Statistics (II): Measures of Central Tendency
2 (N + 1)th N + 1 th
4 2
The position of the second quartile (Q2) = term = term
The position of the third quartile (Q3) = 3 (N + 1) th term.
4
After finding the positions of a quartile, we apply the similar process of computing the
required quartile as like the process of computing median.
Example 5: Find the first quartile (Q1) and the third quartile (Q3) from the data given
below.
18, 24, 20, 19, 23, 34, 28
Solution:
Arranging the data in ascending order,
18, 19, 20, 23, 24, 28, 34
Here, N = 7
The position of the first quartile (Q1) = N + 1 th term = 7+1 th term = 2nd item
4 4
Thus Q1 lies at the 2nd position.
∴ The first quartile (Q1) = 19.
Again, the position of the third quartile (Q3) = 3(N + 1) th term = 6th term
4
Thus, Q3 lies at the 6th position.
∴ The third quartile (Q3) = 28
Example 6: The observations 2x + 1, 3x – 1, 3x + 5, 5x – 1, 38, 7x – 4 and 50 are in
ascending order. If the first quartile is 20, what is the value of x? Also, find
the third quartile of the observations.
Solution: Here,
The given data in ascending order is 2x + 1, 3x – 1, 3x + 5, 5x – 1, 38, 7x – 4, 50
No. of items (n) = 7
N+1 th 7+1 th
4 4
Now, the position of the first quartile (Q1) = term = = 2nd term
The value of 2nd term is 3x – 1. So, the first quartile (Q1) = 3x – 1.
According to question, Q1 = 20
or, 3x – 1 = 20
or, 3x = 21
∴ x = 7
Again, the position of the third quartile (Q3) = 3 n + 1 th 7+1 th
4 4
term = 3 term = 6th term
The value of 6th term is 7x – 4.
∴ The third quartile (Q3) = 7x – 4 = 7 × 7 – 4 = 45.
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Statistics (II): Measures of Central Tendency
Example 7: Compute the first and the third quartiles from the table given below.
Marks 40 50 60 70 80 90
Solution: No. of students 4 6 10 12 5 2
Cumulative frequency distribution table,
Marks (x) No. of students (f) c.f.
40 4 4
50 6 10
60 10 20
70 12 32
80 5 37
90 2 39
Total N = 39
Now, the position of the first quartile (Q1) = N + 1 th term = 39 + 1 th term = 10th term
4 4
In c.f. column, the corresponding value of the c.f. 10 is 50.
∴ The first quartile (Q1) = 50 3 (N + 1) th term = 30th term
and i4ts corresponding value
Again, the position of the third quartile (Q3) = is 70.
In c.f. column, the c.f. just greater than 30 is 32
∴ The third quartile (Q3) = 70
18.6 Mode
Let’s take a set of ordered observations. 2, 5, 5, 7, 7, 7, 7, 10, 10, 10, 15
Here, the observation ‘7’ appears maximum number of times. Therefore, 7 is the
mode of the given observations. Thus, the mode of a set of data is the value with the
highest frequency. A distribution that has two modes is called bimodal. The mode
of a set of data is denoted by Mo.
(i) Mode of discrete data
In the case of discrete data, mode can be found just by inspection, i.e., just by
taking an item with highest frequency.
Example 8: Find the mode for the following distribution,
12, 10, 18, 12, 10, 12, 15, 12, 10
Solution:
Arranging the data in ascending order
10, 10, 10, 12, 12, 12, 12, 15, 18
Here, 12 has the highest frequency.
∴ Mode = 12
Example 9: Find the mode of the given distribution:
Marks 27 30 36 40 45 50
No. of students 4 10 16 9 6 5
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Statistics (II): Measures of Central Tendency
Solution:
Here, the marks 36 has the highest frequency, i.e., 16.
∴ Mode = 36.
(ii) Mode of grouped and continuous data
In the case of grouped and continuous data, the class with highest frequency
is observed and it is taken as the model class. Then, by using the following
formula, mode can be computed.
Mode (M0) = L + f1 – f0 f2 ×c
2f1 –f0 –
Where, L = the lower limit of the model class
f0 = the frequency of the class preceding the model class
f1 = the frequency of the model class
f2 = the frequency of the class succeeding the model class
c = the width of the class interval
Alternatively, mode can also be computed by the following empirical relation.
Mode = 3 Median – 2 Mean
Example 10: Find the model class from the data given below:
x 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50
Solution: f 6 8 12 7 3
Here, 12 is the highest frequency and its corresponding class is 20 – 30.
∴ Model class is 20 – 30.
Example 11: Compute the mode from the data given below:
Age in years 3 – 5 5 – 7 7 – 9 9 – 11 11 – 13 13 – 15
Solution: No. of pupils 10 35 70 35 12 6
Here, 70 is the highest frequency and its corresponding class is 7 – 9.
∴ Model class is 7 – 9.
Now, L = 7, f1 = 70, f0 = 35, f2 = 35 and c = 2
∴ Mode (M0) = L + f1 – f0 f2 × c = 7 + 70 – 35 35 × 2 = 7 + 35 × 2 = 8
2f1 –f0 – 140 – 35 – 70
So, the required mode is 8 years.
18.7 Range
The difference between the largest and the smallest score is called range.
∴ Range = Largest score – Smallest score
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Example 12: The marks obtained by 10 students of class 9 in Mathematics are given
below. Find the range.
78 36 27 95 43
15 69 84 72 51
Here, the highest marks = 95
The lowest marks = 15
∴ Range = highest score – lowest score
= 95 – 15 = 80
EXERCISE 18.2
General section
1. a) Define median of a set of observations.
b) Define the first quartile (Q1) and the third quartile (Q3) of set of observations.
c) Write the formula to find the position of median of a descrete series.
d) Write the formulae to find Q1 and Q3 of a descrete series.
e) Define mode. Write the formula to find mode of a grouped and continuous data.
f) Define range.
2. a) Find the medians of the following sets of data.
(i) 18, 16, 27, 20, 25
(ii) 21, 28, 14, 42, 35
(iii) 15, 30, 35, 25, 20, 45, 40
(iv) 22, 16, 14, 26, 32, 30
(v) 16, 13, 10, 14, 11, 12, 15
b) The weights of five high school students are given below. Find their median weight.
50 kg, 54 kg, 45 kg, 63 kg, 48 kg
c) Find the median age of a group of 7 people whose ages in years are as follows.
47, 61, 13, 34, 56, 22, 8
d) What is the position of the median marks in the table given below?
Marks 18 27 32 40 46
No. of students 2 3 10 9 5
3. a) Find the first quartiles (Q1) of the following sets of data.
(i) 14, 12, 17, 23, 20, 16, 10
(ii) 16, 25, 10, 30, 35, 8, 12
(iii) 40, 20, 30, 10, 16, 12, 8
b) Find the third quartiles (Q3) of the following sets of data.
(i) 15, 9, 21, 33, 27, 39, 45
(ii) 18, 26, 14, 22, 30, 38, 34
(iii) 30, 20, 50, 80, 40, 60, 70
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Statistics (II): Measures of Central Tendency
4. a) If the given data is in ascending order and the median is 40, find the value of x.
20, 30, 3x + 5, 50, 60
2
b) If the given data is in ascending order and the median is 200, find the value of p.
100, 150, 5p + 10, 250, 300
4
c) a + 1, 2a – 1, a + 7 and 3a + 4 are in ascending order. If the median is 17, find the
value of a.
d) 5, k + 3, 2k + 1, 3k – 2, 26 and 31 are in ascending order. If its median is 17, find
the value of k.
5. a) 20, x + 6, 35, 40, 47, 51, 58 are in ascending order. If the first quartile of the data is
24, find the value of x.
b) 110, 2x + 1, 120, 125, 130, 135, and 140 are in ascending order. If the first quartile
3
of the data is 115, find the value of x.
c) 9, 11, 12, 14, 18, 8m + 1, 28 are in ascending order. If the third quartile is 25, find
the value of m.
d) x + 1, x + 3, x + 5, 2x, 3x – 5, 4x – 10 and 3x – 1 are in ascending order. If the upper
quartile of the data is 18, find the value of x.
6. a) Find the modes of the following distributions.
(i) 7, 9, 5, 7, 10, 9, 7, 12
(ii) 15 kg, 21 kg, 17 kg, 21 kg, 28 kg, 21 kg, 15 kg, 21 kg
b) In a class, there are 15 students of 16 years, 14 students of 17 years, and 16 students
of 18 years. What is the modal age of the class?
c) In a factory, the daily wages of 30 labourers is Rs 25, 35 labourers is Rs 36, and
20 labourers is Rs 45. What is the modal wage of the factory?
d) The age of 5000 students of a school are shown in the table. What is its modal class?
Age in years No. of students
4 to 8 1050
8 to 12 2856
12 to 16 1094
e) In a factory, number of labourers and their remuneration are as follows. Find the
modal class.
Remuneration (in Rs) 1500 – 2000 2000 – 2500 2500 – 3000
No. of labourers 220 215 120
f) Find the mode of the following distribution.
(i) a, c, b, b, c, a, c, a, b, c, a, c, (ii) d, f, e, f, e, d, f, d, e, f, d, f
g) Find the modal size of the shoes from the data given below.
Size of shoe 5 6 7 8 9 10
No. of men 10 15 30 25 18 12
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h) Find the mode from the following data:
Daily wages (in Rs) 70 90 110 130 150 170
No. of workers 4 12 15 18 20 12
7. a) The heights of 10 students of class 9 are given below.
95 cm, 110 cm, 120 cm 90 cm, 100 cm,
105 cm, 98 cm, 115 cm, 1 12 cm, 116 cm
(i) What is the height of the tallest student?
(ii) What is the height of the shortest student?
(iii) Find the range of the height.
b) Find the range of the following data.
35, 40, 58, 70, 45, 27, 86, 65
c) The marks obtained by 20 students in Mathematics are given below. Find the
range.
Marks 40 50 60 70 80 90
No. of students 2 3 5 7 2 1
Creative section
8. Find the mode from the following distribution.
a) Age in years 20 – 30 30 – 40 40 – 50 50 – 60 60 – 70 70 – 80
50 40
No. of people 150 200 140 90
b) Daily wages (in Rs) 400 – 500 500 – 600 600 – 700 700 – 800 800 – 900
No. of workers 35 50 30 25 18
9. Find the median marks from the data given below:
a) Marks 24 36 50 65 78
No. of students 2 4 12 11 6
b) Marks 10 15 20 25 30 35
No. of students 3 7 15 12 7 3
c) Wage per hour (in Rs) 45 55 65 75 85 95
No. of workers 20 25 24 18 15 7
d) Weight (in kg) 36 40 25 44 30 49
No. of students 8 10 3 65 4
10. a) Calculate the first quartile (Q1) from the data given below:
Marks obtained 32 36 40 44 48 52
No. of students 259632
b) Compute the third quartile (Q3) from the following table:
Wages per hour (in Rs) 50 60 70 80 90 100
No. of workers 6 10 15 13 8 3
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Statistics (II): Measures of Central Tendency
c) Find the first quartile (Q1) and the third quartile (Q3) from the following distribution:
Ages (in years) 22 27 32 37 42
No. of people 35 42 40 30 24
d) Find the quartiles from the following distribution:
Height (in cm) 90 100 110 120 130 140
No. of students 20 28 24 40 35 18
Project work and activity section
11. a) Measure the height of your any 10 friends and compute the mean, median, and
mode height.
b) Write the number of students in class 4 to class 10 of your school. Collect the data
of monthly fees of each students of these classes. Show these information in table.
