Vedanta Excel in Mathematcs Book -10 Final (2078)

Revision and Practice Time

Set
1. In a survey of 2,000 tourists who visited Nepal, it was found that 250 visited neither

Pokhara nor Janakpur, 1,125 visited Pokhara and 750 visited Janakpur.
(i) Represent the above information in a Venn-diagram.
(ii) How many were there who visited both the places?
(iii) How many were there who visited only one place?
2. In an examination, 80 students secured the 'A+ grade' in English or Mathematics. Out of
them, 50 students obtained the 'A+ grade' in Mathematics and 10 in both the subjects.
(i) How many students have secured the 'A+ grade' in English?
(ii) How many students have secured the 'A+ grade' in only one subject?
(iii) Represent the above information in a Venn-diagram.
3. In a class of 75 students, 30 students liked cricket but not badminton and 25 liked
badminton but not cricket. If 10 students did not like both games, how many students
liked both games? Represent the above information in a Venn-diagram.
4. In an examination, 40 % of students passed in Mathematics only and 30 % passed in
Science only. If 10 % of the students were failed in both the subjects,
(i) What percent of the students passed in both the subjects?
(ii) What percent of the students passed in Mathematics?
(iii) Represent all result in a Venn-diagram.
5. In a survey of 500 farmers, it was found that 75 % were farming crops, 20 % were farming
crops as well as vegetables and 10 % of them were not involved in farming.
(i) How many farmers were involved in vegetable farming?
(ii) How many of them were not farming vegetables?
(iii) Draw a Venn-diagram to represent all the results.

6. In a class of 40 students, 30 % likes Mathematics but not English, 45 % likes English but
not Mathematics and 10 % students like other subjects.
(i) Find the number of students who like English.
(ii) How many students do not like Mathematics?
(iii) Draw a Venn-diagram to represent the above information.

7. In an examination 80% examinees passed in Mathematics, 75% passed in Science, 5%
failed in both subjects and 300 students passed in both subjects.
(i) Draw a Venn-diagram to show the above information.
(ii) Find the number of students appeared in the examination.
(iii) Find the number of students who passed only in Science.

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8. In an examination, 48 % examinees passed in Mathematics, 62 % passed in Science and
20 passed in both the subjects. If none of the examinees failed in both the subjects, find
(i) the number of examinees who failed in Mathematics only.
(ii) The number of examinees who failed in Science only.
(iii) Represent the above information in a Venn-diagram.

9. In a survey a few number of Indian tourists who visited Nepal during 'Visit Nepal 2020',
90% tourists visited Sauraha, 70% visited Muktinath, 10% didn't visit both places and 450
tourist visited Sauraha only.
(i) Illustrate the above information in a Venn-diagram.
(ii) How many tourists were surveyed?
(iii) How many tourists visited Sauraha and Muktinath?

10. In a survey of some farmers in Pakhribas, Dhankuta it was found that 60% of farmers were
involving in vegetable farming, 80% in crop farming, 50% in vegetable and crop farming
and 30 farmers were not involving in both types of farming.
(i) Draw a Venn-diagram to illustrate the above information.
(ii) Find the number of farmers who took part in the survey.
(iii) How many farmers were involving in only one type of farming?

11. In a locality of Sindhupalchowk district, 70 % of people speak Nepali and 30 % speak
Tamang language. Those people who speak Tamang language also speak Nepali and 240
people do not speak both the languages.
(i) How many people can speak Nepali but not Tamang language?
(ii) Represent the above information in a Venn-diagram.

12. In a group of 80 music lovers, 42 liked modern songs and 30 % liked modern songs but not
folk songs.
(i) Find the number of people who liked both types of songs.
(ii) How many of them liked folk songs but not modern songs.
(iii) Represent the above information in a Venn-diagram.

13. In a survey of some people, 60 % were farmers, 50 % were service holders and 10 % were
in other occupations. If 150 people were farmers as well as service holders, find
(i) only the number of farmers.
(ii) How many people were not service holders?
(iii) Represent the above information in a Venn-diagram.

14. In a group of 100 non-veg people, 40 like only chicken and 30 like only fish. If the number
of people who do not like any of the two non-veg items is double of the number of people
who like both items, find the number of people who like at most one item by using a
Venn-diagram.

15. In a group of people, 110 people like nuclear family but not joint family and 40 like joint
family but not nuclear family. If the number of people who like nuclear family is twice the
number of people who like joint family, find the number of people who like only one type
of family by using a Venn-diagram.

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16. Out of 140 students, 50 passed in English and 20 passed in both Nepali and English. The
number of students who passed in Nepali is twice the number of students who passed in
English. Using a Venn-diagram, find the number of students who passed in Nepali only
and who didn’t pass in both subjects.

17. In a survey, 100 people liked winter but not summer and 30 liked summer but not winter.
If the number of people who liked summer is one-third of the number of people who liked
winter, find
(i) the number of people who liked both seasons.
(ii) Represent the above information in a Venn-diagram.

18. Among 300 students of a school, 32 students took part in quiz contest, and 20 took part in
debate competition. Also, the number of students who took part only in debate competition
is half of the number of students who took part only in quiz contest.

(i) Find the number of students who took part in both the programmes.

(ii) Find the number of students who did not take part in both competitions.

(iii) Draw a Venn-diagram to represent the above information.
19. In a survey of people, 120 drink tea and 90 do not drink tea. 150 drink coffee and 75 drink

coffee but not tea.
(i) How many people drink both drinks?
(ii) How many people do not drink both drinks?
(iii) Draw a Venn-diagram to show all the results.
(Hint: n(T) = 120, n(T) = 90, n(C) = 150, no(C) = 75
? n(T ˆ C) = n(C) – no(C). Also, n(T ‰ C ) = n(T) – no(C))

20. In a group of 95 students, the ratio of students who like Mathematics and Science is
4 : 5. If 10 of them like both the subjects and 15 of them like none of the subjects, then by
drawing a Venn-diagram, find how many of them

(i) like only Mathematics? (ii) like only Science?

21. 40 students in a class like Mathematics or Science or both. Out of them 14 like both the
subjects. The ratio of number of students who like Mathematics to those who like Science
is 4 : 5.
(i) How many students like Mathematics?
(ii) How many students like only Science?
(iii) Draw a Venn-diagram to show the above information.

22. In an examination, 60 examinees passed at least one subject Science or Maths. The ratio of
the number of examinees who passed only Science and only Maths is 3 : 2 and 15 passed
both the subjects.
(i) How many students passed Science?
(ii) How many students passed only one subject?
(iii) Represent the above information in a Venn-diagram.

23. In a class, the ratio of the number of students who passed Maths but not Science and those
who passed Science but not Maths is 3 : 5. Also, the ratio of the number who passed both
the subjects and those who failed both the subjects is 2 : 1. If 80 students passed only one
subject, and 100 students passed at least one subject, find

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(i) the total number of students.
(ii) Draw a Venn-diagram to show all results.

24. In the group of some students, it was found that the ratio of the number of football-lover
and basketball-lover is 9 : 10. 20 % of them liked both games, 25 % liked only football and
45 did not like both the games.

(i) Find the total number of students in the group. Then, 25 % of x = 45)
(ii) Illustrate the above information in a Venn-diagram.

(Hint: n(F) = 9x, n(B) = 10x, 9x = 45 %, x = 5 %, ?10x = 50%

25. During the lockdown period by COVID-19, most of the schools of a municipality used
zoom or google meet to run their regular classes virtually. Among 60 schools, 5 schools
used google meet but not zoom. If the number of schools that used zoom only was twice
the number of schools that used both the platforms and thrice the number of schools that
used none of these platforms, find the number of schools that used both the platforms by
using Venn-diagram.

26. Of 300 students of a school, it was found that 20 students use the exercise books of neither
brand A nor brand B. Also, the number of students who use both the brands is twice the
number of students who use brand A only and 30 fewer than the number of students who
use brand B only.

(i) Find the number of students who use the exercise books of both the brands.
(ii) Find the number of students who use the exercise books of only one of these brands.
(iii) Show the result in a Venn-diagram.
27. In a survey, it was found that the ratio of people who like apple but not banana and who
like banana but not apple is 3:4 and the ratio of people who like both and dislike both the
fruits is 1:2. If 66 people like at least one and 83% of the people like at most one of the
fruits, how many people were participated in the survey? Also, show the obtained result i
in a Venn-diagram.

28. In a survey of 60 people, it was found that they love playing either cricket or football or
both or none. If the ratio of the number people who love playing only one to both to none
of these games is 3:2:1 and the ratio of the number of people who love playing cricket only
to football only is 3:2, find:

(i) the number of people who love playing cricket.
(ii) The number of people who love playing football.
(iii) Represent the above data in a Venn-diagram.
29. A survey was conducted in a group of people regarding the use of mask and sanitizer for
prevention from the flu. Half of people said that they use only mask and 10 don’t use mask
at all. Also, 40% of the people said that they use sanitizer and 6 use none of them.

(i) Show the above information in a Venn-diagram.
(ii) How many people were surveyed?
(iii) How many people use both mask and sanitizer?
(iv) How many people use the mask?

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Revision and Practice Time

30. In an examination, 40 % of the examinees passed in Maths, 45 % passed in Science and
55 % passed in Health. If 10 % of them passed in Maths and Science, 20 % in Science and
Health and 15 % in Health and Maths and every student passed at least one subject.
(i) What percent of examinees passed in all the three subjects?
(ii) Represent the above information in a Venn-diagram.

31. In a survey of 100 people, it was found that all of them like at least one of the three
tea brands: Tokla, Muna or Mechi tea. 60 liked Tokla, 32 liked both Tokla and Muna, 22
liked both Tokla and Mechi, 20 liked Muna and Mechi, 15 liked Muna only and 10 liked
Mechi only. Draw a Venn-diagram to represent the above information and by using
Venn-diagram, find,
(i) the number of people who liked all the three brands.
(ii) How many people liked Tokla brand only?

32. In a class of 175 students, 100 are studying Maths, 70 are studying physics, 46 studying
chemistry, 30 studying Maths and Physics, 28 studying maths and chemistry, 23
studying physics and chemistry and 18 studying all the three subjects. Find by using a
Venn-diagram,
(i) how many students are studying only one subject?
(ii) How many students are studying none of these three subjects?

33. A survey regarding mobile games was conducted among 110 youths and found that the
numbers of youths who are addicted in PUBG, Free Fire and Ludo are 55, 40 and 35
respectively. 20 youths are addicted in PUBG and Free Fire, 10 are addicted in Free Fire
and Ludo, 15 are addicted in Ludo and PUBG and 20 play none of these games.
(i) Show the obtained information in a Venn-diagram.
(ii) Find the number of youths who are addicted in exactly one of these games.
(iii) Find the number of youths who are addicted in exactly two of these games.

34. In an examination out of 270 examinees, 85 succeeded in English, 110 succeeded in
Japanese, 15 in English and Japanese, 25 in Japanese and Korean and 20 in Korean and
English. Likewise, 10 examinees succeeded in all three languages and 20 in none of
these languages.
(i) Draw a Venn- diagram to show the result.
(ii) How many examinees were succeeded in at most one of these languages?
(iii) How many examinees were succeeded in at most two of these languages?

35. A school awarded 30 medals in football, 20 in basketball and 45 in volleyball. If 2
students got medals in all these sports when 80 medals were awarded in total, how
many students got the medals in exactly two sports?

Tax and Money Exchange

1. After allowing 10% discount on the marked price of a mobile phone 13% VAT was levied
and sold it. If the difference between the selling price with VAT and selling price after
discount is Rs 585, find the marked price of the mobile.

2. A businessman exchanged Rs 8,40,000 into pound sterling at the rate of £ 1 = Rs 140.
After one day, Nepali currency is revaluated by 5% and he exchanged his pounds into
Nepali currency again. Find his gain or loss.

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3. A business man exchanged Rs 5,50,000 into US dollar at the rate of $1 = NPR 110. After
a week, Nepali currency was revaluated by 10% and he exchanged his dollar into Nepali
currency again. What is his gain or loss?

4. Bishwant bought some Australian dollar of NPR 1,50,000 at the exchange rate of
1 AU$ = NPR 77.02. After a week Nepali currency is devaluated by 5% and he exchanged
his dollars into Nepali currency again. How much profit or loss did he make?

5. A bank buys US dollars at the rate of $ 1 = NPR 112.50 and sells at the rate of
$1 = NPR 113. Rita exchanged NPR 1,69,500 into dollars from the bank. After a week,
Nepali currency is devaluated by 4% and she exchanged her dollars into Nepali currency
again. How much profit or loss percent did she make?

6. Mr. Gupta marks the price of his goods 20 % above the purchase price but allows 10 %
discount to his customers. In this way, he earns a profit of Rs 120. What is the actual
purchase price of the goods?

7. A trader marks the price of an article 50 % above the cost price. He then gives 12 %
discount to the buyers and earns a profit of Rs 16. Find the cost price of the article.

8. A retailer marks the price of an article 40 % above its cost price and gives a discount of
15 % to the customer. If he gains Rs 76, find the marked price of the article.

9. A Palpali cap is marked to sell at a profit of 10 % by a shopkeeper. If he sells it at a discount
of Rs 10.50, there will be a loss of 5 %. Find the cost price of the cap.

10. A shopkeeper sells an article at 8 % discount and gains Rs 40. If he does not allow any
discount, he will gain Rs 60. At what price did he purchase the article?

