Probability
Example 11: The number of match–sticks in each of 20 boxes were counted. The results
are shown in the table given below:
No. of match–sticks 39 40 41 42
Frequency 5843
If one of these boxes is selected at random, what is the probability that
(i) it contains 40 sticks? (ii) it contains more than 40 sticks?
Solution:
(i) Here, n(S) = 20 (ii) Again, n(E) = 4 + 3 = 7
n(E) = 8 ? P(E) = n(E) = 7 .
n(S) 20
P(E) = n(E) = 8 = 2
? n(S) 20 5
Example 12: If the probability of germinating a seed of a pea plant is 0.85, how many
seeds out of 1000 will germinate?
Solution: Now, P(E) = n(E)
Here, n(S) = 1000 n(S)
P(E) = 0.85 or, 0.85 = n(E)
n(E) = ? 1000
n(E) = 850
Hence, the required number of geminated seeds is 850.
Example 13: Bulbs are packed in cartons, each containing 60 bulbs. 500 cartons were
examined for defective bulbs and the results are given in following table.
No. of defective bulbs 01234 5 More than 5
No. of cartons 280 120 45 31 16 62
If one carton is selected at random, what is the probability that:
(i) it has no defective bulb?
(ii) it has defective bulbs less than 3?
(iii) it has defective bulbs more than 4?
Solution:
Here, total number of cartons, n(S) = 500
(i) The number of cartons that have no defective bulbs = n(E1) = 280
n(E1) 280
? P(E1) = n(S) = 500 = 0.56
(ii) The number of cartons that have less than 3 defective bulbs
n(E2) = 280 + 120 + 45 = 445
n(E2) 445
? P(E2) = n(S) = 500 = 0.89
(iii) The number of cartons that have more than 4 defective bulbs = n(E3) = 6 + 2 = 8
n(E3) 8
? P(E3) = n(S) = 500 = 0.016
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Probability
EXERCISE 18.1
General section
1. a) Define probability? What is the probability of an event if it is certain? If the event is
impossible, what is its probability?
b) If the number of favourable outcomes and possible outcomes are n(E) and n(S)
respectively, find the probability of the event ‘E’.
c) What do you mean by sample space in probability? Write the sample space of the
following random experiments.
(i) tossing of two coins simultaneously (ii) tossing of three coins simultaneously
(iii) rolling a die (iv) rolling two dice simultaneously
d) What do you mean by mutually exclusive events? Write down with examples.
e) Define independent and dependent events with examples.
f) What do you mean by equally likely event? Write with examples.
g) Define empirical probability. In what way is it different from theoretical probability?
2. a) What is the probability of getting 4 when a die is rolled once?
b) A bag contains 5 identical balls of green, yellow, blue, red and purple. If a ball is
drawn randomly from the bag, what is the probability of getting blue ball?
c) A marble is drawn from a box containing 6 blue and 9 yellow marbles. What is the
probability of getting blue ball?
d) There are 15 black, 5 green, 10 red and 10 yellow balls in a bag. If a ball is drawn
randomly, find the probability that the ball is green.
e) What will be the probability of not getting 6 when a die is rolled once?
f) When a die is thrown, find the probability that the face turned up may be odd
number only.
g) A card is drawn from a well shuffled pack of 52 cards. What is the probability that
the card drawn will be a king?
h) When a card is drawn from a well shuffled pack of 52 cards, find the probability that
the card will be a red queen.
i) If a die is thrown once, what is the probability of getting a number multiple of 3?
j) In a class of 40 students, 3 boys and 5 girls wear spectacles. If a teacher called one of
the students randomly in the office, find the probability that this student is wearing
the spectacles.
k) In a class of 45 students, there are 20 girls and the rest are boys. Find the probability
of choosing randomly a boy as the monitor.
l) A number card numbered from 1 to 30 is drawn randomly. Find the probability of
getting the card having a prime number.
m) What is the probability of giving birth to a child by a pregnant woman on Monday
only?
n) A box contains 3 red, 5 black and 7 white balls. If a ball is drawn at random, find
the probability of not getting a black ball.
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Probability
3. a) Find the probability of touching the following letters of the word ‘PROBABILITY’ by
closing the eyes.
(i) touching ‘P’ (ii) touching ‘B’ (iii) touching ‘I’
(iv) not touching ‘R’ (v) not touching ‘A’ (vi) not touching ‘B’
b) There are 40 students in a class with roll numbers from 1 to 40. The roll number of
Bhurashi is 18. If a teacher calls only one student with roll number exactly divisible
by 3 to do a problem on blackboard, what is the probability that Bhurashi will be
selected? Also find the probability that she will not be selected.
c) If a card is drawn at random from a deck of 52 cards, what is the probability that the
card
(i) is an ace? (ii) is an ace of spade? (iii) is a black ace?
d) In the adjoining spinner, find the probability of the pointer to stop on
(i) the number 7
(ii) the sectors of odd numbers only
(iii) the sectors of numbers exactly divisible by 3
(iv) the sectors of numbers greater than 5
(v) the sectors of numbers except 7 and 8
(vi) the sectors of numbers whose sum is 10
Creative section
4. a) Find the probability that a number chosen at random from the integers between 5
and 16 inclusive is a multiple of 3 or a multiple of 2.
b) Find the probability that a number chosen at random from the integers between 10
and 20 inclusive is a multiple of 5 or a multiple of 2.
c) From a pack of 52 cards a card is drawn at random. Find the probability of getting
this card a spade or a diamond.
d) When a card is drawn from a well shuffled pack of 52 cards, find the probability of
getting the card a club or a jack.
e) A number card numbered from 1 to 30 is drawn randomly. What is the probability
of getting the card having the number which is the multiple of 5 or 6?
f) From a well shuffled pack of 52 cards, a card is drawn at random. Find the probability
of getting this card a red or non-faced one.
g) A card is drawn at random from a well shuffled pack of 52 cards. Find the probability
that the card will be a black or face card.
h) One card is drawn at random from the number cards numbered from 10 to 21. Find
the probability that the card may be a prime or even numbered card.
i) A number card is drawn at random from the group of number cards numbered from
1 to 20. Find the probability of getting a square or cube numbered card.
5. a) Out of 1000 newly born babies, 562 are girls. What will be the empirical probability
that the newly born baby is a girl?
b) In a survey of 100000 people who are chain–smokers, 100 people were found
suffering from lungs–cancer. What is the probability of the people suffering from
cancer?
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Probability
c) If the probability of germinating a seed of a flower is 0.87, how many seeds out of
1000 will germinate?
d) A dice is thrown 300 times and the record of outcomes is given in the table.
Outcomes 123456
Frequency 45 36 50 55 60 54
Calculate the empirical probability of getting the numbers
(i) less than 3 and (ii) greater than 4.
e) Glass tumblers are packed in cartons, each containing 12 tumblers. 200 cartons
were examined for broken glasses and the results are given in the table below:
No. of broken glasses 0 1 2 3 4 More than 4
Frequency 164 20 9 4 2 1
If one cartoon is selected at random, what is the probability that:
(i) it has no broken glass?
(ii) it has broken glasses less than 3?
(iii) it has broken glasses more than 1?
(iv) it has broken glasses more than 1 and less than 4.
Project work
6. a) Make a few number of peer groups of students in your class. Each group will take a
coin. One student will throw the coin and another student will record the outcome
as head (H) or tail (T) in each peer group.
(i) Throw the coin 10 times and find the empirical probability of getting H or T.
(ii) Throw the coin 20 times and find the empirical probability of getting H or T.
(iii) Throw the coin 50 times and find the empirical probability of getting H or T.
b) Now, compare the empirical probabilities obtained by each group and discuss in the
class.
7. Make groups of 5 friends. Each friend throws a basketball into the basket 10 times and
others keep the record that the ball goes inside or outside the basket.
a) Find the empirical probability of throwing the ball inside the basket by each friend.
b) Compare the probabilities and congratulate the best players.
Objective Questions
1. Which of the following is not an example of random experiment?
(A) Tossing a coin (B) Rolling a die
(C) Drawing a card from well shuffled deck of 52 cards
(D) Throwing a stone from the roof of a house
2. If the number of favourable cases and exhaustive cases of an experiment are n (E) and
n (S), the probability of event E is
(A) n(E) (B) n(S) (C) n(E) (D) n(S) – n(E)
n(S) n(E) n(S) + n(E) n(S)
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Probability
3. The probability of sure event is
(A) 0 (B) 0.5 (C) 1 (D) None of these
4. The probability scale of any event E is
(A) 0 < P (E) < 1 (B) 1 < P (E) < 0 (C) 0 ≤ P (E) ≤ 1 (D) 1 ≤ P (E) ≤ 0
5. For any event E, P ( E ) is equal to
(A) - P (E) (B) 1 – P (E) (C) P (E) – 1 (D) P (E)
6. In a class of 40 students, 3 boys and 5 girls wear spectacles. If the principal called one of
the students at random in the office, the probability of this student wearing spectacles is
(A) 3 (B) 1 (C) 4 (D) 1
40 8 5 5
7. A bag contains 20 identical balls out of which 5 are red, 7 are white and the rest are
green. If a ball is drawn at random, the probability of getting a green ball is
(A) 1 (B) 7 (C) 3 (D) 2
4 20 5 5
8. In a survey of 468 children aged 2 – 3 years, it was found that 117 liked to eat potato chips.
If a child is selected at random, the probability that he/she likes to eat potato chips is
(A) 0.25 (B) 0.50 (C) 0.75 (D) 0.80
9. A coin is flipped to decide which team starts the game. What is the probability that your
team will start the game?
A) 0 (B) 1 (C) 0.5 (D) 1
10. A card is drawn from a well shuffled deck of 52 cards. What is the probability that the
card drawn will be a king?
(A) 1 (B) 1 (C) 3 (D) 12
52 13 13 13
11. When a coin is tossed 50 times, head occurs 20 times. What is the probability of getting
tail?
(A) 0.4 (B) 0.6 (C) 0.8 (D) 0.9
12. A box contains the number cards numbered from1 to 20. If a card is drawn at random,
what is the probability of the getting card with multiple of 5?
(A) 0.2 (B) 0.3 (C) 0.4 (D) 0.5 utd32`q H7
13. If the probability of germinating a seed of a pea plant is 0.85, how many seeds out of 1000
will germinate?
(A) 580 (B) 508 (C) 850 (D) 805
18. In a medical check-up of 1000 of pregnant women, it is estimated that the probability of
giving birth of twins is 0.005, how many pregnant women will have twins?
(A) 5 (B) 50 (C) 500 (D) None
14. A die is rolled once. What is the probability that the digit turned off is a number less
than 3?
(A) 1 (B) 1 (C) 1 (D) 5
3 2 6 6
15. What is the probability that the pointer to stop in the sector of an odd
number?
1 2 3 4
(A) 7 (B) 7 (C) 7 (D) 7
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Revision and Practice Time
Revision and Practice Time
Dear students let's revise the overall contents by practicing the additional problems given
below. You have to do these problems independently as far as possible.
Set
1. In a survey of 100 farmers in Kavrepalanchok district, 50 have animal farming, 80 have
vegetable farming, and every farmer has at least one farming.
(i) How many farmers have both farming?
(ii) How many of them have only one farming?
(iii) Illustrate the above information in a Venn-diagram.
2. In an annual function of a school, 75 students participate either in Science or in
Mathematics exhibitions. 30 of them participate in Science exhibition and 20 participate
in both exhibitions.
(i) How many students participate only in Mathematics exhibition?
(ii) Represent the above information in a Venn-diagram.
3. In a survey of some tourist, 60 % of them visited Pokhara, 70 % visited Solukhumbu,
and 45 % visited both the places during 'Visit Nepal 2020'.
(i) Show the above information in a Venn-diagram.
(ii) What percent of the tourists did not visit Pokhara or Solukhumbu?
4. In an examination, 80% of students got grade A in Science, 90% got grade A in
Mathematics, and every student got grade A at least in one subject. Find the percent of
students who got grade A:
(i) in both subjects
(ii) in Mathematics only
(iii) show the information in a Venn-diagram.
5. In a survey of a group of people, 50% are using cellular data and 60% are using Wi-Fi.
If 30% of them neither use cellular data nor Wi-Fi, then:
(i) represent the above information in a Venn-diagram
(ii) Find the percent of people who are using cellular data as well as Wi-Fi.
6. In a survey of some students, it was found that 60 % of the students were studying
commerce and 40 % were studying science. If 40 students were studying both the
subjects and 10% did not study any of two subjects, by drawing a Venn-diagram,
(i) find the total number of students.
(ii) Find the number of students who were studying science only.
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Revision and Practice Time
7. In a group of 80 students, 25 students liked apple only, 23 liked orange only and 20 of
them did not like both the fruits.
(i) How many students liked apple as well as orange?
(ii) How many students liked apples?
(iii) How many students liked oranges?
(iv) Present the results in a Venn-diagram.
8. In an examination, 60 % examinees failed in Mathematics, 55 % failed in English and
24 failed in both subjects. If none of the examinees passed in both subjects, find
(i) The number of examinees who passed in Mathematics only.
(ii) Represent the above information in a Venn-diagram.
9. In an examination 45 % of the students passed in Mathematics only and 35 % passed
in Science only. If 10 % students failed in both subjects, find:
(i) The percentage of students who passed in both subjects.
(ii) What percentage of the students passed in Mathematics.
(iii) Illustrate the results in a Venn-diagram.
10. 60 students in a class like Mathematics or Science or both. Out of them 15 like both
subjects. The ratio of the number of students who like Mathematics to those who like
Science is 2 : 3.
(i) Find the number of students who like mathematics.
(ii) Find the number of students who like science only.
Represent the above information in a Venn-diagram.
11. In a group of 125 students, the ratio of students who like tea and coffee is 4 : 5. If
20 of them like both the drinks and 10 of them like non of the drinks, by drawing a
Venn-diagram, find:
(i) How many of them like only tea?
(ii) How many of them like only coffee?
12. Out of 80 students of class IX, 25 students like to play volleyball but not basketball, 20
students like to play basketball but not volleyball and each student like to play at least
one of the game.
(i) How many students like to play volleyball as well as basketball?
(ii) How many students like to play volleyball?
(iii) Show the above information in the Venn-diagram.
13. In a survey of 300 youths it was found that 70% of them like motorbike and 55% like
scooter.
(i) Present the above information in a Venn-diagram.
(ii) Find the number of youths who like motorbike and scooter both.
(iii) Find the number of youths who like scooter only.
(iv) Find the number of youths who like only one of these vehicles.
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Revision and Practice Time
14. 27 districts of Nepal have shared their border with India, 15 districts have shared their
border with China and 2 districts have shared their border with India and China both.
(i) How many districts have shared their border with China only?
(ii) How many districts have shared their border with none the countries?
(iii) Draw a Venn-diagram to show the above information.
15. Among 70 patients admitted in a hospital, 11 patients have diabetes and 14 have blood
pressure. If 15 patients have at least one these two diseases, find the number of patients
who have only one of these diseases by drawing a Venn-diagram.