Then compute the mean, median and quartiles of the data.
OBJECTIVE QUESTIONS
Let’s tick (√) the correct alternative.
1. In a discrete data, if Σfx = 66a + 99b and Σf = 6a + 9b, what is the mean ( x ) ?
(A) 6 (B) 9 (C) 11 (D) 15
2. In a data having 9 observations, if the 5th term is 20, what is the value of median?
(A) 9 (B) 20 (C) 5 (D) 34
3. A data has 7 observations. If its 2nd term is 4x + 1, and the value of Q1 is 45, what is the
value of x?
(A) 2 (B) 4 (C) 9 (D) 11
4. Of all mathematics textbooks from different publications in a stationery, a student
has to purchase the textbook which is most liked by students and teachers. What
measure of central tendency would be most appropriate, if the data is provided to her?
(A) Mean (B) Mode C) Median (D) Any of the three
5. The mean of three different natural numbers is 20. If lowest number
is 7, what could be highest possible number of remaining two numbers?
(A) 45 (B) 40 (C) 48 (D) 52
6. In a discrete data, the mean of 100 observations was 60. If the frequency of the
observation 60 is mistakenly written as 25 instead of 15 then what will be exact mean?
(A) 60 (B) 56 (C) 54 (D) 49
7. The mean of a data with 11 observations is 42. If the mean of first 6 observations is 39
and that of last 6 observations is 44, what is the sixth observation?
(A) 35 (B) 36 (C) 38 (D) 43
8. What is the name of the quartile which divides the data below 25%?
(A) Q1 (B) Q2 (C) Q3 (D) both (A) and (C)
9. In what ratio does the upper quartile divide the data arranged in ascending order from
bottom?
(A) 1:2 (B) 1:3 (C) 3:1 (D) 2:1
10. What percent of observations are there below Q3 of a data? 75%
(A) 25% (B) 50% (C) 60% (D)
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Unit 18 Probability
19.1 Probability –Looking back
Classwork-Exercise
1. Answer the following questions as quickly as possible.
a) (i) How many faces are there in a coin?
(ii) Which faces can be surely placed when the coin is tossed?
(iii) Can both faces be shown at once?
(iv) What is the possibility of getting head?
(v) What is the possibility of getting tail?
b) (i) How many faces are there in a die?
(ii) Which faces can be surely placed when a die is rolled?
(iii) What is the possibility of getting 1?
2. Choose the words from the table given below that best describes an event
that has a possibility of: a) 0% b) 50% c) 100% d) 120%
Certain impossible probable improbable
19.2 Probability – Introduction
In our real-life situations, we often talk about the probability of happening of any
events. For example, probability of raining, probability of increasing the price of
petroleum, probability of winning the games or election and so on are a few cases.
Thus, probability refers something likely to occur; however, it is not certain to occur.
Facts to remember
Probability is the numerical measurement of the degree of certainty of the occurrence
of events.
For example, when a coin is tossed, it is 50/50 chance that the head or tail occurs.
1 21.
So, the probability of occurrence of head is 2 and that of tail is also
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Probability
The first foundation of modern mathematical theory of probability
was laid in the seventeenth century by French mathematicians
Pierre de Fermat (1601-1665) and Blaise Pascal (1623-1662) while
contemplating a gambling problem posed by Chevalier de Mere,
a gambler and a member of aristocracy in 1654. James Bernoulli
(1654 – 1705), A. de Moivre (1667 – 1754), and Pierre Simon
Laplace are among those who made significant contributions to
this field. Laplace’s Theorie Analytique des Probabilités, 1812, is considered to be
the greatest contribution by a single person to the theory of probability.
Following are a few important terms which are frequently used in probability. It is
essentially important to have the proper concept of these terms.
Experiments and outcomes:
An action by which an observation is made is called the experiment. Any experiment
whose outcome cannot be predicated or determined in advance is called a random
experiment.
A few examples of random experiments
a) Tossing a coin
b) Rolling a die
c) Drawing a card from a well
shuffled pack of 52 playing cards
The results of a random experiment are called outcomes. For example, while
tossing a coin, the occurrence of head or tail is the outcome.
Sample space:
Each performance in a random experiment is called a trial and an outcome of a trial
is called a sample point. The set of all possible outcomes (i.e., sample points) of a
random experiment is known as sample space. Usually, a sample space is denoted
by S.
For example,
(i) The possible outcomes of a random experiment of throwing a die are 1, 2, 3,
4, 5 or 6.
∴ The sample space, S = {1, 2, 3, 4, 5, 6}
(ii) The possible outcomes of a random experiment of tossing a coin are head (H)
or tail (T).
∴ The sample space, S = {H, T}
(iii) When two coins are tossed simultaneously, the sample space,
S = {HH, HT, TH, TT}
Event:
A sample space S of a random experiment is a universal set. Every non-empty subset
of the sample space S is called an event. For example, while tossing a coin, the
sample space, S = {H, T}. Here, the subsets {H}, {T} and {H, T} are the events of
S. The empty set I is also an event of S, but it is the ‘impossible event’. S is called
the ‘sure event’. An event containing only one element of S is called a simple or
elementary event.
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Probability
For example, If S = {H, T}, then {H} and {T} are the elementary events.
If S = {1, 2, 3, 4, 5, 6}, then {1}, {2}, {3}, {4}, {5}, and {6} are the elementary
events.
Exhaustive cases:
The total number of all possible outcomes of a random experiment is known as
exhaustive cases.
For example, while tossing a coin, S = {H, T}. So, exhaustive cases = 2
While tossing two coins simultaneously, S = {HH, HT, TH, TT} So, exhaustive cases
= 4. Similarly, the exhaustive cases of tossing three coins simultaneously is 8. Thus,
if n denotes the number of tossing coins simultaneously, then the exhaustive cases
can be obtained as 2n.
Mutually exclusive events:
Two or more events of a sample space S are said to be mutually exclusive if the
occurrence of any one event excludes the occurrence of the other events.
For example, while tossing a coin, the occurrence of head excludes the occurrence of
tail or vice versa. So, they are mutually exclusive events.
Furthermore, consider the experiment of throwing a die. Let
A be the event, ‘the number obtained is less than 5’. Then,
A = {1, 2, 3, 4}.
Again, let B be the event, ‘the number obtained is at least 5.
Then, B = {5, 6} Here, A ∩ B = φ.
Thus, the joint occurrence of A and B is an impossible event. In this case the events
A and B are called mutually exclusive events. In general, if A and B are any two
events on a sample space S and A ∩ B = φ, the events A and B are said to be
mutually exclusive.
Independent and dependent events
Two or more events are said to be independent if the occurrence of one of the events
does not affect the occurrence of the other events. For example, in the random
experiment of tossing a coin twice or more, the occurrence of any one event in the
first trial does not affect the occurrence of any event in the second trial.
On the other hand, two or more events are said to be dependent if the occurrence
of one of the events affects the occurrence of the other events. For example, while
drawing a ball in two successive trials from a bag containing 2 red and 3 green
without a replacement, getting any one coloured ball in the first trial affects to draw
another ball in the second trial. So, these are the dependent events.
Equally likely events
Two or more events are said to be equally likely if the chance of occurring any one
event is equal to the chance of occurring other cases. For example, while throwing a
die, the chance of coming up the numbers 1 to 6 is equal. similarly, while tossing a
coin, head (H) or tail (T) are equally likely events.
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Probability
Favourable and unfavourable cases
The outcomes in an random experiment which are desirable (or expected) to us are
called favourable cases and all other cases are unfavourable cases.
For example:
While tossing a coin, S = {H, T}
Here, the favourable number of case of head is 1 and tail is also 1.
While tossing two coins, S = {HH, HT, TH, TT}
The favourable number of case of both of them head is 1 and tail is also 1.
The favourable number of cases of at least one head is 3 and at least tail is also 3.
Some useful facts about playing cards, coin and die
a) Playing cards
There are 52 cards in a packet of playing cards.
There are 26 red and 26 black coloured cards in the packet.
13 Hearts (♥) and 13 diamonds (♦) are the red coloured cards.
13 Spades (♠) and 13 clubs (♣) are the black coloured cards.
There are 4 suits (Hearts, Clubs, Diamonds, Spade), each with 13 cards
(Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King).
There are 12 face cards in the packet. Among them 4 are Jacks (heart,
diamond, spade and club), 4 are Queens and 4 are Kings.
b) Coin
There are two faces of a coin, head (H) and tail (T).
c) Die
There are six faces in a die which are numbered from 1 to 6.
In a fair die, the sum of the numbers turning on the opposite
sides (1 and 6, 2 and 5, 3 and 4) are always equal to 7.
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Probability
19.3 Probability of an event
Let n(S) be the total number of all possible outcomes (Exhaustive number of cases)
of an experiment and n(E) be the favourable number of cases of the sample space S,
then the probability of happening the event (E) is defined as,
P(E) = Favourable number of cases = n(E)
Exhaustive number of cases n(S)
Also, the probability of non–happening of the event is given by P'(E) = 1 – P(E)
19.4 Probability scale
Probability is measured on a scale from 0 to 1. A 0 (zero) probability means there is
no chance of an event happening. A probability of 1 means that it is certain the
event will happen.
In the adjoining spinner, the probability of the pointer to stop on
green is 41. The probability of the pointer to stop on blue is 41.
The probability of the pointer to stop on yellow is 14.
The probability of the pointer to stop on green or yellow is 21.
The probability of the pointer to stop on black is 0, which means not possible.
The probability of the pointer to stop on green, blue, red or yellow is 1.
Similarly, there are 6 blue, 9 red and 5 green marbles of same shape and size in a
bag. When a marble is randomly taken out, then
the probability of getting a blue marble, P (B) is 6 = 0.3
20
the probability of getting a red marble, P (R) is 9 = 0.45
20
and the probability of getting a green marble, P (G) is 5 = 0.25
20
Facts to remember
1. The probability of an event always lies between 0 to 1, but it can never be less
than 0 and more than 1. i.e., for an event E, 0 ≤ P (E) ≤ 1.
2. The probability of a sure event is 1.
3. The probability of impossible event is 0.
Worked-out Examples
Example 1: A bag contains 5 different coloured marbles. If a marble is drawn randomly
from the bag, what is the probability of getting a green marble?
Solution:
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Probability
Here, n(S) = 5
The favourable number of cases, n(E) = 1 ∴ P (E) = n(E) = 1
n(S) 5
1
Hence, the probability of getting green marble is 5 .
