government/community response	Early weather warnings, afforestation, soil conservation and public awareness programmes.	Avalanche forecasting, rescue teams, early warning systems and disaster preparedness programmes.	NDMA; IMD

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Exploring Some More Progressions Class 9 Extra Questions Advanced Maths Chapter 6

During revision, students quickly go through NCERT Class 9 Advanced Maths Book Solutions and Class 9 Advanced Maths Chapter 6 Exploring Some More Progressions Extra Questions and Answers for clarity.

Class 9 Exploring Some More Progressions Extra Questions

Exploring Some More Progressions Class 9 Very Short Question Answer

Question 1.
Find the sum of first 9 terms of the GP 1, 2, 4, 8, …………… .
Answer:
Given, GP is 1, 2, 4, 8, … upto 9 terms.
Here, first term, a = 1 and n = 9.
Now, common ratio, r = \(\frac{2}{1}\) = 2
∴ The sum of the first n terms of a GP,
S_n = \(\frac{a\left(r^n-1\right)}{r-1}\), r ≠ 1
On substituting all the values, we get
S₉ = \(\frac{1\left(2^9-1\right)}{2-1}\) = \(\frac{512 – 1}{1}\) = 511
Hence, the sum of first 4 terms is 15.

Question 2.
Find the sum of the geometric series 3 + 6 + 12 + … + 1536.
Answer:
Hint Given GP is 3 + 6 + 12 + … + 1536.
Here, a = 3, r = \(\frac{6}{3}\) = 2 > 1
Now, use S_n = \(\frac{a\left(r^n-1\right)}{r-1}\), r > 1
= 3069

Question 3.
Find the sum of first 5 terms of the GP 2, 4, 8, 16, …
Answer:
Given, a = 2 and, r = \(\frac{4}{2}\) = 2
Also, n = 5
Now, the sum of first n terms of GP,
S_n = \(\frac{a\left(r^n-1\right)}{r-1}\), r > 1
On substituting all the values, we get
S₅ = \(\frac{2\left(2^5-1\right)}{2-1}\) = 2(32 – 1) = 2 × 31 = 62
Hence, the sum of the first 5 terms is 62.

Question 4.
Find the sum of the first n terms of the GP p, q, \(\frac{q^2}{p}\), \(\frac{q^3}{p^2}\), ……. for \(\frac{q}{p}\) > 1
Answer:
Hint First term = p and common ratio, r = \(\frac{p}{q}\)

Question 5.
Find the value of S_∞ for the GP 4, 2, 1, \(\frac{1}{2}\), … .
Answer:
Given, GP is 4, 2, 1, \(\frac{1}{2}\), …
Here, a = 4 and r = \(\frac{2}{4}\) = \(\frac{1}{2}\)
Since, r < 1
Therefore, the sum of an infinite GP exists.
We know that S_∞ = \(\frac{a}{1 – r}\)
On substituting the values, we get
S_∞ = \(\frac{4}{1-\frac{1}{2}}=\frac{4}{\frac{1}{2}}\) = 8
Hence, the sum to infinity is 8.

Question 6.
Find the sum to infinity of the GP 3, 1, \(\frac{1}{3}\), \(\frac{1}{9}\), … .
Answer:
Given, a = 3 and r = \(\frac{1}{3}\)
Since, r < 1
Therefore, the sum of an infinite GP exists.
We know that S_∞ = \(\frac{a}{1 – r}\)
On substituting all the values, we get
S_∞ = \(\frac{3}{1-\frac{1}{3}}=\frac{3}{\frac{2}{3}}\) = 3 × \(\frac{3}{2}\) = \(\frac{9}{2}\)
Hence, the sum of first 4 terms is \(\frac{9}{2}\).

Question 7.
Does the GP 1, 3, 9, 27, … have a finite sum to infinity?
Answer:
Given, a = 1 and r = \(\frac{3}{1}\) = 3
Now, r = 3 > 1
Therefore, the condition for sum to infinity does not satisfy.
Hence, the GP does not have a finite sum to infinity.

Question 8.
Find the sum to infinity of the GP 2 + 1 + \(\frac{1}{2}\) + \(\frac{1}{4}\) +… .
Answer:
Given, GP is 2 + 1 + \(\frac{1}{2}\) + \(\frac{1}{4}\) + … .
Here, a = 2 and r = \(\frac{1}{2}\)
Since, r < 1
Therefore, the sum to infinity exists.
We know that S_∞ = \(\frac{a}{1 – r}\)
On substituting the values, we get
S_∞ = \(\frac{2}{1-\frac{1}{2}}=\frac{2}{\frac{1}{2}}\) = 4
Hence, the sum to infinity is 4.