11. When a shopkeeper sells his goods at the marked price, he ghaeinpsur3c3h13as%e th. eIfahrteicalell?ows
10 % discount, he makes a profit of Rs 300. At what price did

12. A shopkeeper gained Rs 8 by selling a pen allowing 10 % discount. He would have gained
Rs 20 if discount was not allowed. What was the selling price of the pen?

13. When an article is sold after allowing a discount of 10 % on its marked price, a profit of
Rs 400 is made. If it had been sold without allowing a discount, there would have been a
profit of Rs 700. Find the marked price and the cost price of the article.

14. The marked price of an article is 30 % above its selling price and the cost price is 35 % less
than its marked price. Find the discount percent and gain percent.

15. A man allowed 10 % discount on the fixed price of a piece of land and offered 5 %
commission to the broker. If he received Rs 2,56,500, what was the fixed price of the land?

16. A man allowed 5 % discount on the fixed price of a second-hand car and offered 5 %
commission to the broker. If he received Rs 2,88,800, find the fixed price of the car.

17. The marked price of a cycle is Rs 7,520. What will be the price of the cycle if 15 % VAT is
levied after allowing 5 % discount on it?

18. The marked price of a camera is Rs 3200 and the shopkeeper announces a discount of
8 %. How much does a customer have to pay for buying the camera if 10 % VAT is levied
on it?

19. Allowing 20 % discount on the marked price of a watch, the value of the watch will be
Rs 2,376 when the VAT of 10 % is added. Find its marked price.

20. After allowing 5 % discount on the marked price of a radio, 10 % VAT is charged on it,
then its price becomes Rs 1,672. Find the amount of discount.

21. Mrs. Baral buys a television set with 10 % VAT after getting 15 % discount for Rs 7,480.
What was the actual price of television set and the amount of VAT?

Vedanta Excel in Mathematics - Book 10 304 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Revision and Practice Time

22. After allowing 16 % discount on the marked price of an article and levying 13 % VAT, the
price of the article becomes Rs 9,492. Find the value added tax.

23. An article was purchased for Rs 700. What marked price was fixed so that 7 % gain was
made by selling after giving 1221 % discount. Also find the selling price of the article.

24. A shopkeeper allows a discount of 10 % on the marked price of an article. What was the
marked price of the article which cost him Rs 1,800 and makes a profit of 15 %?

25. The marked price of an article is Rs 5,000. This price is 25 % above the cost price. If the
article is sold by allowing 15 % discount, find the profit percent.

26. A tourist buys an article at 15 % discount on the marked price and he pays 13 % VAT. If
the VAT amounts to Rs 442, find the marked price and the selling price including the VAT.

27. The marked price of an electric item is Rs 1,600 and the shopkeeper allows some discount.
After levying 10 % VAT, if a customer pays Rs 1320 for it, find the discount percent.

28. A shopkeeper selling an article at a discount of 25 % loses Rs 125. If he allows 10 %
discount, he gains Rs 250. Find the marked price and the cost price of the article.

29. A mobile set, after allowing a discount of 10 % on its marked price, was sold at a gain of
20 %. Had it been sold after allowing 20 % discount, there would have been a profit of
Rs 350. Find the cost price of the mobile set.

30. After allowing 20 % discount on the marked price and then levying 10 % VAT, a radio was
sold. If the buyer had paid Rs 320 for VAT, how much would he have got the discount?

31. A man buys an article at 15 % discount on the marked price and he pays 10 % VAT. If the
discount amounts to Rs 600, find the marked price and the selling price including VAT.

32. The marked price of a watch is Rs 2,000. If a shopkeeper allows a discount equal to the
VAT rate and a customer pays Rs 1,980 with VAT, find the discount rate.

33. The marked price of a calculator is Rs 1,000 and a shopkeeper allows discount which is
two times the VAT rate. If a customer pays Rs 836.20 with VAT, find the rates of discount
and VAT.

34. A shopkeeper allows 20 % discount while selling electrical items. How much percent
above the cost price must he mark his items so as to make a profit of 12 % ?

35. After allowing 20 % discount and levying 13 % VAT, an article is sold for Rs 2,260.
Calculate the amount of discount if its rate is reduced to 12 %.

36. The manufacturer marks the price of a photocopy machine at Rs 2,50,000. He sells the
machine to the wholesaler at a discount of 25% on its marked price and the wholesaler
sells it to a supplier at a discount of 15% on its marked price. Also, the supplier spends
Rs 1,500 for transportation and Rs 500 for local tax and sells it to a customer by making a
profit of 8% without any discount and at each stage the VAT rate is 13%.
(i) How much VAT does the wholesaler pay while buying the machine?
(ii) How much VAT does the supplier get back after selling the machine?
(iii) How much VAT should the customer pay while buying the machine?

37. Mr. Chaudhary buys a mobile set for Rs 16,950 with 13% VAT. He sells it by making 10%
profit with 13% VAT.
(i) How much amount of VAT will Mr. Chaudhary get back?

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(ii) How much VAT should the customer pay while purchasing the mobile?
(iii) How much amount should a customer pay for the mobile set with VAT?
38. If the cost price and selling price of a laptop with 13% value added tax are Rs 54,240 and
Rs 62,715 respectively, find the difference between the VAT amounts while buying and
selling the laptop.

Compound interest, Population growth and Depreciation

1. By what percent is the yearly compound interest on Rs 8,000 for 2 years at 5% p.a.
more than the simple interest of the same sum of money for the same interval of time
at the same rate of interest?

2. The compound interest on a certain sum of money in 2 years and 4 years are Rs 4,200
and Rs 9,282 respectively. Calculate the compound interest of the sum in 3 years.

3. The compound interest of a certain sum of money compounded semi-annually in 1
year and 2 years are Rs 16,400 and Rs 34,481 respectively. Find the rate of interest and
the sum.

4. Hari borrowed Rs 1,30,000 from Krishna at the rate of 21% p.a. simple interest and lent
it to Mohan at the same rate of interest compounded annually. How much profit did he
make in 1 year 6 months?

5. Ram borrowed a sum of money at the rate of 5% simple interest for 2 years and he
lent it at the same rate of compound interest for the same duration of time. In this
transaction if he gained Rs 30, find the sum borrowed.

6. Sumnima borrows a certain sum of money at 3% p.a. simple interest and invest the
same sum at 5% p.a. compound interest compounded annually. If she makes a profit of
Rs 1,082 after 3 years, what is the sum she borrowed?

7. Find the difference between compound interest payable annually and half annually for
a sum of Rs 14,000 at the rate of 12% p.a. in 2 years.

8. If the difference between the compound interest and the simple interest on a sum of
money for 2 years at 20% p.a. is Rs 400, find the sum.

9. The simple interest on a sum of money in 2 years is Rs 90 less than its compound
interest. If the rate of interest is 15% p.a., find the sum.

10. The compound interest on a sum of money in 2 years at the rate of 10% p.a. will be
Rs 420 more than simple interest. Find the sum.

11. If the difference between the compound interest compounded half yearly and yearly
on a sum of money for 2 years at the rate of 20% p.a. is Rs 289.20, find the sum.

12. The sum of simple interest and compound interest after 3 years is Rs 504.80 and the
rate of interest is 10% p.a. Find the principal.

13. Mr. Gurung lent altogether Rs 6,000 to Ram and Rahim for 2 years. Ram agrees to pay
simple interest at 10% p.a. and Rahim agrees to pay compound interest at the rate of
8 % p.a. If Rahim paid Rs 50 more than Ram as the interest, find how much did
Mr. Gurung lend to each?

14. The compound interest on a certain sum for 2 years at 10% p.a. is Rs 420. What would
be the simple interest on the same sum at the same rate for the same time?

Vedanta Excel in Mathematics - Book 10 306 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

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15. What will be the compound interest on Rs 5,000 for 2 years at the rate of 20%
compounded per annum? At what time will it fetch the same interest on the same sum
at the same rate of simple interest?

16. A sum of money deposited at 10% p.a. interest compounded half yearly amounts to
Rs 23,152.50 in 121 years. Find the sum.

17. According to the yearly compound interest, in what time will the compound interest
on Rs 4,00,000 at the rate of interest 6.5% p.a. be Rs 53,690?

18. If the compound amount of a sum compounded annually at 10% p.a. for 3 years is
Rs 10,648, find the sum.

19. The compound interest of a sum of money in 1 year and 2 years are Rs 400 and Rs 832
respectively. Find the rate of interest compounded yearly and the sum.

20. A person took a loan of Rs 46,875. if the rate of compound interest is 4 paise per rupee
per year, in how many years will the compound interest be Rs 5,853?

21. At what rate of compound interest per annum will be the compound interest on
Rs 3,43,000 be Rs 1,69,000 in 3 years?

22. If a sum of Rs 62,500 amounts to Rs 70,304 in 121 years according to the compound
interest compounded half yearly. Find the rate of interest.

23. A invested Rs 25,000 for 3 years at the rate of 12% simple interest per annum and
B invested the same amount for the same time at the rate of 10% annual compound
interest.
(i) Calculate the interest received by A and B each.
(ii) Without altering time and rate of interest, how much more or less money should
A have to invest for equal interest?

24. If the compound amounts of a sum of money in 2 years and 3 years are Rs 12,500 and
Rs 13,750 respectively, find the rate of interest.

25. The compound amounts of a sum of money in 2 years and 3 years are Rs 10,580 and
Rs 12,167 respectively. Find the rate of interest and the sum, compounded yearly.

26. A sum of money amounts to Rs 19,360 in 2 years and Rs 23,425.60 in 4 years at the rate
of compound interest annually. Find the rate of compound interest and the sum.

27. The compound amounts of a sum of money in 3 years and 4 years are Rs 3,240 and
Rs 3,888 respectively, find the rate of interest compounded yearly and the sum.

28. What sum of money will yield Rs 6,305 compound interest compounded semi-annually
at 10% p.a. in 121 years?

29. Ashok invests a certain sum of money at 20% per annum, compounded yearly. Gita
invests an equal amount of money at the same rate of interest per annum compounded
half-yearly. If Gita gets Rs 99 more than Ashok in 18 months, calculate the investment
of each of them.

30. Shivaraj has a son Subin of 14 years old and a daughter Archana of 16 years. He divides
Rs 66,300 between his son and daughter and deposits in their own accounts in a bank
at 10% per annum compound interest. If both children receive equal amounts at the
age of 18 years,

i) What is the share of each of them?

(ii) What amount will each receive at the age of 18 years?

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Revision and Practice Time

31. Riya borrowed a certain sum of money from a bank and paid it back in 2 instalments.
If the rate of the compound interest was 4% p.a. and she paid back Rs 6760 annually,
what sum did she borrow? By what percent is the principal of the first year more than
that of the second year? Find it.

32. Sunil borrowed a certain sum from a bank at 5% per annum compound interest. He
cleared the loan by paying Rs. 31,500 at the end of the first year and Rs. 22,050 at the
end of the second year.
(i) Find the sum that he borrowed.
(ii) How much more amount would he need if he cleared the loan only at the end of the
second year?

33. The present population of a town is 66,550. If the annual population growth rate is
10% p.a. What was the population of the town 3 years ago?

34. 2 years ago, the population of a town was 1,20,000. If the rate of growth of population
is 3% p.a., what is the present population of the town?

35. Find the rate of growth of population of a village if its population increased from 12,800
to 14,112 in 2 years.

36. In how many years will the population of a town be 26,901 from 24,400 at the growth
rate of 5% p.a.?

37. The population of a village increases every year by 5%. At the end of two years, the
total population of the village was 10,000 after 1,025 of them migrating to other places.
Find the population of the village in the beginning.

38. In the beginning of 2076 B.S., the population of a town was 1,50,000 and the rate
of growth is 4% p.a. If in the beginning of 2070 B.S., 19,000 people migrated there
from different places, what will be the population of the town in the beginning of
2078 B.S.?

39. The present price of a bus is Rs 40,00,000. After how many years will the price be
Rs 29,16,000 if it's price reduces at 10% p.a. compound depreciation?

40. The current price of a machine is Rs 6,50,000. If it is depreciated at the rate of 10 % per
year, what will the price of the machine be after 3 years?

41. If the cost is depreciated at the rate of 4% p.a., the cost of a motorcycle after 3 years
becomes Rs 1,10,592. Find the original price of the motorcycle.

42. If the value of an article depreciated from Rs 36,000 to Rs 29,160 in two years, find the
rate of depreciation per year.

43. A man bought a computer for Rs 44,100 and after using it for 2 years, he sold it for Rs
40,000. Find the rate of compound depreciation of the computer.

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Revision and Practice Time

44. In BS 2077, a man bought a plot of 4 aana at the rate of Rs 2,00,000 per aana in a rural
municipality and immediately invested Rs 27,00,000 for building a house on it. If the
value of land increases every year by 20% and that of the house decreases every year by
20%, when will the values of the land and house be equal? Find it.

45. Ayush took a loan of Rs 75,000 from a bank at the rate of simple interest. Immediately, he
lent the whole amount to Hishila at the same rate of compound interest. If he gained Rs
120 at the end of 2 years, find the rate of interest. By what percentage was the interest
paid by Hishila more than the interest paid by Ayush?

46. Mr. Gurung bought some secondary shares of a hydro-power company at the rate of
Rs 400 per share. But the value of shares of the company depreciated at the rate of
10% per year for two years and appreciated at the rate of 15% per year for three years.
If Mr Gurung received Rs 2,46,381.75 by selling all his shares of the company after 5
years, how many shares were bought? Also, find his profit or loss from the shares in 5
years.