Example 2: A card is drawn randomly from a pack of 52 cards. Find the probability of
getting a king.
Solution: Here, S = {1, 2, … 52}
Here, n(S) = 52 ∴n(S) = 52 king of
The favourable number of cases, n(E) = 4 E = {king of club, king of diamond,
n(E) 4 1 a king is heart, king of spade}
∴ P(E) = n(S) = 52 = 13
∴ n(E) = 4.
Hence, the probability of getting 1 .
13
Example 3: A box contains 4 red, 5 green and 6 yellow balls. If a ball is drawn at
random, find the probability of not getting a yellow ball.
Solution:
Here, n(S) = 4 + 5 + 6 = 15
Favourable number of cases of getting yellow ball, n(E) = 6
Now, P(E) = n(E) = 6 = 0.4
n(S) 15
∴ The probability of not getting a yellow ball, P'(E) = 1 – 0.4 = 0.6
Example 4: A card is drawn at randomly from a well-suffled pack of 52 cards. What is
the probability that the card so drawn be a red ace?
Solution:
Here, n(S) = 52
Favourable number of cases of red ace, n(E) = 2
n(E) 2 1
∴P(E) = n(S) = 52 = 26 .
Hence, the probability of red ace is 1 .
26
Example 5: In a class of 36 students, there are 15 boys and the rest are girls. Find the
probability of choosing randomly a girl as a monitor.
Solution:
Here, n(S) = 36
The favourable number of cases, n(E) = 36 – 15 = 21
n(E) 21 7
∴ P(E) = n(S) = 36 = 12 .
Hence, the required probability is 7 .
12
Example 6: When a die is thrown once, find the probability of getting an odd number.
Solution:
Here, n(S) = 6
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Probability
The favourable number of cases, n(E) = 3
∴P(E) = n(E) = 3 = 1
n(S) 6 2
1
Hence, the probability of getting an odd number is 2 .
Example 7: From the cards numbered 1 to 20, a card is drawn at random, find the
probability of getting the card of a prime number.
Solution:
Here, n(S) = 20 Here, S = {1, 2, 3, … 20}
∴ n(S) = 20
The favourable number of cases, n(E) = 8 E = {2, 3, 5, 7, 11, 13, 17, 19}
∴ n(E) = 8
∴P(E) = n(E) = 8 = 2
n(S) 20 5
2
Hence, the required probability is 5 .
Example 8: From the cards numbered 1 to 30, a card is drawn randomly. Find the
probability of getting a card having the number which is divisible by 5 or 9.
Solution:
Here, n(S) = 30
The favourable number of cases of divisible by 5 = 6
The favourable number of cases of divisible by 9 = 3
The total number of favourable cases, n(E) = 6 + 3 = 9
n(E) 9 3
∴P(E) = n(S) = 30 = 10
Hence, the required probability is 3 .
10
Example 9: Three unbiased coins are tossed simultaneously. Write down the sample
space and find the probability of getting (i) all heads (ii) at most two heads
(iii) one tail (iv) at most one tail.
Solution:
To find the sample space S,
Here, sample space of tossing the first coin = {H, T}
Sample space of tossing the first
and the second coins = H H T
HH HT
T TH TT
= (HH, HT, TH, TT}
Samplespaceoftossingallthethreecoins= H HH HT TH TT
T HHH HHT HTH HTT
THH THT TTH TTT
∴ n(S) = 8 = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
(i) The number of favourable cases of all heads, n(E) = 1 ∴ P(E) = n(E) = 1
n(S) 8
n(E) 7
(ii) The number of favourable cases of at most two heads, n(E) = 7 ∴ P(E) = n(S) = 8
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(iii) The number of favourable cases of one tail, n(E) = 3 ∴P (E) = n(E) = 3
n(S) 8
n(E) 4 1
(iv) The number of favourable cases of at most one tail, n(E) = 4 ∴P (E) = n(S) = 8 = 2
19.5 Empirical probability (or Experimental probability) 1
2
When a coin is tossed, theoretically the probability of getting head is . The
probability obtained in this way is called theoretical probability.
On the other hand, if a coin is tossed 20 times, theoretically the head should occur
1
2 × 20 times, i.e., 10 times. However, in real experiment it may not happen, i.e., the
occurrence of head may be 6, 9, 11, 15, or any other number of times. The probability
of any event which is estimated on the basis of the number of actual experiments is
known as empirical (or experimental) probability.
If n(S) be the total number of times, an experiment is repeated and n(E) be number
of observed outcomes, the empirical probability is defined as,
P(E) = Number of observed outcomes = n(E)
Total number of times an experiment is repeated n(S)
Example 10: When a coin is tossed 200 times, head occurs 80 times. Find the probability
of (i) head and (ii) tail.
Solution: Alternative process
Here, total number of trials n(S) = 200
P(E) = P(T) = 200 – 80 = 120
n(E) = n(H) = 80 ∴P(T) = P(E) = 120 = 0.6
P(S) 200
n(E) 80
(i) Now, P(E) = P(H) = n(S) = 200 = 0.4
(ii) Again, the probability of tail, P(T) = 1 – 0.4 = 0.6
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
n(E) 8 2
∴ P(E) = n(S) = 20 = 5
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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
EXERCISE 19.1
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.
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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.
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
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(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) 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.
f) 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?
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 and activity section
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.
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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
Let’s tick (√) the correct alternative.
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) nn((SE)) (B) nn((SE)) (C) n(S)n+(E)n(E) (D) n(S) – n(E)
n(S)
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
3 1 4 1
(A) 40 (B) 8 (C) 5 (D) 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
1 7 3 2
(A) 4 (B) 20 (C) 5 (D) 5
8. 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
9. A card is drawn from a well shuffled deck of 52 cards. What is the probability that the
card drawn will be a king?
1 1 3 12
(A) 52 (B) 13 (C) 13 (D) 13
10. 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
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Unit 20 Trigonometry
20.1 Trigonometry - Introduction
Trigonometry is a branch of mathematics that studies relationships between sides and
angles of triangles. The word Trigonometry is derived from Greek, where trigonon
means ‘triangle’ and metron means ‘measure’.
Throughout history, trigonometry has it’s broad applications in many area such as
surveying, celestial mechanics, navigation, and so on.
20.2 Trigonometric ratios A
In the given right–angled triangle, ∠B is a right angle. The longest perpendicular hypotenuse
side, which is the opposite side of right angle is the hypotenuse (opposite)
(h). When we take ∠C as the reference angle, the side opposite C
to it is perpendicular and the side adjacent to it is the base. So, B base
AB is perpendicular and BC is base. (adjacent)
When we take ∠A as the reference angle, BC is the perpendicular
and AB is the base.
The ratios of any two sides of a right–angled triangle with respect base A hypotenuse
to one of the two acute angles are known as Trigonometric Ratios. (adjacent)
The acute angle about which the ratio is formed is called the B perpendicularC
angle of reference. Sine, cosine and tangent are the name of the (opposite)
three main trigonometric ratios.
C
(i) The sine of an angle A, written as sinA, is the ratio of opposite
side (perpendicular) and hypotenuse.
opposite perpendicular p BC
i.e., sinA = hypotenuse = hypotenuse = h = AC
(ii) The cosine of an angle A, written as cosA, is the ratio of adjacent
side (base) and hypotenuse. base b AB ph
adjacent hypotenuse h AC
i.e., cosA = hypotenuse = = =
Bb A
(iii) The tangent of an angle A, written as tanA, is the ratio of
opposite side (perpendicular) and adjacent side (base).
opposite perpendicular p BC
i.e., tanA = adjacent = base = b = AB C
In the above cases, the acute angle A is taken as the angle of reference.
Now, let’s find the trigonometric ratios of taking another angle of h
pA
reference C. b
B
(i) sinC = perpendicular = p = AB
hypotenuse h AC
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(ii) cosC = base = b = BC
hypotenuse h AC
(iii) tanC = perpendicular = p = AB
base b BC
Thus, sine, cosine, and tangent are known as the fundamental trigonometric ratios. The
reference angles are denoted by different Greek letters such as a (alpha), b (beta), y (gamma),
q (theta), f (phi), etc. However, the reference angles are also denoted by English capital
letters like A, B, C, ... etc.
20.3 Relation between trigonometric ratios
p b
We know that sinA = h and cosA = h
∴ sin2A = p2 and cos2A = b2 Thus, sin2A + cos2A = 1
h2 h2 (i) sin2A = 1 – cos2A and sinA =
p2 b2 (ii) cos2A = 1 – sin2A and cosA = 1 – cos2A
Now, sin2A + cos2A = h2 + h2 1 – sin2A
= p2 + b2
h2
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p https://www.geogebra.org/m/uffpj4tr
Also, we have, tanA = b .
Dividing the numerator and the denominator by h, we get
tanA = p/h = sinA Thus, tan A= sinA .
b/h cosA cosA
Worked-out examples
Example 1: From the figure alongside, find the values of sina and tanq in terms of sides
of triangles.
Solution: A
Here, for the reference angle a, hypotenuse (h) = BC P
perpendicular (p) = AB and base (b) = AC a
p AB Q
∴ sina = h = BC
Again, for the reference angle q, in rt ∠ed DPQB,
hypotenuse (h) = PB, perpendicular (p) = BQ and base (b) = PQ
p BQ
∴ tanq = b = PQ
X
Example 2: In the given DXYZ, find the fundamental trigonometric 100 ft.
ratios of reference angle b.
b
Solution: 80 ft.
Y Z
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Here, for reference angle b, hypotenuse (h) = XY = 100 ft.
base (b) = YZ = 80 ft.
perpendicular (p) = XZ = ?
By using Pythagoras theorem, XZ = XY2 – YZ2 = 1002 – 802 = 60 ft.
Now, sinb = XZ (p) = 60 ft. = 35, cosb = YZ (b) = 80 ft. = 4
XY (h) 100 ft. XY (h) 100 ft. 5
and tanb = XZ (p) = 60 ft. = 3
YZ (b) 80 ft. 4
Example 3: Express tanθ in terms of sinθ.
Solution:
We know that, tanθ = sinθ and cosθ = 1 – sin2θ ∴ tanθ = sinθ
cosθ 1 – sin2θ
Example 4: If tanθ = 33 , find the value of cosθ.
56
Solution:
Here, tanθ = 33 or, p = 33 or, p : b = 33 : 56
56 b 56
Let p = 33x and b = 56 x
Then, by using Pythagoras theorem,
h2 = p2 + b2 = (33x)2 + (56x)2 = 1089x2 + 3136x2 = 4225x2 = 65x
Now, cosθ = b = 56x = 56
h 65x 65
Example 5: In the adjoining figure, find the trigonometric ratios of cosθ and tanα.