Question 9.
Evaluate 3^1/2 × 3^1/4 × 3^1/8 × … ∞.
Answer:

Exploring Some More Progressions Class 9 Short Question Answer

Question 1.
Find the first term of GP, whose common ratio is \(\frac{1}{2}\) and the sum of first 6 terms of the GP is \(\frac{63}{64}\).
Answer:
Given, r = \(\frac{1}{2}\) and S₆ = \(\frac{63}{64}\)
Now, the sum of first n terms of a GP,
S_n = \(\frac{a\left(r^n-1\right)}{r-1}\), r < 1
On substituting the given values, we get

Hence, the first term of the GP is a \(\frac{1}{2}\).

Question 2.
Find the sum of the first 12 terms of the GP 0.2, 0.04, 0.008, ……. .
Answer:
Given, a = 0.2
Now, the common ratio,
r = \(\frac{0.04}{0.2}\) = 0.2
Also, n = 12
We know that the sum of first n terms of a GP,
S_n = \(\frac{a\left(1-r^n\right)}{1-r}\), r < 1
On substituting the given values, we get
S₁₂ = \(\frac{0.2\left[1-(0.2)^{12}\right]}{1-0.2}\)
= \(\frac{0.2\left[1-(0.2)^{12}\right]}{0.8}\) = \(\frac{1}{4}\) [1 – (0.2)¹²]

Question 3.
The first term and seventh term of a GP are 729 and 64, respectively. For r > 0, determine S₉.
Answer:
Hint Given, ar⁶ = 64 ⇒ r = \(\frac{2}{3}\) [∵ r > 0]

Question 4.
The ratio of the sum of the first three terms and the sum of the first six terms of a GP is 125 : 152. Find the common ratio.
Answer:
Let a be the first term and r be the common ratio of given GP.
Given, S₃ : S₆ = 125 : 152

Hence, the common ratio is \(\frac{3}{5}\).

Question 5.
Evaluate \(\sum_{k=1}^{11}\left(2+3^k\right)\).
Answer:
Hint \(\sum_{k=1}^{11}\left(2+3^k\right)=\sum_{k=1}^{11} 2+\sum_{k=1}^{11} 3^k\) = 11 × 2 + 3¹ + 3² + … + 3¹¹
= 265741

Question 6.
Evaluate \(\sum_{r=1}^n\left(3^r-2^r\right)\).
Answer:
Hint Let S_n = \(\sum_{r=1}^n\left(3^r-2^r\right)\)
⇒ S_n = (3¹ + 3² + 3³ + … upto n terms) – (2¹ + 2² + 2³ +…upto n terms)

Question 7.
Find the sum of the series
0.5 + 0.55 + 0.555 +… upto n terms.
Answer:
Hint Let S_n = 0.5 + 0.55 + 0.555 + … upto n terms
⇒ S_n = 5[0.1 + 0.11 + 0.111 + … upto n terms]
⇒ S_n = \(\frac{5}{9}\)[\(\frac{9}{10}\) + \(\frac{99}{100}\) + \(\frac{999}{1000}\) + …. upto n terms]
⇒ S_n = \(\frac{5}{9}\) [(1 + 1 + 1 + … upto n terms) – \(\frac{1}{10}\) + \(\frac{1}{10^2}\) + \(\frac{1}{10^3}\) + … upto n terms]
⇒ S_n = \(\frac{5}{9}\left[n-\frac{1}{10} \frac{\left(1-\frac{1}{10^n}\right)}{\left(1-\frac{1}{10}\right)}\right]\)
⇒ \(\frac{5}{9}\) [n – \(\frac{5}{9}\) (1 – 10^-n)]

Question 8.
In a GP, the ratio of the sum to infinity and the first term is 5 : 2. Find the common ratio.
Answer:
Let the first term of the GP be a and the common ratio be r.
Now, the sum of the infinite GP is given by
S_∞ = \(\frac{a}{1 – r}\), r < 1.
According to the question,
S_∞ a = 5 : 2
⇒ \(\frac{S_{\infty}}{a}\) = \(\frac{5}{2}\)
On substituting the value of S_∞, we get
= \(\frac{\frac{a}{1-r}}{\frac{a}{a}}\) = \(\frac{5}{2}\)
\(\frac{1}{ 1 – r}\) = \(\frac{5}{2}\)
⇒ 2 = 5(1 – r) ⇒ 2 = 5 – 5r
⇒ 5r = 5 – 2 ⇒ 5r = 3
⇒ r = \(\frac{3}{5}\)
Hence, the common ratio is \(\frac{3}{5}\).