47. Four business persons started a publication house at Rs 3,00,00,000 with equal shares. If
they incurred a loss of 10% in the first year and earned the profit of 12% in the next two
years, calculate the net profit made by each shareholder in 3 years. By what percentage
was the profit amount so earned more than the loss incurred in the first year?

Mensuration

1. The capacity of a cylindrical water tank is 539 litres. If its height is 1.4 m, find the
radius of its base.

2. If the surface area of a sphere is 22176 sq. cm, find its radius.

3. If the height of a cone is three times the radius of the base and its volume is 512S cm3,
find the radius of its base.

4. If the volume of a hemisphere is 486S cm3, find its radius.

5. a) A cylindrical water tank contains, 3,85,000 litres of water. If its height is 10 m, find
its diameter.

b) A 60 cm high cylinder with 14 cm its diameter is cut vertically into two equal halves.
What is the volume of a half part?

c) If the sum of the radius and height of a right cylinder is 50 cm and the circumference
of its base is 220 cm, find its total surface area and volume.

d) If the radius of the base and the height of a cylinder are in the ratio of 5:7 and the
volume is 550 cubic cm, find the radius of the base of the cylinder.

e) The ratio of the radius of base and the height of a cylinder is 7:8. If the volume of the
cylinder is 9,856 cm3, find the total surface area of the cylinder.

f) The sum of the height and the radius of the base of a cylinder is 34 cm. If the total
surface area of the cylinder is 2,992 cm2, find its volume.

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g) The adjoining solid is a half portion of a cylinder obtained by 35cm
cutting vertically through its height. Find the curved surface area,
total surface area and volume of the solid.

h) The adjoining cylindrical vessel is 70 cm high and the radius 70cm 7cm
of its base is 35cm. If it contains some water up to the height 35cm
of 20 cm, how much water is required to fill the vessel
completely? 20cm

i) 50 circular plates, each of radius 7 cm and thickness 5 mm are placed one above the
other to form a cylindrical shape. Find the volume of the cylinder so formed.

j) How many cubic metres of earth must be dug out to construct a cylindrical well
which is 25 m deep and the radius of the base is 3.5 m?

6. a) If the perimeter of the plane circular surface of a hemisphere is 176 cm, calculate

(i) the total surface area of the hemisphere,
(ii) the volume of the hemisphere.
b) The external and internal radii of a hollow spherical metallic shell are 35 cm and
14 cm respectively. Calculate the volume of metal contained by the shell.

c) A solid metallic sphere of radius 7 cm is cut into two halves. Find the total surface
area of the two hemispheres so formed.

d) How many lead balls each of radius 1 cm can be made from a solid sphere of
diameter 12 cm?

e) A spherical ball of radius 3 cm is melted and recast into three spherical balls. The
radii of two of them are 1.5 cm and 2cm. Find the diameter of the third ball.

f) A cylindrical tub of of radius 16 cm contains water to a certain depth. When a
spherical ball is dropped into the tub, the level of water is raised by 9 cm. Find the
radius of the ball.

g) A cylinder whose height is two-third of its diameter, has the same volume as a
sphere of radius 21 cm. Calculate the curved surface area of the cylinder.

h) The surface area of a ball is twice the area of the curved surface of a cylinder. If the
height and radius of the base of the cylinder be 10.5 cm each, find the volume of the
sphere.

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Revision and Practice Time

i) A hemispherical bowl with radius 21 cm is full of water. h
If the water is poured into a 1m tall jar with the diameter
of the base 28 cm, find the height of the water level in the
jar.

7. The umbrella is made by stitching 8 triangular pieces of cloth, each piece measuring
73.2 cm, 73.2 cm and 26.4 cm. How much cloth is required for the umbrella? If the rate of
cloth is 5 paisa per sq. cm, find the cost of the cloth required to make the umbrella.

8. Once, a teacher gives two rectangular pieces of chart paper each of 66cm long and 44cm
wide to Ram and Sita separately. If Ram rolls the paper along the length and Sita rolls the
paper along the breadth to form the largest cylinders, whose cylinder has the more volume
and by how much?

9. A cylindrical roller made of iron is 1.05 m long. Its internal radius is 50 cm and the
thickness of the iron sheet used in making the roller is 5 cm.
(i) If it takes 400 complete revolutions to level a playground, find the cost of levelling the
ground at Rs 15 per sq. metre.
(ii) If 1 cm3 of iron weighs 7.8 g, find the mass of the roller.

10. A water-well is composed of 30 identical cement rings with 84cm internal diameter,
3.5 cm thickness and 28 cm height each.
(i) If the labour cost for digging out the well is Rs. 400 per cubic meter and the cost of a
cement ring is Rs 850 per ring, estimate the total cost of well construction.
(ii) If the water level is found up to 5 rings below from the top, find the quantity of water
in the well.

11. A cylindrical vessel of height 24 cm and diameter 40 cm is full of water. How many
cylindrical bottles, each of height 10 cm and diameter 8 cm, are required to empty
vessel?

12. A gate has two cylindrical pillars with a hemispherical top in each pillar. The height of
each pillar is 13 ft. and the height of cylindrical part in each pillar is 12.3 ft. find the cost
of colouring the surface of the pillars at Rs 25 per sq. ft.

13. There are 10 cylindrical pillars which are surmounted by the hemispherical ends of same
radius all around in the ground of a temple. The radius of each pillar is 9 inch and total
height 15ft 9 inch. If 1ft 9 inch of each pillar is inside the ground, calculate the cost of
painting the pillars at Rs 15 per sq. ft. (12 inches = 1 ft.)

14. The sum of the inner and the outer curved surfaces of a hollow metallic cylinder of height
21cm is 1056 cm2 and the volume of material used in making it is 1056 cm3. Find its
internal and external radii.

15. On a sunny day, Mina brought a completely spherical watermelon of diameter 12cm.
The volume of juice made from the watermelon is 75% of its total volume. If she pours
the whole juice into the identical cylindrical glasses of height 8cm and radius 3cm and
distributes among her family members at a glass of juice per member, how many members
are there in her family? Find.

16. A cylindrical water tank of diameter 2.1m and height 3.5 m is surmounted by the
hemispherical top. If the tank is being filled by a pipe of diameter 7 cm through which the
water flows at the rate of 3m/s. Calculate the time it takes to fill the tank.

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17. A metal container in the form of a cylinder is surmounted by a hemisphere of the same
radius. If the total internal height of container is 3 m and surface area are 22 m2, estimate
the cost of filling up the container with water at 42 paisa per litre.

18. A right circular cylindrical jar of diameter of base 14cm and height 56cm is full of water.
If 12 identical metallic balls are completely dropped in the jar, 6468 cubic centimetre of
water is still left in the jar, find the radius of each ball.

19. The internal part of a thermos is cylindrical. The volume of the water required to fill the
whole thermos is 2826 cm3 and 1695.6 cm3 of water is required to fill the thermos to a
level which is 10 cm below the top. Find the internal radius and the height of the thermos.
(Use S=3.14cm)

20. 13860 litre of diesel is required to fill a cylindrical tank completely and 10395 litre of
diesel is required to fill it up to 1m below the top. Find radius and height of the tank. Also,
find the wetted surface area of the tank when it is half-filled with diesel.

21. a) The volume of a prism having its base a right-angled triangle is 864 cubic cm. If
b)
the lengths of the sides of the triangle containing the right-angle are 8 cm and

9 cm, calculate the height of the prism.

Given figure is a triangular prism whose volume C C'
is 1800 cubic cm. If AB = 15 cm, BC = 8 cm and 8 cm B'
‘ABC = 90°. Find the length of the prism.
15 cm B A'
A

c) The area of cross section of the adjoining prism is 54 cm2 12cm
and its volume is 2,160 cm3. Find its lateral surface area 15cm
and the total surface area.

d) The area and perimeter of the base of a triangular prism are 24 cm2 and 24 cm

respectively. If the total surface area of the prism is 408cm2, find its lateral surface

area and volume. A A'
e) The lateral surface area of the adjoining triangular
13cm
prism is 2,700 cm2. If the perimeter of its base is B'

54 cm, find its total surface area and volume. B 21cm C'
C

f) In a marriage ceremony of Sujit’s daughter

Deepika, he has to make an arrangement for the

accommodation of 50 people. For this purpose,

he plans to build a tent in the shape of triangular

prism as shown in the figure of length 20m so 20 m
that each person has 4 sq. m of space on the

ground and 15 cu. m. of air to breathe. What

should be the height of the tent? Also, calculate the cost of canvas to build the tent
at Rs 125 per sq. m.

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Revision and Practice Time

22. a) Given figure is a square based pyramid. If the slant height O
of the pyramid is 13 cm and the side of the base 10 cm, 13 cm
determine the volume of the pyramid.

10 cm

b) Find the total surface area of the solid pyramid with square base,
where the length of the side is 16 cm and height is 6 cm.

c) The given solid is a square-based pyramid of side 14 cm. If the 16 cm
length of the edge of its triangular face is 25 cm, find the total 25 cm
surface area of the pyramid.

14 cm

d) The length of the base of a square-based pyramid is 18 cm and its volume is

1,296 cm3. Find the total surface area of the pyramid. O

e) In the adjoining figure, the length of a side of the base of 12 cm
the pyramid having a square base is 12 cm and the total
surface area of the pyramid is 384 cm2, find the volume of
the pyramid.

12 cm

f) The total surface area of a square-based pyramid is 896 cm2 and its slant height is
25 cm. Find the volume of the pyramid.

g) The length of the side of a square-based pyramid is 12 cm. If the lateral surface area
of the pyramid is 240 cm2, find its total surface area and volume.

h) The height and length of side of a square based pyramid are in the ratio 2:3. If the
area of the triangular surfaces of the pyramid is 1500 cm2, find its volume.

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23. a) The adjoining solid is a crystal formed by the combination of
two equal square based pyramid of side 12 cm. If the height of
the crystal is 16 cm, find the total surface area and volume. 16 cm

12 cm

b) In the adjoining combined solid, the uppermost part is the 1.3m 50cm
square-based pyramid of side 1.2 m and the lowermost part
is the square-based cuboid. Calculate the total surface area 1.2m
and volume of the solid.

1.2m

6cm

c) In the given solid, there are two pyramids of the same size 16cm
and shape at the ends of a cubical solid. If the height of one
pyramid is 6 cm and that of the cubical solid is 16 cm, find 28cm
the volume of the whole solid.

d) The adjoining solid is the combination of a square-based 8cm
pyramid and a square-based prism. If the total surface area
of the solid is 960 cm2, find the volume of the solid.

20cm

12cm 12cm

e) If the volume of the combined solid made up of the square- 5cm
based pyramid and a prism is 228 cm3, find the total surface
area of the solid. 5cm

6cm 6cm

24. a) The area of circular base of a cone is 616 cm2 and its slant height is 20.5 cm. Find
the curved surface area and the total surface area of the cone.

b) The perimeter of the circular base of a cone is 132 cm and its vertical height is
28 cm. Find the curved surface area, total surface area and volume of the cone.

c) The circumference of the base of a cone is 44 cm and the sum of its radius and
slant height is 32 cm. Calculate the curved surface area, total surface area and the
volume of the cone.

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Revision and Practice Time

d) The total surface area of a cone whose base radius 5 m is 282 6 sq. metres. Find
the volume of the cone. 7

e) If the total surface area of a cone is 704 cm2 and its slant height is 25 cm, find its
curved surface area and the volume.

f) The volume of a right circular cone is 9,856 cm3. If the circumference of its circular
base is 88 cm, find its curved surface area and the total surface area.

g) The area of the circular base of a right circular cone is 1,386 cm2. If the volume of
the cone is 12,936 cm3, find its curved surface area and the total surface area.

25. a) Find the total surface area and the volume of the following solids.

(i) (ii) 28cm (iii) 20cm
14cm
50cm

25cm 58cm

40cm

68cm

(iv) (v) 42cm (vi)

31 cm 14.4cm 15cm 7cm

14cm 18.6cm 24cm

42cm 56cm

b) In the figure, a right circular cylinder has two conical 2.1 m
ends with radius of the base 42 cm and height 56
cm. If the solid is 2.1 m long, find its total surface
area and volume.

42cm 72cm
143cm
c) In the adjoining solid, one of its ends is a cone
and the other opposite end is a hemisphere with
diameter 42 cm. Calculate the total surface area
and the volume of the solid.

d) In the adjoining figure, a conical vessel 20 cm ?
with diameter of its circular base 20 cm is 30 cm
completely filled with water. If the water is
poured completely into a cylindrical jar of the
same diameter, find the height of water level in
the jar.

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e) A hemispherical bowl with diameter 5 cm is 5 cm
completely filled with water. If the water is
completely poured into a right circular conical 10 cm
vessel of height 10 cm as shown in the figure, find
its radius.

26. A vertical iron pole consisting of a cylindrical portion 3.7 m high and of base diameter
12 cm is surmounted by a cone 8 cm high. If 20 cm of its lower part is inside the earth,
find the total cost of painting its surfaces at 15 paisa per sq. cm. (Use S=3.14)

28. 12 identical metallic spheres; each of having diameter 10 cm are melted and recast a
conical shape of height 15 cm and attached on the top of a temple. Estimate the cost of
painting the surface of the cone with golden colour at Rs 1.50 per square centimetre.
(Use S=3.14)

29. A conical clay of base radius 5cm and height 20cm is reshaped in to a spherical object.
Find the total cost to paint on the surface of the spherical clay at 50 paisa per sq.cm.
(Use S=3.14).

30. A circus tent of diameter 70m is composed up of cylindrical part of height 9 m and
conical part of height 12m above it.

(i) How much canvas is used in building the tent?