Solution:
In rt ∠ed ∆ ABC, using Pythagoras theorem,
AC = AB2 – BC2 = 202 – 162 = 400 – 256 = 144 = 12 20
In rt. ∠ ed ∆ DCA, using Pythagoras theorem,
16
CD = AC2 + AD2 = 122 + 52 = 144 + 25 = 169 = 13
D
Now, cosθ = b = AC = 12 and tanα = p = AC = 12 A q
h CD 13 b BC 16
4 cm
Example 6: In the adjoining figure, ∠ABC = 90°, AB = 3 cm, 3 cm
BC = 4 cm and BD ⊥ AC. Find the values of cosq and
tanq. B C
Solution:
In rt ∠ed ∆ ABC, AC = AB2 + BC2 = 32 + 42 = 9 + 16 = 25 = 5
In ∆ ABC and ∆ BCD,
(i) ∠ABC = ∠BDC [Both are right angles]
(ii) ∠ACB = ∠BCD [Common angle]
(iii) ∠BAC = ∠DBC [Remaining angles]
∴ DABC ~ DBCD
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Now, AB = BC = ABCC [corresponding sides of similar triangles are proportional]
BD CD
3 cm 4 cm 5 cm
or, BD = CD = 4 cm
From 1st and 3rd ratios, we get
3 cm = 5 i.e. BD = 12 cm
BD 4 5
From 2nd and 3rd ratios, we get
in4CrcDtm. ∠=ed54D BCD i.e. reCfDere=nc1e56ancmgle q,
Again, for
BC (h) = 4 cm, CD (p) = 16 cm and BD = 12 cm
5 5
b 12/5 12 3
∴ cosq = h = 4 = 5×4 = 5
tanq = p = 16/5 = 4
b 12/5 3
EXERCISE 20.1
General section
1. Write the trigonometric ratios (sine, cosine, and tangent) with respect to the given angle
of reference in terms of the ratios of sides of the following right angled triangles.
2. Write the values of the trigonometric ratios (sine, cosine, and tangent) to the given angle
of reference.
5 13
9
3. a) From the given figure, find the trigonometric ratios of
sinα and tanθ.
b) In the given triangle, find the trigonometric ratios of
sinβ and cosθ.
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c) Find the trigonometric ratios of tanα and cosθ from the
figure.
P
d) Find the trigonometric ratios of sina and cosq from the figure alongside. a
T
4. a) Express the trigonometric ratio sinA in terms of cosA . Q qR
b) Express the trigonometric ratio cosθ in terms of sinθ.
c) Express the trigonometric ratio tanθ in terms of sinθ.
d) Express the trigonometric ratio tanα in terms of cosα.
e) If sinθ = 3 , find the value of cosθ. f) If cosα = 5 , find the value of sinα.
5 13
g) If tanA = 4 , find the value of sinA. h) If sinA = 5 , find the value of tanA.
3 13
i) If tanθ = 20 , find the value of cosθ. j) If cosθ = 9 , find the value of tanθ.
21 15
k) If sin (90° – α) = BC , write down the ratio of sinα.
CA
l) If cos (90° – θ) = BC , write down the ratio of cosθ.
CA
Creative section
5. a) From the given figure, find the trigonometric ratios
of Sinθ and Tanα.
b) From the given figure, find the trigonometric ratios
of cosα and tanθ.
c) From the given figure, prove that tanα = 5 .
12
d) From the given figure, prove that cosθ = 4
5
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X
4 b 26 ft6 ft
3 W
e) From the adjoining figure, show that tanb = .
Y 24 ft Z
D
A q
O R
6. a) In the adjoining figure, ABCD is a rhombus in which
AC = 6 cm, BD = 8 cm and ∠ADB = θ. Find the values of B C
sinθ and tanθ. aS
P O
Q
b) In the given rhombus PQRS, PR = 24 cm, QS = 10 cm and
∠SPR = a, find the values of cosa and tana,
7. a) In the circle given alongside, O is the centre, M is the mid-point of O
chord AB. If AB = 12 cm and OM = 8 cm, and ∠OAM = q, find the A q B
values of sinq and cosq. M
b) In the figure alongside, O is the cente of circle, P the mid-point of O
the chord XY. If ∠POY = a, XY = 18 cm and OY = 15 cm, find the a
PY
trigonometric ratios of sina and tana, X
8. Prove the following trigonometric identities: sinA
tanA
a) tanA.cosA = sinA b) tanθ.sinθ = 1 – cos2θ c) = cosA
1 – sin2θ
tanα sinβ
d) cosA = 1 – sin2A e) sinα = 1 f) tanβ = 1 – sin2b
tanA 1 – cos2A 1 – sin2α
1 – cos2A h) (1 + tan2θ) cos2θ = 1.
g) 1 – sin2A = tan2A
i) (sinA + cosA)2 = 1 + 2 sinA.cosA j) (sinA – cosA)2 = 1 – 2 sinA.cosA
9. a) If sinA = 3 and cosB = 5 , find the value of sinB + cosA.
5 13
12 9
b) If tanA = 5 and tanB = 12 , find the value of sinA + cosB.
c) If 5 sinA = 3, show that tanA = 15 .
cosA 16
cosq – sinq
d) If 4tanq = 3, show that cosq + sinq = 1
7
e) If 2tana = 3, show that 3sina – 2cosa = 5
3sina + 2cosa 13
f) If sinA – cosA = 0, prove that tanA = 1.
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g) If sinθ = p , show that tanθ = p
q q2 – p2
h) If tanα = p , prove that p sinα + q cosα = p2 + q2 .
q
10. a) In the adjoining figure, AB = 6, BC = 8, ∠ ABC = 90°,
BD ⊥ AC and ∠ ABD = θ, find the value of sinθ.
b) In the given figure, AB = 12, BC = 5, ∠ ABC = 90°, BD ⊥ AC
and ∠ DBC = α, find the value of cosα and tanα.
A
c) In the given figure, AB = 6 cm, AC = 10 cm, ∠ ABC = 90°, 10 cm 6 cm
∠ BAD = ∠ ACB and ∠ADB = q, find the value of sinq and B
tanq. q
D
C
20.4 Values of trigonometric ratios of some standard angles
The angles like 0°, 30°, 45°, 60°, and 90° are commonly known as the standard angles.
The values of trigonometric ratios of these standard angles can also be obtained
geometrically without using the table or a calculator.
(i) Values of trigonometric ratios of the angles 45°
Let PQR be an isosceles right–angled triangle right angled at Q.
Here, PQ = QR = a (suppose)
Now, ∠QRP = ∠QPR
or, ∠PQR + ∠QRP + ∠QPR = 180°
∴
90° + ∠QRP + ∠QRP = 180°
∠QRP = 45°
Again, in rt. ∠ed ∆ PQR, using Pythagoras theorem,
PR = PQ2 + QR2 = a2 + a2 = 2a2 = a 2
Now, in rt. ∠ ed ∆ PQR,
sinR = p = PQ = a a = 1 , cosR = b = QR = a a = 1 , tanR = p = PQ = a = 1
h PR 2 2 h PR 2 2 b QR a
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(ii) Values of trigonometric ratios of the angle 30° and 60°
Let PQR be an equilateral triangle, where PQ = QR = RP = 2a (suppose) and
∠P = ∠Q = ∠R = 60°. PM ⊥ QR is drawn.
As the perpendicular drawn from a vertex to the opposite side bisects the opposite side
as well as the vertical angle in an equilateral triangle,
1
QM = MR = 2 × 2a =a
∠QPM = ∠MPR = 1 × 60° = 30°
2
In rt. ∠ed ∆PQM, using Pythagoras Theorem,
PM = PQ2 – QM2 = (2a)2 – a2 = 3a2 = a 3
Now, in rt. ∠ed ∆ PQM, p QM a 1
sin (QPM) = sin30° h PQ 2a 2
= = = =
cos (QPM) = cos30° = b = PM = a3 = 3
h PQ 2a 2
tan (QPM) = tan30° = p = QM = a a = 1
in rt. ∠ed ∆ PQM, b PM 3 3
Now,
sin (PQM) = sin60° = p = PM = a3 = 3
h PQ 2a 2
cos (PQM) = cos60° = b = QM = a = 1
h PQ 2a 2
tan (PQM) = tan60° = p = PM = a 3 = 3
b QM a
(iii) Values of trigonometric ratios of the angle 0°
Let PQR be a right-angled triangle in which ∠RPQ is a right angle. Suppose, ∠PQR = θ.
Here, when θ tends to be 0°, PR also tends to be 0 and QR tends to be equal to PQ.
∴ When θ = 0 then, PR = 0 and QR = PQ
Now, in rt. ∠ed ∆ PQR,
sinθ = p = PR cosθ = b = PQ tanθ = p = PR
h QR h QR b PQ
∴ sin0° = 0 = 0 ∴ cos0° = PQ = 1 ∴ tan0° = 0 = 0
PQ PQ PQ
(iv) Values of trigonometric ratios of the angle 90°
Let PQR be a right-angled triangle in which ∠RPQ is a right angle. Suppose, ∠PQR = θ.
Here, when θ tends to be 90°, PQ tends to be 0 and QR tends to be equal to PR.
∴When θ = 90°, then PQ = 0 and QR = PR
Now, in rt. ∠ed ∆ PQR,
sinθ = p = PR cosθ = b = PQ tanθ = p = PR
h QR h QR b PQ
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 323 Vedanta Excel in Mathematics - Book 9
Trigonometry
∴ sin90° = PR = 1 ∴ cos90° = 0 = 0 ∴ tan90° = PR =∞
PR PR 0
The values of trigonometric ratios of the standard angles are given in the following table.
Trigonometric ratios Angles
sin
cos 0° 30° 45° 60° 90°
tan 0 1
1 13
1 2 22 0
0 3 11 ∞
2 22
11 3
3
Worked-out examples
Example 1: Find the value of sin230° + sin245° + sin260° .
Solution:
1 2 1 2 3 2
2
Here, sin230° + sin245° + sin260° = 2 + 2 +
= 1 + 1 + 3 = 1+2 +3 = 6 = 112
4 2 4 4 4
Example 2: Prove that 1 + tan30° = 1 + sin60°
1 – tan30° 1 – sin30°
Solution:
1
L.H.S. = 1 + tan30° = 1+ 3 = 3 +1 = 3 +1 × 3 + 1 = ( 3 + 1)2 = 2 + 3
1 – tan30° 1– 1 3 –1 3 –1 3 + 1 ( 3 )2 – (1)2
3
R.H.S. = 1 + sin60° = 1+ 3 = 2+ 3 =2+ 3
1 – sin30° 1– 2 2–1
1
2
∴ L.H.S. = R.H.S. proved.
Example 3: In the given figure, ABC is a right angled triangle and ACDE is a rectangle.
Find the size of BD.
Solution:
Here, in rectangle ACDE , AE = CD = 2 m and ED = AC = 50 m
In rt. ∠ed ∆ ABC,
tanA = p = BC = BC
b AC 50
Vedanta Excel in Mathematics - Book 9 324 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Trigonometry
or, tan30° = BC Vedanta ICT Corner
50 Please! Scan this QR code or
browse the link given below:
or, 1 = BC
3 50 https://www.geogebra.org/m/j8wckkch
or, BC = 50 = 50 × 3 = 50 × 1.732 = 28.87 m.