Question 9.
A ball rebounds to (\(\frac{1}{2}\)) th of the height from which it falls. If it is dropped from a height of 40 m, find th total distance travelled before coming to rest.
Answer:
Given, the ball is first dropped from the height of 40 m.
After striking the ground, it rebounds to half of the previous height.
Therefore, the successive rebound heights are
20, 10, 5, \(\frac{5}{2}\), …… .
This forms a geometric progression with a = 20 and r = \(\frac{1}{2}\).
Since, the ball travels 40 m downward initially. After that, for each rebound, the ball travels upward as well as downward through the same distance.
Hence, the total distance travelled is [40 + 2(20 + 10 + 5 +…)] m.
Now, 20 + 10 + 5 + is an infinite GP.
Since, |r| < 1
Therefore, the sum to infinity exists.
We know that S_∞ =
On substituting the values, we get S_∞ = \(\frac{20}{1-\frac{1}{2}}\) = \(\frac{20}{\frac{1}{2}}\) = 40
Therefore, total distance travelled = 40 + 2(40) = 40 + 80 = 120 m
Hence, the total distance travelled by the ball before coming to rest is 120 m.

Question 10.
Find the nth term of the sequence 6, 14, 26, 42, 62,. .. and hence find its 8th term.
Answer:
The given sequence is 6, 14, 26, 42, 62, ….. .
It can be written as

Let the first elements of three rows be b, a and d, respectively.
Then, b = 6, a = 8 and d = 4
Now, the nth term is given by
t_n = bC(n – 1, 0) + a(n – 1, 1) + dC(n – 1, 2)
= 6 × 1 + 8(n – 1) + 4 × \(\frac{(n-1)(n-2)}{2}\)
= 6 + 8n – 8 + 2(n² – 3n + 2)
= 8n – 2 + 2n² – 6n + 4 = 2n² + 2n + 2
Now, putting n = 8, we get
t₈ = 2(8)² + 2(8) + 2
= 2(64) + 16 + 2 = 128 + 16 + 2 = 146
Hence, the 8th term of the sequence is 146.

Sets Class 9 Long Question Answer

Question 1.
Let S be the sum, P be the product and R be the sum of reciprocals of n term in a GP.
Prove that P²Rⁿ = Sⁿ.
Answer:
Let the GP be a, ar, ar², ar³, ….. , ar^{n – 1}.
Given, S = Sum of n terms
= a + ar + ar² + ar³ + ….. + ar^{n – 1}
= \(\frac{a\left(r^n-1\right)}{r-1}\) [let r > 1] …… (i)
and R = Sum of the reciprocals of n terms


= [a²rⁿr^-1]ⁿ = (a²r^{n – 1})ⁿ = a²ⁿr^{n(n – 1)}
= P² = LHS [from Eq. (iii)]

Question 2.
The sum of an infinite GP is 6 and the sum of the cubes of its terms is 27. Find the first term.
Answer:
Let the first term be a and the common ratio be r, where |r| < 1.
The GP is a, ar, ar², …….. .
Now, the sum of the infinite GP is a
\(\frac{a}{1 – r}\) = 6
Therefore, a = 6 (1 – r) ……… (i)
Also, the squares of the terms form another GP
a³, a³r³, a³r⁶, …….. .
The sum to infinity of this GP is given as 27.
So, \(\frac{a^3}{1-r^3}\) = 27
On substituting a = 6(1 – r) from Eq. (i), we get

⇒ 8 (1 – r)² = 1 + r + r²
⇒ 8(1 – 2r + r²) = 1 + r + r²
⇒ 8 – 16r + 8r² = 1 + r + r²
⇒ 7r² – 17r + 7=0
Using the quadratic formula,