(ii) If the width of the canvas is 2m, calculate the total cost of canvas used at Rs 25 per
meter.

31. An exhibition tent is in the form of a cylinder surmounted by a cone. The diameter of
base is 48m, height of the tent is 19 m and the height of the cylindrical part is 12 m.

(i) If 10% extra canvas is used for folding and stitching, how much canvas is required
to make the tent?

(ii) If the tent is 2.2m wide, find the cost of the canvas at Rs 65 per meter.

32. Two solid spheres of radii 2 cm and 4 cm are melted and recast into a cone of radius 6
cm. What is the height of the cone so formed? Also, find the cost of painting the surface
of the cone at 25 paisa per square centimetre? (Use S=3.14cm)

H.C.F. and L.C.M.

1. Find the H.C.F. and L.C.M. of the following expressions

a) x2 + 2x , x3 – 4x b) 3x2 – 15x , 3x3 – 75x

c) a2 – 6a + 6b – b2, b2 + ab – 6b d) m2 – n2- m - n, m2n – mn2 –mn

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e) p3 – p , p3 + 2p2 + p f) 18a3b – 2ab3 , 27a3b – 18a2b2+ 3ab3

g) (a – b)2 + 4ab, ab2 +a2b h) (x + 2)2 – 8x, 2ax3– 8ax

i) 6x3 – 6x, 9x4 + 9x j) a3b3 – 4ab, a4b4 – 8ab

k) x3 – x2 + x – 1, 2x3 - x2 + x – 2 l) a3 + a2 – 4a – 4, 2a3 - a2 - a + 2

m) 16a4 – 4a2 + 4a – 1, 8a3 + 1 n) 1 + 4x + 4x2 -16x4, 1 – 8x3

o) p4 + 4p2+ 16, p3 + 8 p) 16a4 + 4a2b2+ b4, 8a3 + b3

q) x4 + x2+ 169, x3 + x (x + 13) + 4x2 r) 9k4 + 14k2+ 25, 3k2 (2k+1)+5k(k+2)

s) x3 – 3x2 – x + 3 , x3 – x2 – 9x + 9 t) x3 + 2x2 – x – 2 , x3 + x2 – 4x – 4

u) x2 – 14x – 15 + 16y – y2 , x2 – y2 +2y – 1

v) x2 – 10x – 11 + 12y – y2 , x2 – y2 +22y – 121

w) (1 – x2)(1 – y2) + 4xy, 1 – 2x + y – x2y + x2

x) (x2 – 4)(y2 – 9) – 24xy, 12 – 12x – 4y + x2y + 3x2

y) 18(2x3 – x2 – x), 20 (24x4 + 3x)

z) x6 – y6, x5 + x3y2+xy4

2. Find the H.C.F. and L.C.M. of the following expressions

a) 6(a3 – 4a), 8(a3b – a2b – 6ab), 9(a2b – ab – 2b)

b) 12(4x3y – xy), 15(8x4y2 – xy2), 20(6x2y – 5xy + y)

c) 8m4 +27mn3, 8m3n + 2m2n2 – 15mn3, 4m3n – 9mn3

d) 2x3 – x2 – x, 4x3 – x, 8x4 + x

e) x3 + x2 + x, x4 – x, x7 – x

f) p3 – p2 + p, p4 + p, p5 + p3 + p

g) x3 – y3, x4 + x2y2 + y2, x3 + x2y + xy2

h) y3 – 1, y4 + y2 + 1, y3 + 2y2 + 2y + 1

i) a3 – 1 – 2a2 + 2a, a3 + 1, a4 + a2 + 1

j) x2 - y2 - 2yz – z2, y2 - z2 - 2zx – x2, z2 - x2 - 2xy – y2

k) 9m2 - 4n2 - 4nr – r2, r2 - 4n2 - 9m2 – 12mn, 9m2 + 6mr + r2 - 4n2

l) 9x2 - 4y2 - 8yz – 4z2, 4z2 - 4y2 - 9x2 – 12xy, 9x2 + 12xz + 4z2 - 4y2

m) 8x3 – 1, 16x4 - 4x2 – 4x – 1, 16x4 + 4x2 + 1

n) 1+4x+4x2 – 16x4, 1+2x – 8x3 – 16x4, 1 + 4x2 + 16x4

o) 81x4 – 9x2 – 6x – 1, 81x4 + 27x3 – 3x – 1 , 81x4 + 9x2 + 1

p) a2 – 6a – 7 + 8b – b2, a2 + a – b2 + b, a2 - b2 + 2b – 1

q) x2 + 2x – 15 – 8y – y2, x2 – 3x – 2xy + 3y + y2, x2 + 5x – 5y – y2

3. Ram had a rectangular plot of land. He divided the land along the length to make two
rectangular plots A and B. He sells the plot-A to Lakpa and plot-B to Sushma. If the area
of Lakpa’s plot is (6x2 + 17x + 7) sq. ft. and that of Sushma’s plot is (8x2 + 2x – 1) sq. ft.,
find the breadth of the plots.

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Revision and Practice Time

Simplification of Rational Expressions

Simplify the following rational expressions.

1. 1+ 1 1+ 1 1– 2 2. 1– 2 1– 1 1+ 3
p p–1 p+1 m m–2 m–3

3. 2ab – a–b + a+b 4. 2x + 5 – 3x – 2 + x2 + 1
a2 – b2 a+b a–b x+1 x–1 x2 – 1

5. x2 + y2 – x2 – y2 6. x+1 + x–1 – 1
xy y(x + y) x(x + y) 2x3 – 4x2 2x3 + 4x2 x2 – 4

7. ax2 + b + ax2 – b + 4ax3 8. x+1 + x–1 – 4x
2x – 1 2x + 1 1 – 4x2 x–1 x+1 x2 + 1

9. 2 + 2 + 1
(x – 2) (x – 3) (x – 1) (3 – x) (1 – x) (2 – x)

10. 1 + 2 + 3
(x – 2) (x – 3) (x – 1) (3 – x) (1 – x) (2 – x)

11. x + x–1 – x–3
(x + 3) (x – 1) (x + 3) (2 – x) (2 – x) (x – 1)

12. 1 + 1 + 1
(a – b) (a – c) (b – a) (b – c) (c – a) (c – b)

13. 1 + x2 – 1 + 2 – 1
x2 – 5x + 6 3x x2 – 4x + 3

14. 1 – a2 – 2 + 3 + 3
a2 – 5a + 6 4a a2 – 3a + 2

15. 2 + 2 – a2 + 4 + 12
a2 – 2a – 24 a2 – 3a – 18 7a

16. x–3 + 2x – 1 – 2x + 5
x2 – x – 6 2x2 + 5x – 3 x2 + 5x + 6

17. x–3 + 2(x – 1) 3 – 3(x – 4)
x2 – 7x + 12 x2 – 4x + x2 – 5x + 4

18. c2 a2 + ab + b2 + b2 + bc + c2 + c2 + ca + a2
– c(a + b) + ab a2 – a(b + c) + bc b2 – b(c + a) + ca

19. 4a2 – (3b – 4c)2 + 9b2 – (4c – 2a)2 + 16c2 – (2a – 3b)2
(4c + 2a)2 – 9b2 (2a + 3b)2 – 16c2 (3b + 4c)2 – 4a2

20. (x x+y + y–z – (z z+x
+ y)2 – z2 x2 – (y – z)2 + x)2 – y2

21. y–2 + y2 y +2 4 – 16
y2 – 2y + 4 + 2y + y4 + 4y2 + 16

22. 2x – y – 2x + y + 16x3
4x2 – 2xy + y2 4x2 + 2xy + y2 16x4 + 4x2y2 + y4

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Revision and Practice Time

23. 3a +1 1 + 3a – 1 – 2
9a2 + 3a + 9a2 – 3a + 1 81a4 + 9a2 + 1

24. 1 – 1 + 2a
1 + a + a2 1 – a + a2 1 + a2 + a4

25. 1 – 1 + 2x
1 + x + x2 1 – x + x2 1 – x2 + x4

26. 1 – 3 + 3 – 1 27. x3 – x x2 1 + 1 x + 1
a–1 a a+1 a+2 x–1 + 1– x+1

28. x 1 – 2 1+ x 1 1 – 2 29. x x y + xy + x2 + 2x2y2
–1 2x + + 2x – 1 + x2 – y2 x2 + y2 x4 – y4

30. a + a a b – 6a2 + 8a4
a–b + a2 – b2 a4 – b4

31. 1 + 1 + 2a + 4a3 32. 1 + 1 – 2 + 4
a – 2b a + 2b a2 + 4b2 a4 + 16b4 1–x 1+x 1 + x2 1 – x4

33. 2 – 1 – 3y + xy 34. 2 + 1 + 3a + a
x+y x–y y2 – x2 x3 + y3 1+a a–1 1 – a2 1 + a3

35. 1+a + 1 4a – 1–a + 1 8a 36. x+a – x–a – 4ax – 8a3x
1–a + a2 1+a + a4 x–a x+a x2 + a2 x4 + a4

37. y 2 2 + 2 + 4y + 8y3 38. x x y – 2x2 – xy + x2 + 2x2y2
– y+2 y2 + 4 y4 + 16 + x2 – y2 x2 + y2 x4 – y4

39. 1 + 2 + 4 + 8 40. 1 1 b + 1 2b + 4b3 – 8b7
a+1 a2 + 1 a4 + 1 a8 – 1 + + b2 1 + b4 b8 – 1

41. 1+ b + 2ab – a + 4a3b 42. 1 + m – m n n – m2 n2 + 2m2
a–b a2 + b2 a+b a4 + b4 n + mn – m2 – n2

43. 1 + 1 + 1 – m + 3
m m–1 m+1 m2 – 1 m(m2 – 1)

44. If (a + b + c)–1 = a–1 + b–1 + c–1, show that (a + b + c)–3 = a–3 + b–3 + c–3

Indices
1. Simplify.

a) 2x + 3 – 2x + 2 b) 3x + 2 + 3x + 1 c) 11m – 11m–1
2x + 2 3x + 1 + 3x 5 × 11m–1

d) 6n + 2 + 7 × 6n e) 8n + 2 + 9 × 8n f) a–3b4 × 4 a2b–8
6n + 1 × 8 – 5 × 6n 8n + 1 × 10 – 7 × 8n
2 4 2p–6q–3
g) 3 64x3 ÷ 125y–3 – 2 3
i) (343) 3
j) 6 x5 4 x3 x2 h) (0.0001)4 5 323

k) x –1 x3 x –4 l) 4 512p10q9 ÷

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m) 3 27x12y –3 ÷ 4 16x8y4 n) 4 48x11y9 o) 4 16x8y4
2. Simplify. 3x3y5z –4 3 8x6y3

xa – c a + b . xb – a b +c xc – b c + a q +r ar r +p
xb – c xa – b
a) xc – a . b) (ap . aq)p–q ÷ ar ×
aq ap

11 1

c) 1 y–z 1 y–x z–y d) l +m xl2 × m +n xm2 × n +l xn2
xm2 xn2 xl2
a x–y . a x–z . 1

a x–z

a–b b–c c–a a b c

e) xa2 + b2 × xb2 + bc × xc2 + ac f) ab xb × bc x c × ca xa
x–a 2 b c a
x–ab x–c 2 xa xb
xc

1– b b 1+ a a
a b a–b
a–b×
p2 2p 1
g) (p – y)y – (p – y)y – 1 + (p – y)y –2 h) a –1 b b +1 a
b a a–b
a –b ×

x2 – 1 x× x – 1 y–x 1 a+b 1 a+b
y2 y b a
i) j) a– × b +

1 y× 1 x–y 1 1
x2 x a2 b2
y2 – y+ b2 – b × a2 – a

k) 1 + ax 1 + az – y + 1 + ay 1 + ax – z + 1 + az 1 + ay – x

–y –z –x

l) 1 + 1 + xr – p +1 + 1 + xr – q + 1
xq – p xp – q 1 + xp – r + xq – r

m) x 2 + x 2 – 1 – 1
3 3

x 1 –1 x 1 +1 x 1 +1 x 1 –1
3 3 3 3

3. Solve x+5 x +1
a) 2x – 2x – 2 = 6 b) 3x + 1 – 3x = 54 c) 3 2 = 9 2

d) 2x = 16 e) 5x – 1 × 2x + 2 = 80 f) 33x.9x + 1 = 94x
8x h) 2 × 3x – 1 = 3 × 2x – 1 3

g) 25x 42x+1 = 4x i) 75x – 4 . a4x – 3 = 72x – 3 . ax – 2
2x

j) 32x – 10.3x + 9 = 0 k) 32x – 4.3x + 1 + 27 = 0 l) 4x – 3.2x + 2 + 32 = 0

m) 4x + 128 = 3.2x + 3 n) 5x – 1 + 5 = 5 1 o) 3x + 27 = 12
5x 5 3x

p) 2x + 32 = 12 q) 5x + 5 =6 r) (3x – 1) (3x – 9) = 0
2x 5x

4. a) If a = xq +r . yp , b = xr + P . yq , c = xp + q . yr, show that aq–r br – pcp – q = 1

1 – (ab) – 1 , prove that x3 + 1 + 3x = ab.
3 ab
b) If x = (ab) 3

Vedanta Excel in Mathematics - Book 10 320 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Revision and Practice Time

c) If a = xyp–1 , b = xyq – 1 and c = xyr –1 , prove that aq – r br – p cp – q = 1

d) If 2x =3y = 6 –z, show that 1 + 1 + 1 =0
x y z

e) If m = ax, n = ay and a2 = (my . nx)z , show that xyz = 1

f) If xy = yx, proved that x x =x x – 1.
y y y

xy
g) If 2x = 3y = 6z, prove that z = x + y

h) If 3x = 5y = 75z, show that z = xy
2x + y

i) If a = 2x, b = 2y and aybx = 4, proved that xy = 1

5. a) If abc = 1, prove that: (1 + a + b–1)–1 + (1 + b + c–1)–1 + (1 + c + a–1)–1 = 1

b) If xyz = –1, prove that: (1 – x – y–1)–1 + (1 – y – z–1)–1 + (1 – z – x–1)–1 = 1

c) If p + q + r = 0, prove that: (1 + xp + x–q)–1 + (1 + xq + x–r)–1 + (1 + xr + x–p)–1 = 1

d) If a+b+c = k, show that x2a + x2a + xk – c + x2b + x2b + xk – a + x2c + x2c + xk – b =1
xk – b xk – c xk – a

6. a) 2 + 2– 2 prove that 1 + 2– 1 (ii) 2x(x2 – 3) = 5
3
If x2 – 2 = 23 3, (i) x = 23

b) 2 + 3– 2 prove that 1 – 3– 1 (ii) 3x(x2 + 3) = 8
3
If x2 + 2 = 33 3, (i) x = 33

7. a) Solve: 4x – 3 × 2x + 2 + 32 = 0. Also, use the value of x so obtained to verify the
equation 5x + 53 – x = 30.

b) The area of a window is 4x square meter. The curtain of size 3 m by 2x m is used
in the window. If the area of the curtain is 2 square meters more than the area of
window, find the possible areas of the window.