33 3 3
Now, BD = BC + CD = 28.87 m + 2m = 30.87 m.
Example 4: In the given figure, a ladder rest against a vertical wall A
making an angle of 60o with the ground. If the height of
the wall at which the upper end of the ladder is supported 30ft.
is 30 ft, find the length of the ladder and the distance of 60°
the foot of the ladder from the wall.
BC
Solution:
Here, height of the wall (AC) = 30 ft, angle made by the ladder (θ) = 60o
Length of ladder (AB) =?
Distance between the foot of the ladder and the wall (BC) =?
In right angled triangle ABC,
sin60° = p = AC tan60° = p = AC
h AB b BC
3 30 30
or, 2 = AB or, 3 = BC
or, 3 AB = 60 or, 3 BC = 30
or, AB = 60 × 3 = 60 3 = 20 3 or, BC = 30 × 3 = 30 3 = 10 3
3 3 3 3 3 3
Hence, the length of the ladder is 20 3 ft. and the distance between the foot of the ladder
from the wall is 10 3 ft.
Example 5: A tree of 18 m height is broken by the wind so that its top touches the
ground. If the height of the remaining part of the tree is 6 m, find the
angle made by the broken part of the tree with the ground. A
Solution:
Let, AB be the height of the tree before it was broken, CD be the broken part
of the tree. 18m
Here, AB = 18 m, BD = 6 m ∴ CD = AD = 18 m – 6 m = 12 m D
Let, ∠BCD = θ 6m
In right angled triangle BCD, q B
p BD C
sinθ = h = CD
or, sinθ = 6m
12 m
1
or, sinθ = 2
or, sinθ = sin30° ∴ q = 30°
Hence, the angle made by the broken part of the tree makes an angle of 30o with the ground.
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 325 Vedanta Excel in Mathematics - Book 9
Trigonometry
General section EXERCISE 17.2
1. Evaluate: c) 2 cos45°.tan60°
a) sin60°.cos30° b) sin30°.tan45°
d) sin30° – cos30°.tan30° e) tan245° + cos245° f) sin230° – cos260°
h) 2tan30°.cos30°
g) 1 × sin30° × cos45° k) 2 2 sin45°.cos60° i) 2 tan60°.sin60°
2 3
j) tan260° + 4 cos245° l) sin45° × cos45° × tan45°
m) sin230° + cos260° n) tan30°.tan60°.cos45° o) sin30°.cos60°.tan45°
p) sin60°.cos30°.tan30° q) sin90°.cos45°.tan0° r) tan0° + cos0° + sin0°
s) sin30° (cos30° + sin60°) t) 2 3 sin30°.tan30°
u) 3 .cos60°
sin60°
v) tan60° w) sin30° + sin60° x) 2tan30°
2cos30° cos30° + cos60° 1 – tan230°
y) sin60°.cos30° + cos60°.sin30° z) cos60°.cos30° – sin60°.sin30°
sin60°.cos30° – cos60°.sin30° cos60°.cos30° + sin60°.sin30°
Creative section - A
2. Prove that
a) sin60°.tan60° = sin60° b) sin30°.tan60° = cos30°
c) cos230° + cos260° = 1 d) sin230° + sin260° = 1
e) 2tan30° = tan60° f) 1 2tan30° = sin60°
1 – tan230° + tan230°
g) 2sin60°.cos60° = cos30° h) 1 + tan30° = 1 + sin60°
1 – tan30° 1 – sin30°
3. In the following right angled triangles, find the unknown sizes of the sides.
Z
Y
P RX
4. In the given figure, calculate the length of the side AC.
(sin28° = 0.46)
Vedanta Excel in Mathematics - Book 9 326 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Trigonometry
5. In the following right angled triangles, find the unknown sizes of the acute angles.
FP
E
R
Q
G
Creative section - B
6. a) In the given figure, ABC is a right angled triangle and
ACDE is a rectangle. Find the size of BD.
b) In the figure, PQR is a right angled triangle and PRST
is a rectangle. Find the size of TS.
7. a) An electric pole 15 m high is supported by a wire fixing its one end on the ground
at some distance from the pole. If the wire joining the top of the pole is inclined to
the ground at an angle of 60°, find the length of the wire.
b) A ladder rests on the top of the wall in such a way that it makes an angle of 45°
with the ground. If the height of the wall is 25 ft, find the length of the ladder. Also,
find the distance of the foot of the ladder and the wall.
c) The top of a pine tree broken by the wind makes an angle of 30° with the ground at
a distance of 5 3 m from the foot of the tree. What was the height of the tree before
it was broken?
8. a) A ladder 24 meter long is resting against a wall. If the distance between the foot of
the ladder from the wall is 12 m, find the angle made by the ladder with the floor.
b) The length of the shadow of a 30 3 m high tree is 10m at 2:30 p.m., find the
altitude of sun.
c) A man of height 1.5 m is standing in front of a tower of height 81.5 m. At what
angle should he observe the top of the tower from the distance of 80 m?
Project work and activity section
9. a) Draw two right-angled triangles using two different sets of Pythagorean triplets.
Then write the trigonometric ratios of sine, cosine and tangent in each triangle
taking both the acute angles of the triangles as reference angles.
b) From any two sets of Pythagorean triplets, use only two numbers from each set to
draw perpendicular and base of two right-angled triangles in a graph. Then draw
hypotenuse in each triangle and find the length of hypotenuse with a ruler. Now,
write the trigonometric ratios of sine, cosine and tangent in each triangle.
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 327 Vedanta Excel in Mathematics - Book 9
Trigonometry
10. Draw three right-angled triangles of your own measurements with one of the acute angles
being 30°, 60° and 45° respectively in three triangles. Measure the lengths of sides of each
triangle. Then find the value of the following trigonometric ratios upto three decimal
places.
a) sin30°, cos30° , tan30° b) sin60°, cos60° , tan60° c) sin45°, cos45° , tan45°
OBJECTIVE QUESTIONS
Let’s tick (√) the correct alternative.
1. In a right angled triangle, if θ be a reference angle, the side opposite to θ is
(A) base (b) (B) perpendicular (p) (C) hypotenuse (h) (D) altitude
2. In a right angled triangle, tangent of an angle is the ratio of:
(A) perpendicular and hypotenuse (B) base and hypotenuse
(C) perpendicular and base (D) base and perpendicular
3. In ∆ABC given alongside; what is the trigonometric ratio of sinα? A
(A) 43 (B) 35 (C) 54 (D) 35 3 cm
a
B 4 cm C
4. What is the value of sin2A + cos2A?
(A) -1 (B) 0 (C) 1 (D) 2
5. Which one of the following relations is NOT true?
(A) sinA×cosecA = 1 (B) cosA×secA = 1 (C) tanA×cosA = 1 (D) sin2A+cos2A = 1
6. The value of sinA in terms of cosA is:
(A) 1 – cosA (B) 1 – cos2A (C) 1 – cos2A (D) 1+ cos2A
7. The value of the trigonometric ratio sin45° is
(A) 0 (B) 21 (C) 1 (D) 3
2 2
8. The value of the expression sin30° + cos60°?
(A) 1 (B) 0 (C) 3 (D) 21
2
9. The maximum value of the expression 3sinA + 4cosA is
(A) 7 (B) 4 (C) 3 (D) 0
10. If 2sinθ - 1 = 0 (00 ≤ θ ≤ 00), then the value of θ is
(A) 00 (B) 300 (C) 450 (D) 600
Vedanta Excel in Mathematics - Book 9 328 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
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. If P = {x: x is a prime number between 20 and 40} and Q = {y: y is an odd number
between 20 and 30}, find and show the following operations in Venn-diagrams.
(i) P∪Q (ii) P∩Q (iii) P – Q (iv) Q – P
2. If A = {x: x∈N, 4 ≤ x ≤ 10} and B = {1, 4, 9, 16, 25}, find A∆B.
3. If U = {1, 2, 3, …, 12} is a universal set, A = {factors of 12} and B = {multiples of 3}
are the subsets of U, find the following sets.
(i) A (ii) B (iii) A ∪ B (iv) A ∩ B
(v) A ∪ B (vi) A ∩ B (vii) A – A ∩ B (viii) (A ∪ B) – B
4. If A = {1, 2, 3, 4, 5, 6, 7}, B = {2, 4, 6, 8, 10, 12, 14} and C = {10, 14}. Find and show
the following set operations in Venn-diagrams.
(i) A∪B∪C (ii) A∩B∩C (iii) A – (B ∪ C) (iv) B ∪ (C∩A)
5. P, Q and R are the subsets of a universal set U. If U = {x: x∈W, x≤ 12}, P = {prime
numbers}, Q = {odd numbers less than 10} and R = {natural numbers less than 6},
find:
(i) P ∪ Q (ii) P ∩ Q (iii) Q ∪ R (iv) P ∪ Q (v) P ∩ Q
6. If A = {letters in the word ‘mathematics’} and B = {letters in the word ‘geometry’}.
(i) Find (A – B) ∪ (B – A) and (A∪B) – (A∩B)
(ii) Establish the relation between (A – B) ∪ (B – A) and (A∪B) – (A∩B)
7. If U = {x: x∈N, x≤10}, H = {y: y=2n, n∈W, n≤3} and T= {z: z=3n, n∈W, n≤2}.
(i) Find H – T and T – H (ii) Are H – T and T – H equal?
8. If A = {x : x ≤ 10, x is an odd number} and B = {y : y ≤ 10, y is a prime number}, show
that n(A ∪ B) = n(A) + n(B) – n(A ∩ B).
9. 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.
10. During the lockdown due to pandemic of corona virus, out of 50 schools of a
municipality; 35 schools managed their regular classes virtually using zoom platform,
8 used google meet platforms and 3 schools used both of these platforms.
(i) Draw a Venn-diagram to represent the above information.
(ii) Find the number of schools of the municipality that could not manage their virtual
classes during lockdown.
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 329 Vedanta Excel in Mathematics - Book 9
Revision and Practice Time
11. In a group of 80 students, 50 students like apples, 40 like oranges and 25 of them like
both the fruits.
(i) Present the results in a Venn-diagram.
(ii) How many students like apples only?
(iii) How many students like neither apples nor oranges?
12. In an examination, 60 % examinees passed in Mathematics, 55 % passed in English and
40% passed in both subjects. By using Venn-diagram, find the percentage of students
who failed in both the subjects.
Taxation
1. Mr. Gurung is the proprietor of a garment factory. His annual income is Rs 5,55,000. If
the tax up to Rs 4,50,000 is exempted, 10% tax is charged for the income of Rs 4,50,001
to Rs 5,50,000 and 20% tax is charged for the income of Rs 5,50,001 to Rs 7,00,000
respectively. Calculate the annual income tax that should be paid by him.
2. Mrs. Ghising deposited Rs. 6,00,000 for 3.5 years at 10% p.a. simple interest in her
fixed account at a commercial bank. How much net interest would she get if 5% of
interest is paid as income tax?