16. A survey conducted about the use of social media facebook or twitter or both or none
got a record that 2000 people use facebook only, 500 people use twitter only and 50
use none of these social media. If the number of people who use facebook is twice
that of twitter, by showing in a Venn-diagram, find the number of people who were
participated in the survey.
17. In a group of 75 people it was found that 10 people drink both tea and coffee and 5
people drink neither tea nor coffee. If the number of people who drink tea only is half
of the number of people who drink coffee only.
(i) Draw a Venn-diagram to show the above information.
(ii) Find the ratio of number of people who drink tea and who drink coffee.
18. In a group of 50 teachers it was found that 11 teachers are teaching in secondary level
only, 24 teachers are teaching in basic level only. If the number of teachers who are
teaching in both the levels are twice that of teachers who are teaching in other level,
find the number of teachers who are teaching in both basic and secondary levels by
using a Venn-diagram.
19. In a survey of community, it was found that 50% of the people preferred yoga, 60%
preferred jugging and 10% preferred neither yoga nor jugging. If 200 people preferred
both yoga and jugging, by using a Venn-diagram find:
(i) How many people were participated in the survey?
(ii) How many people preferred only one of these?
20. During the lockdown due to pandemic of corona virus, 80% of the schools of a
municipality managed their regular classes virtually through zoom or google meet
platforms. 35 schools used zoom platform, 8 people used google meet and 3 used both
zoom and google meet platforms.
(i) Draw a Venn-diagram to represent the above information.
(ii) Find the total number of schools of the municipality.
(iii) Find the number of schools that could not manage their virtual classes.
(iv) Find the number of schools that managed virtually using at most one of these
platforms.
Profit and Loss
1. A grocer bought 100 kg of rice for Rs 10,500. He sold 80 kg of rice at Rs 120 per kg and
the remaining quantity of rice at the rate of cost price. Find his profit or loss percent.
2. A shopkeeper sold 5 kg of apples for the cost price of 4 kg of apples. Find his loss
percent.
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Revision and Practice Time
3. Mr. Pariyar purchased a house under construction for Rs 15,00,000 and he spent
Rs 5,00,000 to complete its construction. If he sold the house to Mr. Thapa at 14% profit,
how much did Mr. Thapa pay for the house?
4. Mrs. Yadav sold a mobile set to Mrs. Gurung for Rs 9,660 and gained 15%. What would
be her profit or loss percent if she had sold it for Rs 8,190.
5. Manoj sold a fan for Rs 1,800 at 10% loss. At what price should he sell it to gain 10%?
6. If a grocer sells 20 l of oil for the cost price of 22l, find his profit percent.
7. A retailer bought 1,200 eggs. 200 of them were broken and he sold the remaining eggs
at Rs 9 each and made a loss of 614%. At what price did he purchase each egg?
8. A dishonest shopkeeper has two false balances. One balance weighs 10% more while
buying the goods and other weighs 10% less while selling the goods. Find his gain
percent just by weighing.
9. A crooked shopkeeper sells goods at the cost price. But his 1 kg weight weighs 900 g
only. Find his gain percent.
10. A shopkeeper sells an article at a loss of 15%, but if he had charged Rs 30 more he would
have gained 5%. What is the cost of the article?
11. Sunayana makes a profit of 5% by selling a watch for Rs 1,680. What percent of profit is
made if she sells it for Rs 1,800?
12. A stationer bought 40 story books for Rs 1,600. He sold 10 books for Rs 40 each and 15
books for Rs 45 each. At what price should he sell each of the remaining books to make
15% profit on his outlay?
13. A retailer sold two mobiles at Rs 2,800 each. If he gained 10% on one and lost 10% on
the other, find his gain or loss percent in the whole.
14. Anita buys an article for Rs 120 and sells it to Shova at a profit of 25%. Shova sells it
to Pinky and Pinky sells it for Rs 198 making a profit of 10%. What profit percent did
Shova make?
15. A grocer has some rice of worth Rs 3,000. He sold 13 of it with 10% loss. By how many
percent must the selling price be increased for making 10% profit on the outlay?
16. Dhurmus buys a watch for Rs 500 and sells it to Suntali at a profit of 10%. After a few
days, Suntali sells the watch back to Dhurmus at a loss of 10%. Find the profit made by
Dhurmus.
17. A shopkeeper loses 20% when a radio is sold for Rs 1200. What percent of profit or loss
does he make if it is sold for Rs 1575?
18. If 4% of selling price of a blanket is equal to 5% of its cost price and 6% of selling price
is Rs 20 more than 7% of cost price. Find the profit or loss amount.
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Revision and Practice Time
19. Mr. Rai sold two goats for Rs.5,500 each; gaining 10% on one and losing 10% on the
other. Find his gain or loss percentage on the whole transaction.
20. If Mrs. Shrestha sold a kurtha for Rs. 2,760 at a gain of 15% and a Sari for Rs. 3,240 at a
loss of 10%. Find her gain or the loss percentage on the whole transaction.
21. A shopkeeper bought two rugs for Rs. 15,000. He sold one of them at 10% gain and other
at 10% loss. If the selling prices of both the rugs are equal, find his gain or loss percent
in this transaction.
22. A bag was sold on its marked price at a gain of 20%. But allowing 5% discount, there
would be gained Rs 140. Find the cost price of the bag.
23. Mr. Rai marked the price of a 'Bhojpure Khukuri' to make a profit of 10%. If he had sold
it at a discount of Rs 105, there would have been a loss of 5%. Find the cost price of the
'Khukuri'.
24. The marked price of an article is Rs 2,800 which is 40% above the cost price. If it is sold
by allowing 20% discount, what would be the gain percent?
25. A tradesman marks his goods at such a price that he can deduct 4% for cash and still
makes 20% profit. What is the marked price of an article which costs him Rs 1,400?
26. An article, after allowing a discount of 20% on its marked price was sold at a gain
of 20%. Had it been sold after allowing 25% discount, there would have been gained
Rs 125. Find the marked price of the article.
27. A shopkeeper marks the price of an article Rs 550 and gives the customer a discount of
10%. In this way he gains Rs 75. How many percent will the marked price be more than
the cost price?
28. A shopkeeper gained Rs 8 by selling a pen allowing 10% discount. He would have
gained Rs 20, if discount had not allowed. What was the cost price of the pen?
29. How much should the selling price be fixed for rice bought at Rs 70 per kg to earn a
profit of Rs 6 per kg when 5% discount is allowed?
30. A man buys a watch for Rs 1,200 and marks it in such a way to make a profit of 3331%
after allowing a discount of 20%. Find the marked price.
31. A shopkeeper bought an article for Rs 400. He marked it for sale at a price that would
earn him a profit of 60% on his cost. If he sold the article for Rs 480 allowing some
discount, calculate the marked price of the article and the discount percent.
32. The marked price of a camera is Rs 3,200 and the shopkeeper announces a discount of
8%. How much will a customer have to pay for buying it if 10% VAT was levied on it?
Vedanta Excel in Mathematics - Book 9 310 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Revision and Practice Time
33. If a tourist paid Rs 5,610 for a carved window made of wood with a discount of 15%
including 10% VAT, what is the marked price of the window?
34. After allowing 20% discount on the marked price and then levying 10% VAT, a radio
was sold. If buyer had paid Rs 320 for VAT, how much would he have got the discount?
35. After allowing 25% discount on the marked price and then levying 10% VAT, a cycle
was sold. If the discount amount was Rs 750, how much VAT was levied on the price of
the cycle?
36. The marked price of a mobile set is Rs 6,000. After giving a certain percent of discount
and levying 10% VAT, its price becomes Rs 5,610. What is the discount percent?
37. The marked price of an electric fan is Rs 2,400 and the shopkeeper allows 20% discount.
After levying VAT, if a customer pays Rs 2,208 for it, find the VAT percent.
38. If a tourist bought a statue at a discount of 20% with 8% VAT and got Rs 256 back for
VAT at the airport, what was the marked price of the statue?
39. After getting 20% discount a customer paid Rs 5,062.40 with 13% VAT to buy a watch
from a retailer. If the retailer made a profit of 12%, by how many percent did he mark
the price of the watch above the cost price?
40. The marked price of an item is certain percent above the cost price and it is sold at 10%
discount levying 13% VAT to make a profit of 8%. If a customer pays Rs 702 as VAT, by
how many percent is the marked price above the cost price?
41. The marked price of a bag is Rs 2,500 and 4% discount is allowed to make 20% profit.
By how many percent is the discount to be increased to reduce the profit by 2.5%?
42. A ready-made garments shop allows 20% discount on its garments and still makes a
profit of 20%. What is the marked price of a tracksuit which is bought by the shop-
keeper for Rs.1200?
43. The value of a mobile set of same model in two shops A and B are given below.
Shop-A Shop-B
Marked price = Rs 25,000 Marked Price = Rs 24,000
Discount = 10% Discount = 8%
Which shop is more expensive and by what percentage? Find it.
44. Mrs. Sherpa wishes to buy a bicycle for her son. She finds that shop-A marks the price
of a cycle 40% above the cost price and allows 15% discount and another shop-B marks
the price of cycle of the same model 50% above the cost price and allows 20% discount.
If both the shopkeeper buy the cycles at the same price, in which shop would you
suggest to Mrs. Sherpa to purchase the bicycle? Give reason with calculation.
45. The price of a refrigerator is tagged Rs 40,000. The store allows 10% discount on its
tagged price. How much should a customer pay for it with 13% value added tax? By
what percentage is the VAT amount more than the discount amount?
46. A shopkeeper sold a laptop for Rs 58,986 including 13% value added tax after allowing
same percentage of discount. Find the marked price of the laptop. Also, calculate the
difference between the discount and VAT amount.
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Revision and Practice Time
Commission and taxation
1. Rekha gets Rs 7,000 weekly salary in a departmental store. She gets some additional
money by commission rates. She gets 10% of everything she sells. If Rekha sold
Rs 30,000 worth of items this week, what is her salary for the week?
2. A client tells you he is willing to pay you a 5% commission as long as he gets Rs 4,75,000
on the sale of his home. To comply with his request what would be the minimum listing
price of the home?
3. By selling a piece of land for Rs 25,00,000, a real estate agent receives 1 % commission
on the first 10 lakh and 0.5 % commission on the remaining amount of selling price.
How much money does the owner of the land receive?
4. When a business company increased its profit from 25 % to 35 %, the amount of profit
increased to Rs 13,30,000. If the company decided to distribute 60 % bonus equally
to its 40 employees from the increase amount of profit, how much bonus does each
employee receive?
5. An agent is given 2.5% commission on selling a piece of land for Rs 7,50,000 and 5%
commission for additional amount of selling price above the given fixed price. If the
agent sold the land for Rs 9,00,000, how much commission did he receive and what
sum did the land owner get?
6. The monthly salary of an employee in a departmental store is Rs 12,000 and 1%
commission is given when the monthly sales is more than 6 lakh rupees. If the sale of
the shop in a month is 10 lakh rupees, find the income of the employee in the month.
7. The monthly salary of a sales girl in a cosmetic shop is Rs 11,000 and a certain
commission is given as per the monthly sales. If the sales of a month is Rs 5,00,000 and
the total income of the girl in that month is Rs 21,000, find the rate of commission.
8. A publication house provides 0.5% commission upto the sales of 10 lakh, 1% commission
for 10 to 15 lakh and 2% commission for more than 15 lakhs to its distributors. If the
sales of a distributors is Rs 25,00,000, find his commission.
9. The monthly income of an individual is Rs 38,500 from which 10% of the salary is
deducted as provident fund. If 1% social security tax is charged upto the annual income
of Rs 4,00,000 and 10% tax is charged from Rs 4,00,000 to Rs 5,00,000, how much
income tax should be paid by the individual in a year?
10. The monthly salary of a married couple is Rs 75,000. 10% of the salary is deducted as
provident fund and another 10% is deducted as citizen investment trust. If 1% social
security tax is levied upto the annual income of Rs 4,50,000, 10% tax is levied on
Rs 4,50,000 to 5,50,000 and 20% tax is levied on Rs 5,50,000 to 7,50,000, how much
income tax should the couple pay in a year?
11. According to the tax policy implemented by Inland Revenue Department (IRD),
government of Nepal, 1% social security tax is levied upto the annual income of
Rs 4,50,000 for a married couple. 10% tax is levied on Rs 4,50,000 to Rs 5,50,000, 20% tax is
levied on Rs 5,50,000 to Rs 7,50,000 and 30% tax is levied above Rs 7,50,000 to 20,00,000.
The monthly salary of a married man is Rs 1,80,000. If 10% of his salary is deducted
as provident fund, another 10% is deducted as citizen investment trust and he pay
Rs 20,000 as the premium of his insurance, how much income tax does he pay in a year?
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12. Mr. Limbu bought 750 shares out of 20,000 shares sold by a commercial bank at Rs 150
per share. If the bank earned a net profit of Rs 1,25,00,000 in a year and it decided to
distribute 7.5% dividend to its shareholders, how much dividend will the man receive?
13. Rita bought 1,000 shares out of 25,000 shares sold by a Finance Company. The company
earned a net profit of Rs 96,40,000 and it decided to distribute a certain percent of
profit as dividend. If Rita received Rs 46,272, what percent of profit was distributed as
dividend?
14. Mrs. Magar bought 500 shares out of 50,000 shares sold by a hydro power company.
When the company distributed 15% of its net profit, she received Rs 22,500 as dividend
in a year. Calculate the net profit of the company.
Household Arithmetic
1. The rate of electricity charge upto 20 units is Rs 3 per unit and Rs 6.50 per unit from 21
to 30 units. Find the charge of consumption of 28 units with Rs 50 service charge.
2. The meter reading for the consumption of electricity of a household was 1248 units on
1 Mangisr and 1388 units on 1 Poush. If the customer made the payment of bill on the
25th Poush, calculate the total charge with fine under the following conditions.
Units 0 - 20 21 - 30 31 - 50 51 - 150
Rate of charge per unit Rs 3 Rs 6.50 Rs 8 Rs 9.50
Service charge = Rs 100, payment upto the 30th day from the meter reading 5% extra
fine.
3. The meter box of a house is of 15 A. If the family made the payment of Rs 1,309 with
service charge of Rs 125 on 36th day of meter reading how many units of electricity was
consumed in the month? Calculate it under the following rates.
Units 0 - 20 21 - 30 31 - 50 51 - 150
Rate of charge per unit Rs 4 Rs 6.50 Rs 8 Rs 9.50
Payment upto the 40th day from the meter reading - 10% fine.
4. The following table shows the meter reading of Rahul’s house from Kartik to Magh.
Month 1 Kartik 1 Mansir 1 Poush 1 Magh
Meter reading 05329 05374
05287 05302
He made the payment of the bill within 7 days of meter reading in each month. Answer
the questions on the basis of the following rate of electricity:
Units 0-20 21 – 30 31 – 50
Rate of charge per units Rs 3 Rs 6.50 Rs 8.00
Service charge Rs 30 Rs 50 Rs 50
(i) Find the number of units of consumption of each month.