Surds

1. Simplify. b) 27 – 8 3 + 75 c) 3 27 + 2 12 – 2 3
a) 125 – 45 + 5 e) ( 7 + 5) ( 7 – 5) f) (2 5 + 3 3) (2 5 – 3 3)
d) 3 128 + 3 54 – 3 250

g) ( 2 + 3)2 h) ( 2 + 4 3) ( 12 – 2 3) i) 8 + 18
2
j) 12 + 27 k) 3 20 + 2 125
3 2 45 l) 6 45 – 2 80 – 3 20
2 180
2. Simplify.
b) 3 c) 4 – 15 + 4 1 15
a) 4 7 + 3 2 3+ 2– 5 –
5 2+2 7

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Revision and Practice Time

d) 5 + 3 + 5 – 3 e) 6 21 × 4 3 f) 3 + 2 2 + 3 1 2
5– 3 5+ 3 5 28 – 16 7 + 6 63 +2

g) 5 + 3 – 5 – 3 h) 1+x + 1–x
1+ 1+x 1– 1+x
5– 3 5+3

i) x + x – 1 – x – x – 1 j) x + y + x – y + x + y – x – y
x– x–1 x+ x–1 x+y– x–y x+y+ x–y

3. Solve.

a) 2x + 8 + x = 20 b) x + 1 = 5x – 1

c) 3x2 – 2 + 1 = 2x d) 6x2 + 9x + 4 + 2 = 3x

e) 4x – 3 + 2x + 3 = 6 f) x + 1 + 2x = 7

g) x + 1 + 2x – 5 = 3 h) x + 5 + x + 21 = 6x + 40

i) y–1 – 5 = y +1 j) x –1 = 2 + x +1
y –1 2 x +1 3

k) 5x – 4 = 4 – 5x – 3 l) 5x – 4 = 2 + 5x – 2
5x + 2 2 5x + 2 2

m) 7x – 4 = 6 – 7x + 2 n) x+2 + x+7 = 15
7x + 2 5 x+7

o) x+ x+a = 2a p) 2 x – 9 = 10 + 2 x
x+a x+3 x + 20

q) 7 + 4x + 1 = 6 r) x+1 + x–1 = 10
7 – 4x + 1 x–1 x+1 3

s) x + a = 4 2 + x– a t) y + y – 1 – y = 1
x– a x+ a

4. If a + b + c = 7, solve for x: x – 1 – b – c + x – 1 – c – a + x – 1 – a – b = 3
abc

Simultaneous equations

1. In a dairy, the rate of cow milk is Rs 90 per litre and the rate of buffalo milk is Rs 110 per
litre. If Ajita paid Rs 510 for 5 litres of milk, how many litres of cow milk and buffalo
milk did she purchase?

2. The cost of tickets to enter in the central zoo is Rs 150 for adult and Rs 50 for a child. If
a family paid Rs 700 for 6 tickets altogether. How many tickets were purchased in each
category?

3. Dharmendra’s capital increases by Rs 3,000 when he sells 50 shares of a company-P
and buys 30 shares of company-Q. Similarly, Chhiring’s capital decreases by Rs 4000
when he sells 20 shares of company-P and buys 25 shares of company-Q.

(i) What is the value of each share of company-P?

(ii) What is the value of each share of company-Q?

Vedanta Excel in Mathematics - Book 10 322 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Revision and Practice Time

4. A school has been providing the hostel facility to its students since last year. For this, it
takes a fixed monthly fee and an additional charge as taking food per day in mess. Ram
and Sita have been staying in the hostel since last month. Ram took food for 20 days
and paid Rs 8,000 as hostel charges whereas Sita took the food for 24 days and paid
Rs 8,600 for the last month.
(i) Find the fixed monthly charge of the hostel.
(ii) Find the cost of food per day in the hostel.

5. There are some hens and some cows in Krishna’s agro firm. All the cows and hens are
normal and healthy. If the total number of animals’ heads is 400 and the total number
of legs is 850, find the number of hens and the number of cows.

6. Five years ago, father's age was 4 times his son's age. Now the sum of their ages is 45
years. Find their present ages.

7. Ten years ago, the son's age was twice the daughter's age. Now the son's age is 3 years
more than the daughter's age. Find their present ages.

8. 6 years ago a man's age was six times the age of his son. 4 years hence, thrice his age
will be equal to eight times his son's age. Find their present ages.

9. The ages of two girls are in the ratio of 5 : 7. Eight years ago their ages were in the ratio
of 7 : 13. Find their present ages.

10. 6 years ago, a man's age was 6 times the age of his son. 4 years hence, four times of his
age will be equal to 8 times of his son's age. Find their present ages.

11. The difference of the age of a father and his son is 30 years. After five years, the age of
the father will be twice the age of the son. Find their present ages.

12. 20 years ago, a father was 5 times as old as his son. Now he is 10 years older than two
times the age of his son. Find their present ages.

13. A mother says to her daughter, "5 years ago I was 5 times as old as you were but 10
years hence, I shall be only twice as old as you will be." Find their present ages.

14. The present age of a mother is four times the age of her daughter. If the age of the
daughter after 20 years is equal to the age of the mother before 25 years, find their
present ages.

15. The present age of Kishan is double the age of Radha. If the age of Kishan after 7 years
is equal to the age of Radha after 14 years, find their present ages.

16. Mr. Thapa was four times as old as his son was in 2002 but he was only three times as
old as his son in 2007. What is the year of the son's birth?

17. In 2020, Salim was two times as old as Kedar was. If the ratio of their ages will be 5:3
in 2025, find their years of birth.

18. When the age of Rita was equal to the present age of Shova, she was thrice as old as
Shova was. If the sum of their present ages is 40 years, find their ages.

19. A number of two digits is six times the sum of its digits. If 9 is subtracted from the
number, the digits are reversed. Find the number.

20. A number of two digits is equal to the four times the sum of its digits. If 18 is added to
the number, the digits are reversed. Find the number.

21. A number consists of two digits whose sum is equal to 10. If 36 is subtracted from the
number, the digits are reversed. Find the number.

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 323 Vedanta Excel in Mathematics - Book 10

Revision and Practice Time

22. A number of two digits is 3 more than 7 times the sum of the digits. If the places of the
digits are reversed, the new number so formed is decreased by 27. Find the original
number.

23. A number between 10 to 100 exceeds 4 times the sum of its digits by 9. If the number
is increased by 18, the result is equal to the number formed by interchanging its digits.
Find the number.

24. A two digit number is 3 times the sum of its digits. The sum of the number formed by
reversing its digits and 9 is equal to 3 times the original number. Find the number.

25. The digit at the unit place of a two digit number is four times the digit at tens place. If
the sum of the digits is 10, find the number.

26. A number of two digits is 45 less than the number formed by reversing the digits. If the
digit at ones place exceeds thrice the digit at tens place by 1, find the number.

27. The sum of two times the smaller number and three times the bigger number is 34.
If two times the bigger number is subtracted from the five times the smaller one, the
result is 9. Find the number.

28. Two buses were coming from two villages situated just in the opposite direction. The
average speed of one bus is 8 km/hr more than that of another one and they had started
their journey in the same time. If the distance between the villages is 360 km and they
meet after 4 hours, find their average speed.

29. Koshi bus started its journey from Kathmandu to Dharan at 4 p.m. at the average speed
of 50 km/hr. 1 hour later Makalu bus also started its journey from Kathmandu to the
same destination at the average speed of 60 km/hr. At what time would they meet each
other?

30. In 6 hours Harka walks 6 km more than Dorje walks in 3 hours. In 8 hours Dorje walks
12 km more than Harka walks in 9 hours. Find their speed in km per hour.

31. The area of a rectangular field increases by 11 m2 if its length is increased by 3 m and
breadth is reduced by 1 m. But, the area of the filed decreases by 14 m2 if its length is
reduced by 4 m and breadth is increased by 3 m. Find the length and breadth of the
rectangle.

32. Chameli travels 14 km to her home partly by rickshaw and bus. If she travels 2 km by
rickshaw and remaining distance by bus, she reaches to her home in 30 minutes. On the
other hand, if she travels 4 km by rickshaw and remaining distance by bus, she takes 9
minutes more. Find the speed of the rickshaw and that of the bus.

Quadratic equation

1. When 6 is added to the square of a whole number, the sum is 55. Find the
number.

2. If 4 is subtracted from the square of a positive number, the result is 60. Find the number.
3. If 7 is added to two times the square of a natural number, the sum is 57. Find the

number.
4. When 40 is subtracted from thrice the square of a positive number, the result is 68. Find

the number.
5. If 8 is subtracted from one-third of square of a positive number then the result is 19.

Find the number.
6. If 7 is added to two-third of square of a positive number then the sum is 31. Find the

number.

Vedanta Excel in Mathematics - Book 10 324 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Revision and Practice Time

7. When a positive number is added to its square, the sum becomes 56. Find the number.

8. When a positive number is subtracted from its square, the result is 90. Find the number.

9. The product of two consecutive natural numbers is 56, find the numbers.

10. When 100 is subtracted from five times the square of a natural number, the result is
equal to square of two times the number. Find the number.

11. Find two consecutive even numbers whose product is 80.

12. The product of two consecutive odd numbers is 255, find the numbers.

13. If the difference between the squares of two consecutive even numbers is 44, find the
numbers.

14. If the sum of squares of two consecutive odd numbers is 202, find the numbers.

15. If a natural number is treble the other and their product is 48, find the numbers.

16. A natural number exceeds two times another natural number by 3 and their product is
90, find the numbers.

17. The product of two natural numbers is 300. If one number is three- fourth the other,
find the numbers.

18. Divide 10 into two parts so that their product will be 24.

19. The sum of two numbers is 12 and their product is 35, find the numbers.

20. The difference of two numbers is 3 and their product is 54, find the numbers.

21. If 5 is subtracted from a number, the result is 50 times the reciprocal of the number,
find the number.

22. The difference between two numbers is 2 and the sum of their reciprocal is 5 . Find the
numbers. 12

23. The product of ages of a boy 5 years ago and 5 years hence is 200. Find his present age.

24. Gita is 6 years older than Sita and Rita is 6 years younger than Sita. If the product of
ages of Gita and Rita is 288, find the present age of Sita.

25. The present ages of father and son are 37 years and 8 years respectively. How many
years ago, the product of their ages was 96? Find it.

26. The present ages of elder and younger brothers are 14 years and 9 years respectively.
How many years ago, the product of their ages was 300? Find it.

27. The difference of the ages of two sisters is 5 years and the product of their ages is 84.
Find the present ages of two sisters.

28. A brother is 4 years younger than his sister and the product of their present ages is 77.
Find their present ages.

29. The product of present ages of two brothers is 160 and 4 years ago, elder brother was
twice as old as his younger brother. Find their present ages.

30. The sum of the present ages of elder and younger brother is 28 years and the product of
their ages is 187. Find their present ages.

31. A brother was 5 years old when his sister was born. The product of ages of numerical
value is equal to six times the sum of their ages. Find the present ages of the brother and
sister.

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 325 Vedanta Excel in Mathematics - Book 10

Revision and Practice Time

32. The sum of present ages of father and his son is 50 years. After 10 years, the product of
their ages will be 1000. Find their present ages.

33. The sum of present ages of a mother and her daughter is 45 years. 5 years ago, the
product of their ages was 124. Find their present ages.

34. The present age of mother is equal to the square of the age of her daughter after one
year. If the age of daughter in 10 years hence will be 1 year less than the age of her
mother in 10 years ago, find their ages.

35. The product of present ages of mother and son is 400. If they both live on till the son
becomes as old as mother at present, the sum of their ages will be 110, find their present
ages.

36. The product of present ages of father and son is 1500. When the father’s age was equal
to the present age of the son, the sum of their ages was 40, find their present ages.

37. The sum of digits of a two digit number is 7 and their product is 12. Find the numbers.

38. In a two-digit number, the product of digits is 24. If 18 is subtracted from the number,
the digits are reversed. Find the number.