3. The monthly income of an unmarried individual is Rs 50,000. If 1 % social security
tax is charged up to the annual income of Rs 4,00,000. Then 10% and 20% taxes are
charged for the next incomes of Rs 1,00,000 and Rs 2,00,000 respectively. Calculate the
annual income tax paid by the individual.
4. Mrs. Bista is a secondary level English teacher. Her monthly salary is Rs 39,990 with
allowance of Rs 2,000. She gets a festival expense which is equivalent to her basic
salary of one month and 10% of her salary excluding allowance and festival expense
is deducted as provident fund. If 1% social security tax is levied upon the income of
4,50,000, 10% and 20% taxes are levied on the next incomes of Rs 1,00,000 and up to
Rs 2,00,000 respectively, how much income tax should she pay in a year?
5. Mr. Rawal is the Branch Manager of a bank. His monthly salary is Rs 95,000 and 10% of
his salary is deducted as provident fund. He pays Rs 24,520 as the premium of his life
insurance. If 1% social security tax is levied upon the income of 4,50,000, 10%, 20%
and 30% taxes are levied on the next incomes of Rs 1,00,000, Rs 2,00,000 and up to Rs
12,50,000 respectively, how much income tax should he pay in a year?
6. The present rate of income tax is fixed by Inland Revenue Department (IRD) is given
below.
For an individual For couple
Income slab Tax rate Income slab Tax rate
Up to Rs 4,00,000 1% Up to Rs 4,50,000 1%
Rs 4,00,001 to Rs Rs 5,00,000 10% Rs 4,50,001 to Rs Rs 5,50,000 10%
Rs 5,00,001 to Rs Rs 7,00,000 20% Rs 5,50,001 to Rs Rs 7,00,000 20%
The monthly salary of Miss Jirel, a cashier in a bank, is Rs 30,000 and her annual
income is equivalent to his salary of 15 months. Similarly, the monthly salary of Mrs.
Dharel, a married civil servant is Rs 40,000 and her annual income is equivalent to her
salary of 13 months including festival expense. According to the above rates of tax,
who pays the more income tax and by how much? Find it.
Vedanta Excel in Mathematics - Book 9 330 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Revision and Practice Time
7. 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?
8. 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?
9. 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?
10. 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?
11. 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?
12. 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.
13. 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?
14. 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?
15. 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?
16. 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 shopkeeper
for Rs.1200?
17. The value of a mobile sets of same models in two shops A and B are given below.
Shop-A: Marked price = Rs 25,000, Discount = 10% and VAT=13%
Shop-B: Marked Price = Rs 24,000 Discount = 8% and VAT= 13%
Which shop is more expensive and by what percentage? Find it.
18. Mrs. Sherpa wishes to buy a bicycle for her son. She visits two shops nearby her house
and finds that Mr. Gupta marks the price of a cycle 40% above the cost price, he allows
15% discount and levies 13% VAT. Also, Mr. Thakur marks the price of cycle of the
same model 50% above the cost price, he allows 20% discount and levies 13% VAT. If
both the shopkeepers bought the cycles at Rs 2,000, in which shop would you suggest
to Mrs. Sherpa to purchase the bicycle? Give reason with calculation.
19. 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?
20. 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.
21. A wholesaler sold a generator for Rs 3,50,000 to a retailer. The retailer spent Rs 6,000
for transportation and Rs 4,000 for the local tax. If the retailer sold it at a profit of Rs
20,000 to a customer, how much did the customer pay for it with 13% VAT?
22. Annapurna supplier purchased some building materials for Rs 7,77,000 and sold
them at 5% profit to Dhaulagiri supplier. The Dhaulagiri supplier spent Rs 10,000 for
transportation and Rs 4,000 for the local tax and sold at a profit of 10% to a customer.
How much did the customer pay for the materials with 13% VAT?
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 331 Vedanta Excel in Mathematics - Book 9
Revision and Practice Time
Commission, Bonus and Dividend
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. 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?
5. 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.
6. 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.
7. 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
sale of a distributor is Rs 25,00,000, find his commission.
8. Mr. Bhujel is a bike dealer who purchases a bike for Rs 3,00,000. He intends to sell the
bike for Rs 3,50,000. However, he gives a discount of 3% on this price to the buyer and
a commission of 3% to a broker. How much profit does he make?
9. Mr. Kumal decides to sell 10 annas of a land he owns at the price of Rs 4,00,000 per
anna. He promises a commission of 2% on the sale of the land to a broker. If he gives a
discount of 2% on the original price to the buyer, find his income from the sale of the
land, after paying the commission to the broker.
10. A cement factory made a net profit of Rs 3,60,00,000 in the last year. The management
of the factory decided to distribute 16% bonus from the profit to its 120 employees.
(i) By what percent should the bonus be increased so that each employee can receive
Rs 60,000?
(ii) What should be the profit of the company so that it can provide Rs 75,000 to each
employee at 20% bonus?
11. 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?
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.
Vedanta Excel in Mathematics - Book 9 332 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Revision and Practice Time
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 - 100 101 - 250
Rate of charge per unit Rs 3 Rs 6.50 Rs 8 Rs 9.50 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 - 100 101 - 250
Rate of charge per unit Rs 4 Rs 6.50 Rs 8 Rs 9.50 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?
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:
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 333 Vedanta Excel in Mathematics - Book 9
Revision and Practice Time
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. An umbrella is made by stitching 8 triangular pieces of cloth, each piece measuring 60
cm, 63 cm and 39 cm. how much cloth is required for the umbrella? If the cost of 1 cm2
is Rs 0.50, find the total cost of cloth.
2. An umbrella is made by stitching 8 triangular pieces of cloth, each measuring 50 cm,
20 cm and 50 cm. If the rate of the cloth is Rs 0.75 per sq. cm, find the cost of the cloth
required to make the umbrella.
3. Mr. Rudraman had a triangular field ABC as shown in the C
figure. He partitioned it into three smaller triangular plots 52 m 25m 39 m
ABD, ACD and BCD. He shared the plot ABD to his son, the 33m D
plot BCD to his daughter and donated the plot ACD to build 34m
a health post. A
(i) What is the area of plot that is shared to his son in Aana? 65 m
B
(ii) What is the area of plot that is shared to his daughter in Kattha?
(iii) What is the area of plot that is donated to build the health post in Dhur?
4. 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.
Vedanta Excel in Mathematics - Book 9 334 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Revision and Practice Time
5. 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.
6. 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
7. 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.
8. 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.
9. 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.
10. 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.
30cm
30cm
11. The following figure shows a closed victory stand made of wood. 30cm
(i) Find its volume
(ii) Find its lateral surface area. 30cm 1 30cm
(iii) Find its total surface area. 2 3 40cm
12. Mr. Rai manages the accommodation for 30 guests in his
daughter’s birthday ceremony. For this purpose, he plans to make a tent in the shape
of triangular prism of length 20 m in such a way that each person has the space of 4
square meters on the floor and 8 cubic meters to breathe. What is the height of the tent?
Find it.
13. The area and perimeter of the base of a triangular prism are 24 cm2 and 24 cm
respectively. If the total surface area of the prism is 408cm2, find its lateral surface area
and volume.
14. The capacity of a cylindrical water tank is 539 litres. If its height is 1.4 m, find the
radius of its base.
15. If the surface area of a sphere is 22176 sq. cm, find its radius.
16. If the volume of a hemisphere is 48π cm3, find its radius.
17. A cylindrical water tank contains, 3,85,000 litres of water. If its height is 10 m, find its
diameter.
18. A 60 cm high cylinder with 14 cm its diameter is cut vertically into two equal halves.
What is the volume of a half part?
19. If the sum of the radius and height of a right cylinder is 50 cm and the circumference
of its base is 220 cm, find its total surface area and volume.
20. If the radius of the base and the height of a cylinder are in the ratio of 5:7 and the
volume is 550 cubic cm, find the radius of the base of the cylinder.
21. The ratio of the radius of base and the height of a cylinder is 7:8. If the volume of the
cylinder is 9,856 cm3, find the total surface area of the cylinder.
22. A cylindrical roller made of iron is 1.05 m long. Its internal radius is 50 cm and the
thickness of the iron sheet used in making the roller is 5 cm.
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Revision and Practice Time
(i) If it takes 400 complete revolutions to level a playground, find the cost of levelling
the ground at Rs 15 per sq. metre.
(ii) If 1 cm3 of iron weighs 7.8 g, find the mass of the roller.
23. The sum of the height and the radius of the base of a cylinder is 34 cm. If the total
surface area of the cylinder is 2,992 cm2, find its volume.
24. 50 circular plates, each of radius 7 cm and thickness 5 mm are placed one above the
other to form a cylindrical shape. Find the volume of the cylinder so formed.
25. How many cubic meters of earth must be dug out to construct a cylindrical well which
is 25 m deep and the radius of the base is 3.5 m? 35cm
26. The adjoining cylindrical vessel is 70 cm high and the radius of its 70cm 20cm
base is 35cm. If it contains some water up to the height of 20 cm,
how much water is required to fill the vessel completely?
27. How many liters of petrol can be hold in the tanker with cylindrical tank of radius 1.5
m and length 3.5 m?
28. When three metallic spheres of radii 2 cm, 12 cm and 16 cm are melted and recast a
single sphere, what will be the radius of the single sphere?
29. Calculate the surface area and volume of a spherical globe of height 21 cm.
30. How many lead balls each of radius 1 cm can be made from a solid sphere of diameter
12 cm?
31. A cylindrical tub of radius 16 cm contains water to a certain depth. When a spherical
ball is dropped into the tub, the level of water is raised by 9 cm. Find the radius of the
ball.
32. A cylinder whose height is two-third of its diameter, has the same volume as a sphere
of radius 21 cm. Calculate the curved surface area of the cylinder.
33. The surface area of a ball is twice the area of the curved surface of a cylinder. If the
height and radius of the base of the cylinder be 10.5 cm each, find the volume of the
sphere.
34. A hemispherical bowl with radius 21 cm is full of water. If the
water is poured into a 1m tall jar with the diameter of the base
28 cm, find the height of the water level in the jar.
h
Sequence and Series
1. Rewrite the series 3+9+27+81+243 in sigma notation.
2. Each chemical compound contains Carbon atoms (C) and Hydrogen atoms (H). A few
of the chemical compounds are represented in the diagrams below.
H HH H HH
|| | || |
H C H , H C C H , H C C C H , ...
|
|
|
|
|
|
|
|
|
| || |||
H HH HHH
(i) Draw one more diagram in the same pattern.
(ii) Find the nth term formula of the sequence of number of Hydrogen atoms (H).
(iii) Find the number of Hydrogen atoms (H) in 10th diagram.
Vedanta Excel in Mathematics - Book 9 336 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Revision and Practice Time
3. Consider a sequence 21, 18, 15, … , –33.
(a) How many terms are there in the sequence?
(b) Which term of this series is zero?