(ii) Find the electricity charge of each of the month.
(iii) By what percentage was the payment of Poush more than that of Mansir?
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5. A household has a meter of capacity 5A. The meter reading of 1 Jestha, 1 Asar and
1 Shrawan of the household was 2450 units, 2486 units and 2550 units respectively. If
the payment of Jestha and Asar was made only on the 4th Shrawan, find the total charge
of electricity.
Units 0-20 21 – 30 31 – 50 51 – 100
Rate of charge per units Rs 3 Rs 6.50 Rs 8.00 Rs 9.50
Service charge Rs 30 Rs 50 Rs 50 Rs 75
The rules of rebate/fine:
From meter reading within 7 days 8th – 15th days 16th – 30th days 31st – 40th days
Rebate/fine 5% fine 10% fine
2% rebate -
6. The previous reading of the local calls of a telephone line is 2052 and the current reading
is 2276. Answer the following questions:
a) Find the number of calls
b) If the minimum charge upto 175 calls is Rs 200 and charge for each additional call
is Re 1, how much will be charge with 10% TSC and 13% VAT?
7. A household made 585 telephone calls in a month. If the minimum charge upto 175
calls is Rs 200 and the charge for each additional call is Re 1, find the charge to be paid
with 10% TSC and 13%VAT.
8. The minimum charge of telephone calls upto 175 calls is Rs 200. The charge for each
extra call is Re 1. If a man paid Rs 559.35 with 10% TSC and 13% VAT for his telephone
bill, find the number of calls made.
9. The minimum charge of consumption of 10 units of water is Rs 110 and the charge for
each additional unit is Rs 25 per unit. If a household consumed 36 units of water in a
month, find the charge of consumption of water with 50% sewerage service charge.
10. Water is supplied in a hotel by a ppaiipdewoifthsiizneth34e ". If the hotel consumed 112 units of
water in a month and the bill was fifth month of the bill issued, find the
amount required to clear the bill. (Minimum charge for 27 units is Rs 1,490, additional
charge is Rs 40 per unit, sewerage service charge 50%.)
11. Prakash hired a taxi and travelled 5 km. The minimum fare of Rs 14 appeared immediately
after the meter was flagged down. Then, the fare went on at the rate of Rs 7.20 per 200
metres. An additional waiting charge of Rs 7.20 per 2 minutes was charged for the
waiting of 10 minutes during the journey. Calculate the total fare paid by the man.
12. Mr. Janak paid Rs 86 taxi fare after travelling a certain distance. If the minimum fare is
Rs 14 and the additional fare per 200 metres is Rs 7.20, find the distance travelled by
her.
13. Ganesh hired a taxi and travelled a certain distance. He paid the total fare of Rs 372
including the additional waiting charge for 5 minutes. If the minimum fare is Rs 21, the
fare per 200 metres is Rs 10.80 and the waiting charge is Rs 10.80 per 2 minutes find the
distance travelled by him.
Mensuration
1. Ram draws a triangle with the side lengths 30 cm, 28 cm and 26 cm. Sita draws a
parallelogram whose base is 28 cm and the certain height. If the area of the triangle and
parallelogram are equal, what is the height of the parallelogram drawn by Sita?
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2. Amar has two square shaped handkerchiefs. If the sum of the area of both the
handkerchiefs is 2225cm2 and the difference between their perimeters is 60 cm, find
the sides of the handkerchiefs.
3. Binaya has a semi-circular protractor of perimeter 36 cm. What is its area?
4. The length of a square room is 6 m. What will be the cost of carpeting its floor with the
carpet of width 1.8 m at Rs 150 per metre?
5. A wire in the form of a rectangle 25.6 cm long and 18.4 cm wide is bent and reshaped
into the form of a circle. Calculate the change in area in percent.
6. Manish is cycling in such a way that the wheels of the cycle are making 210 revolutions
in a minute. If the diameter of the wheel is 80 cm, calculate his speed in kilometer per
hour.
7. A rectangular garden is 20 ft long and 15 ft wide. If the area of a path of uniform width
constructed all around inside it is 174 square ft, find the width of the path.
8. A rectangular photo frame has a photo of length 20 cm and 10 cm and the boarder of
uniform width. If the area covered by the boarder is 400 cm2, find the width of the
boarder of the frame.
9. A kitchen room is 20 ft long and 15 ft broad. How many tiles of 12 inch × 9 inch are
required to pave the floor? (12 inch = 1 ft)
10. A rectangular field is twice as long as its breadth. A path of uniform width 5 ft. is
running inside the field. If the cost of constructing the path at Rs 45 per sq. ft. is
Rs 76,500, find the area of the field.
11. A temple has a rectangular base 20 m long and 10 m broad. A path 1 m wide surrounding
the temple is to be paved by marbles each of size 250 sq. cm. If the cost of each marble
is Rs 75, find the cost of paving the marbles.
12. A road 2 m broad surrounds a square park 4225 m2 in area. Find the cost of paving
marbles each of 25 cm by 8 cm in size on the road at Rs 50 per piece of marble.
13. A room is 12 m wide and 4 m high. If the cost of colouring its walls at Rs 17 per sq.
metre is Rs 3536, find the cost of plastering its floor and ceiling at Rs 25 per sq. metre.
14. The length of a rectangular park is 60 m and its breadth is 40 m. Two crossing paths
running across the middle of the garden and a path all around inside it have the uniform
width of 2 m.
(i) Find the cost of paving the paths by the stones of the size 20 cm by 15 cm at
Rs 12,000 per 1000 stone.
(ii) Find the cost of covering the empty space with grass at Rs 75 per square meter.
15. A square room is 6.5 m high. If the cost of papering its walls at Rs 25 per sq. metre is
Rs 6500, find the cost of carpeting its floor at Rs 64.50 per sq. metre.
16. The cost of carpeting a room which is 5 m high and the length twice its breadth is
Rs 36,000 at the rate of Rs 500 per sq. metre. Find the cost of plastering its walls and
ceiling at the rate of Rs 40 per sq. metre.
17. The height of a square room is one-third of its length. If the cost of plastering its four
walls and ceiling at Rs 36 per sq. m is Rs 18,900, find the cost of carpeting its floor at
Rs 75 per sq. m
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18. A metal cube of edge 6 cm is melted and formed into three smaller cubes. If the edges
of two smaller cubes are 3 cm and 4 cm respectively, find the edge of the third smaller
cube.
19. The external dimensions of a closed wooden box are 27 cm, 19 cm and 11 cm. If the
thickness of the wood in the box is 1.5 cm, find
(i) the volume of the wood in the box.
(ii) The cost of the box if wood costs Rs 5.50 per cm3.
(iii) The number of 4 cm solid cubes that can be placed into the box.
20. The cost of carpeting a square room at Rs 125 per sq. metre is Rs 8000. If the room
contains 288 cu. m of air, find the cost of painting its walls at Rs 25 per sq. metre.
21. The ratio of the length and the breadth of a room is 3:2 and its volume is 576 m3. If the
cost of paving marbles on its floor at Rs 150 per sq. metre is Rs 14400, find the cost of
painting its 4 walls and ceiling at Rs 10 per sq. metre.
22. Mr. Rai built a house having two rectangular rooms of same width and height after the
destruction of his old house by the earthquake. The first room having two windows
each of size 4 ft ×4.5 ft and two doors each of size 3 ft ×6.5 ft is 18 ft long, 12 ft wide and
9 ft high, the second room having a window of size 4.5 ft ×4.5 ft and a door common to
the first room is 16 ft long.
(i) Find the cost of carpeting both the rooms at Rs 150 per sq. ft.
(ii) Find the cost of plastering the walls inside the rooms at the rate of Rs 80 per sq. ft.
23. Dipak is a farmer in a village. He constructed a poly-house
to grow vegetables as shown in the given figure.
(i) Find the volume of the poly-house. 5m
(ii) Find the plastic required for the construction of
14m 30m
poly-house and the cost of plastic at Rs 5 per
square meter.
Algebraic Expressions b) a2 + b2 – c2 +2ab
d) 16x4 – 4x2 + 4x – 1
1. Resolve into factors. f) x2 – (a + b) x + ab
h) a4b-4 + a2b-2 + 1
a) x2 + 2xy + y2 – 9 j) b2(b + 1) – d2 (d + 1)
c) p2 – q2 – r2 – 2qr l) 16m4 + 8m3 – 2m – 1
e) a2 + b2 – c2 – d2 – 2ab – 2cd n) (a2 –5a)2 + 10 (a2 –5a) + 24
g) (1 – a2) (1 – b2) + 4ab p) x3 – y3 – z3 – 3xyz
i) x8 + x4y4 + y8
k) p3 + 2p2 + 2p + 1
m) 64 – 144x + 108x2 –27x3
o) a3 + b3 + c3 – 3abc
2. Simplify:
a) x2 – 4a2 – x2 + 2ax – 8a2
x2 – 2ax x2 – 4a2
b) (a b+c c) + (b c+a a) + (c a+b b)
– b) (a – – c) (b – – a) (c –
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c) a2 – 1 + 2 + a2 – 1 + 6 + a2 – 1 + 15
3a 5a 8a
d) (a a +b c2 + a2 b– c c)2 + b2 c+a a)2
+ b)2 – – (b – – (c +
e) x+y – x–y + 4xy f) 1 1 a + 1 2a + 4a3
x–y x+y x2 + y2 + + a2 1 – a4
g) a2 + b2 – b2 – a2 h) 1 + 1 + x2 – 1 + 2x x4
ab a (a + b) b (a + b) x 1 – x + x2 1 + x2 +
i) a2 + ac – a2 – c2 c3 – 2c j) a2 – (b – c)2 + b2 – (a – c)2 + c2 – (a – b)2
a2c – c3 a2c + 2ac2 + a2 – c2 (a + c)2 – b2 (a + b)2 – c2 (b + c)2 – a2
Indices
1. Simplify.
xa(b – c) xa c xa – c a–b xb – a b–c xc – b c–a
xb(a – c) xb x– b x– c x– a
a) × b) × ×
c) xb b+c–a xc c+a–b xa a + b – c d) xa 1 xb 1 xc 1
xc xa xb ca
× × ab × bc ×
xb xc xa
xp + q r–p xq + r p–q xp + r q–r f) (am . an)m – n ap n + p × ap p+m
an am
e) xp – q × xq – r × xr – p
1+ x ×xx–y 1 – y xy–y
y x
g) a2 – 2a + 1 h)
(a – x)x (a – x)x – 1 (a – x)x – 2 y +1 ×xx–y x – 1 xy–y
x y
i) ab xba × bc xbc × ca xac j) 1 xb1 × 1 x1a × 1 x1c
xba xbc xac xbc 1 ab xb1 ca x1a
c
111
k) 1 + xb – a + xc – a + 1 + xc – b + xa – b + 1 + xa – c + xb – c
111
l) 1 + ax – y + az – y + 1 + ay – z + ax – z + 1 + xz – x + xy – x
2. a) If a + c = 2 and xz = y, show that xb – c. yc – a.za – b = 1
by
b) If abc = 1, prove that: (1 + a + b–1)–1 + (1 + b + c–1)–1 + (1 + c + a–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
x2a x2b x2c
d) If a+ b+c = p, show that x2a + xp–b + x p–c + x2b + xp–c + xp–a + x2c + xp–q + x p–b = 1
332 2
e) If x2 + 2 = + 3– 3 , show that 3x (x2 + 3) = 8.
f) If x = 1 – (ab)– 1 , prove that abx (x2 + 3) = a2b2 – 1
3
(ab) 3
g) If p x = q y = r z and xyz = 1, prove that p + q + r = 0.
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3. Solve. b) 9x + 1 = 32x + 1 + 54
a) 4x + 2 = 22x + 1 + 28
c) 4x – 2 + 2 × 22x – 1 = 1 116 d) 32x – 1 – 2 × 9x – 1 = 9
e) 3x + 3x + 1 + 3x +2 + 3x + 3 = 40 f) x 2 + 2x 1 + 2x = 20
g) 7x – 2 × 23x – 2 = 2 h) 35x – 4 × a4x – 3 = 32x – 3 × ax – 2
i) 2x + 1 = 414 7 j) 2x – 1 + 21 – x = 221
2x
4. a) If ax = by and b = a2, show that x – 2y = 0.
b) If a = 9x, b = 9y and aybx = 81, show that xy = 1.
c) If xmxn = (xm)n, prove that xm(n – 2) × xn(m – 2) = 1.
d) If xa = yb = zc and y3 = xz, show that 3 = 1 + 1c .
b a
Simultaneous Equations
1. Solve each pair of simultaneous equations.
a) x+ 3 = 4 (x + 3) b) x + 1 = 5 (y + 1)
x- 3 = 2 (y + 8) x – 4 = 3 (y + 4)
c) 10x + y = 4 (x + y) + 3 d) 10x + y = 4 (x + y)
10x + y + 27 = 10y + x 10y + x – 10 = 10x + y
e) 2 (x – 3) + 3 (y – 5) = 0 f) 3x + 5y = 7x + 3y = 4
5 (x – 1) + 4 (y – 4) = 0 4 5
g) x + y – 2 = y + x – 3 = 6 h) 2 + 5 = 11 , 5 + 1 = 27
3 2 3x 4y 12 4x 2y 20
i) 4 – 3 =–1 j) 2x 3 y + 2 = 13
x+y x–y + 2x – y 42
12 + 5 =4 2 – 2 = – 2
x+y x–y 2x + y 2x – y 63
2. a) The sum of two numbers is 55. If the greater number is 5 more than the smaller one,
find the numbers.
b) The sum of two numbers is 62 and their difference is 26. Find the numbers.
c) If two times the sum of two numbers is 50 and three times their difference is 45,
find the numbers.
d) The total cost of a book and a pen is Rs 210. If the pen is cheaper than the book by
Rs 90, find the cost of each item.
e) Three years ago the sum of the age of a father and his son was 48 years and three
years hence father's age will be three times that of this son. Find their present ages.
f) Two years ago, father's age was nine times the son's age but three years later it will
be five times only. Find their present ages.
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g) The ages of two girls are in the ratio 5 : 7. Eight years ago their ages were in the ratio
of 7 : 13. Find their present ages.
h) A son was born when his father was 28 years old. If the sum of their ages in
2068 B.S. was 44 year, find their ages in 2078 B.S.
i) A number of two digits exceeds four times the sum of its digits by 3. If 36 is added
to the number, the digits are reversed. Find the number.
j) The sum of digits in a two digit number is 11. The number formed by interchanging
the digits of that number would be 45 more than the original number. Find the
original number.