Question 3.
The sum of an infinite GP is 6 and the sum of the sum of the squares of its terms is 12. Find the sum of the cubes of the terms of the GP.
Answer:
Let the first term of the GP be a and the common ratio be r, where r < 1.
Now, the sum of the infinite GP,
\(\frac{a}{1 – r}\) = 6
Therefore, a = 6 (1 – r) ………… (i)
Also, the squares of the terms form another GP
a², a²r², a²r⁴, …….. .
The sum to infinity of this GP is given as 12.
So \(\frac{a^2}{1-r^2}\) = 12
On substituting the value of a from Eq. (i),
we get \(\frac{[6(1-r)]^2}{1-r^2}\) = 12
⇒ \(\frac{36(1-r)^2}{(1-r)(1+r)}\) = 12
⇒ \(\frac{36(1-r)}{1+r}\) = 12
⇒ 3(1 – r) = 1 + r
⇒ 3 – 3r = 1+ r
⇒ 2 = 4r
⇒ r = \(\frac{1}{2}\)
Now, substituting r = \(\frac{1}{2}\) in Eq. (i), we get
a = 6(1 – \(\frac{1}{2}\)) = 6 × \(\frac{1}{2}\) = 3
Therefore, the GP is 3, \(\frac{3}{2}\), \(\frac{3}{4}\), ……. .
Now, the cubes of the terms form the GP
27, \(\frac{27}{8}\), \(\frac{27}{64}\), …………. .
Here, a = 27 and r = (\(\frac{1}{2}\))³ = \(\frac{1}{8}\)
Therefore, the sum to infinity of the cubes ,
S_∞ = \(\frac{576}{1-\frac{1}{2}}=\frac{576}{\frac{1}{2}}\)= 27 \(\frac{8}{7}\) = \(\frac{216}{7}\)
Hence, the sum of the cubes of the terms of the GP is \(\frac{216}{7}\).

Question 4.
Write the rational number corresponding to the decimal expansion \(0.3 \overline{56}\).
Answer:
Given, decimal expansion is \(0.3 \overline{56}\).
In series form, it can be written as
\(0.3 \overline{56}\) = 0.3 + 0.056 + 0.00056 + 0.0000056 + … + ∞.

Question 5.
The mid-points of the sides of a square of side 24 cm are joined to form another square. This process is repeated indefinitely. Find the sum of the areas of all the squares formed.
Answer:
The side of the first square is 24 cm.
Therefore, the area of the first square = (24)² = 576 cm²
Now, when the mid-points of the sides of a square are joined, a new square is formed inside it.
Since, the area of the new square is half of the area of the previous square.
Therefore, the areas of the successive squares form the GP 576, 288, 144, 72, … .
Here, a = 576 and r = \(\frac{288}{576}\) = \(\frac{1}{2}\) .
Since, |r| < 1
Therefore, the sum to infinity exists.
We know that S_∞ = \(\frac{a}{1 – r}\)
On substituting the values, we get
S_∞ = \(\frac{576}{1-\frac{1}{2}}=\frac{576}{\frac{1}{2}}\) = 576 × 2 = 1152
Hence, the sum of the areas of all the squares formed is 1152 cm².

Question 6.
The sum of an infinite GP is 57 and the sum of their cubes is 9747. Find the GP.
Answer:
Let a be the first term and r be the common ratio of an infinite GP.
Given, the sum of the infinite GP = 57
⇒ \(\frac{a}{1 – r}\) = 57 ………. (i)
and sum of the cubes = 9747
⇒ a³ + a³r³ + a³r⁶ + … = 9747
⇒ \(\frac{a^3}{1-r^3}\) = 9747 ………….. (ii)
On dividing the cube of Eq. (i) by Eq. (ii), we get
\(\frac{a^3}{(1-r)^3} \cdot \frac{\left(1-r^3\right)}{a^3}\) = \(\frac{(57)^3}{9747}\)
⇒ \(\frac{1-r^3}{(1-r)^3}\) = 19
⇒ \(\frac{1+r+r^2}{(1-r)^2}\) = 19 [∵ a³ – b³ = (a – b)(a² + b² + ab)]
⇒ 1 + r + r² = 19(1 + r² – 2r) [∵ (a – b)² = a² + b² – 2ab]
⇒ 18r² – 39r + 18 = 0
⇒ (3r – 2) (6r – 9) = 0
⇒ r = \(\frac{2}{3}\) or r = \(\frac{3}{2}\)
⇒ r = \(\frac{2}{3}\)
[∵ r ≠ \(\frac{3}{2}\) because -1 < r < 1 for an infinite GP]
On putting r = \(\frac{2}{3}\) in Eq. (i), we get
\(\frac{a}{1-\left(\frac{2}{3}\right)}\) = 57
⇒ 3a = 57
⇒ a = 19
Hence, the required GP is 19, 19 × \(\frac{2}{3}\), 19 × (\(\frac{2}{3}\))², ….
i.e. 19, \(\frac{38}{3}\), \(\frac{76}{9}\), …. .

Question 7.
After striking a floor a certain ball rebounds (\(\frac{4}{5}\))th of the height from which it falls.
Find the total distance that it travels before coming to rest, if it is gently dropped from a height of 120 m.
Answer:
Given, a ball dropped from the height of 120 m. After dropping 120 m, height after first rebound
= \(\frac{4}{5}\) × 120 =96 m
After dropping 96 m, height after second rebound
= \(\frac{4}{5}\) × 96 = 76.8 m
After dropping 76.8 m, height after third rebound
= \(\frac{4}{5}\) × 76.8 = 61.44
Continuing this way, we get a geometric series
96 + 76.8 + 61.44 + … ,
where first term, a = 96
and common ratio, r = \(\frac{4}{5}\).
The sum of an infinite geometric series is given by
S_∞ = \(\frac{a}{1 – r}\), r< 1.
So, S_∞ = \(\frac{96}{1-\frac{4}{5}}=\frac{96}{\frac{1}{5}}\) = 480
Since, after the initial drop of 120 m, each rebound covers equal upward and downward distances. Thus, total distance = 120 + 2 × 480 = 1080 m
Hence, the total distance travelled by the ball before coming to rest is 1080 m.

Question 8.
A trophy is to be made out of waste material in the form of equilateral triangle as base, on this base another equilateral triangle is kept so that its vertices are mid-point of sides of the triangle, again another equilateral triangle is kept in the second equilateral triangle obtained in the same manner and process continues. Now, sides of each equilateral triangle are decorated with green ribbon to give natural environmental look. If side of equilateral triangle at the base is 30 cm. Find the total length of ribbon required to decorate the trophy.
Answer:
Hint By the given condition, we have perimeter of first triangle = 90 cm,
perimeter of second triangle = 45 cm,
perimeter of third triangle = \(\frac{45}{2}\) cm and so on.

∴ Sum of perimeters of triangles forms an GP
90 + \(\frac{90}{2}\) + \(\frac{90}{4}\) + \(\frac{90}{8}\) … upto ∞.
Here, a = 90 and r = \(\frac{1}{2}\)
Now, use sum of an infinite GP,
S_∞ = \(\frac{a}{1 – r}\), r < 1
= 180 cm

Question 9.
Find the 8th term and the sum of the first 8 terms of the sequence 2, 7, 15, 26, 40, … .
Answer:
Let the given sequence be 2, 7, 15, 26, 40, ….. .
Now, let us form the table of differences.

We observe that the second differences are same.
If the first element in each row is denoted by b, a, d, respectively.
Then, b = 2, a = 5 and d = 3
Now, the nth term of the sequence is given by
t_n = b . ^{n – 1}C₀ + a . ^{n – 1}C₁ + d . ^{n – 1}C₂
On substituting the values, we get
t_n = 2 . ^{n – 1}C₀ + 5 . ^{n – 1}C₁ + 3 . ^{n – 1}C₂

Hence, the 8th term is 100.
Now, the sum of the first n terms is given by
S_n = b . ⁿC₁ + a . ⁿC₂ + d . ⁿC₃
On substituting the values, we get
S_n = 2 . ⁿC₁ + 5 . ⁿC₂ + 3 . ⁿC₃
Therefore,

= 16 + 140 + 168
= 324
Hence, the sum of the first 8 terms is 324.

Question 10.
Find the nth term and the sum of the first n terms of the series 5 + 15 + 32 + 58 + 95 + … .
Answer:
Let the given sequence be 5, 15, 32, 58, 95, …
Now, let us form the table of differences.