4. If the third and the ninth terms of an A.P. are 4 and -8 respectively, find the corresponding
series.
5. In an arithmetic sequence, the fourth term is equal to 3 times its first term and seventh
term exceeds twice the third term by 1. Find the sequence.
6. A metered taxi shows Rs 21 at the time of stating and then runs up by Rs 54 for each
additional kilometer travelled during 9:00 pm to 6:00 am. If a man started a journey of
10 km by the taxi at 11:00 pm, how much fare did he pay for journey?
7. After a knee surgery, the trainer told to Mrs. Chhetri to return to her jogging program
slowly. He suggested jogging for 12 minute each day for the first week. Each week,
thereafter, he suggested that she has to increase that time by 6 minute per day. How
much weeks will it be before she is up to jogging 60 minute per day?
8. A person saves Rs.10 on first day, Rs.20 on second day, Rs.40 on third day and so on.
How much amount will he save on the 10th day?
9. Mr. Gurung went on a 4 days hiking trip. Each day he walked 10% more than the
distance he walked on the day before. If he walked 6655 m in the last day, what is the
distance he walked on the first day of the trip?
1. Resolve into factors. Factorisation
a) x2 + 2xy + y2 – 9 b) a2 + b2 – c2 +2ab c) p2 – q2 – r2 – 2qr
d) 16x4 – 4x2 + 4x – 1 e) a2 + b2 – c2 – d2 – 2ab – 2cd f) x2 – (a + b) x + ab
g) (1 – a2) (1 – b2) + 4ab h) a4b-4 + a2b-2 + 1 i) x8 + x4y4 + y8
j) b2(b + 1) – d2 (d + 1) k) p3 + 2p2 + 2p + 1 l) 16m4 + 8m3 – 2m – 1
m) 64 – 144x + 108x2 –27x3 n) (a2 –5a)2 + 10 (a2 –5a) + 24 o) a3 + b3 + c3 – 3abc
p) x3 – y3 – z3 – 3xyz
H.C.F. and L.C.M.
1. Find the H.C.F. and L.C.M. of the following expressions
a) x2 + 2x , x3 – 4x b) 3x2 – 15x , 3x3 – 75x
c) a2 – 6a + 6b – b2, b2 + ab – 6b d) m2 – n2- m - n, m2n – mn2 –mn
e) p3 – p , p3 + 2p2 + p f) 18a3b – 2ab3 , 27a3b – 18a2b2+ 3ab3
g) (a – b)2 + 4ab, ab2 +a2b h) (x + 2)2 – 8x, 2ax3– 8ax
i) 6x3 – 6x, 9x4 + 9x j) a3b3 – 4ab, a4b4 – 8ab
k) x3 – x2 + x – 1, 2x3 - x2 + x – 2 l) a3 + a2 – 4a – 4, 2a3 - a2 - a + 2
m) 16a4 – 4a2 + 4a – 1, 8a3 + 1 n) 1 + 4x + 4x2 -16x4, 1 – 8x3
o) p4 + 4p2+ 16, p3 + 8 p) 16a4 + 4a2b2+ b4, 8a3 + b3
q) x4 + x2+ 169, x3 + x (x + 13) + 4x2 r) 9k4 + 14k2+ 25, 3k2 (2k+1)+5k(k+2)
s) x3 – 3x2 – x + 3 , x3 – x2 – 9x + 9 t) x3 + 2x2 – x – 2 , x3 + x2 – 4x – 4
u) x2 – 14x – 15 + 16y – y2 , x2 – y2 +2y – 1
v) x2 – 10x – 11 + 12y – y2 , x2 – y2 +22y – 121
w) (1 – x2)(1 – y2) + 4xy, 1 – 2x + y – x2y + x2
x) (x2 – 4)(y2 – 9) – 24xy, 12 – 12x – 4y + x2y + 3x2
y) 18(2x3 – x2 – x), 20 (24x4 + 3x) z) x6 – y6, x5 + x3y2+xy4
2. Find the H.C.F. and L.C.M. of the following expressions
a) 6(a3 – 4a), 8(a3b – a2b – 6ab), 9(a2b – ab – 2b)
b) 12(4x3y – xy), 15(8x4y2 – xy2), 20(6x2y – 5xy + y)
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Revision and Practice Time
c) 8m4 +27mn3, 8m3n + 2m2n2 – 15mn3, 4m3n – 9mn3
d) 2x3 – x2 – x, 4x3 – x, 8x4 + x e) x3 + x2 + x, x4 – x, x7 – x
f) p3 – p2 + p, p4 + p, p5 + p3 + p g) x3 – y3, x4 + x2y2 + y2, x3 + x2y + xy2
h) y3 – 1, y4 + y2 + 1, y3 + 2y2 + 2y + 1 i) a3 – 1 – 2a2 + 2a, a3 + 1, a4 + a2 + 1
j) x2 - y2 - 2yz – z2, y2 - z2 - 2zx – x2, z2 - x2 - 2xy – y2
k) 9m2 - 4n2 - 4nr – r2, r2 - 4n2 - 9m2 – 12mn, 9m2 + 6mr + r2 - 4n2
l) 9x2 - 4y2 - 8yz – 4z2, 4z2 - 4y2 - 9x2 – 12xy, 9x2 + 12xz + 4z2 - 4y2
m) 8x3 – 1, 16x4 - 4x2 – 4x – 1, 16x4 + 4x2 + 1
n) 1+4x+4x2 – 16x4, 1+2x – 8x3 – 16x4, 1 + 4x2 + 16x4
o) 81x4 – 9x2 – 6x – 1, 81x4 + 27x3 – 3x – 1 , 81x4 + 9x2 + 1
p) a2 – 6a – 7 + 8b – b2, a2 + a – b2 + b, a2 - b2 + 2b – 1
q) x2 + 2x – 15 – 8y – y2, x2 – 3x – 2xy + 3y + y2, x2 + 5x – 5y – y2
3. Ram had a rectangular plot of land. He divided the land along the length to make two
rectangular plots A and B. He sells the plot-A to Lakpa and plot-B to Sushma. If the area
of Lakpa’s plot is (6x2 + 17x + 7) sq. ft. and that of Sushma’s plot is (8x2 + 2x – 1) sq. ft.,
find the breadth of the plots.
Indices
1. Simplify.
a) 2x + 3 – 2x + 2 b) 3x + 2 + 3x + 1 c) 11m – 11m–1
2x + 2 3x + 1 + 3x 5 × 11m–1
d) 6n + 2 + 7 × 6n e) 8n + 2 + 9 × 8n f) a–3b4 × 4 a2b–8
6n + 1 × 8 – 5 × 6n 8n + 1 × 10 – 7 × 8n
3 2
g) 3 64x3 ÷ 125y–3 –2 h) (0.0001) 4 i) (343) 3
5 323
j) 6 x5 4 x3 x2 k) x –1 x3 x –4 l) 4 512p10q9 ÷ 4 2p–6q–3
m) 3 27x12y –3 ÷ 4 16x8y4 n) 4 48x11y9 4
3x3y5z –4
o) 16x8y4
3 8x6y3
2. 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 xa 1 xb 1 xc 1
ca
× × d) ab × bc ×
xc xa xb xb xc xa
xp + q r–p xq + r p–q xp + r q–r ap n+p ap p+m
xp – q xq – r xr – p an am
e) × × f) (am . an)m – n ×
1+ x ×xx–y 1 – y xy–y
y x
g) a2 – 2a + 1 2 h)
(a – x)x (a – x)x – 1 (a – x)x – y +1 ×xx–y x – 1 xy–y
x y
xba xbc xac 111
xba xbc xac xb xa xc
i) ab × bc × ca j) 1 x1c × 1 xb1 × 1 x1a
bc ab ca
Vedanta Excel in Mathematics - Book 9 338 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Revision and Practice Time
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
3. 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.
Simultaneous Equations
1. Solve each pair of simultaneous equations.
a) x+ 3 = 4 (x + 3) b) 10x + y = 4 (x + y) + 3
x- 3 = 2 (y + 8) 10x + y + 27 = 10y + x
3x + 5y 7x + 3y
c) 2 (x – 3) + 3 (y – 5) = 0 d) 4 = 5 = 4
5 (x – 1) + 4 (y – 4) = 0
y – 2 x – 3 2 5 11 5 1 27
e) x + 3 = y + 2 = 6 f) 3x + 4y = 12 , 4x + 2y = 20
2. In a dairy, the rate of cow milk is Rs 90 per litre and the rate of buffalo milk is Rs 110
per litre. If Ajita paid Rs 510 for 5 litres of milk, how many litres of cow milk and buffalo
milk did she purchase?
3. The cost of tickets to enter in the central zoo is Rs 150 for adult and Rs 50 for a child. If
a family paid Rs 700 for 6 tickets altogether. How many tickets were purchased in each
category?
4. There are some hens and some cows in Krishna’s agro firm. All the cows and hens are
normal and healthy. If the total number of animals’ heads is 400 and the total number
of legs is 850, find the number of hens and the number of cows.
5. Five years ago, father's age was 4 times his son's age. Now the sum of their ages is 45
years. Find their present ages.
6. Ten years ago, the son's age was twice the daughter's age. Now the son's age is 3 years
more than the daughter's age. Find their present ages.
7. 6 years ago a man's age was six times the age of his son. 4 years hence, thrice his age
will be equal to eight times his son's age. Find their present ages.
8. The ages of two girls are in the ratio of 5 : 7. Eight years ago their ages were in the ratio
of 7 : 13. Find their present ages.
9. 6 years ago, a man's age was 6 times the age of his son. 4 years hence, four times of his
age will be equal to 8 times of his son's age. Find their present ages.
10. The difference of the age of a father and his son is 30 years. After five years, the age of
the father will be twice the age of the son. Find their present ages.
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 339 Vedanta Excel in Mathematics - Book 9
Revision and Practice Time
11. 20 years ago, a father was 5 times as old as his son. Now he is 10 years older than two
times the age of his son. Find their present ages.
12. A mother says to her daughter, "5 years ago I was 5 times as old as you were but 10
years hence, I shall be only twice as old as you will be." Find their present ages.
13. The present age of a mother is four times the age of her daughter. If the age of the
daughter after 20 years is equal to the age of the mother before 25 years, find their
present ages.
14. The present age of Kishan is double the age of Radha. If the age of Kishan after 7 years
is equal to the age of Radha after 14 years, find their present ages.
15. Mr. Thapa was four times as old as his son was in 2002 but he was only three times as
old as his son in 2007. What is the year of the son's birth?
16. In 2020, Salim was two times as old as Kedar was. If the ratio of their ages will be 5:3
in 2025, find their years of birth.
17. When the age of Rita was equal to the present age of Shova, she was thrice as old as
Shova was. If the sum of their present ages is 40 years, find their ages.
18. A number of two digits is six times the sum of its digits. If 9 is subtracted from the
number, the digits are reversed. Find the number.
19. A number of two digits is equal to the four times the sum of its digits. If 18 is added to
the number, the digits are reversed. Find the number.