Quadratic Equation b) 12x2 – cx – 20c2 = 0
d) x2 (a – b) – x (a + b) + 2b = 0
1. Solve these equations:
a) (2x – 3) (3x – 8) = (3x – 2) (x + 4)
c) px2 + qx + r = 0
e) x2 – 2 x = 32 f) 4 – x 5 2 = 3
3 x–1 + x
g) x–3 – x+3 = 667 h) x+3 – 1–x = 4 1
x+3 x–3 x–2 x 4
i) 1 – 1 + x 1 2 =0 j) 20 + x2 = 11 – 3x2 + 5
x–1 x–2 + 15 10
k) 1 a – 1 a = 8 l) 1 + x 1 1 1– 1 = 7
3x – 3x + 5a + x–1 8
m) a + 1 + x = 1 + 1 + 1 n) ax2 + bx + c = ax + b
b a b x px2 + qx + r px + q
o) 1 – b) + 1 c) = (x – 1 c) + (a + 1 + a)
(a + b) (x (c + a) (x – b) (x – b) (c
2. a) The sum of two numbers is 11 and the sum of their squares is 65. Find the numbers.
b) The difference of two numbers is 2 and the difference of their squares is 20. Find
the numbers.
c) In a two-digit number, the product of the digits is 18 and their sum is 9. Find the
numbers.
d) The sides of a right angled triangle are (x – 1) cm, x cm and (x + 1) cm. Find the
length of each side.
e) If the perimeter of a rectangular garden is 90 m and its area is 500 sq. metre, find its
length and breadth.
3. Prove that the roots of quadratic equation ax2 + bx + c = 0, a z 0 are x = – b ± b2 – 4ac .
2a
Also, use this formula to find the roots of the equation 2x2 – x – 3 = 0.
Ratio and proportion
1. a) Find the ratio of H.C.F. and L. C. M. of 12 and 18.
b) Find the ratio of H.C.F. and L.C.M. of 30, 40, 60.
2. a) The ratio between two numbers is 3 : 4. If their L.C.M. is 180, find the numbers.
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b) The ratio between three numbers is 4 : 5 : 6. If their L.C.M. is 600, find the numbers.
c) Two numbers are in the ratio of 2 : 3 and the difference of their squares is 320. Find
the numbers. (Hint: (3x)2 – (2x)2 = 320)
d) If twice A is equal to thrice B and four times B is equal to five times C, find A : C.
e) If 0.3 of a number is equal to 0.18 of another number, find the ratio of the numbers.
3. a) What number should be subtracted from each of the numbers 54, 71, 75 and 99
so that the remainder may be proportional?
b) Find two numbers such that the mean proportional between them is 28 and the
third proportional to them is 224.
(Hint: xy = 28, i.e. xy = 784, i.e. y = 784 and x : 784 = 784 : 224)
x x x
c) Find two numbers such that the mean proportional between them is 18 and the
third proportional to them is 144.
d) If y is the mean proportional between x and z, prove that (xy + yz) is the mean
proportional between (x2 + y2) and (y2 + z2).
(Hint: y = xz , i.e. y2 = xz
Mean proportional between (x2 + y2) and (y2 + z2) = (x2 + y2) (y2 + z2)
e) If three quantities are in continued proportion, prove that the first is to the third is
the duplicate ratio of the first to the second.
4. Write down any four properties of proportion with examples. If a : b = c : d, show that
(a – b) : (a + b) = (c – d) (c + d)
5. a) If a : b : : c : d show that
(i) (a + b) : (c + d) = a2 + b2 : c2 + d2
(ii) (a – b) : (c – d) = 7a2 – 8b2 : 7c2 – 8d2
(iii) (a2 + c2) : (ab + cd) = (ab + cd) : (b2 + d2)
b) If x : a : : y : b : : z : c, show that
(i) x3 + y3 + z3 = 3xyz
a3 b3 c3 abc
(ii) (a ax – by y) + (b by – cz z) + (c cz – ax x) = 3
+ b) (x – + c) (y – + a) (z –
c) If a = c =ef , prove that: (ab + cd + ef)2 = (a2 + c2 + e2) (b2 + d2 + f2)
6. a) b d
If (a2 + b2) (x2 + y2) = (ax + by)2, show that x = by.
a
b) (a2 + b2) : (m2 + n2) = ab : mn, show that a : b = m : n
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c) If ay – bx = cx – az = bz – cy , prove that x = y = z
p q r a b c
Hint: c (ay – bx) b (cx – az) a (bz – cy)
cp = bq = ar
Sum of antecedents
Then each ratio = Sum of consequents
d) If 4a + 9b = 4c + 9d , by using componendo and dividendo property, prove that
=4adc–. 9b 4c – 9d
a
b
e) If (11a + 7b) (11c – 7d) = (11c + 7d) (11a – 7b), show that a : b = c : d.
f) If (a + 3b + 2c + 6d) (a – 3b – 2c + 6d) = (a + 3b – 2c – 6d) (a – 3b + 2c – 6d), prove
that a : b = c : d.
g) If (p + q)2 = p , show that q is the mean proportional between p and r.
(q + r)2 r
Geometry - Triangle
1. In the given figure, BOC = 128° and OB and OC are bisectors
of ABC and ACB respectively. Find the measure of BAC.
128°
2. In the adjoining figure AB = AC, BO and CO are the bisectors A C
of DBC and BCE respectively and BAC = 64°. Find the 64°
measure of BOC. B
DO E
D
3. In the given figure ABC is an equilateral triangle. BA is produced A
to D so that BA = AD. Find the value of ACD.
4. In the figure given alongside, MNO = 115°, QOP = 75° and BC
RQO = 140°, prove that MN // RP.
MN
115°
O
140° 75°
RQP
A
5. In the adjoining triangle, D is the mid-point of BC and AD A BC.
Prove that ∆ ABC is an isosceles triangle.
B DC
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Revision and Practice Time R S
6. In the adjoining figure, prove that PQ // RS. O Q
7. In the given rectangle PQRS, M is the mid-point of RS. P
Prove that PQM is an isosceles triangle.
8. In the given figure, prove that QS is the perpendicular bisector of
PR.
9. In the figure alongside. ABCD is a square X, Y and Z are the points D Z C
on the sides AB, BC and CD respectively such that AX = BY = CZ. Y
Prove that XYZ is an isosceles triangle.
AX B
PT Q
10. In the given figure, PQ // SR. ST and RT bisect PSR and R
SRQ respectively. Prove that: PQ = PS + QR.
C
S Q
A
11. In the adjoining 'ABC, the bisectors of B and C meet at O. Q
O
OP A BC and OQ A AB. Prove: P
P
(i) OP = OQ (ii) OA bisect A
B
12. In the figure alongside, QS is median to side PR of 'PQR. T
QS is extended to T such that QS = ST. Prove that: S
RA
(i) PT = QR (ii) PT // QR
13. In the adjoining figure, BX A AC, CY A AB and BX = CY. Prove that Y X
AB = AC. AC
B
14. In the given figure, AB = AC. BO and CO are the bisectors of
ABC and ACB respectively. Prove that AO is the bisector of O
BAC. BC
Vedanta Excel in Mathematics - Book 9 322 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Revision and Practice Time
P
15. In the given figure, PQ = PR and QO = RO. PO is produced O R
to meet QR at S. Prove that QS = SR. B
S
Q A D
16. In the adjoining figure, AB = CD and AD = BC. O
Prove that BO = OC.
C
AP
17. In the given figure, AB // PQ and AC // PR.
If BQ = CR, prove: B QC R
(i) AB = PQ (ii) AC = PR
F
18. In the adjoining figure ABCD and DEFG are squares. Prove that G E
AE = CG. A
D
A B C
R
19. In the given figure, BA A AC, RQ A PQ, C
AB = QR, and BP = CR. Prove that AC = PQ. B P
20. In the figure alongside, OAD = ODA and A Q
OBC = OCB. Prove that AB = DC. O D
B A C
B C
21. In the given figure, AB = BC, MB = BN, AB A BC, and
MB A BN. Prove that AM = CN. M
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Vedanta Excel in Mathematics - Book 9
Revision and Practice Time
AX
22. In the figure given alongside, AB = BC,
ABX = CBY and BXC = BYA. Prove that B O
(i) OA = OC (ii) OX = OY
(Hint: ∆ BAY # ∆ BCX, Then, BAO = BCO. Join A, C.
BAC = BCA, CAO = ACO) CY
23. In the figure alongside, PQRS is the square. U is the mid-point PT S
R
of PQ, RUT = 90°, TU and RQ are produced to meet at V. U T
Prove that TR = PT + PQ. VQ A
R R
24. In the given figure , PQ = SR, OP = OS and
OQ = OR. QP and RS are produced to meet at E. S
Prove that TQ = TR.
O
P
Q
B
25. In the given figure, OA = OB and AP = BQ. Q
Prove that QX = PX. X
(Hint: Join A, B. ∆ PAB # ∆ QBA, Then ∆ APX # ∆ BQX) OP
26. In the figure alongside, M is the mid-point of BC, A
Q is the mid-point of MR and AB // NM // CQ.
Prove that (i) PR = 3 PM (ii) AB = 4CQ N
P C
BM
Q
PQ
27. In the adjoining figure, AC = BC, PCA = QCB and A O B
PBA = QAB. Prove that ∆ OPQ is an isosceles triangle. C
(Hint: ∆ QAC # ∆ PBC)
A
28. In the given ∆ ABC, AE is the bisector of BAC. O
If BO A AE and BM = MC, prove that OM // AC.
B EM C
Vedanta Excel in Mathematics - Book 9 324 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Revision and Practice Time
A
29. In the adjoining figure, AD is the bisector of BAC,
BN A AD and BM = MC.
Prove that MN = 1 (AC – AB) C N B
2 MD
30. In the adjoining trapezium PQRS, X and Y are the mid- P Q
X Y
points of PS and QR respectively.
Prove that XY = 1 (PQ + SR).
2
(Hint: Join Q, X and produce it to meet RS produced at T. Then, S R
∆ PQX # ∆ STX) S
P
31. In the adjoining figure PQRS and MQNO are rectangles. M O
Q
Prove that (i) MQ = 1 SR (ii) MN = 1 QS A NR
2 2 B
32. In the figure alongside, ABCD is a trapezium. M and
N are the mid-points of AD and BC respectively. M PQ N
Prove that AQ = QC and BP = PD.
DC
33. In triangle ABC, O is any point within it. P, Q and R are on OA, OB and OC respectively,
such that P is mid-point of OA, PQ // AB and QR // BC. Prove that PR // AC.
34. In the given figure, PMN = PRQ. P
N
Prove that PM.QR = PR.MN. M
Q R
A
34. In the adjoining figure, AB // CE, CA A AB and
DC
ED A BC. Prove that AB.DE = CD.AC. B
35. In the given figure, PQ // XY // RS. Prove that, E R
(i) ∆ QXY ~ ∆ QRS
(ii) ∆ SYX ~ ∆ PQS P
(iii) ∆ PQX ~ ∆ XRS X
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Vedanta Excel in Mathematics - Book 9
Revision and Practice Time
P S
T
36. In the given figure, diagonals of a quadrilateral PQRS are intersected
at T at right angle. Prove that PQ2 + SR2 = QR2 + PS2.
37. In the given figure. AB // MN // DC. If AB = x, DC = y and Q R
1 1 1z. A M D
MN = z. Prove that: x + y = z y
x
N C
B Z
W O
38. In the figure alongside, O is any point interior to the rectangle.
Prove that OX2 – OY2 = OW2 – OZ2.
X Y
D
Parallelogram
1. In the adjoining quadrilateral ABCD, A = C and B = D. A C
Show that ABCD is a parallelogram.
B
D C
2. In the given quadrilateral ABCD, AO = OC and BO = OD. O
Prove that ABCD is a parallelogram.
AB
3. In the adjoining parallelogram ABCD, P and Q are the
mid-points of the sides AD and BC respectively. Prove that
BP and QD trisect the diagonal AC at X and Y respectively.
(Hint: PD = BQ and PD // BQ. ? PB = DQ and PB // DQ
So, PBQD is a parallelogram. Then ' AYD # ' BCX,
DY = BX. Then, in ' AYD and ' BCX apply mid-point theorem)
4. In the given parallelogram BEST, M and N are the mid-points T S
of the sides BE and SE respectively. If the diagonal TE and BS N
intersect at O, prove that OMEN is a parallelogram. O
B ME
Vedanta Excel in Mathematics - Book 9 326 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Revision and Practice Time
AB
5. In the figure alongside, ABCD is a quadrilateral in which OA = OC, O
OB = OD and AOB = AOD = DOC = COB = 90°. Prove
that ABCD is a rhombus.
DC
6. In the given figure, XP and ZQ are perpendicular Z Y
to the diagonal WY of the parallelogram WXYZ. P
Prove that XP = ZQ. Q
X
W
H G Q
B
7. In the adjoining figure, EFGH is a P
parallelogram and P is the mid-point of FG. F
EP and HG are produced to meet at Q. Prove E A
that HQ = 2HG.
8. In the given parallelogram ABCD, P is the mid-
point of DC and BP bisects ABC. Prove that AP D PC
bisects DAB. Also show that APB = 90°.
AB
9. In the adjoining parallelogram ABCD, AE is the bisector of D EC
BAD. Prove that DE = BC.
AP B
10. In the adjoining figure, ABCD is a rectangle and P, Q, R and S Q
C
S are the mid-points of AB, BC, CD and DA respectively. D R
Prove that PQRS is a rhombus. B
PA CQ
11. In the given figure, ABCD is a parallelogram. The diagonal
AC is produced to P and Q such that AP = CQ. Prove that
PD // BQ.
D
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 327 Vedanta Excel in Mathematics - Book 9
Revision and Practice Time
XS R
12. In the figure alongside, PQRS is a parallelogram. The diagonal
QS is produced to X and Y. If PX // YR, prove that SX = QY.
P QY
Y
13. In the given figure, WXYZ is a parallelogram. ZP bisects Z
WZY and XQ bisects WXY. Prove that PXQZ is a Q
parallelogram. P
X
W
R
14. In the adjoining figure, PQRS is a rhombus. Prove that S Q
PQS = SQR, QRP = PRS, RSQ = QSP and
SPR = QPR.
P
D GC
15. In the given square ABCD, E, F, G and H are the mid-points
of AB, BC, CD and DA respectively. Prove that EFGH is also a H
square. F
AE B
A
16. In the figure given alongside, ABC is an equilateral B C
triangle. The sides AB and AC are produced to M and N N
respectively. BP and CP are the bisectors of MBC and P
NCB respectively. Prove that ABPC is a parallelogram.
M
SB R
17. In the adjoining figure, PQRS is a parallelogram and AB
bisects PR. Prove that AO = OB. O
P AQ
18. In a parallelogram ABCD, a line segment PQ meets AB at P, CD at Q and diagonal BD
at O. If OP =OQ, prove that PQ bisects BD.
Vedanta Excel in Mathematics - Book 9 328 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Revision and Practice Time
DQ C
19. In the given figure, ABCD is a parallelogram. If P and Q A
are the mid-points of the sides AB and DC respectively, PB
prove that RC = 2 AQ.