39. A number of consists of two digits whose product is 21. If 36 is subtracted from the
number, the digits are interchanged. Find the number.

40. In a two-digit number, the product of digits is 20 and if 9 is added to the number, the
digits will be inversed. Find the number.

41. The product of digits in a two digit number is 15. The number formed by interchanging
the place of digits is 18 more than the original number. Find the number.

42. In a positive number of two digits, the ten’s place digit exceeds the unit’s place digit by
5. If the product of digits is 36, find the number.

43. A number between 10 and 100 is equal to four times the sum of the digits. If the product
of digits is 18, find the number.

44. A two digit number is four times the sum and three times the product of its digits. Find
the number.

45. In a number of two digits, the square of the sum of its digits is 81. If 9 is subtracted from
five times the number, the digits are reversed. Find the number.

46. The square of sum of the digits of a two-digit number is 49. If 13 is added to three times
the number, the digits are reversed. Find the number.

47. The breadth of a rectangular field is 3 m less than its length. If the area of the field is 88
sq. m, find its perimeter.

48. A room is 5 m longer than its width. If the area of the floor is 150 sq. m, find the length
and breadth of the room.

49. The perimeter of a rectangular garden is 46 m and it area is 126 sq. m. Find the length
and breadth of the garden.

50. The area of a rectangular ground 1,200 square metres and its perimeter is 140 metres.
By what percentage should the length of the ground be decreased so that it reduces to a
square?

51. The area of a rectangular field 3,000 ft. and its perimeter is 220 ft. By what percentage
should the breadth of the filed be increased so that it becomes a square?

52. A bus travels a journey of 360 km at a uniform speed. If the speed of the bus had been
5 km/hr more, it would have been taken 1 h less for the same journey. Find the speed of
the bus.

Vedanta Excel in Mathematics - Book 10 326 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Revision and Practice Time

53. Due to heavy rain, an aeroplane started late by 1 hour from Kathmandu. The pilot
decided to increase the speed of the aeroplane by 100 km/hr from its usual speed to cover
a journey of 1200 km in the same time. Find the usual speed of the aeroplane.

54. 40 marbles are equally distributed among a certain number of children. If there were 2
children less, each would have received 1 more marble. Find the number of children
and the number of marble received by each child.

55. Rs 1,000 was equally distributed among a certain number of students. If there were 5
students more, each would have received Rs 10 less. Find the number of students and
the amount received by each student.

Area of triangles and quadrilaterals P ST U
1. a) In the given figure, if QT = 12 cm and TA = 5 cm, find the area A

of rectangle PQRS. R
AD
Q
F
b) In the given figure, DE A BC. If BC = 10 cm and DE = 8 cm, find B 10cm C E 8cm
the area of 'ABF.
AB
c) In the given figure, ‘AMB = 90°, BE = 14 cm and the area of
parallelogram ABCD is 42 cm2, find the length of AM. M
DC
E
T

d) In the given figure, PQRS is a parallelogram and QM = TM. If P M S
the area of PQRS is 96 square cm, find the area of 'QRT.

QR

e) In the given figure, PQRS is a rhombus in which PR = 16 cm P Q
and QS = 10.5 cm. Find the area of 'PQT.

TS R

AD

f) In the given figure, ABCD is a square. E is the mid-point of BC
and BD = 24 cm; find the area of 'ABE.

BEC

2. Show that the line segment joining the midpoints of a pair of opposite sides of a
parallelogram divides it into two equal parallelograms.

3. Parallelograms PQRS and QRTU stand on the same base QR and between the same
parallel lines PT and QR. Prove that:

(i) 'PQU # 'SRT (ii) Area of PQRS = Area of QRTU

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 327 Vedanta Excel in Mathematics - Book 10

Revision and Practice Time

4. Rectangle ABCD and parallelogram EBCF stand on the same base BC and between the
same parallel lines AF and BC, prove that:

(i) 'ABE # 'DCF (ii) Area of rectangle ABCD = Area of EBCF

5. Prove that the area of triangles ABC and DBC standing on the same base BC and
between AD//BC area equal.

6. Prove that the parallelograms standing on the equal base and between the same parallel
lines are equal on area.

7. If XY//MN, P, Q and R are the points on XY such that PM//QN, prove that the area of

triangle RMN is half of the area of quadrilateral PMNQ. D QC
O
8. In the given figure, ABCD is a parallelogram in which diagonals AC
and BD intersect at O. A line segment through O meets AB at P and DC PB
1
at Q. Prove that: area of trap. APQD = 2 area of parm ABCD.

A

9. In the given quadrilateral ABCD, M is the mid-point of D C
AC. Prove that: M

area of quad. ABMD = area of quad. DMBC.

AB

10. Prove that the diagonals of a parallelogram divides it into four triangles of equal area.

B

11. In the given figure, ABCD is a quadrilateral. BE is drawn A
parallel to AC and it meets DC produced at E. Prove that: area
of 'ADE = area of quad. ABCD.

DC E

12. In the adjoining figure, PQ // BC. If BX // CA and A Q
CY // BA meet the line PQ produced at X and Y P
respectively, prove that: X Y

area of 'ABX = area of 'ACY. BC

13. ABCD is a parallelogram and X is any point within it. Prove that the sum of ' XAB and
' XCD is equal to half of the parallelogram.

14. ABCD is a parallelogram. X and Y are any points on CD and AD respectively. Prove that
' AXB and ' BYC are equal in area.
SR

15. In the given parallelogram PQRS, SM and QN are perpendicular N
to PR. Prove that SM = QN. M

P Q
P

16. In 'PQR, A and B are the mid-points of the sides PQ and PR A B
respectively. D and C are two points of QR such that AD//BC. D CR
1
Prove that ABCD = 2 ' PQR. Q

Vedanta Excel in Mathematics - Book 10 328 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Revision and Practice Time

P Q
N
17. In the given figure, PQRS is a trapezium where MN is the M R
median of the trapezium and PQ // MN // SR. Prove that
(i) Area of ' PSN = Area of ' QRM Q
N
(ii) ' QRM = 1 trap. PQRS. S
2 R
T
P
M
18. In the given trapezium PQRS, PQ // SR. M and N are the M S
mid-points of diagonals PR and QS respectively. Prove that
' PNR = ' QMS. S
P

19. In the given parallelogram PQRS, M is any point on the Q
side PS. QM and RS are produced to meet at T. Prove that
'QRT = quad. PQST.

20. In the given figure, PQRS and PABC are the parallelograms P A QR
with equal area. Prove that SA // BR. S R

21. In the trapezium ABCD, AB // DC and P is the mid- CB
point of BC. D
1 C
Prove that ' APD = 2 trap. ABCD. P
B
A Q
T
P
S
22. In the adjoining figure, it is given that PQ // RS and Q
area of ' PRS = area of ' QRT. Prove that RQ // ST.
PD
R

23. In the adjoining diagram, ABCD is a parallelogram. Prove that, A
Area of ' APQ = Area of ' PDC.

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 329 BC
Vedanta Excel in Mathematics - Book 10

Revision and Practice Time A

24. In the given ' ABC, D, E, F and G are the mid-points of F E O
BC, AD, BE and CF respectively. B G C
Prove that ' ABC = 8 ' EFG.
P D

25. In the figure, A, B and C are the mid-Points of PR, QA A
BD

and QR respectively. D is any point of AR. Prove that, Q C R
P R
' PQR = 8 ' BCD.

26. In the adjoining figure, PRST is a parallelogram and PT is T S

the median of ' PQR. Prove that, ' PQT = ' STR. Q D

A C

27. In the given quadrilateral ABCD, AO = OC. S
Y
Prove that 2 ' ABD = quad. ABCD. O R

B

P

28. In the figure alongside, PQRS is a parallelogram. X and Y are

any points on QR and RS respectively such that XY // QS. Prove

that, Area of ' PQX = Area of ' PSY. Q X

29. In a parallelogram ABCD, P is any point on the diagonal AC. Prove that :
(i) Area of 'ABP = Area of 'APD (ii) Area of 'BPC = Area of 'CPD

30. In a parallelogram BEST, the diagonal BS is produced to a point L. Prove that 'BTL and
'BLE are equal in area.

31. If AB//DC, BC//AD and a point M on AD is the mid-point of side EC of a triangle BEC.
Prove that the area of triangle BEC is double the area of triangle ADC.

32. In parallelogram PQRS, QR is produced to T such that QR = RT. The line PT cuts RS at
O. Prove that the triangle POQ is twice as large as the triangle SOT.

Construction
1. Construct a quadrilateral ABCD in which AB = BC = 5.5 cm, CD = DA = 4.5 cm and

³A = 600. Also, construct 'ADE equal in area to the quadrilateral ABCD.
2. Construct a 'QRT which is equal in area to the quadrilateral PQRS having PQ = 5 cm,

QR = RS = 5.5 cm, RP = 7 cm and ³PRS = 300.

Vedanta Excel in Mathematics - Book 10 330 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Revision and Practice Time

3. Construct a quadrilateral MNOP in which NO = MN = 4.2 cm, OP = PM = 5.2 cm and
³NOP = 750. Also construct 'PQO which is equal in area to the quadrilateral MNOP.

4. Construct a quadrilateral ABCD in which AB = 4.5 cm, BC = 5.5 cm, CD = 5.7 cm, AD
= 4.9 cm and diagonal BD = 5.9 cm. Also, construct 'DAE whose area is equal to the
given quadrilateral.

5. Construct a quadrilateral PQRS in which PQ = 5 cm, QR = 4.5 cm and RS = PS = PR
= 6 cm. Also, construct 'QRT equal in area to the given quadrilateral PQRS.

6. Construct a triangle ABC having sides a = 6.4 cm, b = 6 cm and c = 5.6 cm. Also,
construct another triangle having one side 7 cm and equal in area to the 'ABC.

7. Construct an equilateral triangle PQR having each side 5.2 cm. Also, construct another
triangle QRT having ³TQR = 750 and equal in area to the 'PQR.

8. Construct an isosceles triangle ABC having base (BC) = 6 cm and altitude (AD)
= 4.5 cm. Also, construct another isosceles triangle BCD having BC = CD and equal in
area to the 'ABC.

9. Construct a triangle ABC having sides a = 6.4 cm, b = 5.8 cm and c = 5.2 cm. and
construct parallelogram having area equal to the triangle ABC and having an angle 600.

10. Construct a parallelogram BIKE having BE = 7.5 cm and equal in area to the triangle in
which AB = 7.1 cm, ³BAC = 600 and AC = 5.7 cm.

11. Construct a triangle ABC in which BC = 6 cm, ³A=900 and ³C =300 then construct a
rectangle having area equal to the given triangle ABC.

12. Construct a parallelogram ABCD in which AB = 6 cm, BC = 4 cm and ³BAD = 450.
Also, construct a triangle APQ having one angle 600 and equal in area with parallelogram
ABCD.

13. Construct a parallelogram ABCD in which AB = 5 cm, AD = 6 cm and diagonal
BD = 6 cm. Also, construct another parallelogram equal in area to the parallelogram
ABCD having that the measurement of one angle 450.

14. Construct a rhombus PQRS in which diagonal PR = 6cm and diagonal QS = 8cm. Also
construct a ' PSA whose area is equal to the area of the rhombus PQRS.

15. Sketch a rectangle ABCD and parallelogram BCEF standing on the same base BC and
between AE and BC. Complete the construction with AB = 4 cm, AE = 6.5 cm and
AF = 1.5 cm.

Circle

1. Find the sizes of unknown angles in the following figures.

a) b) c) d)

A R C
Ox Oyx
O A x 58° O
PQ 54°
x 80° y 20°
C AB
BC
B

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Revision and Practice Time

e) P f) g) D C h) D C
x 130° B
R y A
2x x
Q 50° O S x O
O Q
x y 57° 95°
AB
P
R

i) D C j) D k) B l) A
6x 20° x
A 3x Oy B xR A
O
25°
30° C B O 52° C
43° Dx E
A yC E D
B E
o) A x 50° D
m) C A n) A p) E46° A D
xC
D x By

68° x O O
E
B 33° D BC
B C E

q) r) s) D E t) AF
80° (x+20)°

A C DB

O B A O B xB C (x–20)°
A 40° 130° x F E
CE C DE
x
D v) A x 118° D w) D xE x) E
O 75° 120° 40° D xC
u) B 10° EA
C C
A 124° B
92° AO

x C BB
D

y) z) za) D zb)
x
X Z C P
x
A M x
O
D 50° Y O 150° AB
Ex O 55° B
A 25°
67° B N CD
C ze)
D
zc) E zd) O zf)
30° C A 40° 30° D
x A x B CF
A OB x x 78° E
OB BC
C 30° D A D

Vedanta Excel in Mathematics - Book 10 332 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Revision and Practice Time

2. Find the lengths asked in each of the following figures.

a) b) c) d)

O O A O
6cm B
9 cm M Z 4cm 9cm R6cm
Y 8 cm X C 8 cm D P Q
P 12 cm Q
Find YZ. Find AB.
Find MQ. Find PQ.

3. Find the sizes of known angles in the following figures.

a) b) c) C

C 31°
TO
45°O z B O
xy N x x

60° A Q B 60° Q R
TA 32° R P
P

d) C e) A f) C
x
T O Ox E O
26° 80° E
A P F B y 28° D B
B C 62° x
AD

g) A h) D i)
35°
B O B x B
56° C
C OF
GO

x x 50°
ED
TA N

j) P k) B l) B
E E
Tx
134° O Ax O75° D A 40F° x O

Q F D
C C

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 333 Vedanta Excel in Mathematics - Book 10

Revision and Practice Time

4. a) If the points P, Q and R lie on the circumference of a circle with centre C, prove
that angle QCR is twice the angle QPR.

b) If A, B, C and D are the concyclic points, prove that the angles ACB and ADB
are equal.

c) Prove that the opposite angles of a cyclic quadrilateral are supplementary.