We observe that the third differences are same.
If we denote the first elements of each row by b, a, d, d₁ respectively.
Then, b = 5, a = 10, d = 7 and d₁ = 2
Now, the nth term of the sequence is given by
t_n = b . ^{n – 1}C₀ + a . ^{n – 1}C₁ + d . ^{n – 1}C₂ + d₁ . ^{n – 1}C₃
On substituting the values, we get
t_n = 5 . ^{n – 1}C₀ + 10 . ^{n – 1}C₁ + 7 . ^{n – 1}C₂ + 2 . ^{n – 1}C₃
Therefore,
t_n = 5 + 10(n – 1) + 7 . \(\frac{(n-1)(n-2)}{2}\) + 2 . \(\frac{(n-1)(n-2)(n-3)}{3 \times 2}\)
= 5 + 10(n – 1) + \(\frac{7}{2}\)(n – 1) (n – 2) + \(\frac{1}{3}\)(n – 1) (n – 2) (n – 3)
= 5 + 10n – 10 + \(\frac{7}{2}\)(n² – 3n + 2) + \(\frac{1}{3}\)(n³ – 6n² + 11n – 6)

Exploring Some More Progressions Class 9 Case Based Questions

Question 1.
A student draws a square with a side length of 20 cm. By joining the mid-points of its sides, another smaller square is formed inside it. This process is repeated infinitely, creating a sequence of nested squares, where the area of each new square is exactly half of the previous one.
(i) Identify the first term (a) and the common ratio (r) for the sequence of the areas.
(ii) (a) If the process stops after the 10th square, find the total area of the squares formed.
Or
(b) Write the specific mathematical expression for the sum of the areas of the first n squares.
(iii) If the squares are drawn infinitely, use the formula S_∞ = \(\frac{a}{1 – r}\) to calculate the total area.
Answer:
Given, the side length of the first square is 20 cm.
So, the area of the first square = 20² = 400 cm²
Now, each new square has half the area of the previous square.
Therefore, the sequence of areas forms a GP, 400, 200, 100, 50, … .
(i) Thus, the first term and common ratio are a = 400 and r = \(\frac{1}{2}\).

(ii) (a) If the process stops after the 10th square, then the finite GP sum formula used is

Or
(b) The sum of the areas of the first n squares,

(iii) If the squares are drawn infinitely, then we use
S_∞ = \(\frac{a}{1 -r}\), r < 1.
On substituting a = 400 and r = \(\frac{1}{2}\), we get
S_∞ = \(\frac{400}{1-\frac{1}{2}}=\frac{400}{\frac{1}{2}}\) = 800
Hence, the total area of all the infinitely many squares is 800 cm².

Question 2.
A ball is dropped from a height of 27m. On every bounce, it reaches a height exactly \(\frac{2}{3}\) of its previous height.
(i) Calculate the height of the first bounce and the second bounce in digits.
(ii) What is the total vertical distance the ball travels (only counting the upward bounces) during the first 3 bounces?
(iii) (a) If we only look at the downward drops starting from 27 m, what is the infinite sum of all these downward distances?
Or
(b) If the ball was dropped from a height, where the first bounce was 18 m and the total distance of all bounces (to infinity) was 54 m, what would be the common ratio (r)?
Answer:
Given, a ball is dropped from a height of 27 m.
On every bounce, it rises to \(\frac{2}{3}\) of its previous height.
Thus, the heights of successive bounces form a geometric progression with a = 27 and r = \(\frac{2}{3}\).
(i) First Bounce 27 × \(\frac{2}{3}\) = 18
Therefore, the height of the first bounce is 18 m.
Second Bounce
18 × \(\frac{2}{3}\) = 12
Therefore, the height of the second bounce is 12 cm.

(ii) The heights of the upward bounces are 18m, 12m, 8m.
Now, 8 = 12 × \(\frac{2}{3}\)
Therefore, the total upward distance travelled during the first 3 bounces is 18 + 12 + 8 = 38 m.
Hence, the required total upward distance is 38m.

(iii) (a) The downward distances are 27m, 18m, 12m, 8m, … .
This is an infinite GP with a = 27 and r = \(\frac{2}{3}\).
Since, |r|< 1, the sum to infinity exists.
Using the formula, S_∞ = \(\frac{a}{1 – r}\)
S_∞ = \(\frac{27}{1-\frac{2}{3}}=\frac{27}{\frac{1}{3}}\) = 27 × 3 = 81
Hence, the infinite sum of all downward distances is 81 m.
Or
(b) Given, first bounce height = 18 m
and total distance of all bounces to infinity = 54 m
The bounce heights form the GP
18, 18r, 18r², …… .
Therefore, S_∞ = \(\frac{18}{1 – r}\)
Given that, \(\frac{18}{1 – r}\) = 54
Now, 18 = 54 (1 – r)
⇒ 18 = 54 – 54r
⇒ 54r = 36
⇒ r = \(\frac{36}{54}\) ⇒ r = \(\frac{2}{3}\)

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