R
20. In the given figure, P, Q, R and S are the mid-points of P A D
AB, BC, CD and AD respectively. Prove that PQRS is a Q
parallelogram. S
C
B
R
A
P
21. In the adjoining figure, AB = AC, PB = CM and PQ // AM. B QO C
Prove that PM and QC bisect each other. M
Geometry - Circle
1. In the given figure, O is the centre of the circle. If AB = 8 cm, A O
C MB
CD = 6 cm, OA = 5 cm, AB // CD and ON A CD, find the length of ND
MN.
A B
2. AB and CD are two parallel chords of the adjoining circle with centre
O and they are 23 cm apart. If AB = 16 cm and CD = 30 cm, find the C O D
radius of the circle.
3. In the given figure, O is the centre of the circle. If PQ = RS, prove that R P
OM bisects PMS. M O
4. In the adjoining figure, O is the centre of two concentric circles. If Q
the chord MN intersects the smaller circle at R and S, prove that S
MR = SN.
O
MR SN
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 329 Vedanta Excel in Mathematics - Book 9
Revision and Practice Time
5. O is the center of the circle given alongside. E and F are the mid- B
E
points of the equal chords AB and CD respectively. Prove that A
O
(i) OEF = OFE (ii) BEF = DFE C FD
6. In the adjoining figure, OAB is an isosceles triangle and a circle O
with O as the centre cuts AB at C and D. Prove that AC = DB.
AC DB
7. In the given figure, O is the center of the circle. If OS A PQ and QR is O R
S P
the diameter of the circle, prove that OS // PR and PR = 2OS. Q
AC
8. In the figure alongside, MN is the diameter of a circle with centre O. M OD N
If BD = CD, prove that OAD = OCD. B
P
9. In the given figure, PQ is the diameter of a circle with centre O. If O
PQ A AB, prove that AOM = BOM. A MB
Q
AO C
MN
10. In the adjoining figure, equal chords AB and CD of a circle with E
centre O, cut at right angle at E. If M and N are the mid-points of AB DB
and CD respectively, prove that OMEN is a square.
11. In a circle the chords AB and AC are equidistant from the centre O.
Prove that the diameter AD bisects BAC and ADC.
12. Prove that equal chords of a circle subtend equal angles at the centre.
13. In the figure alongside, A and B are the centres of two intersecting C
circles. If CD intersects AB perpendicularly at P, prove that A M
(i) CM = DN (ii) CN = DM PB
N
D
Vedanta Excel in Mathematics - Book 9 330 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Revision and Practice Time
Trigonometry D
90°–D
1. From the given figure, find the trigonometric ratios of sinD and C
cosT.
90°–T
A B
2. From the given figure, find the values of sinT, cosT, tanT, D D C
T
sinD cosD and tanD 12 cm 3 cm
A 4 cm B
3. In the following figures, find the values of sinT and cosD.
a) A b) P
T
12 cm 3 cm D 25 cm 20 cm
B 12 cm Q
4 cm
T C D
R
4. In the adjoining figure, O is centre of circle. AB the diameter and A
X is the mid-point of the chord PQ. If AB = 20 cm, PQ = 16 cm and O
XTQ
PQB = T, find the value of tanT.
B
1 sinT – 2cosT 3 P
2 sinT + 2cosT 5
5. If tanT = , show that = –
6. If ntanD = m, prove that msinD – ncosD = m2 – n2
msinD + ncosD m2 + n2
C
7. In the given ' ABC, B = 90° and CAB = 45°. If AB = x cm, 45° A
prove that cos45° = 1 . x cm
2
A
B
8. In the adjoining equilateral ' ABC, AD A BC and AC = 2a
units. Prove that sin60° = 3 .
2
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 331 BC
Vedanta Excel in Mathematics - Book 9
Revision and Practice Time
P
9. In the given figure, PQR is an equilateral triangle in which
1. x x
3 R
PQ = QR = RP = x and PS A QR. Prove that tan30° =
Q S
x
A
10. In the given figure, ABC = 90° and CA = 2BC. Find the measure
of BAC.
B C
A
11. In the adjoining figure, ABC is a right-angled triangle, where
B = 90°, AB = 3 cm and AC = 3 2 cm, find the size of T.
B C
A
12. In the given figure, AED is a right-angled triangle and BCDE is a 55ft
rectangle. If AB = 55 ft, BC = 50 ft, CD = 5 ft and ADE = T find the 5ft
value of T. E TD
B 50ft C
P 30° Q
T
13. In the figure alongside, PQRS is a rectangle. If PQT = 30°,
SR = 2 3 cm and QTR = 90°, find the measurement of QR.
S R
14. In the given figure, ABCD is a square and CPM = 90°. D C
30°
If CM = 4 cm, PD = 3 cm, find the side of the square. P
15. Prove that AM B
a) cos230° – sin230° = cos60°
2sin230° b) cos230° – sin230° = 1 –
c) tan60° – tan30° = tan30° d) sin30° + sin60° = sin90°
1 + tan60°.tan30° cos30° + cos60°
Vedanta Excel in Mathematics - Book 9 332 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Revision and Practice Time
16. a) If D = 30°, prove that sin2D = 2sinD.cosD
b) If T = 45°, prove that cos2T = 1 – tan2T
1 + tan2T
c) If E = 30°, prove that tan2E = 2 tanE
1 – tan2E
d) If A = 60°, B = 30°, prove that tan (A – B) = tanA – tanB
1 + tanA.tanB
e) If A = 60°, B = 30°, prove that sin (A + B) = sinA.cosB + cosA.sinB.
f) If T = 30°, prove that cos2T = cos2T – sin2T
17. a) If 4sin2A = 1, what is the value of A in degrees (0° d A d 90°)?
b) If sinT = cosT, find the value of T (0° d T d 90°)?
c) If 3tanD = 3 , find the value of D (0° d D d 90°).
d) If 2 3 cos T = 3, find the value of T (0° d D d 90°).
Statistics
1. a) Change the following frequency distribution of 'inclusive' type into the frequency
distribution of the 'exclusive' type.
Class 1 – 10 11 – 20 21 – 30 31 – 40 41 – 50
No. of students 3 5 8 6 2
b) The mid-values of a frequency distribution are 5, 15, 25, 35, 45 and 55. Find the
class size and the class intervals.
(Hint: class size = 15 – 5 = 10
10
Lower limit of the first class = 5 – 2 = 0
Upper limit of the first class = 5 + 10 = 10
2
So, class intervals are 0 – 10, 10 – 20, ...)
c) The mid-values of a frequency distribution are 4, 8, 12, 16, 20, 24, 28, 32, 36 and 40.
Find the class size and the class intervals.
2. a) Group the following data into the class intervals of the given lengths in a cumulative
frequency table. Then, show the data in Histogram. Also, draw the 'less than' and
'more' ogives in separate graph papers.
30, 22, 15, 18, 17, 23, 16, 33, 32, 28,
17, 35, 19, 21, 24, 31, 29, 19, 30, 16,
19, 22, 16, 27, 26, 22, 31, 29, 32, 18
(i) class interval of length 3
(ii) class interval of length 5
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 333 Vedanta Excel in Mathematics - Book 9
Revision and Practice Time
b) In a survey of 1200 students of different schools, a question was asked. How many
hours do they study in a week? The response of the students is shown in the table
given below.
No. of hours (x) No. of students
0<x<5 125
0 < x < 10 250
0 < x < 15 350
0 < x < 20 300
0 < x < 25 175
Represent the above data in histogram and 'Less than' and 'more than' ogive.
3. The adjoining pie chart represents the answers
of 1080 students who like to study different Geography
subjects in future.
a) How many students like to study maths? 71° Science
b) How many students like to study geography? 55°
c) What percent of the students like to study English
English? 90°
Maths
4. a) The mean height of 5 boys is 53 inches. If a boy
of 59 inches joins in the group, what is their mean height?
b) If the mean of 4 different numbers is 20 and the mean of 6 different numbers is 30.
Find the mean of those 10 numbers.
c) If the mean of 2, 6, x, 12 and 10 is 8, find the median of the data.
d) The mean of 5 numbers is 27. If one number is excluded, their mean will be 25. Find
the excluded number.
5. Find the mean, median and mode from the following data:
a)
Marks 10 20 30 40 50 60
obtained
No. of 8 10 15 12 10 5
students
b)
Wages 500 600 700 800 900 1000
(in Rs)
No. of 2 4 8 12 10 6
workers
Vedanta Excel in Mathematics - Book 9 334 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Revision and Practice Time
6. a) The mean of the data given below is 54. Find the value of k.
x 10 30 50 70 90
f 7 k 10 9 13
b) The mean of the following data is 18.8. Find the value of p.
x 5 11 17 23 29
f 6 3 12 11 p
c) Given that mean is 40 and N = 51, find the missing frequencies in the following
data:
x 10 20 30 40 50 60
f 2 3 - 21 - 5
d) The mean of the data given below is 14. Find the missing frequencies if their sum is 10.
x 5 10 15 20 25
f7-8-5
e) The mean of the data given below is 27 and the missing frequencies are equal. Find
them.
x 15 20 25 30 35 40
f 4 - 10 7 - 3
7. Find the first quartile (Q1), the second quartile (Q2), and the third quartile (Q3) from the
data given below:
a)
x 10 20 30 40 50 60
f 8 10 15 12 10 5
b) Present the following data in a cumulative frequency table , then compute the first,
the second, and the third quartiles.
15, 10, 20, 5, 10, 15, 20, 25, 20, 15, 10,
25, 20, 30, 10, 15, 20, 25, 15, 20, 5, 30, 20
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 335 Vedanta Excel in Mathematics - Book 9
Revision and Practice Time
Probability
1. In a well-shuffled pack of 52 playing cards, find the probability that a card drawn at
random is a diamond.
2. From a pack of playing cards, the king of Heart is removed. The remaining cards are well
-shuffled. What is the probability that a card drawn at random is a heart?
3. From a pack of playing cards, two cards are taken, which are not aces. They are not re-
placed, and the remaining cards are then well shuffled. What is the probability that the
next card drawn is an ace?
4. Three athletes A, B and C are to run a race. B and C have equal chances of wining, but
A is twice as likely to win as either. Find the probability of each athlete wining.
5. From a well-shuffled pack of 52 cards, one card is drawn. Find the probability that it is
(i) a king, (ii) the queen of hearts,
(iii) a diamond, (iv) either the queen of hearts or the jack of spades,
(v) either a two or a three, (vi) either a two or a spade,
(vii) not the ace of spade, (viii) not a club,
(ix) not a diamond, (x) either a king, a queen or a jack.
6. A boy writes down at random a whole number larger than 1 and smaller than 11. Find
the probability that it is
(i) odd (ii) even (iii) prime
(iv) a factor of 12 (v) a perfect square (vi) a power of 2
7. A man has 3 pairs of black socks and 2 pairs of brown socks. If he dresses hurriedly in
the dark, find the probability that
(i) the first sock he puts on is brown,
(ii) the first sock he puts on is black,
(iii) after he has put on a black sock, he will then put on another black sock,
(iv) that after he has first put on a brown sock, the next sock will also be brown.
8. A bookshelf contains 10 detective stories, 9 historical novels and 7 books on sport. A
man selects one at random. What is the probability that it is a book on sport?
What is the probability that the next person to pick from this Shelf (before the first book
has been replaced) select at random
(i) a book on Sport, (ii) a detective story, (iii) a historical novel?
9. Two numbers are chosen at random from 1, 2, 3. What is the probability that their sum
is odd?
Vedanta Excel in Mathematics - Book 9 336 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Answers
1. Sets
Exercise : 1.1
1. a) A – B = {x : x A, but x B} b) A B = {x : x A and x B}
c) B – A = {x : x U, but x (B – A)} d) A B = {x : x A or x B}
e) A – B = {x : x U, but x (A – B)} f) A B = {x : x U, but x (A B)}
g) B – A = {x : x B, but x A} h) A = {x : x U, but x A}
i) B = {x : x U, but x B} j) A B = {x : x U, but x (A B)}
2. a) Q – P b) Q c) P Q d) P Q e) P – Q f) P Q g) P – Q h) P Q i) P
3. a) A B b) A B c) A B d) A B
e) A – B f) Y – X g) (A B C) h) P Q R
4. a) (i) A B = {1, 2, 3, 5, 6, 7, 9} , A B = {4, 8, 10}
(ii) A B = {1, 3}, A B = {2, 4, 5, 6, 7, 8, 9, 10}
(iii) A – B = {5, 7, 9}, A – B = {1, 2, 3, 4, 6, 8, 10}
(iv) B – A = {2, 6}, B – A = {1, 3, 4, 5, 7, 8, 9, 10}
b) (i) P Q R = {1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15}
(ii) P Q R = {6} (iii) P Q R = {7, 11, 13, 14}
(iv) P Q R = {1, 2, 3, 4, 5, 7, 8, ..., 15} (v) (P Q ) R = {3, 6, 12}
(vi) (P Q) R = {2, 3, 4, 6, 9, 12, 15}
5. a) and b show to your teacher.
6. a) P Q = {1, 2, 3, 4, 5, 6, 8} P Q = {7, 9, 10}
PQ PQ
b) P Q = {2, 4} P Q = {1, 3, 5, 6, 7, 8, 9, 10}
PQ PQ
c) P – Q = {1, 3, 5} P – Q = {2, 4, 6, 7, 8, 9, 10}
PQ PQ
d) Q – P = {6, 8} Q – P = {1, 2, 3, 4, 5, 7, 9, 10}
PQ PQ
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 337 Vedanta Excel in Mathematics - Book 9
Answers f) P Q = (7, 9, 10)
e) P Q = {1, 3, 5, 6, 7, 8, 9, 10} PQ
PQ
7. a)
A B C = {1, 2, 3, 4, 5, 6, 7, 9, 11, 12, 15} A B C = {8, 10, 13, 14}
b) A B C = {3} A B C = {1, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}
c) (A B) C = {3, 6, 9} (A B) C = {1, 2, 4, 5, 7, 8, 10, 11, 12, 13, 14, 15}
d) A (B C) = {1, 3, 5, 7, 9} A B C) = {2, 4, 6, 8, 10, 11, 12, 13, 14, 15}
e) (A – B) C = {3, 6, 9, 11, 12, 15} (A – B) C = {1, 2, 4, 5, 7, 8, 10, 13, 14}
f) A B – C) = {1, 2, 3, 4, 5, 7, 9, 11} A (B – C) = {6, 8, 10, 12, 13, 14, 15}
8. and 9. Verify and show to your teacher. 338 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Vedanta Excel in Mathematics - Book 9
Answers
1. Show to your teacher. Exercise : 1.2
2. a) 9 b) 3 c) 4 d) 5 e) 2 f) 4 g) 6 h) 5 i) 1 j) 2
3. a) (i) 53 (ii) 12 (iii) 8 (iv) 25 b)
(i) 70%
b) (i) 85% (ii) 20% (iii) 35% (iv) 30% (ii) 57%
(iii) 13%
c) (i) 45 (ii) 40 (iii) 75 (iv) 13 (iv) 30%
d) (i) 82% (ii) 29% (iii) 67% (iv) 44%
4. a) A B b) MN 5. a) M
24 12 15 (i) 60 N
(ii) 40 20 13%
(iii) 20 40
30 25 35 57%
10 14 (iv) 15 15 30%
6. a) (i) b) (i) 7. a) (i)
BX YT
A N
23 9 13 35% 20% 45% 200 200 150
(ii) 9 9 (ii) 20%, 35%, 45% 50
b) (i) (ii) 50 (iii) 200 (iv) 150
NC
325 450 525 c) (i) 35 d) (i) C A
200 F M
100 35 115 70% 10% 20%
(ii) 200 (ii) 215 (ii) 10% (iii) 90%
8. a) (i) 145 b) (i) c) (i)
BA TC IC
135 245 165 275 325 175 25% 20% 40%
55 125 15%
(ii) 325 (iii) 175
(ii) 300 (ii) 300
9. a) (i) 57 FC b) (i) AB 10. a) (i) Y
(ii) 7 20 7 30 (ii) 45 80 45 75 X
(iii) 27 (iii) 125
(iv) 37 (iv) 120 15000 350 23650
18
1000
(ii) 1000 (iii) 38,650
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 339 Vedanta Excel in Mathematics - Book 9
Answers
b) (i) 14 (ii) 850 11. a) (i) 625 (ii) 125 (iii) 385 b) (i) 35 (ii) 19
(iii) (iii)
B (iv) P S M
A S
483 36 367 260 125 240 26 9 19
14 125
12. and 13. Complete your project work in a group or individually. Discuss the outcomes in the class.
2. Profit and Loss
Exercise : 2.1
S. P. S. P.
1. a) S.P. = C.P. + P% of C.P. b) S.P. = C.P. – L% of C.P. c) C.P. = (100 + G)% d) C.P. = (100 – L)%
2. a) profit Rs 20, 10% b) loss Rs 55, 10% c) S.P. Rs 810, Rs 60 d) S.P. Rs 444, Rs 111
e) C.P. Rs 960, Rs 96 f) C.P. Rs 800, Rs 80 g) C.P. Rs 500, 12% h) C.P. Rs 860, 15%
3. a) Rs 909, Rs 5,454 b) Rs 192, Rs 9,408 4. a) Rs 1,850, Rs 185 b) Rs 750, Rs 120
5. a) 5% profit b) profit 14% c) profit 14% d) 25% e) profit 6%
6. a) Rs 1,62,250 b) Rs 147 7. a) Rs 6,400 b) Rs 65 per kg c) Rs 50 d) Rs 60
8. a) profit 14% b) loss 9% 9. a) profit 5% b) profit 4%
10. a) Rs 10,032 b) Rs 19,320 11. a) Rs 19.20 b) Rs 100
12. a) Rs 700 b) Rs 40,000 c) Rs 62,500 c) pRrso4fi8t6331% d) Rs 1,600, Rs 2,400
b)
13. a) profit 3% b) Rs 400, Rs 600, 4% loss
14. a) Rs 1,848 b) Rs 700 15. a) Rs 3,933
16. Complete your project work in a group or individually. Discuss the outcomes in the class.
Exercise : 2.2
1. Define the given terms yourself and discuss in the class.
discount amount q–p
2. a) Rs x – Rs y b) c = a – b% of a c) M. P. × 100% d) S.P. + VAT e) p × 100%
3. a) Rs 28 b) Rs 399 c) Rs 300 d) Rs 1,800
4. a) Rs 1,808 b) 13% c) Rs 1,260 d) Rs 1,100
5. a) (i) Rs 70 (ii) Rs 100.10 (iii) Rs 870.10 b) Rs 1,118.70
6. a) Rs 3,051 b) Rs 4,972 c) Rs 2,92,896 7. a) Rs 2,400 b) Rs 20,000 c) Rs 66,000
8. a) Rs 80 b) Rs 5,000 9. a) Rs 3,456 b) Rs 819 c) Rs 510
10. a) 10% b) 20% 11. a) 13% b) 13% 12. a) Rs 8,800 b) Rs 50,000
13. a) (i) Rs 3,200 (ii) Rs 4,000 b) Rs 1,500, Rs 1,230 14. a) Rs 10,500 b) Rs 46,980
15. a) Rs 120 b) Rs 1,600 c) Rs 600 16. a) 25% b) 3313% 17. a) 6% b) 2% 18. a) 10% b) 10%
19. Complete your project work in a group or individually. Discuss the outcomes in the class.
3. Commission and Taxation
Exercise : 3.1
1. a) Rs 37,500 b) Rs 1,62,900 c) 5.5% 2. a) Rs 9,180 b) Rs 23,12,500 c) 12%
3. a) Rs 70,930 b) Rs 33,250 c) (i) Rs 1,215 (ii) Rs 3,625 (iii) Rs 6,642
4. a) (i) Rs 30,300 (ii) Rs 6,50,000 b) Rs 23,100 c) 1%
5. a) Rs 22,500 b) (i) Rs 34,560 (ii) 2% (iii) Rs 62,50,000 c) Rs 20,832
Exercise : 3.2
1. a) Rs 3,054 b) Rs 3,876 2. a) Rs 3,627 b) Rs 11,100 c) Rs 1,22,000 d) Rs 5,01,600
3. a) Rs 9,666 b) Rs 3,911.40 c) Rs 11,150 d( Rs 44,000
4. Complete your project work in a group or individually. Discuss the outcomes in the class.
Vedanta Excel in Mathematics - Book 9 340 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Answers
Exercise : 3.3
1. a) Rs 960 b) (i) Rs 2,304 2. a) 20% b) 20% 3. a) Rs 1,12,00,000 b) Rs 48,60,000
4. a) Rs 36,210 b) Rs 1,26,360 5. a) 22% b) 27% 6. a) Rs 1,48,00,000 b) Rs 1,25,40,000
7. Complete your project work in a group or individually. Discuss the outcomes in the class.
4. Household Arithmetic
Exercise : 4.1
1. b) 145 units c) 4 units d) t = xy + z 2. a) Rs 30 b) Rs 30 c) Rs 75
d) Rs 84 3. a) Rs 162 b) Rs 149 4. a) Rs 142.50 b) Rs 175
c) Rs 136 5. a) Rs 82.32 b) Rs 119.70 c) Rs 187 d) Rs 306.25 6. a) Rs 1,097.50
b) (i) Rs 1,075.55 (ii) Rs 1,097.50 (iii) Rs 1,152.38 (iv) Rs 1,207.25 (v) Rs 1371.88
7. a) Rs 1,102.50, 6.67% b) Rs 3231.25, 21.6% 8. a) 90 units b) 130 units 9. a) Rs 320 b) Rs 321.20
Exercise : 4.2
1. a) Rs 621.50 b) 15.54 c) Rs 124.30 2. a) 500 calls b) 485 calls
3. a) Rs 960.30 b) Rs 9,882 c) (i) Rs 2,342.55 (ii) Rs 2,415 (iii) Rs 2,898 (iv) Rs 3,622.50
4. a) Rs 446 b) Rs 885 5. a) 5 km b) 10 km
6. and 7. Complete your project work in a group or individually. Discuss the outcomes in the class.
5. Mensuration
Exercise : 5.1
1 1 A
1. a) A = s(s – a) (s – b) (s – c) b) A = 3 p2 c) 2 x × y d) 2 h(p + q) e) A = w(l + b – w) f) N = a
4
2. a) (i) 84 cm2 (ii) 48 cm2 (iii) 12 3 cm2 (iv) 105 cm2 b) 48 cm2 c) 72 cm2
d) 26 cm2 e) 20.25 cm2 f) 40 cm2 g) 59.4 cm2 h) 154 cm2
3. a) 80 cm2 b) 564 m2 c) 254 m2 d) 462 cm2 e) 88 cm2 f) 147 cm2
g) 509 m2 h) 42 cm2 4. a) 1,700 m2 b) Rs 10,80,000 c) Rs 1,62,000
5. a) 7.8 m, 3.71 m2 b) 225 cm2 c) 224 cm2 d) (i) 22 cm (ii) 50 cm (iii) 154 cm2
(iv) 98 cm2 (v) 56 cm2 e) 360 cm, 5075 cm2
6. a) 60 m, Rs 6,600 b) 8 pc, Rs 6,000 c) Rs 12,375 d) 6,75,000, Rs 6,41,250
7. a) 690 m2, Rs 89,700 b) (i) Rs 69,480 (ii) Rs 8,36,400 c) (i) Rs 1,60,500 (ii) Rs 51,660
8. a) (i) 2,400 sq.ft (ii) 800 (iii) Rs 84,000 (b) (i) 616 m2 (ii) 7,700 (iii) Rs 2,69,500
c) (i) Rs 1,70,000 (ii) Rs 1,24,322.50 d) Rs 2,88,000 9. a) Rs 45,815 b) Rs 10,903.20
c) 10.5 m, 4935 m2 d) 14 cm, 77 cm2 10. a) Rs 2,583 b) (i) Rs 73,500 (ii) Rs 5,96,750
11. and 12. Complete your project work in a group or individually. Discuss the outcomes in the class.
Exercise : 5.2
1. a) 80 m2, 80 m2, 180 m2, 340 m2 b) 120 m2, 120 m2, 264 m2, 504 m2
2. a) (i) 300 m2 (ii) 300 m2 (iii) 455 m2 (iv) 1055 m2 b) (i) 288 m2 (ii) 240 m2
(iii) 528 m2 c) 209 m2 3. a) Rs 14,520 b) 162 m2 4. a) Rs 16,900 b) Rs 18,900
c) Rs 11,520 5. a) Rs 8,100 b) Rs 47,595 6. a) 5m b) Rs 11,520 c) Rs 7,200
d) 5 m e) 6 m 7. a) Rs 10,800 b) Rs 9,200 c) Rs 72,000 d) Rs 20,250
Exercise : 5.3
1. Answer the given questions yourself and discuss in the class. Then show to your teacher.
2. a) 540 cm3 b) 480 cm2 c) 255 cm2 d) 648 cm2, 810 cm3 3. a) 150 cm2
b) 64 cm3 c) 6 cm, 516 cm2 d) 4 cm e) 50 4. a) 96 cm2, 288 cm2, 480 cm2, 576 cm3
b) 38 cm2, 170 cm2, 246 cm2, 190 cm3 c) 48 cm2, 204 cm2, 300 cm2, 288 cm3
d) 32 cm2, 150 cm2, 214 cm2, 160 cm3 e) 24.47 cm2, 120 cm2, 168.94 cm2, 146.82 cm3
f) 600 cm2, 3,600 cm2, 4,200 cm2, 18,000 cm3 5. a) (i) 2,16,000 cm3, 60 cm (ii) 21,600 cm2
b) (i) 10 m2 (ii) 3,000 l (iii) Rs 12,000 c) (i) 960 (ii) 10 cm
c) 20 cm d) 27 7. a) 9.6 l
6. a) 6 cm b) 20 cm b) 10 cm, 4 l
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 341 Vedanta Excel in Mathematics - Book 9
Answers
Exercise : 5.4
1. a) 22.8 m3 b) 81.6 m3 2. a) 50,000 b) 20,800
3. a) (i) 27,500 (ii) Rs 4,67,500 b) 25,380, Rs 4,56,840 4. a) 5 m b) 15 m
5. a) Rs 10,800 b) 4 m c) Rs 6,480 d) Rs 6,336 e) 4.5 m 6. a) Rs 8,100 b) 4.2 m
7. Complete your project work in a group or individually. Discuss the outcomes in the class.
6. Algebraic Expressions
Exercise : 6.1
1. a) x (x + 1) b) x (x + 2) c) (x+ 2) (x + 1) d) (a + 3) (a + 3)
2. a) 2x (a + 2b) b) 2p(p – 3) c) 3ab (2a + 3b) d) 2px (x – 2 + 3p) e) 3x2y2 (2x + 3y – 1)
f) (2x + 3y) (a + b) g) (3a – 1) (x – y) h) (x + 3) (x + 2) i) (t – 1) (2t – 1)
3. a) (a + b) (x + y) b) (m + n) (p – q) c) (a + b) (a + c) d) (x2 + y2) (m – n) e) (x – 2) (y + 3)
f) (x + 4) (x + 3) g) (p – 8) (p – 1) h) (4x – 1) (4x – 1) i) (a – c) (a – b) j) (x – y) (x + 3)
k) (pr – q) (qr – p) l) (x + y + z) (y + z)
4. a) a b) 3x c) 2p d) x+y e) x+1 f) x–3
2b 2y p+1 3y x–3 x+2
1. a) (3x + 2) (3x – 2) Exercise : 6.2 c) 3a (4x + 5y) (4x – 5y)
b) (5ab + 1) (5ab – 1)
d) (x2 + y2) (x + y) (x – y) e) xy(4x2 + 9y2) (2x + 3y) (2x – 3y) f) (25a2 + 16b2) (5a + 4b) (5a – 4b)
g) (2 + m – n) (2 – m + n) h) (1 + a – b) (1 – a + b) i) (4 + 5p – 5q) (4 – 5p + 5q)
j) 3(a – b) (a + b) k) (a + b + c) (a + b – c) l) (p + q + r) (p – q – r)
m) (a + b – 2) (a – b + 2) n) (4a2 + 2a + 1) (4a2 – 2a – 1) o) (x + y) (ax – ay – 1)
p) (a – b) (a + b – x)
2. a) (x2 + xy + y2) (x2 – xy + y2) b) (a2 + a + 1) (a2 – a + 1)
c) (3x2 + 2xy + y2) (3x2 – 2xy + y2) d) (m2 + 2mn + 4n2) (m2 – 2mn + 4n2)
e) (2x2 + 6xy + 7y2) (2x2 – 6xy + 7y2) f) (p + q) (p – q) (3p + 5q) (3p – 5q)
g) (2y + 1) (2y – 1) (3y + 2) (3y – 2) h) (a2 + 2a + 2) (a2 – 2a + 2)
i) (8x2 + 4x + 1) (8x2 – 4x + 1) j) (10a2 + 15a + 9) (10a2 – 15a + 9)
k) (13b2 + 29bx + 31x2) (13b2 – 29bx + 31x2)
l) ( a2 + a + 1) ( a2 – a + 1) m) ( p2 + q2 + 1) (pq22 + q2 – 1)
b2 b b2 b q2 p2 p2
n) ( x2 + 3x + 1) ( x2 – 3x + 1) o) ( b2 + 3b – 3) ( b2 – 3b – 3)
y2 y y2 y d2 d d2 d
3. a) 96 cm2 b) 216 cm2 c) 299 m2 d) 392 m2 e) 544 sq.ft. f) 756 m2
4. a) (x + y + 5 (x – y + 1) b) (a + b – 2) (a – b – 8) c) (p + q – 14) (p – q + 2)
d) (x2 + y – 5) (x2 – y + 13) e) (3a + 4x – 4) (3a – 4x – 6) f) (25y + z – 2) (25y – z + 18)
g) (4p – 8q + 3r) (4p – 10q – 3r) h) (5x + 3y – z) (5x – 7y + z)
5. a) (ac + ab – bc + bd) (ac – ad + bc + bd) (b) (xy + x + y – 1) (xy – x – y – 1)
c) (3p + 2q + pq – 6) (3p + 2q – pq + 6) d) (30 – xy + 10x + 3y) (30 – xy – 10x – 3y)
6. Complete your project work in a group or individually. Discuss the outcomes in the class.
Exercise : 6.3
1. a) x2 + 5x + 6 = (x + 3) (x + 2) b) (x2 + x – 6) = (x + 3) (x – 2)
c) x2 – 7x + 12 = (x – 4) (x – 3) d) 2x2 + 7x + 3 = (2x + 1) (x + 3)
e) 2x2 + x – 6 = (2x – 3) (x + 2) f) 2x2 – 5x + 3 = (2x – 3) (x – 1)
2. a) (2x + y) (4x2 – 2xy + y2) b) (1 + 3a) (1 – 3a + 9a2) c) 2t (4t – 1) (16t2 + 4t + 1)
d) y(x – 4y) (x2 + 4xy + 16y2) e) (a2 + b) (a4 – a2b + b2) f) (4x2y – 5) (16x4y2 + 20x2y + 25)
Vedanta Excel in Mathematics - Book 9 342 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Answers
g) (a + 2) (a – 2) (a2 + 2a + 4) (a2 – 2a + 4) h) (2x + y) (2x – y) (4x2 – 2xy + y2) (4x2 + 2xy + y2)
i) (a + b + 1) (a2 + 2ab + b2 – a – b + 1) j) (x – 1) (x2 + x + 7)
k) – (x + 3y) (7x2 + 6xy + 3y2) l) (p + 1 ) (p2 – 1 + 1 ) m) ( a – b ) ( a2 + b2 + 1)
p p2 b a b2 a2
1 1
n) (3x – 4y) (9x2 + 12xy + 16y2) o) (2 + 3a) (4 – 6a + 9a2) p) x(x + x ) (x2 – 1 + x2 )
q) p(p + 1 ) (p – 1 ) (p2 – 1 + 1 ) (p + 1 + p1 ) (p – 1 + 1 )
p p p2 p
3. a) (x + 1) (x + 3) b) (x – 8) (x + 1) c) (a – 15) (a – 12) d) (x + 2) (2x + 3)
e) (p – 3) (3p + 2) f) (x – y) (2x + 5y) g) (a – b) (3a – 13b) h) abx (3a + 5b) (3a – b)
k) (x + y + 1) (2x + 2y + 7)
i) ( 3a + 4) ( 4a – 5) j) ( x – 3y ) ( x + y ) m) (2a + 3) (2a – 3) (2a2 + 1)
b b y x y x
l) (x – y – 2) (3x – 3y – 4)
n) (x + 2) (x2 – 2x + 4) (2x3 + 1) o) (x – 3) (x2 + 3x + 9) (3x3 + 2)
4. a) a–2 b) x + 3 c) 2a – 1 d) a–3
a–3 x + 5 a–2 a2 – 3a + 9
e) x+4 f) x+2 g) 1 h) (x + 2) i) x 1 1
x2 + x + 1 x–2 a+b +
5. a) (i) (x + 5) m, (x + 3) m (ii) (4x + 16) m b) (i) (x + 8) m, (x + 5) m (ii) (x2 + 9x + 18) sq. m.
c) (2x2 + 13x + 6) sq. m.
6. Complete your project work in a group or individually. Discuss the outcomes in the class.
Exercise : 6.4
1. a) H.C.F. = x + 2, L.C.M. = (x + 2) (x – 1) (x + 1)
b) H.C.F. = a – 5, L.C.M. = (a – 5) (a – 4) (a + 4)
c) H.C.F. = 2x – y, L.C.M. = (2x – y) (x + y) (x – y)
d) H.C.F. = a + b, L.C.M. = (a + b) (a – b) (a2 – ab + b2)
2. a) ax (x + 2) b) (x – 1) c) (a2 + 2a + 4) d) (x – 1) e) (x + 2)
f) (x + 2) g) (m2 + m + 1) h) x (2x + 1) i) (x2 + xy + y2) j) (a2 – ab + b2)
3. a) (x + 1) (x2 – 4) b) (x – 1) (x – 2) (x – 3) c) (x + 1) (x3 – 1) d) 3 (x2 – 9)
e) (x – 3) (4x2 – 25) f) (x – y) (x4 + x2y2 + y4) g) x (x2 – 9) (8x2 + 26x + 21)
h) (8x3 + y3) (8x3 – y3) i) (x – 1) (x – 2) (x – 3) j) 2x (x3 + 27)
4. a) 7 b) 2y + 3z c) xy (2a + 3b) d) 1 e) 3(x 4 3) f) a – 2b
y+2 2 (x – y) + a + 3x
5. a) (a + b) b) 1 c) 1 d) 4a e) (x – 1 – 1)
p+2 x+y 4a2 – 1 2) (x
f) (y – 3) 5 – 1) g) 0 h) 0 i) x2 + xy + y2 j) 0
(2y
8x xy(x2 + y2) x+y 4y –3
k) x2 – 4 l) x2 – y2 m) x2y2 n) a – 1 o) (y2 – 4) (y + 3)
p) 2a3 q) 2x3 r) 2b s) x+4
a2 – b2 x2 – 4 (a + b) (a – b)2 x2 – 1
t) 2 – 7x u) 1 – 5x
(x – 2) (x + 2)2 (x + 1) (x – 1)2
6. a) 2 b) 1 1 c) 2x d) 6xy e) 2 f) 2x3
x (x + 1) (x + 2) x2 – x+2 x2 – y2 x2 (x2 – 4) a4 – x4
7. a) (x – 3 – 5) b) (a – 3) (a 5 4) (a – 5) c) (x – 1 – 3) d) (x 3x – 7 3)
3) (x – 2) (x – 2) (x –
e) (y + 2y – y) 8. a) 2(x + y) b) 2 (a – 2) c) 2 (x – 3)
3)2 (3 x2 + xy + y2 (a2 – 2a + 4) (x2 – 3x + 9)
4
d) 1 + a2 + a4 9. a) 1 b) 1
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 343 Vedanta Excel in Mathematics - Book 9
Answers
7. Indices
Exercise : 7.1
1 1 1
1. a) am + n b) 1 c) 1 d) 2 2. a) 8 b) 5 c) 121 d) 125 e) 16 f) 2
g) 81 h) 14 i) 1 j) 25 k) 7 l) 2 m) 1 n) 2 o) 10 p) 3
16 13 2 5 2 3
2 2 9 16 2 5
3. a) 1 b) 1 c) 5 d) 3 4. a) 4a2x2 b) 25p2q2 c) 5 d) 7
5. a) 1 b) 1 c) 1 d) 1 e) 1 f) 1 g) 1 h) xy
ab x2 + y2
1 2
7. a) 6 b) 1 c) 2 d) 2 e) 5 f) 1
8. a) 15a2b b) a2 c)1 d) 2p e) 6m2n f) (a 1 b)2 g) 1 y)2 h) (a – b)
q + (2x –
9. a) 1 b) 1 c) 1 d) 1 e) 1 f) 1 g) 1 h) 1 i) x 2a j) a a+b
10. a) 1 b) 1 c) 1 y b
d) 1 e) 1 f) 1
Exercise : 7.2
n 1
1. a) x = p + q – r b) y = m + p – n c) m = n –f)1– d) 2 e) 0 f) 2
e) 4 3 g) –1
2. a) 2 b) 1 c) –4 d) –1 2 h) –1 i) 6
q) 1
j) –2 k) 2 l) 3 m) 1 n) 9 o) 3 p) 2 r) – 2 s) 1 t) 1
3. a) 4 b) 7 2 c) –2 5 2
d) 1
1
4. a) 3 b) 7 c) –3 d) 8 e) 4 f) 5 g) – 2 h) 2
5. a) 3 b) 3 c) 0 d) 1 e) 0 f) –2 g) 1 h) 2
i) –4 j) – 3 k) 1 l) 1 m) –2 n) 3 o) 2
6. a) –2 b) 1 c) 5 d) 0 3 1
e) 2 f) 3
9. a) ±2 b) ±2 c) 2, 3 d) 1, 2
e) 0, 1 f) 1, 2
10. Complete your project work in a group or individually. Discuss the outcomes in the class.
8. Simultaneous Linear Equation
Exercise : 8.1
1. a) x = 3, y = 2 b) x = 2, y = –2 c) x = –1, y = 13
d) x = –3, y = –2 e) x = 2, y = 0 f) x = 0, y = –3
2. a) x = 3, y = 2 b) x = –2, y = –2 c) x = 0, y = –4 d) x = –2, y = –1
3. a) x = 5, y = 3 b) x = –3, y = –4 c) x = – 1, y = 3 d) x = 2, y = 5
e) x = 5, y = 9 f) x = 1, y = – 2 g) x = 5, y = 3 h) x = 3, y = 2
i) x = 8, y = 3 j) x = 4, y = 3 k) x = 4, y = 3 l) x = –1, y = –2
m) x = 4, y = 1 n) x = 6, y = 2 o) x = –4, y = –3 p) x = 5, y = 2
Exercise : 8.2
1. a) 1 b) –4 c) y = –2, x = 2 d) 4 e) 6
2. a) (5, 2) b) (1, 4) c) (–3, 1) d) (3, 5) e) (5, 2) d) x = 2 , y = 1
3. a) x = 3, y = 2 b) x = 4, y = 1 c) x = 1, y = 3 h) x = 3, y = –2
f) x = 2, y = –1 g) x = –1, y = 1
e) x = 3 , y = –1
i) x = 5, y = 0 j) x = 1, y = 0 k) x = 1, y = 1 l) x = 1, y = 1
m) x = 2, y = 3 n) x = 3, y = –2 o) x = 10, y = 3
4. a) x = 3, y = 6 b) x = 2, y = 10 c) x = 1, y = –4 d) x = 6, y = 4
e) x = 2, y = –1 f) x = 1, y = 4 g) x = 2, y = 1 h) x = 5, y = 2
Vedanta Excel in Mathematics - Book 9 344 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Answers
i) x = 3, y = 2 j) x = 3, y = –2 k) x = 4, y = 1 l) x = 3, y = 0
m) x = 5, y = 7 n) x = 4, y = 6 o) x = 5, y = 5
5. a) 18, 12 b) Rs 500, Rs 750 c) umbrella - Rs 500, bag - Rs 1,500
6. a) Rs 7, Rs 25 b) adult - 3, children - 2 c) Rs 300, Rs 1,200 7. a) 27 yrs., 9 yrs.
b) 32 yrs., 8 yrs. c) 39 yrs., 9 yrs. d) 38 yrs., 8 yrs. 8. a) 34 b) 64 c) 27
9. Complete your project work in a group or individually. Discuss the outcomes in the class.
9. Quadratic Equation
Exercise : 9.1
1. a) ±1 b) –2, 3 c) 3, – 1 d) 25 , 2 e) 0, 1 f) 0, 4
2 5 2 3
2. a) ±3 b) ±2 4 d) ±52
g) ±58 h) ±170 c) ± 3 e) ±6 f) ±9
i) ±3 j) ±43 k) ±65 l) ±87
d) x2 – x – 12 = 0
3. a) x2 – 8x + 15 = 0 b) x2 – x – 6 = 0 c) x2 + 6x + 8 = 0
4. a) 0, 4 b) 0, 5 c) 0, 4 d) –1, –2 e) –3, –4 f) 1, 2
g) 3, 5 h) –3, 4 i) 1, 1 j) 4, – 2 k) –3, 7 l) 7, 3
2 3 5 2
1 1 11 3
5. a) 4, 3 b) 3, 2 c) 1, 2 d) 2, 3 e) 4, 2 f) 1, – 20
g) 3, – 1 h) –1, 2 i) ±8 j) ±15 k) 0, 4 l) 5, 23
2 3 7
9 1 1 113 1
6. a) 4, –1 b) 4 , – 4 c) 2 , d) 1, – 5 7. a) 6 cm, 8 cm b) 25 cm, 24 cm, 7 cm
8. a) 3, 4 b) 6, 9 c) 4, 6
Exercise : 9.2
1. a) 1, –3 b) 1, –7 c) 9, –1 d) 14, –4 e) 16 , – 7 f) 1, 1
6 5
2. a) 2, –3 b) 3, 4 c) –2, 5 d) 3, 7 e) 5, –4 f) –8, 3 1
3. a) 2, 1 b) 2, –4 c) 2, –5 d) 3, –1 2
e) 3, 4 f) –7, 4 g) , 3
h) –2, 1 i) – 2 , 3 j) 3, 1 k) –7 , 3 l) –a , a m) 2, 4 n) 1 , 1
3 3 2 3 5 5 3 3 3
11 40 1
o) –1, 5 p) 4, 2 q) 6, 13 r) 1, – 4
4. a) a = 2, b = 3, x = 1, 2 b) p = 5, q = 2, x = 1, 4 c) a = 2, h = 1, k = 3, x = –3, 2
Exercise : 9.3
1. Answer the questions yourself and discuss in the class. Then show to your teacher.
1 4 3
2. a) –1, –2 b) –2, –3 c) 2, –3 d) 2, 2 e) 2, 3 f) 2 , 5
3. a) ±2 b) ±3 c) ±4 d) ±52 e) 0, 1 f) 0, 3 g) 0, – 1 h) 0, – 2
2 3
1 7
4. a) 1, 2 b) 2, 3 c) 1, –3 d) 2, –4 e) 0, 5 f) 0, –14 g) –1, – 2 h) 3, – 3
i) –2, – 5 j) 5, – 5 k) 1, 6 l) 9 , – 5 5. a) 4, 4 b) 9 ± 33 c) 2, –3 d) ±9
3 2 g) 0, 4 10 6 3 2
5 4 1
e) 5, 2 f) 3, 3 h) 0, 2
6. a) 7, 3 b) 9, 12 c) 30, 6 d) 30, 22 e) 9, 7
7. Complete your project work in a group or individually. Discuss the outcomes in the class.
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 345 Vedanta Excel in Mathematics - Book 9
Answers
10. Ratio and Proportion
Exercise : 10.1
1. Complete the answer yourself and show to your teacher.
2. a) 2 : 5 b) 2 : 1 c) 10 : 3 d) 5 : 6 e) 10 : 3 3. a) 1 : 2 b) 4 : 7 c) 2 : 3
4. a) 9 : 16 b) 25 : 4 c) 36 : 49 5. a) 2 : 5 b) 4 : 3 c) 7 : 6
6. a) 8 : 27 b) 125 : 64 c) 343 : 216 7. a) 3 : 2 b) 4 : 5 c) 6 : 7
8. a) 8 : 3 b) 5 : 9 c) 11 : 4 9. a) 1 < 2 b) 3 > 4 c) 3 < 4
10. a) (i) 3 : 10 (ii) 3 : 4 : 10 2 3 5 7 10 5
b) (i) 4 : 7 (ii) 4 : 6 : 7
11. a) 3 : 2 b) 5 : 4 c) 25 : 9 d) 3 : 4 e) 2 : 1 f) 3 : 1
12. a) (i) 2 : 1 (ii) 3 : 5 b) (i) 0 (ii) 3 : 10 c) 16 d) 2
17 3
13. a) Rs 1,500, Rs 3,750 b) 324 g, 216 g c) 20°, 60°, 100° d) 36°, 72°, 108°, 144° e) 77, 55
14. a) 49 b) 27 c) 12 d) 765 g 15. a) 5 b) 10 c) 7 d) 27
16. a) 16, 12 b) 24, 60 c) 7 d) 5 17. a) 16 years, 24 years b) 24 years, 20 years
18. a) 10, 15 b) 30, 35 19. a) 5 l b) 10 l
Exercise : 10.2
1. a) 24 b) 28 c) 9 d) 15 e) 16 f) 75 g) 15 i) 2 2. a) 2 b) 2 c) 10 d) 2
11. and 12. Complete your project work in a group or individually. Discuss the outcomes in the class.