5. a) Verify experimentally that the circumference angle PQR is half of the angle
central angle PSR standing on the same arc PR of a circle with centre S. (Two
circles having radii at least 3 cm are necessary)

b) Explore experimentally the relationship between the opposite angles of a cyclic
quadrilateral ABCD. (Two circles of radii at least 3 cm are necessary)

6. In the given figure prove that PS
X
(i) ' PXR a ' QXS
RQ
(ii) PX = SX
RX QX

AB

7. Given figure is a circle in which ‘ WAY = ‘ XBZ. W Z
Prove that, WZ // XY. X Y
U
8. In the adjoining figure, if PQ // RS, prove that, T
‘ PTR = ‘ QUS.
P Q
9. In the given figure, O is the circumcentre of ' ABC R S
and OD A BC. Prove that ‘ BOD = ‘ BAC.
A

O

B DC

10. In the given figure, AB and CD are two chords of A D
a circle which are intersecting each other at P such O PB
that AP = CP. Prove that AB = CD. C

11. In a circle with centre O, two chords PQ and RS intersect at a point X, prove that
³POR + ³QOR = 2³PXR.

Vedanta Excel in Mathematics - Book 10 334 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Revision and Practice Time

A

12. In the given figure, two chords AB and CD intersect at right

angle at X. Prove that, chords (AC + BD) = chords (AD + BC) C X
D

13. In the adjoining figure, O is the centre of the circle. If B
³OCA = ³ODB, prove that DOAAB. D

C

AOB

A

14. In the given figure, P and Q are the mid-points of AB and AC P E FQ
respectively. Prove that AE = AF.

BC

15. In the adjoining figure, ABCD is a parallelogram. The A EB
inscribed circle cuts AB at E and CD at F. Prove that,
‘ EFD = ‘ ABC. F C
A
D

16. In the given diagram, AE is a diameter. If AD A BC, B DC
prove that ' ABD a ' ACE.
E C
17. In the given figure, APB = CQD. Prove that AC // BD.
A Q
P D

B

18. In the figure given alongside, AB and CD are two parallel A B
chords. Prove that AC = BD . C D

19. In the figure, O is the centre of the circle and the E
chords AB = BC = CD = DE. Prove that AD = BE. D

OC

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 335 AB
Vedanta Excel in Mathematics - Book 10

Revision and Practice Time

20. In the given figure, NPS, MAN and RMS are straight lines. Prove P S
that PQRS is a cyclic quadrilateral. AM

N R
Q
21. In the adjoining figure, PQ = RS. Prove that QT = ST P
and PT = RT. R
Q T
P R
S S
22. In the given figure, PQSR is a cyclic quadrilateral. PR
T
and QS are produced to meet at T. If PQ // RS. prove

that TP = TQ. Q

23. In the adjoining figure, PQRS is a cyclic quadrilateral. P Q
PQ and SR are produced to meet at T. If QT = RT, T
prove that (i) PS // QR (ii) PR = QS.
R
S
Y Z
24. In the given figure, O is the centre of the circle. X
Prove that ‘XOZ = 2 (‘XZY + ‘YXZ)
O

25. In the adjoining figure, AB = CD and BE = AD. E
Prove that ADBE is a parallelogram. AB
DC

26. In the given figure, PS // QT and SR = ST. PS
Prove that ‘QPS + ‘PST = 180°. Q RT

27. In the figure given alongside, O is the centre of the circle, OB

AB is the angular bisector of ‘OAC. Prove that, D
(i) OB // AC ii) ' ACD ~ ' OBD AC

Vedanta Excel in Mathematics - Book 10 336 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Revision and Practice Time

P

28. In the given figure, T is a point on the side QS of ' PQS such that Q

PQ = PT. If PT is produced to meet the circumference of the circle T
at R, prove that ST = SR.
S
R

ACB

29. In the figure, two circles with centres P and Q intersect at C P Q
and D. Prove that ACB is a straight line.

D

30. In the given figure, O is the centre of the circle. If AB // CD, prove A E B
that ‘AOC = 2 ‘BED. O D

C

31. In the adjoining figure, ABCD is a cyclic quadrilateral in which A B
AB // DC. Prove that (i) AD = BC (ii) AC = BD. C

D

32. In the adjoining figure, O is the centre of the circle. O
If PQ A OA, PR A OB and QC = RB, prove that AP = PB. QR

CB
P

33. In the given figure, two circles intersect at the points A and D. If AB X
ABCD and AXYD are cyclic quadrilaterals, prove that BC // XY. DC Y

34. In the given figure, APABC, BRAAC, CQAAB and O is the A
orthocenter of 'ABC. Prove that ³OPQ and ³OPR are Q OR
equal. BPC

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 337 Vedanta Excel in Mathematics - Book 10

Revision and Practice Time

35. In the figure, if D, E and F are the mid-points of sides A E
AB, AC and BC respectively and AG A BC. Prove that D F
DEFG is a cyclic quadrilateral.

BG C
T
36. In the given figure; PQ = RT and SQ the bisector of ³PQR. P S
Prove that SQT is an isosceles triangle. Q R

37. a) If AC and BD are two equal chords on the opposite sides of diameter AB of a circle
with centre O, prove that AC // BD.

b) Two equal circles intersect each other at A and B. If a straight line PQ is drawn
through A to touch the circumference of one circle at P and the other at Q, prove that
BP = BQ.

c) Two circles intersect at A and B. If AC and AD are the respective diameters of the
circles, prove that C, B, D are collinear.

d) Prove that any cyclic parallelogram is a rectangle.

e) If two sides of a cyclic quadrilateral are parallel, prove that the remaining two sides
are equal and the diagonals are equal.

f) If two non parallel sides of a trapezium are equal, show that it is cyclic.

g) Prove that the quadrilateral formed by angle bisectors of a cyclic quadrilateral ABCD
is also cyclic,

h) ABC is an isosceles triangle in which AB = AC. If D and E are the mid-points of AB
and AC respectively, prove that the points B, C, D and E are concyclic.

i) D and E are any two points on equal sides AB and AC of an isosceles triangle ABC
such that AD = AE. Prove that B, C, D, E are concyclic.

j) In an isosceles triangle PQR, PQ = PR. If the bisectors of ‘Q and ‘R meet PR at S
and PQ at T, prove that the points Q, R, S, T are concyclic.

k) Points P, Q, R and S are concyclic such that arc PQ = arc SR. If the chords PR and
QS are intersecting at a point M, prove that:

(i) area of 'PQM = area of 'SMR (ii) chord PR = chord QS

l) PQRS is a cyclic quadrilateral. If the bisectors of ‘QPS and ‘QRS meet the circle
at points A and B respectively, prove that AB is the diameter of the circle.

Vedanta Excel in Mathematics - Book 10 338 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Revision and Practice Time

Trigonometry

1. a) A tower on the bank of a river is of height 40 m high and the angle of elevation of
the top of the tower from the opposite bank is 300. Find the width of the river.

b) What will be the length of shadow of a tree 15 m high on the ground when the sun’s
altitude is 450? Find.

c) A poacher is targeting to a dove sitting on top of the 28 ft. tall pole which is situated
in front of him. If the poacher fixes his catapult at the angle of 600 and does not miss
his target, how far is the poacher from the foot of the pole?

d) A pillar of height 60 m height was fixed at the centre of circular meadow. The angle
of elevation of the top of the pillar was found to be 300 when observed from a point
of the circumference of the meadow. Find the radius of the meadow.

2. a) The angle of elevation of the top of a tower from a point on the ground, which is 30
m away from the foot of the tower, is 300. Find the height of the tower.

b) A tree casts a shadow of length 8 3 m on the ground when sun’s elevation is 600.
Find the height of the tree.

3. a) The angle of elevation of the top of a tree from the roof of the house is 300. If the
heights of the house and the tree are 6m and 18m respectively, find the distance
between the house and the tree.

b) A man 5 ft. tall observes the top of a temple and finds the angle elevation 600. If the
height of the temple is 47 ft., find the distance between the man and the temple.

c) A 1.5 m tall woman is standing in front of 41.5 m high tree. When observing the top
of the tree, an angle of elevation of 450 is formed with the eyes. Find the distance
between the tree and the woman.

d) From the roof of the house 6 m high, the angle of elevation of the top of tower 66 m
high is observed to be 600. Find the distance between the house and tower.

e) An observer finds the angle of elevation of the top of a pole to be 600. If the height
of the pole and the observer are 25 3 m and 3 m respectively, find the distance
between the observer and the pole.

f) At the centre of a circular pond, there is a pole of 11.62 m height above the surface of
the water. From the point on the edge of the pond, a man of 1.62 m height observed
the angle of elevation of the top of the pole and found to be 300. Find the diameter
of the pond.

4. a) A man 1.5 m tall observes the angle of elevation of the top of an electric pole and
finds to be 300. If the distance between the man and the pole is 12 m, find the height
of the pole.

b) From the roof of a house 6 m tall, the angle of elevation of a tower was observed and
found to be 300. If the distance between the house and tower was 14 3 m, find the
height of the tower.

c) A man of 2 meter height observes the angle of elevation of the top of the tree and
finds to be 600. If the distance between the man and the tree is 45m, find the height
of the tree.

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 339 Vedanta Excel in Mathematics - Book 10

Revision and Practice Time

d) A man observes the top of a pole of 51 m height, situated in front of him and finds
the angle of elevation to be 300. If the distance between man and the pole is 86 m,
find the height of the man.

e) A woman observes a bird sitting on the top of a tree in front her and finds the angle
of elevation to be 600. If the distance between woman and the tree is 30m and the
height of tree is 53.76 m, find the height of the woman.

f) The angle elevation of the top of a tower observed from the roof of a house situated
on the same horizontal plain is found to be 450. If the height of the tower is 60 m
and the house is 40 m away from the tower, find the height of the house.

g) From a point on the edge of a circular pond, a person observes the angle of elevation
of the top of a pole standing at the centre of the pond and finds to be 600. If the
diameter of the pond is 20 m and the height of the pole above the surface of water
is 19 m, find the height of the person.

5. a) The angle of depression of a car parked on the road from the top of a 150 m high
tower is 300. Find the distance of the car from the foot of the tower.

b) From the top of a temple 20 m high, the angle of depression of a pigeon sitting
on the ground is observed and found to be 600. How far is the pigeon from the
basement of the temple?

c) A dog sitting on the ground is 60m away from the house. If the angle of depression
of the dog from the roof of the house is found to be 300, find the height of the house.

d) A pilot of an aeroplane finds the angle of depression to the top of radar tower to be
300. At this time, if the horizontal distance of the aeroplane from the radar tower is
500 3 m, find the height of the plane.

6. a) From the top of a tower 40 m high, the angle of depression of the top of a temple
10 m high on the same level of ground was observed and found to be 300. Find the
distance between the tower and the temple.

b) The heights of a house and a tree are 20 metre and 5 metre respectively. If a man
observes the top of the tree from the roof of the house and finds the angle of
depression to be 60°, find the distance between the house and the tree.

c) From the top of a cliff 51 m high, the angle of depression of the top of a pole of
height 15 m, situated in front of the cliff, was observed and found to be 450. Find
the distance between the cliff and the pole.

7. a) The angle of depression of the top of a tree as observed from the roof of the house
30 ft. high is found to be 30°. If the distance between the house and tree is 10 3 ft.,
find the height of the tree.

b) The horizontal between two towers of different heights is 50 m. The angle of
depression of the top of the first tower as seen from the top of the second tower is
300. If the height of the second tower is 45 m, find the height of the first tower.

c) From the top of a tower, the angle of depression of the roof of the house 10 m high
and 40 m away from the tower was observed and found to be 300. Find the height
of the tower.

Vedanta Excel in Mathematics - Book 10 340 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Revision and Practice Time

d) From the roof of a house, the angle of depression of the top of the tree 20 ft. high
was found to be 600. If the distance between the house and the tree is 10 3 ft., find
the height of the house.

8. a) A wire is tied at the top of an electric pole which is 12 m high. The wire is stretched
and its other end is fastened on the ground. If the wire makes and angle of 300 with
the ground, find the length of the wire from the ground to the top of the pole.

b) A ladder, leaning against the wall of height 10 m, touches the top of the wall and
makes an angle of 600 with the ground. Find the length of the ladder.

9. a) A boy who is 1.5 m tall is flying a kite. When the length of string of the kite is 200
m and it makes and angle of 300 with the horizon, find the height of the kite from
the ground.

b) The thread of a kite makes an angle of 600 with the horizon when a man of height
2 m flying a kite. If the length of thread is 100 3 m, what is the height of the kite
from the ground?

c) A girl of height 1.4 m is flying a kite from the roof of a house 33 m high. If the length
of the string of the kite is 80 2 m and makes an angle of 450 with the horizon, find
the height of the kite from the ground.

d) On the roof of a house 10 m high, a 1.2 m tall girl is flying a kite and the kite is at a
height of 128.2 m above the ground. If the string of the kite makes an angle of 30 0
with the horizon, find the length of the string.