20. A number consists of two digits whose sum is equal to 10. If 36 is subtracted from the
number, the digits are reversed. Find the number.
21. A number of two digits is 3 more than 7 times the sum of the digits. If the places of the
digits are reversed, the new number so formed is decreased by 27. Find the original
number.
22. A number between 10 to 100 exceeds 4 times the sum of its digits by 9. If the number
is increased by 18, the result is equal to the number formed by interchanging its digits.
Find the number.
23. A two digit number is 3 times the sum of its digits. The sum of the number formed by
reversing its digits and 9 is equal to 3 times the original number. Find the number.
24. The digit at the unit place of a two digit number is four times the digit at tens place. If
the sum of the digits is 10, find the number.
25. A number of two digits is 45 less than the number formed by reversing the digits. If the
digit at ones place exceeds thrice the digit at tens place by 1, find the number.
26. The sum of two times the smaller number and three times the bigger number is 34.
If two times the bigger number is subtracted from the five times the smaller one, the
result is 9. Find the number.
27. Two buses were coming from two villages situated just in the opposite direction. The
average speed of one bus is 8 km/hr more than that of another one and they had started
their journey in the same time. If the distance between the villages is 360 km and they
meet after 4 hours, find their average speed.
28. Koshi bus started its journey from Kathmandu to Dharan at 4 p.m. at the average speed
of 50 km/hr. 1 hour later Makalu bus also started its journey from Kathmandu to the
same destination at the average speed of 60 km/hr. At what time would they meet each
other?
29. In 6 hours Harka walks 6 km more than Dorje walks in 3 hours. In 8 hours Dorje walks
12 km more than Harka walks in 9 hours. Find their speed in km per hour.
30. The area of a rectangular field increases by 11 m2 if its length is increased by 3 m and
breadth is reduced by 1 m. But, the area of the filed decreases by 14 m2 if its length is
reduced by 4 m and breadth is increased by 3 m. Find the length and breadth of the
rectangle.
31. Chameli travels 14 km to her home partly by rickshaw and bus. If she travels 2 km by
rickshaw and remaining distance by bus, she reaches to her home in 30 minutes. On the
other hand, if she travels 4 km by rickshaw and remaining distance by bus, she takes 9
minutes more. Find the speed of the rickshaw and that of the bus.
Vedanta Excel in Mathematics - Book 9 340 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Revision and Practice Time
Geometry-Triangle
1. Verify experimentally that the exterior angle of a triangle is equal to sum of two opposite
interior angles. (Two triangles of different shapes and sizes are necessary)
2. Verify experimentally that the sum of any two sides of a triangle is greater than the
third side. (Two triangles of different shapes and sizes are necessary)
3. Explore experimentally the relationship between the angle opposite to the longer and
shorter sides of the triangle. (Two triangles of different shapes and sizes are necessary)
4. Experimentally verify that the bisector of the vertical angle of an isosceles triangle is
perpendicular bisector of the base. (Two triangles of different shapes and sizes are
required)
5. An electric pole of height 36 ft. high above the ground is supported 45 ft 36 ft
by two equal wires of length 45 ft. each fixing its top to the ground. ?
If the foot of the pole and the places at which another ends of wires
are fixed, are on a same straight line, find the distance between the
two places on the ground.
6. The diameter of a circular lake is 198 m. A temple is situated on an island at the centre
of the lake. If the height of the temple above the water surface is 20 m, what is the
distance between the top of the temple from ends of the diameter of the lake?
7. In ∆ABC, if AB = AC, AD ⊥ BC, AB = (3x + 1) cm, AC = (3y – 2) cm, BD = (y + 2) cm
and CD = (x + 3) cm, find the length of AD.
8. In triangle ABC, CX ⊥ AB and BY ⊥ AC. If CX and BY intersect at O, prove that ∠A and
∠BOC supplementary.
A
9. In the adjoining figure, BX ⊥ AC, CY ⊥ AB and BX = CY. Y X
Prove that AB = AC. B C
AP
10. In the given figure, AB // PQ and AC // PR.
If BQ = CR, prove:
(i) AB = PQ (ii) AC = PR
B QC R
A
11. In the given figure, BA ⊥ AC, RQ ⊥ PQ,
AB = QR, and BP = CR. Prove that AC = PQ. B C R
P
A Q
C
12. In the given figure, AB = BC, MB = BN, AB ⊥ BC, and B
MB ⊥ BN. Prove that AM = CN. M
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Vedanta Excel in Mathematics - Book 9
Revision and Practice Time PQ
O
13. In the adjoining figure, AC = BC, ∠ PCA = ∠ QCB and
∠ PBA = ∠ QAB. Prove that ∆ OPQ is an isosceles AC BR
triangle. P
X
(Hint: ∆ QAC ≅ ∆ PBC)
QY S
14. In the given figure, PQ // XY // RS. Prove that,
(i) ∆ QXY ~ ∆ QRS
(ii) ∆ SYX ~ ∆ PQS
(iii) ∆ PQX ~ ∆ XRS
15. In the given figure. AB // MN // DC. If AB = x, DC = y and A D
1 1 1z. y
MN = z. Prove that: x + y = xM
z
BNC
Geometry-Parallelogram
1. Prove that the straight line segments that join the ends of two equal and parallel line
segments towards the same sides are also equal and parallel.
2. Prove that the straight line segments that join the ends of two equal and parallel line
segments towards the opposite sides bisect each other.
3. Prove that the opposite angles and sides of a parallelogram are equal. A B
4. Prove that the diagonals of a parallelogram bisect each other.
5. In the figure alongside, ABCD is a quadrilateral in which D O
OA = OC, OB = OD and ∠ AOB = ∠ AOD = ∠ DOC = ∠ COB Z
= 90°. Prove that ABCD is a rhombus. C
Y
6. In the given figure, XP and ZQ are perpendicular P
to the diagonal WY of the parallelogram WXYZ. Q X
A B
Prove that XP = ZQ. WP
7. In the given figure, ABCD is a parallelogram. The diagonal
8. In the figure alongside, PQRS is a parallelogram. The diagonal D CQ
QS is produced to X and Y. If PX // YR, prove that SX = QY. XS R
Vedanta Excel in Mathematics - Book 9 P QY
342 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Revision and Practice Time
Geometry-Construction
1. Construct a quadrilateral ABCD in which AB = 5.1 cm, BC = 5.4 cm, CD = 4.8 cm,
AD = 6 cm and the diagonal BD = 5.9 cm.
2. Construct a quadrilateral ABCD in which AB = 4.7 cm, BC = 5 cm, CD = 4.3 cm,
AD = 4 cm and ∠ABC = 60°.
3. Construct a quadrilateral PQRS in which PQ = 4.5 cm, QR = 5.5 cm, SP = 3.8 cm,
diagonal PR = 6.5 cm and diagonal QS = 5.8 cm.
4. Construct a quadrilateral MATH in which MA = 5 cm, AT = 4.7 cm, MH = 4.3 cm,
∠A = 50o and ∠M = 70o.
5. Construct a quadrilateral BEST in which BE = 5 cm, ES= 3.9 cm, ∠B = 60o, ∠E = 100o
and ∠S = 70o.
6. Construct the trapezium ABCD in which AB = 5.8 cm, BC = 4 cm, AD = 4.5 cm,
∠BAD = 60 o and AB//DC.
7. Construct the trapezium PQRS in which PQ = 4.2 cm, SP = 3.5 cm, ∠QPS = 50o,
∠PQR = 80o and PQ // SR.
8. Construct the trapezium ABCD in which AB = 6 cm, BC = 5.4 cm, diagonal BD = 5.4
cm, ∠ABD = 45o, and AB // DC.
9. Construct the trapezium WXYZ in which WX= 5.7 cm, diagonal WY= 7.6 cm,
YZ = 6.5 cm, XY= 4.5 cm and WX//ZY.
10. Construct the trapezium ABCD in which AB = 5.5 cm, BC = 5 cm, CD = 4 cm, AD =
4.5 cm, and DC//AB.
Geometry-Circle
1. In the given figure, O is the centre of the circle. If AB = 8 cm, O
CD = 6 cm, OA = 5 cm, AB // CD and ON ⊥ CD, find the length of A MB
MN.
CN D
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 AB
radius of the circle.
CO D
3. In the given figure, O is the centre of the circle. If PQ = RS,
prove that OM bisects ∠ PMS. P
R
MO
Q
S
4. In the adjoining figure, O is the centre of two concentric circles. O
If the chord MN intersects the smaller circle at R and S, prove
that MR = SN. MR SN
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 343 Vedanta Excel in Mathematics - Book 9
Revision and Practice Time B
E
5. O is the center of the circle given alongside. E and F are the mid- AO
points of the equal chords AB and CD respectively. Prove that
(i) ∠ OEF = ∠ OFE (ii) ∠ BEF = ∠ DFE
CF D
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
AC
7. 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
8. In the given figure, PQ is the diameter of a circle with centre O. If O
PQ ⊥ AB, prove that ∠ AOM = ∠ BOM. A MB
Q
9. In the adjoining figure, equal chords AB and CD of a circle with A O C
centre O, cut at right angle at E. If M and N are the mid-points of AB MN
and CD respectively, prove that OMEN is a square. E
DB
10. In a circle the chords AB and AC are equidistant from the centre O. Prove that the
diameter AD bisects ∠ BAC and ∠ ADC.
11. Prove that equal chords of a circle subtend equal angles at the centre. C
12. In the figure alongside, A and B are the centres of two intersecting A M
circles. If CD intersects AB perpendicularly at P, prove that PB
N
(i) CM = DN (ii) CN = DM
D
Statistics
1. a) The ages (in years) of members of 5 families of a community are given below.
20, 30, 5, 10, 35, 50, 80, 40, 10, 40, 30, 60, 90, 50, 60,
20, 35, 30, 5, 40, 10, 20, 35, 25, 60, 40, 20, 25, 30, 25
(i) Present this individual series into a discrete series with tally bars.
Vedanta Excel in Mathematics - Book 9 344 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
Revision and Practice Time
(ii) Construct a cumulative frequency table.
(iii) How many people are older than 20 years?
b) The speeds of the vehicles of a highway during 10 a.m. to 12 noon were recorded
as follows:
Speed in km/hr < 10 < 20 < 30 < 40 < 50 <60 <70
Number of vehicles 5 20 35 48 70 85 100
(i) Form the frequency distribution table.
(ii) Construct the more than cumulative frequency distribution table.
2. Present the following data by drawing (i) a histogram (ii) a frequency polygon.
Weight (in kg) 0-20 20-40 40-60 60-80 80-100
Number of people 10 15 22 20 18
3. Find the mean, median and mode from the following data:
a)
Marks obtained 10 20 30 40 50 60
No. of students 8 10 15 12 10 5
b)
Wages (in Rs) 500 600 700 800 900 1000
No. of workers 2 4 8 12 10 6
4. 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
Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 345 Vedanta Excel in Mathematics - Book 9
Revision and Practice Time
5. 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
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 9 346 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur
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