11. Geometry- Triangle
Exercise : 11.1
1. List the different types of angles separately in a table. Discuss in the class and show to your
teacher.
2. a) Only one b) Only one c) True, transitive axiom d) 30° e) 135°
Exercise : 11.2
1, 2, 3. Answer the questions yourself and discuss in the class. Then show to your teacher.
4. a) 55° b) 50°, 100° c) 35° d) 45°, 15° e) 30°, 60°, 90° f) 90°, 60°, 30° g) 90°, 70°, 20°
5. a) 30°, 60°, 90° b) 60°, 50° c) 90°, 140° d) 28°, 112° e) x = a = b = y = 60°
f) 80°, 120°, 160° g) 38°, 122° h) 30°, 20° i) 30°, 50° j) 50° k) 60°, 60°, 60°
l) 100°, 40°, 140° m) 48°, 48°, 48° n) 36°, 19°, 35° o) 40°, 75° p) 40° q) 55°, 165° r) 21°
6. a) 90° b) 40°, 100° c) 90° e) 130°
Exercise : 11.3
1. a) (i) longest side AB shortest side AC (ii) longest side EF shortest side DE
(iii) longest side PQ shortest side PR
b) (i) Greatest angle Y Smallest angle Z (ii) Greatest angle A Smallest angle B
(iii) Greatest angle F Smallest angle E
c) Longest side BC Shortest side AB d) Longest side QR Shortest side PQ
e) Greatest angle Y Smallest angle X
2. a) S.S.S., A = P, B = R, C = Q b) R.H.S., DF = ZX, F = Z, E = Y
c) S.A.S., BC = QR, C = R, B = Q d) A.S.A., F = S, EF = RS, GF = TS
3. a) S.S.S., a = 70°, x = 50°, b = y = 60° b) A.S.A., x = 6.4 cm, y = 5.8 cm, a = 30°
6. and 7. Complete your project work in a group or individually. Discuss the outcomes in the class.
Vedanta Excel in Mathematics - Book 9 346 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Answers
Exercise : 11.4
1. Answer the questions yourself and discuss in the class. Then show to your teacher.
2. a) 70°, 70°, 40° b) 130° c) 60°, 60° d) 20° e) 110° f) 69°, 69°, 111°
g) 70°, 140° h) 65°, 130°, 65° i) 37°, 106° j) 60°, 90° k) 30°, 60° l) 70°, 40°
3. a) 75°, 75°, 30° b) 60°, 70°, 40°, 80° c) 110°
5. a) x = 2, y = 3 b) x = 4 cm, y = 3 cm 6. b) 75°
13. Complete your project work in a group or individually. Discuss the outcomes in the class.
1 Exercise : 11.5
2
1. a) MN = BC b) QY = YR c) 6.4 cm d) 2.9 cm 2. a) 5.2 cm, 30° b) 30°, 95°, 55°, 55°, 30°, 30°
c) 30°, 130° 3. a) 30 cm b) 18 cm 4. a) 4 cm, 6 cm b) 5 cm, 9 cm
12. Geometry: Similarity
Exercise : 12.1
1. a) 8 cm b) 2 cm c) 10 cm, 9 cm d) 12 cm e) 2 cm
2. 4 cm, 9 cm, 4.5 cm 3. a) 4.5 cm b) 7.5 cm, 4.5 cm c) 9 cm, 5 cm d) 3.2 cm, 2.4 cm e) 6 cm
4. a) 6 cm b) 6 cm c) 2 m
Exercise : 12.2
1. Answer the questions yourself and discuss in the class. Then show to your teacher.
2. a)(i) 3, 4, 5 (iii) 6, 8, 10 (iv) 9, 12, 15 (vii) 8, 15, 17 (viii)4.5, 6, 7.5
c) 6, 8, 10; 9, 12, 15; 12, 16, 20 4. c) 9.6 cm, 12.8 cm 6. a) 6 m b) 13 m
3. a) 8 cm, 4.8 cm b) 5 cm, 4 cm c) 8 cm, 9 cm
8. Complete your project work in a group or individually. Discuss the outcomes in the class.
13. Parallelogram
Exercise - 13.1
1. Answer the questions yourself and discuss in the class. Then show to your teacher.
2. a) 10 cm b) 45° c) 40° d) 30°
3. a) w = y = 2x = 72°, z = 3x° = 108° b) a = 4y = 80°, b = 5y = 100°
c) p + 10° = 2p – 50° = 70°, p + q = r = 110° d) x = 20°, 2x = 40°
e) y = 65°, x = 50° f) x = 125° g) a = 40°, b = 140° h) x = 70°, y = 110°
4. a) 70° b) 70° c) 120° d) 25°, 65°
5. a) 140° b) 33° c) 60°, 120° d) 62°, 28° e) 25°, 115° f) 30°, 60° g) 65°
6. a) x = y = 4 cm b) x = 15, y = 5 c) a = 5, b = 1 d) 8 cm e) p = 3 cm, q = 6 cm
7. a) 75°, 15° b) 15°, 75°
14. Complete your project work in a group or individually. Discuss the outcomes in the class.
14. Circle
Exercise - 14.1
1, 2, 3. Answer the questions yourself and discuss in the class. Then show to your teacher.
4. a) 12 cm b) 8 cm c) 24 cm d) 4 cm 5. a) 2 cm b) 10 cm
6. a) 6 cm b) 14 cm c) 6 cm 7. a) (i) 1 cm (ii) 7 cm b) 2.5 cm c) 13 cm
15. a) 10 m b) 48 ft. c) 160 m
16. Complete your project work in a group or individually. Discuss the outcomes in the class.
15. Construction
Complete the construction of quadrilaterals yourself and compare with your friends. Then show to
your teacher.
7. Complete your project work in a group or individually. Discuss the outcomes in the class.
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 347 Vedanta Excel in Mathematics - Book 9
Answers
16. Trigonometry
Exercise - 16.1
1. a) sinT = AB , cosT = BC , tanT = AB b) sinD = PQ , cosD = PR , tanD = PQ
AC AC BC QR QR PR
c) sinE = YZ , cosE = ZX , tanE = YZ 2. a) sinT = 3 , cosT = 4 , tanT = 3
XY XY ZX 5 5 4
b) sinD = 3 , cosD = 4 , tanD = 3 c) sinE = 12 , cosE = 5 ,tanE = 12
5 5 4 13 13 5
3. a) sinD = CQ , tanT = CA b) sinE = BY , cosT = AC
PC AB BX BC
c) tanD = PR , cosT = PR d) sinD = QPRR , cosT = TR
PQ QR QR
4. a) 1 – cos2$ b) 1 – sin2T c) SinT d) 1 – cos2D e) 4 f) 12
1 – sin2T cosD 5 13
g) 4 h) 5 i) 21 j) 4 k) sinD = AB l) cosT AB
5 12 29 3 CA CA
5. a) sinT = 5 , tanD = 3 b) cosD = 3 , tanT = 12 6. a) 3 , 3 b) 12 , 5
13 4 5 5 5 4 13 12
7. a) 4 , 3 b) 3 , 3 9. a) 112 b) 112 10. a) 3 b) 12 , 5 c) 4 , 4
5 5 5 4 65 65 5 13 12 5 3
3 1 Exercise - 16.2 3 1
4 2 2 4
1. a) b) c) 3 d) 0 e) f) 0 g)
h) 1 i) 1 j) 5 k) 1 l) 1 m) 1 n) 1 2
2 2
o) 1 p) 3 q) 0 r) 1 s) 3 t) 1 u) 1
4 4 2
v) 1 w) 1 x) 3 y) 2 z) 0 3. a) 15 cm, 8.7 cm b) 8 cm,
11.3 cm 4. 6.4 cm 5. a) 6.4 cm, 7.7 cm b) 15.96
c) 13.9 cm, 6.9 cm
cm, 5.47 cm c) 5.76 cm, 19.05 cm
6. a) 60°, 30° b) 30°, 60° c) 45°, 45° 7. 61.5 m 8. 65.82 m 9. 80 m, 58.56 m 10. 136.6 m
11. and 12. Complete your project work in a group or individually. Discuss the outcomes in the class.
17. Statistics
Exercise - 17.1
Answer the questions yourself and discuss in the class. Then show to your teacher.
8. Complete your project work in a group or individually. Discuss the outcomes in the class.
Exercise - 17.2
1. b) 108° c) 36° d) 150°
2. a) b) House rent c) Clothing
Salary Food 140° Saving 110°
Business 110° 100° 105°
Others Food
82° 40° 70°
Others
Agriculture
House 40° 75°
rent Education
80° 48° 80° Miscellaneous
d) e) f)
Officer Manager Addidas Oil
100° 120° Coal 126°
Nike 117° 90° 18°Others
Sweeper Natura7l2g°asH3y16d8°r°oNuclear
108° Converse
Clerk 48° 32° 18°
60° Security Goldstar
guard 81° Fila
36°
Vedanta Excel in Mathematics - Book 9 348 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Answers
3. a) Food Rent Education Transportation Miscellaneous
b) 6750 3375 2700 2250 1125
Salaries Raw Materials Fuel Extras
1,50,000 1,20,000 50,000 40,000
c) (i) 120 (ii) 80 (iii) 360 d) (i) 6720 (ii) Y = 4800
e)
Cotton Nylon Polyester Others
25 % 15% 40% 20%
12.5 kg 7.5 kg 20 kg 10 kg
4. Complete your project work in a group or individually. Discuss the outcomes in the class.
Exercise - 17.3
1. a) (i) 50 (ii) 20 (iii) 20 b) (i) 30 (ii) (10 - 20) (iii) 10
c) (i) 20 (ii) 16 d) (i) 75 (ii) 30
4, 5, 6. Draw the ogive curves yourself and discuss in the class. Then show to your teacher.
Exercise - 17.4
1. a) 34 b) Rs 450 c) 13 years d) 67 2. a) 10 years b) 7 c) 7
3. a) 120 b) 27, 93 c) 5 d) 25
4. a) 66 b) 10.8 years c) 63.44 kg d) 62.2 cm
5. a) 28 b) 7 c) (i) 35 (ii) 100 d) (i) 11 (ii) 50
6. a) 11 b) 20 c) 5 d) 10
Exercise - 17.5
1. Answer the questions yourself and show to your teacher.
2. a) (i) 20 (ii) 28 (iii) 30 (iv) 24 (v) 13 b) 50 kg c) 34 years d) 32
3. a) (i) 12 (ii) 10 (iii) 10 b) (i) 39 (ii) 34 (iii) 70 4. a) 25
b) 158 c) 6 d) 7 5. a) 18 b) 172 c) 3 d) 7
6. a) (i) 7 (ii) 21 kg b) 18 years c) Rs 36 d) 8 to 12 e) Rs 1,500 to Rs 2,000
f) (i) c (ii) f g) 7 h) Rs 150 7. a) 34.5 years b) Rs 542.85 8. a) 50 b) 20 c) Rs 65 d) 40 kg
9. a) 36 b) 80 c) 27, 37 d) Q1 = 100 cm, Q2 = 120 cm, Q3 = 130 cm
10. Complete your project work in a group or individually. Discuss the outcomes in the class.
18. Probability
Exercise - 18.1
n (E)
1. a) P {a certain event} = 1 P {an impossible event} = 0 b) P (E) = n(S)
c) (i) {HH, HT, TH, TT}
(ii) {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
(iii) {1, 2, 3, 4, 5, 6}
(iv) {(1, 1), (1, 2), (1, 3) (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2),
(3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5),
(5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}
d), e), f) and g ) Answer the questions yourself and discuss in the class. Then show to your teacher.
2. a) 1 b) 1 c) 3 d) 1 e) 5 f) 1 g) 1
6 5 5 8 6 2 13
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 349 Vedanta Excel in Mathematics - Book 9
Answers
h) 1 i) 1 j) 1 k) 5 l) 1 m) 1 n) 2
26 3 5 9 3 7 3
3. a) (i) 1 (ii) 2 (iii) 2 (iv) 10 (v) 10 (vi) 9
11 11 11 11 11 11
b) 1 , 12 c) (i) 1 (ii) 1 (iii) 1 d) (i) 1 (ii) 1 (iii) 1
13 13 13 52 26 8 2 4
(iv) 3 (v) 3 (vi) 3 4. a) 2 b) 7 c) 1 d) 4
8 4 4 3 11 2 13
e) 1 f) 23 g) 8 h) 5 i) 1
3 26 13 6 4
5. a) 0.562 b) 0.001 c) 870 d) (i) 0.27 (ii) 0.38 e) (i) 0.82 (ii) 0.965 (iii) 0.08
6. and 7. Complete your project work in a group or individually. Discuss the outcomes in the class.
Revision and Practice Time
1. (i) 30 AV Set SM
(ii) 70 20 30 50 10 20 45
2. (i) 45
3. i) P (ii) 15% 4. (i) 70% (ii) 20% (iii)
S
P S
15% 45% 25% 10% 70% 20%
15%
5. (i) N (ii) 40% 6. (i) 400 PS
M (ii) 120 50% 10% 30%
10% 40% 20%
7. (i) 12 30% 8. (i) 64 10%
(ii) 37 AO MS
(iii) 35 25 12 23 72 24 64
20
9. (i) 10% 10. (i) 30 11. (i) 40
(ii) 55% M S (ii) 30 M S (ii) 55 T C
45% 10% 35% 15 15 30 40 20 55
10% 10
Vedanta Excel in Mathematics - Book 9
350 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
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