10. a) A tree 24 m high is broken by the wind so that its top touches the ground and makes
an angle of 300 with the ground. Find the length of broken part of the tree.

b) A tree of 40 m high is broken by the wind so that its top touches the ground and
makes an angle of 600 with the ground. Find the length of broken part of the tree.

c) If the top of a tree broken by the wind makes an angle of 450 with the ground at a
distance of 10 2m from the foot of the tree, find the height of the tree before it was
broken.

d) A tree broken due to thunderstone forms a right angled triangle with the ground.
The broken part of the tree makes an angle of 600 with ground. If the top of the tree
is 15 m far from its foot, how tall was the tree before it was broken?

11. a) A man observes the top of tower of 50 3 m height from 150 m far from the foot of
the tower. Find the angle of elevation of the top of the tower.

b) Find the angle of elevation of the sun when the height of a tree and length of its
shadow are 10 m and 10 m respectively.

c) A man of height 5 ft. tall observes a bird sitting on the top of a tree which is situated
in front of him. If the height of the tree is 55 ft. and the distance between the man
and the tree is 50 ft., find the angel of elevation.

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 341 Vedanta Excel in Mathematics - Book 10

Revision and Practice Time

d) The diameter of a circular pond is 48 m and a pillar of height 30 3 m is fixed at
the centre of the pond. If the length of the pillar inside the water surface is 5 3 m,
what will be the angle elevation when a person of 3 m tall observes the top of the
pillar from the bank of the pond?

12. a) A man observes the top of a tree of height 7 m from the roof of a house 22 m high.
b) If the distance between the tree and the house is 15m, find the angle of depression
made by the man.
13. a)
b) A boy, on the top of 30 m high view tower, observes a 2 m tall girl standing on ground.
If the girl is 48.49 m far from the foot of the tower, find the angle of depression of
14. a) the girl from the eyes of the boy.
b)
c) A man is 1.6 m tall and the length of his shadow is 80 cm. Find the length of
shadow of a building 35 m tall at the same time of the day.
15. a)
b) The length of shadow of a house 30 m high is 30 3m at 3:00 p.m., find the length
of shadow of the tree 20 m tall at the same time.
16. a)
The shadow of a tree on the ground is found to be 20 m longer when the sun’s
altitude is 450 than it is 600. Find the height of the tree.

Find the height of a house if the angles of elevation of its top changes from 300 to
600 as the observer advances 50 3m towards its base.

From an aeroplane flying vertically over a straight road, the angles of depression of
two consecutive kilometer stones on the same sides are 300 and 450. At what height
is the aeroplane from the ground?

A house is 28 m high. The angle of elevation of the top of a tower just in front of it
from the top of the house is 300 and from the foot of the house is 600. Find the height
of the tower and the distance between the house and tower.

From the top of a building 40 m high, the angle of elevation and angle of depression

of the top and bottom of a tower are observed to be 450 and 600 respectively. Find

the height of the tower. A

In the given 'ABC; ³BAC = 1030, ³ACB = 320, 4 2 cm 103° C
AB = 4 2 cm and BC = 9 cm. Find the area of the 'ABC. 32°

B 9 cm

b) Find the area of 'ABC in which a = 14 cm, b = 10 3 cm and ³ACB = 600.

P

c) From the information given in the figure, find the area of 4x16 cm
'PQR. 2x 3x
Q 17 cm
R

d) Find the area of 'MNP in which MN = 16 cm, ³NMP = 750 and ³MNP = 300.

Vedanta Excel in Mathematics - Book 10 342 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Revision and Practice Time

e) In the given 'ABC is 90 sq. cm. If ³BAC = 700, ³ABC = 500 A
70°
and BC = 12 3 cm, find the length of AC. B C
50°

f) In 'DEF, DE = 6 2 , ³DEF= 450 and the area is 24 cm2, find the measurement of EF.

P

g) In the given figure; PQ = 8cm, QR = 14 cm and the area is 12 cm 8 cm 14 cm R
28 3 cm2, find the size of ³PQR.

Q

h) In 'ABC, AB = 8 cm and BC = 12 cm. If the area of ABC is 24 2 cm2, find the
measure of ³ABC.

A D

17. a) In the adjoining figure, ABCD is a rhombus. If AB = 12
cm and ‘ ABC = 60°, find the area of the rhombus.

60° C
B

A D

b) The area of the given parallelogram is 96 cm2. If
‘ ABC = 60° and BC = 12 cm, find the value of AB.

60° C
B 12 cm

c) The adjoining figure is a parallelogram ABCD of area A B
180 sq. cm. If AD = 18 cm and ‘ ADC = 30°, find the
length of DC. 18 cm C
Q
30°
D 30°

d) In the given figure, PQRS is a rhombus. If the area of P R
the rhombus is 16 cm2 and ‘ PQR = 30°, find the S
length of SR.

e) The adjoining figure is a rhombus ABCD whose area is A D
C
128 3 sq.cm. If ‘ABC = 60°, find the length of a side of 60°
the rhombus.

B

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 343 Vedanta Excel in Mathematics - Book 10

Revision and Practice Time

A D

f) The area of the given parallelogram ABCD is 48 sq.cm. 8 cm
If AB = 8 cm, BC = 12 cm, find the measure of ‘ ABC.

B 12 cm C

D A

g) In the adjoining figure, ABCD is a rhombus whose area is
18 3 cm2. If BC = 6 cm, calculate the value of ‘BAD.

C 6 cm B

A

18. a) Find the area of the given quadrilateral ABCD in 30° D
which AB = BC, AD = CD = 6 cm, BD = 10 cm and B

‘ BDC = 30°.

C

b) Find the area of the given quadrilateral ABCD in which A D
‘ DBC = 60°, AB = AD, BC = CD = 8 cm and AC = 12 cm. 60° O

B

c) In the given figure, ' OQP is an equilateral C
triangle in which OQ = 6 cm, MNOQ is a ON
rhombus. Find the area of the figure.

PQ M

d) In the given figure, ABCD is a trapezium. A B
If CD = 2AB, find the area of the trapezium. 10 cm

e) From the information given in the figure alongside, 30° C
find the area of the shaded region. D 12 cm

A D
30°
5 3 cm
60° E 6 cm
B
8 cm C

Vedanta Excel in Mathematics - Book 10 344 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Revision and Practice Time

f) In the given figure, BCDE is a parallelogram and E is A FD
the mid-point of AB. If AE = 3 cm, EF = 4 cm and E
‘ B = 60°, find the area of parallelogram and ' ABC. C
D
60°
B

g) In the given figure, area of the quadrilateral ABCD is A
24 cm2. Find the length of AD.
4 3 cm
4 cm
60°
B 6 cm C

19. a) In the given figure PQ = 12 cm, QR = 16 3 cm, P
RS = 8cm, ' PQR = 60° and ‘PSR = 30°. If
12 cm
' PQR = 4 ' PSR, find the length of PS. Q 60°

16 3 cm R 30° S
8 cm

b) In the given figure, ABCD is a quadrilateral. AC A

is a bisector of ‘ BCD. If BC = 16 cm, AC = 10 cm, 10 cm D
‘ACD = 30° and ' ABC = 2 ' ACD, find the length of 30°
CD.
B 16 cm C

c) In the given figure, AE = CD = 4 cm, EC = 6 cm, A 4 cm E

and BD = 8 cm. If the area of quadrilateral ABDE is 6 cm

24 cm2, find the value of ‘C and the area of triangle

CDE. B 8 cm D 4 cm C

Statistics

1. Compute the mean of the following frequency distribution.

a) Class interval 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60
Frequency 7 5 6 12 8 2

b) Wages (in Rs) 40 – 50 50 – 60 60 – 70 70 – 80 80 – 90 90 – 100

No. of workers 3 6 12 7 8 4

c) Marks 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60 60 – 70

No. of students 2 5 7 6 3 2

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Revision and Practice Time

d) Marks 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60 60 – 70

No. of students 3 12 20 17 23 5

2. Find the arithmetic mean by using 'deviation' method of the following data.

a) Marks 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50

No. of students 3 8 12 7 2

b) Wages (Rs) 30 – 40 40 – 50 50 – 60 60 – 70 70 – 80

No. of workers 50 54 85 45 50

3. Find the arithmetic mean by using 'step deviation' method of the following data.

a) Age (in years) 18 – 24 24 – 30 30 – 36 36 – 42 42 – 48 48 – 54

No. of workers 6 8 12 8 4 2

b) Weekly 200–300 300–400 400–500 500–600 600–700 700– 800
earning (Rs)
No. of 40 20 15 25 10 5
workers

4. a) Construct a frequency table of class interval of 10 from the given data and find the
mean.
22, 30, 58, 50, 38, 16, 40, 27, 35, 45,
56, 48, 42, 37, 41, 28, 18, 44, 36, 49,
62, 37, 29, 39, 32, 20, 43, 52, 40, 55

b) Construct a frequency table of class interval of 5 from the given data and compute
the mean.
24, 42, 65, 50, 40, 27, 18, 7, 30, 60, 37, 28, 20, 10,
15, 22, 41, 54, 36, 35, 16, 51, 17, 32, 12, 21, 39, 58,
34, 53, 38, 33, 16, 9, 23, 31, 40, 50, 10, 23, 41, 13

5. a) If the mean of the following table is 34, find the value of x.

Marks obtained 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60

No. of students 5 15 20 x 20 10

b) Find the missing frequency if the mean of the following distribution is 35.5.

x 5 – 15 15 – 25 25 – 35 35 – 45 45 – 35 55 – 65
f 4 6 10 k 6 5

c) The arithmetic average of the following data is 41.5, find the value of p.

Marks obtained 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60 60 – 70

No. of students 3 4 p 15 3 5

Vedanta Excel in Mathematics - Book 10 346 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur

Revision and Practice Time

d) The mean of the given data is 37 and N = 30, find the missing frequencies.

C.I. 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60
f 26––3

6. Find the median of the data given in each of the following tables.

a) Wages (Rs) 30 – 40 40 – 50 50 – 60 60 – 70 70 – 80

No. of workers 50 54 85 45 30

b) Marks obtained 40 – 50 30 – 40 20 – 30 10 – 20 0 – 10

No. of students 2 7 12 9 1

c) Marks 35 – 45 45 – 55 55 – 65 65 – 75 75 – 85

No. of students 7 8 10 9 6

d) Wages (Rs) 50 – 60 50 – 70 50 – 80 50 – 90 50 – 100
No. of workers 2 5 11 16 20

e) Ages 20–25 20–30 20–35 20–40 20–45 20–50 20–55 20–60
No. of 50 120 220 400 550 670 740 800
persons

f) Wages (Rs) 100–110 100–120 100–130 100–140 100–150 100–160

No. of works 5 11 14 18 25 30

7. a) Construct a cumulative frequency table of class interval 10 and compute the
median.
25, 36, 44, 15, 52, 33, 68, 37, 50, 29, 34, 46,
16, 43, 24, 56, 70, 32, 22, 40, 58, 48, 27, 43, 36

b) Construct a cumulative frequency table of class interval 5 and find the median.
21, 35, 48, 27, 40, 32, 47, 38, 31, 50, 39,
36, 44, 28, 33, 30, 48, 37, 26, 34, 46, 24,
49, 31, 29, 38, 43, 32, 46, 40

8. a) The median of the following data is 35. Find the missing frequency.

C.I. 20 – 25 25 – 30 30 – 35 35 – 40 40 – 45 45 – 50
f 258x45

b) If the median of the following distribution is 32, find the value of k.

Marks obtained 5 – 15 15 – 25 25 – 35 35 – 45 45 – 55 55 – 65
No. of students 5 8k971

c) The median of the following data is 50. If N = 100, find the missing frequencies.

C.I. 0 – 20 20 – 40 40 – 60 60 – 80 80 – 100
f 14 – 26 – 16

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Revision and Practice Time

9. Find the first or lower quartile (Q1) from the data given below.

a) Marks obtained 0 – 10 10 – 20 20–30 30–40 40–50 50–60
10 4 2
No. of students 6 8 12

b) Age (in years) 10 – 20 20 – 30 30 – 40 40 – 50 50–60
No. of people 6 9 10 8 7

c) Class interval 20–30 30–40 40 – 50 50 – 60 60–70 70 – 80 80 – 90

Frequency 5 12 18 10 6 3 2

d) Marks obtained 0 – 6 6 – 12 12–18 18–24 24–30 30–36
6 5 8 7 9
No. of students 9

e) Marks obtained 0 – 20 0 – 40 0 – 60 0 – 80 0 – 100
44 66 78 88
No. of students 21

f) Wages (in Rs) 100 – 125 100 – 150 100 – 175 100 – 200 100 – 225
No. of workers 3 7 13 18 20

10. Find the third or upper quartile (Q3) from the following data.

a) Marks obtained 30 – 40 40 – 50 50 – 60 60 – 70 70 – 80 80 – 90
7 5
No. of students 5 8 10 15

b) Class interval 50–60 60–70 70–80 80–90 90–100 100– 110 110–120

Frequency 8 10 16 14 10 5 2

c) Weight (in kg) 5 – 10 10 – 15 15 – 20 20 – 25 25 – 30
Numbers 7 8 10 9 6

d) Wages (in Rs) 150–200 200–250 250–300 300–350 350–400 400–450

No. of workers 5 7 8 10 7 3

e) Marks obtained 10–20 10–30 10–40 10–50 10–60 10–70 10–80
23 26
No. of students 3 8 12 17 21

f) Age (in years) 0 – 10 0 – 20 0 – 30 0 – 40 0 – 50 0 – 60
No. of people 8 18 30 50 60 64

11. a) The first quartile of the following data is 16. Find the value of p.

Marks obtained 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50

No. of students 2 5 x 4 2

Vedanta Excel in Mathematics - Book 10 348 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur