Example 3 : The vertices of ∆PQR are P(4, 2), Q(0, 2) and R(2, 6). Enlarge ∆PQR by using
3
enlargement E (2, 2); 2 and write down the coordinates of the image ∆P'Q'R'. Also,
draw this enlargement on the same graph: Y
P'
Solution: Here, the vertices of ∆PQR are P(4, 2), Q(0, 2) and R(2, 6).
R
Now, enlarging the vertices of ∆PQR by E (1, 2); 3 , we have
2
(x, y) E[(a, b); k] (kx + a(1 – k), ky + b(1 – k)]
∴ P(4, 2) E[(2, 2); 3/2] P' 3 × 4 + 2(1 – 23) , Q' (2, 2) P'
2 X
X' O
P' 3 × 2 + 2(1 – 3 ) = P'(6 – 1, 3 – 1) = P'(5, 2). Y'
2 2
∴ Q(0, 2) E[(2, 2); 3/2] P' 3 ×0 + 2(1 – 3 , 3 × 2 + 2 1 – 3 = Q'(0 – 1, 3 – 1) = Q'(–1, 2)
2 2 2 2
∴ R(2, 6) E[(2, 2); 3/2] R' 3 × 2 + 2(1 – 3 , 3 × 6 + 2 1– 3 = P'(3 – 1, 9 – 1) = R'(2, 8)
2 2 2 2
Hence, the coordinates of the vertices of the image ∆P'Q'R' are P'(5, 2), Q'(–1, 2) and R'(2, 8).
Drawing this enlargement on the graph as alongside:
Example 4 : If the point A(2, a) is enlarged with centre as the origin and scale factor 3 to form an
image A'(b, 12), find the values of a and b.
Solution: Here, A(2, a) E[O; 3] A'(b, 12)
But we have, A(2, a) E[O; 3] A'(3 × 2, 3 × a] = A'(6, 3a)
∴ A' (b, 12) = A'(6, 3a) ∴ b = 6 and 3a = 12 or, a = 4.
Hence, a = 4 and b = 6.
Example 5 : An enlargement maps the point A(4, 0) onto A'(6, –1) and the point B(4, 2) onto B'(6, 3),
find the centre and scale factor of enlargement by using graph. Y
O
Solution: Draw the points A(4, 0) and B(4, 2) and their images A'(6, –1) B'
and B'(6, 3) on the graph as shown in the adjoining figure. B
Join AA' and BB' and produce them. They intersect at the X' C(3, 1) X
point C(3, 1), which is the centre of enlargement.
A
A'
Again, measure the length of CA, CA', CB and CB'. Y' TRANSFORMATION
Here, CA' = 2CA and CB' = 2CB. Hence, the scale factor of the enlargement is 2.
Example 6 : Prove that the enlargement E[O; –1] with centre at origin and scale factor –1 is
equivalent to the half turn about the origin.
Solution: Let (x, y) be a point then enlarge it by E[O; –1] with centre of origin.
We have (x, y) (x, y) E[O; –1] (kx, ky) = [(–1)x, (–1)y] = (–x, –y)
Again, transform (x, y) half turn about the origin, then
(x, y) E[O; 180o] (–x, –y)
Hence, this completes the given statement.
Enlargement and Reduction 301
EXERCISE - 7.4
1. (a) Define enlargement.
(b) What is reduction?
(c) When k < 1 in an enlargement, what happens the image of an object?
(d) Write down the coordinates of the image of a point A(a, b) under E[(a, b); k].
2. Transform the following figures by the given enlargement.
AA P LO X D
A
OB B CQ OR MN Y OZ BC
O E[O; 1 12] E[O; 3] E[C, –3]
E[O; 2] E[O; 3] E[O; –2]
3. Find the coordinates of the images of the points A(2, 1), B(–2, 0), C(–4, –2) and D(2, –3) under the
following conditions, where E indicates enlargement.
(a) E[(0, 0); 2] (b) E[(0, 0); 32] (c) E[(0, 0); 1]
(d) E[(1, 1); 2] (e) E[(–1, 2); –2] (f) E[(1, –2); –3]
4. (a) The vertices of ∆ABC are A(2, 4), B(6, 2) and C(4, 4). Enlarge ∆ABC about the centre as
origin with scale factor 1 1 . Find the coordinates of the vertices of image of ∆ABC and draw
2
∆ABC and its image on the same graph.
(b) Enlarge the parallelogram PQRS with vertices P(0, 1), Q(4, 1), R(6, –1) and S(2, –1) through
the centre (0, 0) with scale factor 3 and then write the coordinates of the vertices of image
parallelogram P'Q'R'S'. Draw the above enlargement in the same graph.
5. (a) P(1, 0), Q(–3, 0), R(–2, 5) and S(0, 4) are the vertices of the quadrilateral PQRS. Find the
coordinates of the image of the quadrilateral PQRS for the enlargement with centre (1, 2) and
scale factor –2.
TRANSFORMATION (b) Transform the unit square ABCD with vertices A(0, 0), B(1, 0), C(1, 1) and D(0, 1) by the
enlargement E[(–1, 3), 2]. Write down the coordinates of the vertices of the image square
A'B'C'D' and represent the above transformation on the graph.
6. (a) If A'(a, 4) is the image of the point A(3, b) under the enlargement about the origin with scale
factor 2, then find the values of a and b.
(b) If the point P(3, b) moves to the point (a, –10) by the enlargement of E[(1, 2); 3] then find the
values of a and b.
(c) Find the pre-image of the point (4, –3) under an enlargement with the centre at(0, 3) and scale factor 2.
7. (a) An enlargement maps the points A(2, 3) onto A'(6, 9) and the point B(–1, 4) onto B'(–3, 12).
Find the centre and the scale factor of the enlargement.
302 Illustrated Optional Mathematics-9
(b) In enlargement maps point P(2, 3) onto P'(2, 5) and Q(6, 4) onto Q'(14, 8). Find the centre and
the scale factor of the enlargement.
(c) On a graph paper, draw the ∆PQR whose vertices are P(3, 2), Q(4, 3) and R(3, 4) on the
same graph paper. Draw the ∆P'Q'R', the image of ∆PQR by an enlargement with the vertices
P'(6, 1), Q'(4, –1) and R'(6, –5). Find its centre of enlargement and the scale factor.
8. (a) Prove that the enlargement with the scale factor 1 and centre (0, 0) is an identity transformation.
(b) Show that the enlargement of a point (2, 3) with centre (0, 0) and scale factor 2 is equivalent
2
to the transformation of the same point by the vector 3 .
ANSWERS
1. and 2. Show to your teacher. (b) A'(3, 3/2), B'(–3, 0), C'(–6, –3), D'(3, –9/2)
(d) A'(3, 1), B'(–5, –1), C'(–9, –5), D'(3, –7)
3. (a) A'(4, 2), B'(– 4, 0), C'(–8, – 4), D'(4, –6) (f) A'(–2, –11), B'(10, –8), C'(16, –2), D'(–2, 1)
(c) A'(2, 1), B'(–2, 0), C'(– 4, –2), D'(2, –3) (b) P'(0, 3), Q'(12, 3), R'(18, –3), S'(6, –3)
(e) A'(–7, 4), B'(1, 6), C'(5, 10), D'(–7, 12) (b) A'(1, –3), B'(4, –3), C'(4, 0), D'(1, 0)
4. (a) A'(3, 6), B'(9, 3), C'(6, 6) (c) (2, 0)
5. (a) P'(1, 6), Q'(9, 6), R'(7, – 4), S'(3, –2)
6. (a) 6, 2 b) 7, –2 (b) [(2, 2); 3] (c) [(4, 5/3); –2]
7. (a) (0, 0), 3
Project Work
(i) Enlarge diagrammatically a point P(x, y) through the centers (0, 0) and (a, b) with scale factor k
separately.
(ii) Write the properties of reflection, rotation, translation and enlargement.
(iii) Reflect a point P(x, y) in the line x = 1 and then reflect its image in the line x = 3. Compare
4
the image of the same point under translation by the vector 0 . What do you find? Write your
conclusion.
TRANSFORMATION
Enlargement and Reduction 303
STATISTICS 8UNIT
Estimated Teaching Periods: 12
Competency
To describe the partition value, dispersion and its coefficient.
At the end of this unit, the students will be able to: Learning Outcomes
find the partition values : quartiles, deciles and percentiles of individuals and discrete data.
introduce dispersion.
compute the quartile deviation, mean deviation from mean, mean deviation from median and
standard deviation and their coefficients of individual and discrete data.
What do you learn?
Partition Values of Individual and Discrete Data
� Quartiles � Deciles � Percentiles
Dispersion of Individual and Discrete Data
� Quartile deviation and its coefficient
� Mean deviation from mean and median and its coefficient
� Standard deviation and its coefficient
Are you ready?
� Identifing individual, discrete and continuous data
� Finding total frequency
� Calculating mean, median and mode of data
� Identify the highest and lowest items in data
� Computing quartiles of data
Specification Grid
STATISTICS Unit Chapter Cognitive Knowl- Under- Applica- Higher Total Total
Domain edge standing tion ability Ques- Marks
Topic tions
1 mark 2 marks 4 marks 5 marks
8 Statistics Partition Values - 1 2 - 3 10
Dispersion
81.1 PARTITION VALUES OF INDIVIDUAL
AND DISCRETE DATA
8.1 (A) Quartiles from Individual and Discrete Data
Learning Objectives
At the end of this topic, you will able to:
find the quartiles of the given individual and discrete data.
The variate values which divide the given data (frequency distribution) into lower upper 100% Q2
lower upper75% Md
four equal parts are called quartiles. There are three quartiles which divide the For Q3 50%
25%
data into four equal parts. They are (i) first/lower quartile (Q1), (ii) second/ 0%
middle quartile (Q2) and (iii) third/upper quartile (Q3) where Q1 < Q2 < Q3.
The first quartile Q1 lies at the first 25% of the data. The second quartile
lies at 50% and third quartile lies at 75% of the data. So, the middle/second
quartile Q2 is the median of the data. As well Q1 is the first/lower 25% or the
last/upper 75% and Q3 is the lower 75% or the upper 25% of the data. Every
data should be arranged in ascending or descending order of magnitude for
calculating quartiles as in median.
Calculation of Quartiles
For Individual Data
The first quartile (Q1) divides the data in the ratio of 1:4. So, the first quartile Q1 is calculated by the
position of 1 of (N + 1)th term after arranging the data in ascending order.
4
N + 1 th
i.e., Q1 = 4 term
The second quartile (Q2) divides the data in the ratio of 2:4 = 1:2. So, the second quartile Q2 is calculated
by the position of half of (N + 1)th term.
2(N + 1) th
i.e., Q2 = 4 term = Median (Md)
The third quartile (Q3) divides the data in the ratio of 3:4. So, the third quartile Q3 is calculated by the
3
position of 4 of (N + 1)th term.
3(N + 1)th
i.e., Q3 = 4 term
N + 1th 3(N + 1)th
It is noted that if the values of 4 item and 4 item are in the decimal form, i.e., n.25 or n.75
where n is a whole number, then Q1 = the position of whole number part item + 0.25 or 0.75 times the STATISTICS
difference between that part and its forward part/item.
i.e., Q1 = The position of the nth item + 0.25 or 0.75 × {(n + 1)th item – nth item}
Similarly, Q3 = The position of the nth item + 0.25 or 0.75× { (n + 1)th item – nth item}
Partition Values of Individual and Discrete Data 305
For Discrete Data
In the discrete data, we can calculate the first quartile (Q1), the second quartile(Q2) and third quartile (Q3)
same in the individual data as follows:
N + 1 th
Q1 = Value of 4 term
2 (N + 1)th N + 1 th
Q2 = Value of 4 = 2 term = Median (Md)
3 (N + 1)th
Q3 = Value of 4 term
N + 1 th 3 (N + 1)th
It is noted that the values of 4 and 4 terms are compared just greater in the cumulative
frequency (cf) of the data.
Consider the discrete data,
x x1 x2 x3 …… xn
fn
f f1 f2 f3 …… cfn
cf cf1 cf2 cf3 ……
N+1 th (N + 1)th
4 4
If cf2 < item or item < cf3, the required quartile is x3.
Example 1 : Find the first quartile from the given data: 13, 14, 7, 8, 9, 12, 15.
Solution: Here, arranging the given data in ascending order,
7, 8, 9, 12, 13, 14, 15.
The number of items ( N) = 7
Now, we have
N + 1 th
The first quartile (Q1) = The position of 4 item
7 + 1 th
= The position of 4 item
= The position of 2th item = 8.
Example 2 : Find the third quartile from the given data: 18, 25, 20, 23, 26, 30.
Solution: Here, arranging the given data in ascending order,
18, 20, 23, 25, 26, 30
The number of items ( N) = 6
Now, we have 3 (N + 1)th
The third quartile (Q3) = The position of 4 1)th item
(6 +
3
= The position of 4 item
STATISTICS = The position of 5.25th item.
= The position of 5th item + 0.25×(6th item – 5th item)
= 26 + 0.25×(30 – 26)
= 26 + 0.25×4 = 26 + 1 = 27.
306 Illustrated Optional Mathematics-9
Example 3 : Find the quartiles of the data below.
x 25 20 30 35 40 45
f 345242
Solution: Making cumulative frequency (cf) table in ascending order,
x 20 25 30 35 40 45 Total
f 4 3 5 2 4 2 N = 20
cf 4 7 12 14 18 20 –
Now, we have.
N + 1 th
The first quartile (Q1) = The value of 4 item
20 + 1 th
= The value of 4 item
= The value of 5.25th item, which is greater than 4 and less than 7.
∴ Q1 = 25. 2 (N + 1)th
The second quartile (Q2) = The value of 4 item
2 (20 + 1)th
= The value of 4 item
= The value of 10.5th item = 30.
The third quartile (Q3) = The value of 2 (N + 1)th
item
3 4 + 1)th
(20
= The value of 4 item
= The value of 15.75th item = 40.
EXERCISE-8.1 (A)
1. (a) What is the first quartile? Write down the formula to calculate Q1.
(b) What percentage does the upper quartile divide the given data?
(c) Write the formula to calculate the third quartile of a data having N terms.
(d) Write the position of the first quartile in a data having 15 terms.
(e) Which quartile is equal to median?
2. Find the first quartile and third quartile from the following data.
(a) 4, 7, 9, 12, 10, 13, 18. (b) 21, 23, 28, 33, 20, 24, 27, 35.
(c) 18, 19, 24, 22, 24, 37. (d) 4, 8, 12, 20, 18, 17, 15, 25, 23.
3. (a) If 2, 2x – 1, 9, 3x, 15, 4x + 2, 22 are in ascending order and the first quartile of the data is 7,
find the value of x.
(b) If 25, 8a + 5, 18, 14 5a – 1, 3a + 1, 2a are in descending order and its third quartile is 7, find STATISTICS
the value of a.
4. Find the quartiles of the following series.
(a) x 10 20 30 40 50
f
3 4 2 51
Partition Values of Individual and Discrete Data 307
(b) Marks 20 25 27 29 30
Frequency 5 3 2 5 2
(c) Ages (in yrs.) 16 17 15 14 10 11
No. of students 10 9 8 7 6 11
(d) Lengths (in cm) 90 110 100 120 125 115 123
3 2 6
No. of Sticks 4 3 4 5
15 Total
5. Calculate the lower and upper quartiles from the data given below: x 50
(a) x 57 8 11 13
f 14 12 8 47
(b) x 10 20 15 30 25 35 40 Total
f 4 7 4 p 8 7 3 40
6. Find the values of Q1 and Q3 from the following data:
(a) x 5 10 15 20 25 30 35 40
cf 2 5 9 11 17 20 25 32
(b) x 12 16 20 24 28 32 36
cf 12 21 32 37 45 50 58
ANSWERS
2. (a) 7, 13 (b) 21.5, 31.75 (c) 18.75, 27.25 (d) 10, 21.5 3. (a) 4 (b) 2
4. (a) 20, 30, 40 (b) 20, 27, 29 (c) 11, 15, 16 (d) 100, 120, 123
5. (a) 5, 11 (b) 20, 35 6. (a) 15, 35 (b) 16, 28
8.1 (B) Deciles from Individual and Discrete Data
Any one of the variate values that divides the given data into ten equal parts is called a decile. There are
nine deciles which divide the given data into ten equal parts. They are :
D1 D2 D3 D4 D5 D6 D7 D8 D9
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
STATISTICS First decile (D1) that divides the data at 10% or one-tenths of the data.
Second decile (D2) that divides the data at 20% or two-tenths of the data.
Third decile (D3) that divides the data at 30% or two-tenths of the data.
..................................................................................................................
Ninth decile (D9) that divides the data at 90% or nine-tenths of the data.
It is noted that the fifth decile is equal to the median or the second quartile. i.e., D5 = Md = Q2
308 Illustrated Optional Mathematics-9
Calculation of Deciles
To calculate the deciles of the given data, it should be arranged in ascending order of magnitude.
For individual Data
The first decile (D1) divides the data in the ratio 1:10. So, the first decile D1 is calculated by the position
1 (N + 1)th term on the ascending arrangement of data.
of 10 of
i.e. D1 = (N + 1)th term.
10
pTohseitsioecnoonfd12d0eocfile(N(D+2)1)dthivteidrmes the data in the ratio 2:10. So, the second decile D2 is calculated by the
on ascending arrangement of the data.
i.e. D2 = 2(N + 1)th term
10
oTfhe130thoirfd(Nde+cil1e)t(hDte3)rmdivoindaessctehneddinatga in the ration 3:10. So the third decile D3 is calculated by the position
arrangement of the data.
i.e. D3 = 3(N + 1)th
10
Similarly, the ninth decile (D9) divides the data in the ration 9:10. So, the ninth decile D9 is calculated by
9 1)th term on ascending arrangement of the data.
the position of 10 of (N +
i.e. D9 = 9(N + 1)th
10
In general, the value of the nth decile (Dn) = n(N + 1)th term
10
Where, n = 1, 2, 3, ......... 9 and
N = Total number of terms.
It is noted that if the values of n(N + 1)th item are in decimal form.
10
i.e. n.ab, Where n ∈ W and a, b are both positive digits.
Then, D1 = Position of nth item + 0.ab {(n +1)th item – nth item}
For Discrete Data
In the discrete data, we can calculate the deciles D1, D2, D3 ...... D9 same in the individual data as follows:
D1 = Value of N+1 th term D2 = Value of 2(N + 1) th term
10 10
D3 = Value of 3(N + 1) th term D4 = Value of 4(N + 1) th term
10 10
D5 = Value of 5(N + 1) th term D6 = Value of 6(N + 1) th term
10 10
STATISTICS
D7 = Value of 7(N + 1) th term D8 = Value of 8(N + 1) th term
10 10
D9 = Value of 9(N + 1) th term
10
Partition Values of Individual and Discrete Data 309
In general,
Value of nth decile (Dn) = value of n(N + 1) th term
10
Where, n = 1, 2, 3, ............. 9 and
N = Total number of terms or sum of frequencies (Σf)
Considering the discrete data.
x x1 x2 x3 ............ xn
f f1 f2 f3 ............ fn
cf cf1 cf2 cf3 ............ cfn
If, cf2 < n(N + 1)th term and n(N + 1)th < cf3. The required nth decile is x3.
10 10
Example 1 : Calculate the third decile of the given data: 4, 6, 10, 12, 9, 7, 15, 11, 17
Solution: Here, arranging the given data in ascending order,
4, 6, 7, 9, 10, 11, 12, 15, 17
Number of terms (N) = 9
Now, we know that
Third decile (D3) = Position of 3(N + 1) th item
10
3(9 + 1)
= Position of 10 th item
= Position of 3 × 10 th item
10
= Position of 3rd item
=7
Example 2 : Find the seventh decile from the given data: 31, 25, 27, 38, 59, 28, 29, 40, 30, 45, 52
Solution: Here, arranging the given data in ascending order.
25, 27, 28, 29, 30 31, 38, 40, 45, 52, 59
Number of terms (N) = 11
Now, we have
Seventh decile (D7) = Position of 7(N + 1) th item
10
7(11 + 1)
= Position of 10 th item
= Position of 7 × 12 th item
10
STATISTICS = Position of 8.4th item
= Position of 8th term + 0.4 (position of 9th item – position of 8th item)
= 40 + 0.4 (45 – 40)
= 40 + 0.4 × 5 = 42
310 Illustrated Optional Mathematics-9
Example 3 : Find the 4th decile from the given data:
x 25 30 20 15 10 35 40
f 4325736
Solution: Making cumulative frequency (cf) table in ascending order,
x 10 15 20 25 30 35 40 Total
Σf = 30
f 7524336
–
cf 7 12 14 18 21 24 30
Here, Σf = N = 30
Now, we have
The value of 4th decile (D4) = Value of n(N + 1)th item
10
= Value of 4(30 + 1)th item
10
= Value of (4 × 31)th item
10
= Value of 12.4th item
= 20 [ 12 < 12.4 < 14]
EXERCISE - 8.1 (B)
1. (a) What is 7th decile?
(b) In which percentage does fourth decile divide the given data?
(c) If a data has N terms, write the formula to find the eighth decile?
(d) Write down the position of third decile in a data having 27 terms.
2. Find the second, fifth and eighth deciles of the given data.
(a) 18, 29, 38, 49, 58, 61, 65, 72, 80
(b) 50, 40, 70, 30, 20, 80, 90, 10, 60
(c) 14, 18, 20, 21, 16, 10, 14, 30, 34, 38, 28, 24, 32, 39, 48, 50, 52, 62, 65
(d) 30, 28, 35, 40, 50, 60, 70, 82, 100, 110, 120, 150, 130, 140, 180, 170, 160, 190, 200
3. Calculate the third, sixth and eighth deciles from the data given below:
(a) 4, 7, 9, 18 (b) 3, 12, 15, 25, 35, 48
(c) 82, 78, 85, 92, 95, 98, 28, 58
(d) 49, 82, 78, 89, 35, 30, 20, 29, 78, 85, 92, 58, 28, 48, 18, 10, 20, 48
4. Evaluate the second, fourth and seventh deciles from the following data.
(a) Marks 12 15 25 26 28 30
No. of students 1 2 5 4 7 3 STATISTICS
(b) Ages (in yrs.) 3 4 5 6 7 8 9
No. of children 4 6 3 2 5 7 4
Partition Values of Individual and Discrete Data 311
(c) Height (in inches) 32 35 45 55 58 62
No. of children 12 14 17 15 10 13
(d) Length (in cm) 5 16 17 19 28 30 35
No. of children 3 4 5 6 4 2 3
5. Calculate D4, D6 and D8 from the table given below:
(a) Marks obtained 15 32 18 25 27 30 41
No. of children 3 4 6 8 4 2 14
(b) x 7824319
f 12 13 14 17 12 19 28
(c) x 3 4 5 6 7 8 9 10
cf 4 7 8 12 15 18 22 25
(d) x 10 30 50 40 20 90 80 70
cf 3 12 21 18 7 35 31 26
ANSWERS
2. (a) 29, 58, 72 (b) 20, 50, 80 (c) 16, 30, 50 (d) 40, 110, 170
3. (a) 5.5, 9, 18 (b) 12.3, 27, 42.8 (c) 72, 87.8, 98 (d) 28.7, 52.6, 89.3
4. (a) 25, 26, 28 (b) 4, 5, 8 (c) 35, 45, 55 (d) 16, 17, 28
5. (a) 25, 32, 41 (b) 4, 7, 9 (c) 6, 8, 9 (d) 40, 70, 80
8.1 (C) Percentiles from Individual and Discrete Data
Any one of the variate values that divides the given data into hundred equal parts is called a percentile
of the given data. There are ninety nine percentiles which divide the given data into hundred equal parts
they are:
P1 P2 P3 P25 P26 P27 P50 P51 P52 P53 P75 P76 P77 P99 P100
0% 1% 2% 3% 25% 26% 27% 50% 51% 52% 53% 75% 76% 77% 99% 100%
STATISTICS First percentile (P1) that divides the data at 1%
Second percentile (P2) that divides the data at 2%
Third percentile (P3) that divides the data at 3%
.............................................................................
Twenty fifth percentile (P25) that divides the data at 25%
Twenty sixth percentile (P26) that divides the data at 26%
..........................................................................................
312 Illustrated Optional Mathematics-9
Fiftieth percentile (P50) that divides the data at 50%
Fifty first percentile (P51) that divides the data at 51%
..........................................................................................
Seventy fifth percentile (P75) that divides the data at 75%
..........................................................................................
Ninety ninth percentile (P99) that divides the data at 99%
It is noted that:
P50 = D5 = Q2 = Md
Calculation of Percentiles
To calculate the percentiles of the given data, it should be arranged in ascending order of magnitude.
For Individual Data
pTohseitfiiornstopfe1rc01e0notifle(N(P+1)1d)ithvtiedrems the data in the ratio 1:100. So the first percentile P1 is calculated by the
on ascending arrangement data.
i.e. P1 = Position of (N + 1)th term
100
The second percentile (P2) divides the data in the ratio 2:100. So, the second percentile (P2) is calculated
2 of (N + 1)th term on ascending ordered data.
by the position of 100
i.e. P2 = Position of 2(N + 1)th term
100
Similarly,
Third percentile (P3) = Position of 3(N + 1)th term
100
Twenty fifth percentile (P25) = Position of 25(N + 1)th term
100
Fiftieth percentile (P25) = Position of 50(N + 1)th term
100
Seventy fifth percentile (P75) = Position of 75(N + 1)th term
100
Ninety ninth percentile (P99) = Position of 99(N + 1)th term
100
In general,
Value of nth percentile (Pn) = Position of n(N + 1)th term
100
Where, n = 1, 2, 3 ................... 99 and
N = Total number of terms STATISTICS
It is noted that if the values of n(N + 1)th term are in decimal form i.e., n.ab, where n∈W and a, b are
positive digits, then 100
Pn = Position of nth term + 0.ab ((n + 1)th term – nth item)
Partition Values of Individual and Discrete Data 313
For Discrete Data
In the discrete data, we can calculate the percentiles P1, P2, P3 ........ P50, P51, ...... P75, P76, ...... P99 same in
the individual data as follows:
P1 = Value of (N1+001)thterm P2 = Value of 2(N10+01)thterm
P3 = Value of 3(N10+01)thterm ................................................
P50 = Value of 50(N + 1)thterm P75 = Value of 75(N + 1)thterm
100 100
P76 = Value of 76(N + 1)thterm ................................................
100
P99 = Value of 99(N + 1)th term
100
In general,
Value of nth percentile (Pn) = Value of n(N + 1)th term
100
Where, n = 1, 2, 3, ..........., 99
N = Total number of term or sum of frequencies (Σf)
Considering the discrete data,
x x1 x2 x3 ............ xn
f f1 f2 f3 ............ fn
cf cf1 cf2 cf3 ............ cfn
If, cf3 < n(N + 1)th term and n(N + 1)th < cf4. The required nth percentile is x4.
100 100
Example 1 : Find the seventeenth and seventy fifth percentiles from the given data: 28, 48, 57, 61,
64, 68, 72, 78, 82, 92, 94
Solution: Here, the given ascending ordered data is: 28, 48, 57, 61, 64, 68, 72, 78, 82, 92, 94
Number of terms (N) = 11
Now, we know that
Seventeenth percentile (P17) = Position of 17(N + 1)th term
100
= Position of 17 (11 + 1)th term
100
= Position of (17 × 12)th term
100
STATISTICS = Position of 2.04th term
= Value of 2nd term + 0.04(Value of 3rd term – Value of 2nd term)
= 48 + 0.04(57 – 48)
= 48 + 0.04 × 9 = 48 + 0.36 = 48.36
314 Illustrated Optional Mathematics-9
Again,
Seventy fifth percentile (P75) = Position of 75 (N + 1)th term
100
= Position of 75 (11 + 1)th term
100
= Position of (75 × 12)th term
100
= Position of 9th term
= 82
Example 2 : Calculate the fifty sixth percentile from the data given below:
Marks (x) 47 32 35 52 62 59 65
6
Frequency (f) 4 7 4 3 2 5
Solution: Making cumulative frequency (cf) table in ascending order,
x 32 35 47 52 59 62 65 Total
f 7 4 4 3 5 2 6 Σf = 31
cf 7 11 15 18 23 25 31 –
Here, Total number of terms (N) = 31
Now, we have
Fifty sixth percentile (P56) = Value of 56(N + 1)th term
100
= Value of 56(31 + 1)th term
100
= Value of (56 × 32)th term
100
= Value of 17.92nd term
= 52 [ 15 < 17.92 < 18]
EXERCISE-8.1 (C)
1. (a) Define fifty seventh percentile. STATISTICS
(b) In which percentage does thirty sixth percentile divide the given data?
(c) Write the formula to calculate the seventy fourth percentile of a data having N terms.
(d) Write down the position of forty ninth percentile of a data having 32 terms.
2. (a) Calculate the twentieth percentile of the data : 8, 9, 12, 15, 18, 21, 24, 28 30
(b) Find the thirtieth percentile of the given data : 85, 92, 97, 102, 110, 125, 135, 150, 155
(c) Find the fifth percentile from the given data : 3, 6, 8, 9, 12, 15, 17, 21, 24, 27, 30
(d) Calculate the sixty fourth percentile from the given data: 12, 14, 28, 37, 38, 40, 41, 48, 50, 52,
55, 59, 60, 62, 66, 68, 72, 74, 78, 82, 85, 87, 92, 95
Partition Values of Individual and Discrete Data 315
3. Calculate the twenty eighth, forty fourth, seventy first and eighty percentiles from the following data:
(a) 78, 38, 48, 58, 98, 28, 88, 68, 18 (b) 44, 38, 22, 56, 30, 34, 58, 14
(c) 125, 135, 155, 115, 145, 185, 175, 195, 205, 255, 235, 225, 215
(d) 85, 25, 75, 23, 17, 57, 28, 38, 57, 68, 29, 62, 70, 45, 38, 48, 27, 18, 20
4. Find the eighth, thirty sixth, fifty first and eighty seventh percentile from the given data:
(a) x 74 78 82 86 90 94 99
f 3765865
(b) Marks 62 69 78 79 82 95
No. of students 3 4 8 4 6 7
(c) Height (in m) 5 7 9 11 13 15 16
No. of houses 3 4 5 7 3 4 5
(d) Length (in ft) 25 30 36 42 45 50
No. of trees 13 14 28 24 34 10
5. Calculate thirteenth, forty seventh, seventy ninth and ninety eighth percentiles from the following data:
(a) x 60 50 10 20 30 70 40
cf 34 27 8 12 18 36 25
(b) x 27 35 25 15 38 48 49 17
cf 24 28 15 9 34 41 48 12
(c) Marks 18 27 32 47 48 59
No. of students 4 5 8 12 18 24
(d) Height (in cm) 95 100 112 118 122 125
No. of students 3 45736
ANSWERS
2. (a) 9 (b) 97 (c) 1.8 (d) 68
3. (a) 36, 52, 79, 88 (b) 26.6, 33.84, 48.68, 56.4 (c) 144.2, 176.6, 214.4, 227 (d) 26.2, 36.2, 58, 68
4. (a) 78, 82, 86, 99 (b) 62, 78, 79, 95 (c) 5, 9, 11, 16 (d) 25, 36, 42, 45
5. (a) 10, 30, 60, 70 (b) 15, 27, 48, 49 (c) 18, 47, 59, 59 (d) 100, 118, 125, 125
Project Work
STATISTICS Record the length of pencils with you and your friends. Find the following partition values by arranging
in ascending order:
(a) Median (b) Quartiles (c) 3rd Decile (d) 8th Decile
(e) 9th percentile (f) 24th percentile (g) 55th percentile (h) 87th percentile
316 Illustrated Optional Mathematics-9
81.21 DISPERSION OF INDIVIDUAL AND
DISCRETE DATA
8.2 (A) Quartile Deviation from Individual and Discrete Data
Learning Objectives
At the end of this topic, the students will be able to:
• introduce the quartile deviation.
• compute the quartile deviation and its coefficient of the individual and discrete series.
Introduction
The quartile divides the data into four equal parts. There are three quartiles such as first quartile (Q1),
second quartile (Q2) and third quartile (Q3) which divide the data into four equal parts. In this section, we
discuss the difference between Q1 and Q3. The quartile deviation is a measure of dispersion based on Q1
and Q3. The difference between Q3 and Q1 is called inter quartile range and the semi-inter quartile range
is called the quartile deviation (QD).
i.e., Inter quartile range = Q3 - Q1 Quartile deviation (QD) = 1 (Q3 - Q1).
Coefficient of QD 2
The quartile deviation is only an absolute measure of dispersion. For comparative study of variability of
two distributions we need a relative measure which is known as coefficient of quartile deviation and is
given by,
Q3 – Q1 Note: Calculation median from quartiles:
CQD = Q3 2 Q1 = Q3 – Q1 1) Median (Md) = Q2 = Q1 + Q3
+ Q3 + Q1 2
2 2) Median (Md) = QD + Q1 and Q1 = Md – QD
3) Median (Md) = Q3 – QD and Q3 = Md + QD
Merits and demerits of QD
The quartile deviation has the following merits and demerits as given below:
Merits Demerits
1. QD is quite easy to understand and calculate 1. QD is not based on all the observations.
2. It is rigidly defined 2. It is affected considerably by fluctuations of
sampling.
3. It is not coefficient by extreme rules 3. It is not suitable for further mathematical treatment STATISTICS
4. It is better measure than range 4. It is not a better reliable measure of variability
5. It can be calculated for open end classes.
Dispersion of Individual and Discrete Data 317
Formula for Computing Q1, Q2 and Q3
The formulae of Q1, Q2 and Q3 are important for computing QD and its coefficient which are given below:
Quartile Individual Series Discrete Series
Q1 The value of N+ 1 th item The value of N+ 1 th item
4 4
Q2 The value of 2(N + 1)th item The value of 2(N + 1)th item
4 4
Q3 The value of 3(N + 1)th item The value of 3(N + 1)th item
4 4
Note: For computing Q1, Q2, or Q3, the data should be in ascending or descending order.
Example 1 : Find the inter-quartile range, the quartile deviation and its coefficient from the given
data: 62, 52, 69, 78, 73, 82, 95.
Solution: Here, arranging the given data in ascending order, we get
52, 62, 69, 73, 78, 82, 95
The no. of items (N) = 7
Now, we have
First quartile (Q1) = N+ 1 th item
4
= 7 + 1 th item
4
= 8 th item
4
= 2th item = 62
Third quartile (Q3) = 3(N + 1)th item
4
= 3(7 + 1)th item
4
=3× 8 th item
4
= 6th item = 82
∴ The inter quartile range = Q3 – Q1 = 82 – 62 = 20
STATISTICS Quartile deviation (QD) = 1 (Q3 – Q1) = 1 × 20 = 10
2 2
Coefficient of QD = Q3 – Q1 = 82 – 62 = 20 = 0.139.
Q3 + Q1 82 + 62 144
318 Illustrated Optional Mathematics-9
Example 2 : Calculate the coefficient of quartile deviation from the given data:
x 15 20 25 30 35 40 45 50
f7 3 8 7 6 2 9 4
Solution: Here, making cumulative frequency table,
x 15 20 25 30 35 40 45 50 Total
f 7 3 8 7 6 2 9 4 N = 46
cf 7 10 18 25 31 33 42 46 –
Now, we have 3(N + 1)th
4
First Quartile (Q1) = N+ 1 th item Third Quartile (Q3) = item
4
46 + 1 th 3(46 + 1)th
= 4 item = 4 item
= 47 th item =3× 47 th
4 4
item
= 11.75th item = 35.25th item
= 25 = 45
∴ The coefficient of quartile deviation (CQD) = Q3 – Q1 = 45 – 25 = 20 = 0.286.
Q3 + Q1 45 + 25 70
Example 3 : The index number of prices of bank shares (X1) and Hydro-power share (X2) in a given
first six months of a year are given below:
Month Jan Feb March April May June
x1 356 330 310 290 250 231
x2 576 600 609 719 815 850
State which is more variable by using coefficient quartile deviation:
Solution: Here,
For series X1 : 356, 330, 310, 290, 250, 231
Arranging the series in ascending order : 231, 250, 290, 310, 330, 356
The number of bank shares (N) = 6
Now, we have 3(N + 1)th
Third Quartile (Q3) = 4 item
First Quartile (Q1) = N+ 1 th item
4 3(6 + 1)th
= 4 item
= 6 + 1 th item
4 21 th
= 4 item
7 th
= 4 item
= 5.25th item
= 1.75th item = 330 + 25% of (356 – 330)
= 231 + 75% of (250 – 231) = 330 + 6.50 = 336.50 STATISTICS
= 231 + 14.25 = 245.25
∴ Coefficient Quartile Deviation (CQD1) = Q3 – Q1 = 336.50 – 245.25 = 91.25 = 0.157
Q3 + Q1 336.50 + 245.25 581.75
Dispersion of Individual and Discrete Data 319
For series X2 : 576, 600, 609, 719, 815, 850
The number of the hydro power shares (N) = 6
Now, we have
First Quartile (Q1) = N+ 1 th item
4
= 6 + 1 th item
4
= 7 th item
4
= 1.75th item
= 576 + 75% of (600 – 576) = 576 + 18 = 594
3(N + 1)th
Third Quartile (Q3) = 4 item
3(6 + 1)th
= 4 item
= 21 th item = 5.25th item
4
= 815 + 25% of (850 – 815) = 815 + 8.75 = 823.75
∴ Coefficient Quartile Deviation (CQD2) = Q3 – Q1 = 823.75 – 594 = 229.75 = 0.162
Q3 + Q1 823.75 + 594 1417.75
Since the CQD1 of X1 is less than the CQD2 of X2, so, the hydro-power shares (X2) is more
variable.
EXERCISE-8.2 (A)
STATISTICS 1. (a) What is dispersion?
(b) Define quartile deviation.
(c) Write the formula to calculate the coefficient of quartile deviation.
(d) Write the relation between QD, Q1 and Md.
2. Find the inter-quartile range, quartile deviation and coefficient of quartile deviation from the
following data:
(a) Salary (Rs.) : 350, 340, 370, 330, 400, 430, 450
(b) Share stock (Rs.) : 125, 175, 150, 200, 250, 210
(c) Height (ft.) : 3.8, 4.5, 4.9, 5.0, 5.1, 5.2, 5.6, 5.7
(d) Length (cm) : 18, 25, 20, 14, 22
3. Calculate the quartile deviation and its coefficient from the given data:
(a) Obtained marks 35 42 49 59 65 72
No. of Students 2 3 7 4 3 1
320 Illustrated Optional Mathematics-9
Income (Rs.) 210 225 250 290 310 350 400
(b) No. of Persons
3 5 4 6 2 5 5
Length (cm) 10.2 14 15.4 18.1 18.3 19.1 19.5
(c) No. of Sticks 2 4
2 5 3 1 3
50 60
4. Find the quartile deviation and its coefficient from the given data below: 35 30
Wages (Rs.) 10 80 30 40 70 20
(a) No. of workers 20 45 85 160 70 55
(b) Size (cm) 4 8 10 12 28 20 32 24 36
No. of Shoe 6 10 18 27 35 40 48 56 60
Height (cm) 25 35 55 65 45 75 95 85
(c) No. of plants (f) 10 5 12 5 2 7 2 4
5. (a) Compare the following two series by using the coefficient of quartile deviation: 170
75
Height (cm), (A) 150 153 158 161 165 167
Weight (kg), (B) 45 48 56 63 67 72
State which has more variability by using the coefficient of quartile deviation.
(b) The daily records of a week of sales in the department store (X1) and the foot path market (X2)
are given below:
Week Sunday Monday Tuesday Wednesday Thursday Friday Saturday
(In '000 Rs.) X1 250 170 350 225 175 215 450
(In '000 Rs.) X2 150 145 170 105 225 185 230
State which distribution has more variation by using the coefficient of quartile deviation.
6. (a) Find the value of the third quartile if the values of the first quartile and inter quartile range are
95 and 90 respectively. Also, find the QD and CQD.
(b) If the third quartile and quartile deviation of the distribution are 150 and 60 respectively, then
find the value of its first quartile and the coefficient of quartile deviation.
(c) If the quartile deviation and its coefficient of a data are 80 and 0.40 then find its quartiles.
ANSWERS
2. (a) 90, 45, 0.117 (b) 76.25, 38.125, 0.21 (c) 0.9, 0.45, 0.089 (d) 8.5, 4.25, 0.21
(c) 2.55, 0.254
3. (a) 5, 0.093 (b) 62.5, 0.217 (c) 20, 0.364
4. (a) 15, 0.33 (b) 6, 0.231 (c) 120, 280
5. (a) B is more variability (b) X1 is more variability STATISTICS
6. (a) 185, 45, 0.32 (b) 30, 0.67
Dispersion of Individual and Discrete Data 321
8.2 (B) Mean Deviation from Individual and Discrete Data
Learning Objectives
At the end of this topic, the students will be able to:
find the mean deviation from mean and its coefficient.
find the mean deviation from median and its coefficient.
Introduction
Mean Deviation (MD) means the average of the absolute values taken from the deviations (differences)
of each item from the central value (mean, median or mode). So, it is also called average deviation.
Mean deviation taken from the mean gives the best result than median and mode. In actual practice,
mean deviation is calculated either from mean or from median. The average of the absolute deviation of
each item from mean or median or mode is called mean deviation from mean or median or mode. While
calculating the mean deviation, if the deviations are taken from mean, it is called mean deviation from
mean and if the deviations are taken from the median, it is called mean deviation from median. Because
of the sum of the deviation from median is less than that of from mean, the value of the mean deviation
from mean is always greater than the value calculated from median. So, generally, the mean deviation is
computed from mean in statistics. The median deviation is also called absolute deviation.
Calculation of Mean Deviation
The mean deviations from the mean and median are calculated according to the types of data.
(i) Mean Deviation in Individual Data
Let X1, X2, X3, ………, Xn are the variate values and X and Md be its arithmetic mean and median
respectively. Then the mean deviation is computed from the following formulae;
(a) MD from mean = Σ | X– X| (b) MD from median = Σ | X– Md |
N N
(ii) Mean Deviation in Discrete Data
Consider the discrete data as;
Item (x) X1 X2 X3 ……… Xn
Frequency (f) f1 f2 f3 ……… fn
Let X and Md be the arithmetic mean and median of the data respectively, then the mean deviation
is computed from the following formulae;
(a) MD from mean = Σf | X– X| (b) MD from median = Σf | X– Md | , where Σf = N.
Σf Σf
STATISTICS Example 1 : Calculate the mean deviation from mean and its coefficient for the data given below:
25, 32, 29, 38, 42, 26.
Solution: Given, 25, 32, 29, 38, 42, 26
322 Illustrated Optional Mathematics-9
No. of items (N) = 6
Now, we know that
Mean (X) = Σf = 25 + 32 + 29 + 39 + 38 + 42 + 26 = 192 = 32.
N 6 6
Making table;
X X–X |X – X|
25 – 7 7
32 0 0
29 – 3 3
38 6 6
42 10 10
26 – 6 6
Σ |X – X| = 32
∴ Mean Deviation from mean (MD) = Σ | X– X| = 32 = 5.33,
N 6
and Coefficient of MD from mean = MD from Mean = 5.33 = 0.17.
X 32
Example 2 : Compute the mean deviation from median and its coefficient from the data given below:
40, 48, 46, 52, 39, 60, 62, 58.
Solution: Here, arranging the given data in ascending order, we get
39, 40, 46, 48, 52, 58, 60, 62
No. of terms (N) = 8
N+1 th 8+1 th 9 th 4.5th
2 2 2
∴ Median (Md) = term = term = term = term
= 4th term + 5th term = 48 + 52 = 100 = 50.
2 2 2
Now, making table;
X 39 40 46 48 52 58 60 62 Total
X – Md – 11 – 10 –4 –2 2 8 10 12 Σ|X – Md|
|X – Md| 11 10 4 2 2
8 10 12 = 59
∴ Mean deviation from median (MD) = Σ | X– Md | = 59 = 7.38,
N 8
and Coefficient of MD from median = MD from median = 7.38 = 0.15.
Md 50
Example 3 : Calculate the mean deviation and its coefficient from the mean for the following data:
Ages (in years) 5 6 7 8 9 10 STATISTICS
No. of students 82 3 2 5 3
Solution: Making the table;
Dispersion of Individual and Discrete Data 323
Ages (in years) No of stds. fX |X – X| f|X – X|
(X) (f)
5 8 40 2.13 17.04
6 2 12 1.13 2.26
7 3 21 0.13 0.39
8 2 16 0.87 1.74
9 5 45 1.87 9.35
10 3 30 2.87 8.61
ΣfX = 164 Σf|X – X| = 39.39
Σf = N = 23
Here, N = 23, ΣfX = 164.
∴ Mean (X) = Σfx = 164 = 7.13.
N 23
Again, Σf|X – X| = 39.39
Mean deviation from mean (MD) = Σf | X– X| = 39.39 = 1.71,
N 23
Coefficient of MD = MD from Mean = 1.71 = 0.24.
X 7.13
Example 4 : Find the mean deviation from median and calculate its coefficient from the given data:
x 13 17 20 23 26 30
f 2 4 3 2 35
Solution: Making table;
X f cf |X – Md| f|X – Md|
13 2 2 10 20
17 4 6 6 24
20 3 9 3 9
23 2 11 0 0
26 3 14 3 9
30 5 19 7 35
Σf = N = 19 Σf|X – Md| = 97
Here, N = 19
N+ 1 th 19 + 1 th 10th
2 2
∴ Median (Md) = term = term = term = 23
Now, we have
Mean deviation from median (MD) = Σf | X – Md | = 97 = 5.11,
N 19
STATISTICS and Coefficient of MD = MD from median = 5.11 = 0.22.
Md 23
324 Illustrated Optional Mathematics-9
EXERCISE - 8.2 (B)
1. (a) What is mean deviation from mean?
(b) Write down the formula to calculate the mean deviation from median of the given individual
data.
(c) Write down the formula to find the coefficient of mean deviation from mean of the given
discrete data.
2. Find the mean deviation from the mean and its coefficient from the following data:
(a) 3, 7, 9, 10, 14, 17 (b) 388, 297, 100, 102, 105, 109
3. Compute the average deviation from the median from the following data and also find its coefficient:
(a) 12, 10, 14, 19, 16, 15, 11 (b) 100, 109, 118, 127, 117, 125
4. Calculate the mean deviation from mean and its coefficient from the following data:
(a) X 5 8 10 13 18 19 22
f 2543136
(b) Price (X) 105 120 135 150 165 180
No. of Books (f) 4 10 5 6 7 8
5. Compute the mean deviation from the median and its coefficient from the following data:
(a) Age 29 32 35 38 41
Women 6 7 4 53
(b) X 3 5 7 9 11 13
cf 5 7 12 16 19 20
ANSWERS (b) 106, 0.58 3. (a) 2.43, 0.17 (b) 7.33, 0.06
(b) 22.76, 0.16 5. (a) 3.48, 0.11 (b) 2.5, 0.36
2. (a) 3.67, 0.37
4. (a) 5.583, 0.4
8.2 (C) Standard Deviation from Individual and Discrete Data STATISTICS
Learning Objectives
At the end of this topic, the students will be able to:
find the standard deviation and its coefficient.
Introduction
The concept of standard deviation (SD) was introduced by Karl Pearson in 1823. SD is good measure of
dispersion. It is also known as root-mean square deviation. It is denoted by the Greek small letter σ (read
as sigma). It measures the absolute dispersion. So, it is the positive square root of the average of square of
the deviation taken from the mean.
Dispersion of Individual and Discrete Data 325
Calculation of Standard Deviation
Standard deviation is calculated according as the types of the data in the following methods:
(i) Standard Deviation in Individual Data
Let X1, X2, X3, ……., Xn are the variants then the SD is calculated by using any one of following
formulae;
(a) Direct Method: SD (σ) = ΣX2 – ΣX 2
N N
(b) Actual Mean Method: SD (σ) = Σ(X – X)2 i.e., Σd2
N N
(c) Assumed Mean Method: SD (σ) = Σd2 – Σd 2
N N
where, d = X – A, A = Assumed Mean.
(ii) Standard Deviation in Discrete Data
Consider the discrete data as;
Item (X) X1 X2 X3 ….. Xn
Frequency (f) f1 f2 f3 ….. fn
The SD is calculated by using anyone of the following formulae;
(a) Direct Method: SD (σ) = Σfx2 – Σfx 2
N N
(b) Actual Mean Method: SD (σ) = Σf(X – X)2 i.e., Σfd2
N N
(c) Assumed Mean Method: SD (σ) = Σfd2 – Σfd 2
N N
where, d = X – A, A = Assumed Mean.
Coefficient of Standard Deviation
The relative measure of dispersion based on the standard deviation is called the coefficient of standard
SD σ
deviation. Therefore, the coefficient of SD = Mean = X
Variance
The variance is the square of the standard deviation. So, it is defined as follows;
Variance (Var.) = Square of SD = σ2
Then its computational formulae are given below:
Σ (X – X )2 ΣX2 ΣX 2
N N N
(a) Var.(X) = = – for individual series.
Σf(X – X )2 ΣfX2 ΣfX 2
N N N
(b) Var.(X) = = – for discrete series.
STATISTICS Coefficient of Variation
The relative measure of dispersion based on the SD in the form of percentage is called the coefficient of
σ
variation. Then the Coefficient of Variation, CV = X × 100%
Note: In comparison between two data, say A and B, the data has less CV than that of B. A is more
consistent and B has more variability.
326 Illustrated Optional Mathematics-9
Example 1 : Calculate the standard deviation of the data given below. Also, find its coefficient:
24, 13, 21, 19, 26, 29.
Solution: Here, 24, 13, 21, 19, 26, 29. No. of item (N) = 6
∴ Mean (X) = ΣX = 24 + 13 + 21 + 19 + 26 + 29 = 132 = 22.
Now, making table; N 6 6
X 24 13 21 19 26 29 Total
X–X 2 –9 –1 –3 4 7 -
(X – X)2 4 81 1 9 16 49 Σ(X – X)2 = 160
∴ Standard deviation (σ) = Σ(X – X)2 = 160 = 26.67 = 5.16
N 6
and its Coefficient (CSD) = σ = 5.16 = 0.23.
X 22
“Alternatively”
Making Table;
X 24 13 21 19 26 29 ΣX = 132
X2 576 169 441 361 676 841 ΣX2 = 3064
Here, N = 6, ΣX = 132, ΣX2 = 3064.
Now, we have
ΣX2 ΣX 2 3064 132 2
N N 6 – 6 = 510.67 – 484 = 26.67 = 5.16,
SD(σ) = – =
Coefficient of SD = σ = 5.16 = 0.23.
X 22
Example 2 Compute the standard deviation and its coefficient from the given data:
X 5 10 15 20 25 30
f 2 3 3 2 52
Solution: Let assumed mean, A = 20
Xf fX d = X – A fd fd2
5 2 10 – 15 – 30 450
10 3 30 – 10 – 30 300
15 3 45 – 5 – 15 75
20 2 40 0 0 0
25 5 125 5 25 125
30 2 60 10 20 200
Σf = 17 ΣfX = 310 Σfd = – 30 Σfd2 = 1150
Here, N = Σf = 17, Σfd = – 30, fd2 = 1150
Now, we have
Σfd2 Σfd 2 1150 – 30 2
N N 17 17
Standard Deviation (SD) = – = – STATISTICS
= 67.65 – 3.11 = 64.54= 8.03,
and its coefficient (CSD) = σ = 8.03 = 0.44.
X 310
17
Dispersion of Individual and Discrete Data 327
EXERCISE-8.2 (C)
1. (a) Write the formula to find the standard deviation of an individual data.
(b) Write down the formula to calculate the coefficient of standard deviation of a discrete data.
(c) Write any one difference between mean deviation and standard deviation.
(d) What is the coefficient of variation?
2. Calculate the standard deviation and its coefficient from the following data:
(a) 15, 17, 21, 25, 27, 30 (b) 27, 45, 57, 62, 77, 84, 92, 97
(c) 4, 14, 16, 6, 8, 10, 17, 21 (d) 10, 16, 18, 20, 24, 26
3. Find the standard deviation and its coefficient from the following data:
(a) Marks 12 17 23 24
No of students 3 4 5 4
(b) Weight (in kg) 18 19 20 21 22 23
No. of students 6 7 4 3 21
(c) Marks 10 20 25 30 35 40
No. of students 1 5 10 12 8 4
(d) Wages (Rs.) 120 130 140 150 160 170
9
No. of students 5 9 14 5 8
70
4. Calculate the variance and the coefficient of variation from the following data: 2
(a) 10, 12, 14, 19, 18, 27, 23, 29 (b) 22, 25, 30, 35, 40, 45, 48
(c) X 10 20 30 40 50 60
f 353543
(d) X 10 20 30 40 50
f
8 12 15 9 6
ANSWERS (b) 22.69, 0.34 (c) 5.55, 0.46 (d) 5.25, 0.28
(b) 1.44, 0.07 (c) 6.53, 0.23 (d) 10.86, 0.34
2. (a) 5.35, 0.24 (b) 84, 26.19% (c) 330.24, 48.33% (d) 152.03, 43.11%
3. (a) 4.58, 0.23
4. (a) 42, 34.11%
Project Work
Record the length of your and your friend’s spam and make frequency distribution table. Find the
STATISTICS following partition values and dispersions:
(a) 3rd Quartile (b) 4th Decile (c) 38th Percentile
(d) 73rd percentile (e) Inter Quartile Range (f) Quartile Deviation
(g) Mean Deviation from Mean (h) Mean Deviation from Median (i) Standard Deviation
328 Illustrated Optional Mathematics-9
GLOSSARY - 1 (ALGEBRA) Pre-image of Function: An element in the domain of
a function.
Ordered Pair: A pair of numbers/objects arranging
in ordered form which is separated by comma (,) and Vertical Line Test of Function: Identifying a function
enclosed by round or small brackets ( ). by drawing a vertical line. If the vertical line intersects
the graph of a relation at once, it is a function. Otherwise,
Components of Ordered Pairs: Numbers/objects of an it is not function.
ordered pair.
Value of Function/Functional Value: If f (x) is a
X-component: A component in the first position of an function and (a, b) in f, then we write f(a) = b, where
ordered pair. f(a) is called the value of the function f(x) at x = a.
Y-component: A component in the second position of Onto Function: Function having equal range and
an ordered pair. codomain.
Equal Ordered Pair: Two ordered pairs having equal Into Function: Function having its range as proper
corresponding components. If in two ordered pairs (x, y) subset of its co-domain of a function.
and (a, b), x = a and y = b, then (x, y) = (a, b) and if (x,
y) = (a, b) then the corresponding components of equal One to One: Every elements of range having single pre-
ordered pairs are equal. i.e., x = a and y = b. image in a function.
Cartesian Product: A set of all possible ordered pairs Many to One: At least one element of range having
of two non-empty sets A and B. It is denoted by A×B and more than one pre-image in a function.
is defined by A×B = {(x, y): x∈A, y∈B}.
One-One and Onto: Function satisfying both one to
Cross Product: Same meaning of Cartesian product. one and onto functions.
Roster Form: Representation of elements/components Many-One and Onto: Function satisfying both many
or ordered pairs in the form of set. to one and onto functions.
Set-builder Form: Representation of elements in the One-One and Into: Function satisfying both one to one
form of rule. and into functions.
Tree Diagram: Diagram with branches for showing the Many-One and Into: Function satisfying both many to
objects. one and into functions.
Mapping/Arrow Diagram: Diagram with ovals/circles Constant: Quantity which has only one value.
for showing the components/objects by arrow-heads.
Variable: Quantity which has more than one value.
Lattice Diagram: Diagram for showing the ordered
pairs in the graph or square grid. Algebraic Expression: Expression which is connected
by sign of basic fundamental operations of arithmetic
Relation: A subset of the Cartesian product of two (+, –, ×, ÷).
non-empty sets. The relation from A to B is denoted by
R:A → B where R ∈ A × B and is defined by R = {(x, y}: Coefficient: Repeated number of the same objects in
x∈A, y∈B}. It is read as "x is related to y" or "x is the addition, which multiplies to the object.
relation of y" and is written as (x, y)∈R or xRy.
Numerical Coefficient: Real number, which multiplies
Domain of Relation: A set of x-components of a relation to the variable.
R. The domain D is defined as, D = {x: (x, y) ∈ R}.
Literal Coefficient: Letter, which multiplies to the real
Range of Relation: A set of y-components of a relation number or variable.
R. The range Re is defined as, Re = {y : (x, y) ∈ R}.
Non–negative Integer: Integer, which is either positive
Inverse of Relation: A new relation obtained by or zero that is not negative number.
interchanging the domain (x-components) and the
range (y-components) of a relation R to each other. It is Term: Multiplicative or division form of constant and
denoted by R-1. The inverse of relation R = {(x, y), x∈ A, variable or only constant or only variable.
y ∈ B} is R-1 = {(y, x); y ∈ B and x ∈ A}.
Polynomial: An algebraic expression having non-
Function: Relation in which every element of the negative integer as the power of variable in each term.
domain is associated with a unique element of the co-
domain. It is denoted by f:A→B or A→B for two non- General or Standard Form of Polynomial: Polynomial
empty sets A and B. If x is the variable of the domain, its of n degree with variable x, which is in the form as,
range is denoted by f(x), g(x), h(x), etc.
anxn–1 + an – 1 xn – 1 + an – 2 xn – 2 + .….. + a2 x2 + a1 x1
Domain of Function: A set of all elements of the first + a0 x0, where an ≠ 0 and a0, a1, a2,……, an–2, an–1, an are
set of a function. constants, n is non-negative integer.
Co-domain of Function: A set of all elements of the Standard Polynomial: Polynomial in general or
second set of a function. standard form.
Range of Function: A set of all the mapped elements Degree of Polynomial: The highest sum of exponents
of the second set with the element of the first set of a of the variables of the terms of a polynomial.
function.
Monomial: A polynomial having only one term.
Image of Function: An element in the range of a function
329
Binomial: A polynomial having only two terms. Arithmetic Sequence: A sequence having the same
difference (backward difference) first time. The linear
Trinomial: A polynomial having only three terms. sequence is also the arithmetic sequence.
Multinomial: A polynomial having more than three terms. Geometric Sequence: A sequence having the same ratio
first time.
Constant Polynomial: A polynomial having only one
constant term or zero degree polynomial. General Term: The specific term of a sequence. It is a
rule of the sequence, which helps to find any term of the
Linear Polynomial: A polynomial of the first degree sequence. The general form of the sequence is given as
polynomial. t1, t2, t3,………….., tn, ………. ,
Here, the nth term is the general term of the sequence.
Quadratic Polynomial: A polynomial of the second
degree polynomial. nth Term of Linear Sequence: The general term (or nth
term or rule) of a linear sequence is dn + (a – d) or a +
Cubic Polynomial: A polynomial of the third degree (n –1)d, where a, b ∈ Q and n ∈ N. It is the first degree
polynomial. algebraic expression in the variable n.
Bi-quadratic Polynomial: A polynomial of the fourth General Term of Quadratic Sequence: The general
degree polynomial. term (nth term or rule) of a quadratic sequence is an2 + bn
+ c, where a, b, c ∈ Q and n ∈ N. It is the second degree
Zero Polynomial: A polynomial in which all the algebraic expression in the variable n.
numerical coefficients are zero.
nth Term of Geometric Sequence: The general term (nth
Equal Polynomial: Two polynomials with equal term or rule) of a quadratic sequence is arn-1, where a∈ Q
coefficients and equal degree of their corresponding terms. and n ∈ N.
Closure Property: Sum of polynomials having the same Series: The representation of the terms of a sequence in
degree of addend polynomials. the sum or addition.
Commutative Property: Same sum of polynomials Finite Series: Series having finite number of terms.
when added from forward or backward.
Infinite Series: Series having infinite number of terms.
Associative Property: Same sum of three polynomials
when added in any arrangement. Ascending Series: A series whose every term, except
first term, is increased than the previous term.
Additive Identity Property: When any polynomial is
added with zero (0), it remains the same. Here, 0 is called Descending Series: A series whose every term, except
addetive identity. first term, is decreased than the previous term.
Additive Inverse Property: When any polynomial is Linear Series: A series which has the same differences
added with its negative form, their sum is zero (0). The at once or has equal first differences.
addetive inverse of a is - a.
Quadratic Series: A series which has the same difference
Sum (or Addition) of Polynomials: Addition of the at the second times or has equal second difference.
coeffici-ents of like terms of the polynomials.
Arithmetic Series: A series having the same difference at
Difference (or Subtraction) of Polynomials: Subtraction first step. The linear sequence is also the arithmetic series.
of the coefficients of like terms of the polynomials.
Geometric Series: A series having the same ratio at first
Sequence: Set of numbers (objects) with specific rule. step.
Term of Sequence: The number (object) of a sequence. Progression: A sequence/series of numbers in which there
is a constant relation between any two consecutive terms.
Common Difference: The same or constant difference
of any two consecutive (successive) terms of a sequence. Arithmetic progression (AP): A sequence (series) of
It is obtained by subtracting the term from just preceding numbers in which there is a constant difference between
term. any two consecutive terms.
∴ Common Difference (d) = Backward Term - Geometric progression (GP): A sequence (series) of
Forward Term (Preceding Term) numbers in which there is a constant ratio between any
two consecutive terms.
Finite Sequence: Sequence having finite number of terms.
Sigma Notation (Using of Summation): A series
Infinite Sequence: Sequence having infinite number of representing in the form of summation. The sum of the
terms. series is denoted by Sn = t1 + t2 + t3 + t4 + t5 + ........ + tn
and its sigma
Ascending Sequence: A sequence whose every term is
increased than the previous term. i
Descending Sequence: A sequence whose every term, (summation) notation is symbolized as ∑[tn] . It is read as
except first term, is decreased than the previous term. n=1
Linear Sequence: A sequence which has the same "summation of tn's when n runs from 1 to i".
differences at the first time or has equal first differences. Partial Sum: The sum of finite terms of AS. The nth term
of AS in terms of Sn is given by tn = Sn – Sn-1 where, n∈N.
Quadratic Sequence: A sequence which has same
differences at the second times or has equal second
difference.
330
GLOSSARY - 2 (LIMIT) Limit of Sequence: A specific value that the term of
infinity sequence "tends to" or "approaches".
Sequence: A list of numbers or objects that are in Inscribed Polygon: A polygon whose vertices are
order. placed on a circumference of a circle.
Numerical Sequence: A sequence formed by Regular Polygon: A polygon having equal sides
numbers. and angles.
General Term of Sequence: A specific term of a n-gon: A polygon having specific number of sides.
sequence.
Rational Number: A number in the form of p th Figure Sequence: A sequence that is formed as the
where, p, q ∈ Z (integer) and q ∉ 0. It is terminatiqng specific pattern of figures.
and repeating or recurring decimal number.
Series: The addition of the terms of the
Fractional Number: A number in the form of fraction corresponding sequence.
or rational number.
Infinite Series: The sum form of the terms of the
Irrational Number: A number that is not rational. It corresponding infinite sequence.
is not terminating and repeating or recurring decimal
number. Absolute Value: The non-negative value of variable
without regard to its sign.
Real Number: A union of rational and irrational
number. Limiting Value of Infinite Series: Approaching
specific value of the sum of all terms of an infinite
Decimal Number: A non-integer that is represented series.
by using a dot called decimal point.
Function: A relation between a set of inputs and
Scientific Notation: A special way of writing a set of permissible outputs with the property that
numbers that are represented in tenths decimal part each input is related to exactly one output.
and multiple of 10.
Domain: A set of inputs of a function.
Infinity: A concept describing something without
any bound or larger than any natural number. It is Range: A set of outputs of a function.
symbolised by ∞. It has no end.
Limit of Function: A value of a function near a
Infinite Number: An indefinitely large number particular input. For f(x) a real function, the limit of
f as x approaches infinity is L, denoted
Countable Infinite Number: An infinitely large
number that is countable. lim f(x) = f(a) or L.
x→a
Positive Infinite Number: A number that goes to
infinite large number at right side of number line. It is It is read as "the limit of f(x) as x tends to a is equal
symbolised by + ∞. to L or f(a)".
Negative Infinite Number: A number that goes to Tend or approach: To come nearer to equal.
infinite large number at left side of number line. It is
symbolised by – ∞. Left Side Limit: A limit of f(x) in which x tends to
a from the left. It is denoted by
Midpoint: A point that halves the given line segment
or object. x lim f(x) = f(a) or L.
→ a–
Limit: A number such that the value of the given
function that remains arbitrarily close to this number Right Side Limit: A limit of f(x) in which x tends
when the independent variable is insufficiently close to a from the right. It is denoted by
to a specified point or is sufficiently large.
lim
x → a+ f(x) = f(a) or L.
331
GLOSSARY - 3 (MATRIX) Double Transpose Property : Transpose of transports
of a matrix is the matrix itself i.e., (AT)T = A.
Matrix: Arrangement of numbers/objects in rectangular
form enclosed by round or square brackets. Transpose Over Addition: Transpose of sum of the
matrix as is the sum of their transpose. i.e., (A + B)T =
Entries/Components: The objects of a matrix. AT + BT
Row: Entries in horizontal array of a matrix. Multiplication of a matrix by a scalar:
Column: Entries in vertical array of a matrix. Multiplication by a scalar to each component of the
a b
Order: Number of rows and column of a matrix in the given matrix. Let A = c d be a matrix and k be a
form of cross (×).
scalar, then the mkualtiplkibcation by k to A is given by
Raw matrix: Matrix with only one row. a b
kA = k c d = kc kd and the order of the matrix
Column matrix: Matrix with only one column.
after multiplication is the same as the initial matrix.
Null matrix: Matrix with all entries as zeros.
Properties of multiplication of a matrix by a scalar:
Square matrix: Matrix with equal number of rows and Let A and B be two matrixes and m and n two scalar
columns. quantities, then the multiplication of a matrix by a scalar
quantity satisfies the following properties:
Rectangular matrix: Matrix having unequal number of
rows and columns. (i) (m + n)A = mA + nA.
Diagonal matrix: Square matrix with non-zero elements (ii) m(A + B) = mA + mB, where A and B must have
in the main or leading diagonal and others as zero. the same order.
Scalar matrix: Diagonal matrix with the same elements (iii) IA = A = AI, where I is an identity matrix.
in the main or principal or leading diagonal.
(iv) If n = 0, then nA = O, where O is a null matrix of the
Identity matrix: Scalar matrix with elements 1's in the
main diagonal. order of A.
Equal matrices: Two matrices with the same order and Multiplication of two matrices: Let A = a1 b1 and
equal corresponding entries. a2 b2
c1 d1 2×2
Triangular matrix: Square matrix with all elements c2 d2 2
above or below the main diagonal as zeros. B= ×2 be two matrices. Then the multiplication
Upper triangular matrix: Square matrix with elements of the matrices A and B are given by,
below the main diagonal as zeros.
A×B= a1 b1 × c1 d1 = a1c1 + b1c2 a1d1 + b1d2 .
Lower Triangular Matrix: Square matrix with elements a2 b2 c2 d2 a2c1 + b2c2 a2d1 + b2d2
above the main diagonal as zeros. The number of column of A and the number of rows
Symmetric Matrix: Square matrix which is itself when of B are equal and also the order of the multiplication
interchanging rows and columns.
of matrices A × B has the number of rows of A × the
Matrix Addition: Adding corresponding components of
the same ordered matrices. number of column of B. If Am×p and Bp×n are two matrices
then Am×p × Bp×n = (A × B)m×n = Cm×n, where C = A × B.
Matrix Subtraction: Subtracting corresponding
components of the same ordered matrices. Conformable for multiplication of two matrices:
Additive Identity: Matrix which doesn't change the Two matrices are conformable for multiplication (or
given matrix when added. i.e. null matrix of the same
order as the given matrix. multiplication of two matrices is defined) if the number
Additive Inverse: Matrix which makes null matrix of of columns in the pre-matrix (first matrix) is equal to the
same order of the given matrix when added.
number of rows in the post matrix (second matrix).
Closure Property in Addition: When two or more
matrices of the same order are added, their sum will be Properties of matrix multiplication: There are the
the same order as given matrices. following properties of matrix multiplication for
the matrices A, B and C; (i) Associative property: If
Commutative Property in Addition: If two matrices A (AB)C and A(BC) are compatible then (AB)C = A(BC).
and B of the same order are added, A + B = B + A. (ii) Distribution property of matrix multiplication over
addition: If A(B + C) and AB + AC are compatible, then
Associative property in Addition: If there are three A(B + C) = AB + AC. (iii) Existence of identify matrix
matrices A, B and C of the same order, (A + B) + C = A + in multiplication: If A is a matrix of m×n order and I,
(B + C). another matrix of the same order such that AI = A = IA,
then I is called multiplication identity of A.
Transpose of Matrix: Matrix obtained by interchanging
the rows and columns of the given matrix A, is denoted
by A' or AT or At.
332
GLOSSARY - 4 (COORDINATE GEOMETRY)
Axes: Number lines which divide the coordinate plane Square: Rectangle with equal adjacent sides or rhombus
into four equal quadrants. Singular form of axes is axis. with any one right angle in which diagonals are equal and
bisect perpendicularly and also bisect vertical angles by
x-axis: Horizontal axis XOX', which is represented by each diagonal.
the line y = 0.
Kite: Convex quadrilateral with equal opposite adjacent
y-axis: Vertical axis YOY', which is represented by the sides in which the main (vertical) diagonal bisects the
line x = 0. second diagonal perpendicularly and the main diagonal
is bigger than the second diagonal.
Quadrants: Sectors of coordinate plane separated by
axes (x-axis and y-axis). There are four quadrants/ Circle: Locus of a moving point equidistant from a fixed
quarters. point in which all radii are equal.
(i) The area covered by XOY is called first quadrant Equidistant: Having equal distances from the points.
in which both coordinates (abscissa and ordinate), are
positive values. Path: Trace/road in which an object moves.
(ii) The area covered by X'OY is called second quadrant Locus: Path of a moving point under certain rule or
in which abscissa (x-value) has negative and ordinate condition.
(y-value) has positive).
Fundamental Principle of locus:
(iii) The area covered by X'OY' is called third quadrant in
which both coordinates are negative values. (i) When a point satisfies on the equation, the point lies
in the locus of the equation.
(iv) The area covered by XOY' is called fourth quadrant
in which abscissa is positive value and ordinate is (ii) When a point lies on the locus, the point satisfies
negative value. the equation of the locus.
Abscissa (x-coordinate): The value represented by Straight Line: Locus of a moving point obtained by the
x-axis, which is positive in right and negative in left of linear equation.
the coordinate plane.
Linear Equation: Equation with degree 1 or power of
Ordinate (y-coordinate): The value represented the variable is always unit.
by y-axis, which is positive upward and negative in
downward of the coordinate plane. Inclination of Line: Angle made by the line in anti-
clockwise direction with x-axis, denoted by θ.
Coordinate Geometry (Analytic geometry): Branch of
the mathematics related to the coordinate plane. Horizontal Line: Line parallel to the x-axis or
perpendicular to the y-axis whose angle of inclination is
Distance Between Two Points: Length of any two 0o.
district points (x1, y1) and (x2, y2) which is computed by
the formula, d = (x2 – x1)2 + (y2 – y1)2. Vertical Line: Line perpendicular to the x-axis or
parallel to the y-axis whose angle of inclination is 90o.
The distance from the origin to the given point (x1, y1) is
computed by the formula, d = x12 + y12 . ObliqueLine: Line neither parallel to x-axis nor parallel
to y-axis whose angle of inclination is between 0o and
Scalene Triangle: Triangle with non-equal sides. 90o.
Isosceles Triangle: Triangle with any two equal sides. Negative line: Line parallel to x-axis but in negative or
opposite direction whose angle of inclination is exactly
Equilateral Triangle: Triangle with equal sides. 180o.
Right- angled Triangle: Triangle with one angle as right Slope of Line: Tangent of angle θ, where θ is the angle
which satisfies the Pythagoras relation h2 = p2 + b2. of inclination of the line, denoted by m. So, m = tanθ =
(tayxhxn--e1iyin,nltytpieen1ror)cecienaepppnttatd(((sxRRs(1euxi,ssn2ye,))t1yh)w2ro)oh,nuetghinthe.tnhtheitesliosnrleoigppieans,issietmss sth=loropxy22ue––gihsyx11mtwan=odpxywo11 hifneotnsr
Isosceles Right Triangle: Right triangle with equal legs. Parallel Lines: Lines whose slopes are equal in which
the distances between them are constant.
Parallelogram: Quadrilateral with opposite parallel
sides in which opposite sides and angles are equal and Collinear or Coincident Lines: When a line contains
diagonals bisect each other. three points and the slopes of the line segments joining
the first two points and last two points are equal.
Rectangle: Parallelogram with any one angle as right in
which diagonals are equal.
Rhombus: Parallelogram with equal adjacent sides in
which diagonals bisect perpendicularly and also bisect
vertical angles.
333
Concurrent: Condition when three or more lines Reduction to double-intercept form: When the general
intersect at single point such as medians or perpendiculars
drawn from the vertices to their opposite bases of triangle equation ax + by + c = 0 changes into double-intercept
is concurrent. x y x y
form a + b = 1, it becomes –c + –c = 1
Equation of Straight Line Parallel to x-axis: Equation ab
of straight line which is equidistant from the x-axis Therefore, the x-intercept (a) = –coecfofincsietanntto(fcy) (a) and
as y-intercept (b), i.e., y = b, that is with constant
y-coordinate. the y-intercept (b) = – constant (c) .
coefficient of y(b)
Equation of straight line parallel to y-axis: Reduction to perpendicular or normal form:
Equation of straight line which is equidistant from the When the general equation ax + by + c = 0 changes into
y-axis as x-intercept (a), i.e., x = a, that is with constant perpendicular or normal form x cosα + ysinα = p,
x-coordinate.
it becomes,
Standard form of equation of straight line: Equation ± a + b2 x + ± b b2 y + ± c =0
of straight line ax + by + c = 0. There are three standard a2 a2 + a2 + b2
forms as follows:
(i) Slope-intercept form Therefore, α = cos-1 ± a or, sin-1 ± b
(ii) Double-intercept form a2 + b2 a2 + b2
(iii) Perpendicular or Normal form
| |and c .
p= ± a2 + b2
General form of equation of a straight line: Equation
in the form of ax + by + c = 0, where a, b and c are Distance from a point to the given line: The distance
arbitrary constants.
fyo1r)mtoultah,edg=ivaexn1 line ax + by +
| |from a point (x1, + by1 + c . c = 0 is
the a2 + b2
Equation of straight line in slope-intercept form: measured by
Equation of straight line in the form of y = mx + c, where
m is the slope of the line and c, the y-intercept. If the line Equation of straight line in point-slope form: The
passes through origin, its equation will be y = mx. If the
line is parallel to x-axis or slope of the line is zero, its equation of a straight line passing through the point (x1,
equation will be y = c. y1) and having the slope m (tan θ) is given by y – y1 =
m(x – x1). If θ = 0o or, m = 0, it becomes y = y1, which
Equation of straight line in double-intercept form: is parallel to x-axis. If θ = 90o, it becomes x = x1, which
x y is parallel to y-axis. The line passes through the origin, it
Equation of a straight line in the form of a + b = 1,
becomes y = mx, which doesn't make any intercept.
where a and b be are the intercepts on x-axis and y-axis Equation of straight line in two-point form: The
respectively. If the line is parallel to x-axis, then the equation of the straight line passing through two points
y2 – y1
equation of the line becomes y = b. If the line is parallel (x1, y1) and (x2, y2) is given by y – y1 = x2 – x1 (x – x1).
to y-axis, then the equation of the line becomes x = a. Area of triangle in coordinate plane: Area occupied
Equation of straight line in perpendicular or normal by the triangle with three coordinate points, which is
form: Equation of a straight line in the form of xcosα +
ysinα = p, where α is the angle made by the perpendicular computed by using the formula,
drawn from the origin to the given line and p be the length x1 x2 x3 x4 x1 –
of the perpendicular. If the angle is 0o, the equation of the ∆= 1 y1 y2 y3 y4 y1 +
line becomes x = p, which is parallel to y-axis. If the 2
angle α is 90o, the equation of the line becomes y = p,
which is parallel to x-axis. = 21(x1y2 – x2y1) + (x2y3 – x3y2) + (x3y1 – x1y3).
When a vertex of triangle is a origin, the formula
1
becomes 2 |(x1y2 – x2y1)|. If the area of triangle is zero,
Reduction of general equation in standard form: the given three vertices or points are collinear.
Changing the general equation ax + by + c = 0 of the Area of quadrilateral in coordinate plane: Area of
straight line into its standard form as,
x y quadrilateral with knowing four vertices is computed
y = mx + c, a + b =1 or xcosα + ysinα = p.
x1 x2 x3 x4 x1 –
Reduction to slope-intercept form: When the general by 1 y1 y2 y3 y4 y1 +
2
equation ax + by + c = 0 changes into slope intercept
a c 1 |(x1y2 x1y4)|
form y = mx + c, it becomes y = – b x– d . Therefore, = 2 – x2y1) + (x2y3 – x3y2) + (x3y4 – x4y3) + (x4y1 –
the slope of the line (m) = – coefficient of x (a) and
coefficient of y (b)
y-intercept = – constant (c) .
coefficient of y (b)
334
GLOSSARY - 5 (TRIGONOMETRY)
Trigonometry: Branch of mathematics studying the Polygon: Close geometric figure bounded by three or more
measurement of parts of triangle. line segments.
Measurement of angle: Amount of angle between the initial Interior angle of polygon: Angle inside the polygon made
line and revolving line, denoted by ∠ or .
by two sides. Measure of interior angle in regular polygon,
Positive angle: Angle when the revolving line rotates in θ = n – 2 × 180o.
anti-clockwise direction. n
Exterior angle of Polygon: Angle outside the polygon made
Negative angle: Angle when the revolving line rotates in
clockwise direction. by its one side and producing its adjacent side. Measure of
360o
exterior angle in regular polygon, α = n .
Sexagesimal system, English or British system: System of Regular Polygon: Polygon in which all sides are equal. The
measuring angle in degree, minute and second i.e., dividing
a right angle into 90 equal parts. angles of the regular polygon are also equal.
Sexagesimal degree or Degree: Unit of measuring angle Irregular Polygon: Polygon which is not regular or has
unequal sides and unequal angles.
when a right angle is divided into 90 equal parts, denoted
by (o). Clock: Machine which takes the standard time for specific
place or country.
∴ 1 right angle = 90o and angle in a circle = 360o:
Hands of Clock: Needles of clock which move left to right,
Sexagesimal minute: Unit of measuring angle when the called clockwise direction.
angle of 1o is divided into 60 equal parts, denoted by (').
∴ 1o = 60' Hour hand: Thick and short hand of a clock which rotates
Sexagesimal second: Unit of measuring angle when the completely once, it takes 12 hours or half day. Therefore,
angle of 1' is divided into 60 equal parts, denoted by ('').
two rotations of hour hand take a day and it takes angular
∴ 1' = 60''. position of 360o at once rotation.
Centesimal or French system: System of measuring angle ∴ 1 day = 24 hours, 1hr = 360o = 30o.
in grade, minute and second. i.e., dividing a right angle into 12
100 equal parts. Minute hand: Middle and long hand of clock which rotates
at once, it takes 60 minutes or 1 hours and it takes angular
360o
Centesimal grade: Unit of measuring angle when a right position of 360o. So, 1 minute takes 60o = 6o.
angle is divided into 100 equal parts, denoted by (g).
∴ 1 hour (hr) = 60 minutes (mins).
∴ 1 right angle = 100g. Second hand: Thin and long hand of clock which rotates at
Centesimal minute: Unit of measuring angle when the once, it takes 60 seconds or 1 minute. 360o
angle of 1g is divided into 100 equal parts, denoted by ('). 60o
∴ 1g = 100' ∴ 1 min = 60 seconds. So, 1 second takes = 6o.
Length of Arc: Measure of arc of a circle which makes an
Centesimal second: Unit of measuring angle when the angle at centre, is denoted by l and is given by l = θr units.
angle of 1' is divided into 100 equal parts, denoted by ('').
∴ 1' = 100''. Central angle: Angle made by arc length of a circle to its
l
centre, is denoted by θ and is given by θc = r .
Grade measure: Measurement of angle in grade (g) only. Radius of Arc: Measure of constant distance from the centre
Degree measure: Measurement of angle in degree (o) only. to the arc and is denoted by r and is given by r= l units.
θ
Circular measure: Measurement of angle in radian (c) only.
Pendulum: Hanging stick or big needle of wall clock, which
Radian system: System of measuring angle in radian or πc.
πc oscillates at once in extreme positions and it takes equal
i.e., 1dirvigidhitnagngalreig=hπt2canogr laenignlteo=22πecq,uwahl epraertπs.=
∴ c angles on both sides of vertical line.
d
in circle. Trigonometric ratios: Ratios of any two sides among
Pi (π): Mathematical constant of the ratio of perpendicular (p), base (b) and hypotenuse (h) of a right-
the circumference of a circle to its diameter. angled triangle. There are six trigonometric ratios for any
i.e., π = c = 3.14159265 …. ≈ 22 (approximately) but in acute angle θ. They are as follows:
d 7
p = sine angle of θ (sin θ), b = cosine angle of θ (cos θ),
radian system of measurement of angle, πc = 180o = 200g. h h
1 radian (1c): Angle subtended at the centre of circle and its p = tangent angle of θ (tan θ), b = cotangent angle of θ (cot θ),
18π0o. b p
arc equal to its radius. It has constant value
i.e., 1c = h = secant angle of θ (sec θ), h = cosecant angle of θ (cosec θ),
180o . b p
π
Relation between degree, grade and circular measures: 1 Fundamental or basic trigonometric ratios: First three
=1π81c000, g1=g =π21c90o trigonometric ratios sin θ, cos θ and tan θ.
right angle = 90o
∴ 1o = Reciprocal relations: Relations of two opposite
10g = = πc , 1c = 180o = 200g . trigonometric ratios such as sin θ × cosec θ = 1, cos θ × sec
9 200 π π θ = 1 and tan θ × cot θ = 1. i.e., the product of reciprocal
trigonometric ratio is always 1.
335
Quotient relations: Relations in the form of division/ signs in fourth quadrant, A means "all" in which all ratios
are positive sign in first quadrant, S means "sin and cosec",
quotient of two trigonometric ratios such as; which are positive sign in second quadrant and T means ''tan
and cot'' which are positive sign in third quadrant and the
tan θ = sin θ and cot θ = cos θ . other ratios in respective quadrants are negative sign.
cos θ sin θ
Trigonometric ratio of angle greater than 90o: When
Pythagorean or squared relations: Relation computing that trigonometric ratio of any angle greater than
90o, at first identify the number of 90o in the angle as (n × 90o
obtained by Pythagoras theorem (h2 = p2 + b2). ± θ). When n is odd, the ratio will be changed on the rule as
sin → cos, tan → cot, sec → cosec and vice versa. Similarly,
There are mainly three types of Pythagorean or when n is even, the ratio is not changed and the sign of the
ratio is arranged on the basis of CAST rule.
squared relations. They are: (i) sin2 θ + cos2 θ = 1
(ii) sec2 θ – tan2 θ = 1 (iii) cosec2 θ – cot2 θ = 1
Conversion of trigonometric ratios: Process of conversion
of one trigonometric ratio into another.
Identity: Mathematical statement which is true for any Trigonometric function: Function involving trigonometric
value of the variable, specially angle. ratios.
Equation: Mathematical statement which is true for some Trigonometric curve: Free hand curve obtained from the
particular values of the variable. trigonometric function.
Trigonometric identity: Identity involving the Sine function: Function with sine ratio, which gives
trigonometric ratios. negative quarter turned symmetric curve passing through
the origin whose maximum and minimum values are + 1 and
Trigonometric equation: Equation involving trigonometric -1 respectively.
ratio.
Standard angles: Angles whose trigonometric values is Cosine function: Function with cosine ratio, which gives
easily computed by geometric interpretation such as 0o, negative quarter turned symmetric curve not passing through
30o, 45o, 60o and 90o only, these are basic for computing all the origin whose maximum and minimum values are +1 and
-1 respectively.
angles which are multiplied by the standard angles.
Tangent function: Function with tangent ratio, which
Values of trigonometric ratios of 0o: Real values of all occurs one cycle between - π/2 and π/2 that is repeated
trigonometric ratios of standard angle 0o such as; every π along x-axis. There are vertical asymptotes at each
end of the cycle whose maximum and minimum values are
sin 0o = tan 0o = 0; cos 0o = sec 0o = 1; undefined. The graph of y = tanθ
cot 0o = cosec 0o = ∞ (undefined).
Values of trigonometric ratios of 30o: Real values of all Compound angle: Sum or Difference of two or more than
two angles. If A and B are two angles then the angles (A + B)
trigonometric ratios of standard angle 30o such as; and (A – B) are the compound angles.
sin 30o = 12; cos 30o = 3 1 Trigonometric ratios of compound angles (A + B) and (A
2 = tan 30o = 3 – B): If A and B are two angles then the trigonometric ratios
of the compound angles (A + B) and (A – B) as given below:
Values of trigonometric ratios of 45o: Real values of all
trigonometric ratio of standard angle 45o such as; 1. cos (A + B) = cos A.cos B – sin A.sin B
sin 45o = cos 45o = 12, tan 45o = cot 45o = 1;
cosec 45o = sec 45o = 2 . 2. cos (A – B) = cos A . cos B + sin A . sin B
3. sin (A + B) = sin A . cos B + cos A . sin B
Values of trigonometric ratios of 60o: Real values of all 4. sin (A – B) = sin A . cos B – cos A . sin B
trigonometric ratios of standard angle 60o such as; 5. tan (A + B) = tan A + tan B
– tan A . tan B
sin 60o = 3 , cos 60o = 21, tan 60o = 3, 1 tan A – tan B
2 + tan A . tan B
13, sec 60o = 2, cosec 60o = 2 6. tan (A – B) = 1
3
cot 60o = . 7. cot (A + B) = cot A . cot B – 1
cot B + cot A
Values of trigonometric ratios of 90o: Real values of all cot A . cot B + 1
8. cot (A – B) = cot B – cot A
trigonometric ratios of standard angle 90o such as; sin 90o
= cosec 90o = 1, cos 90o = cot 90o = 0, sec 90o = tan 90o = ∞ Trigonometric ratio of compound angles (A + B + C): If
A, B and C are three angles then the trigonometric ratios of
Solution of right-angled triangle: Finding the unknown the compound angles (A + B + C) as given below:
parts of right-angled triangle by using known parts i.e., sides
and angles. 1. sin (A + B + C) = cos A . cos B . cos C(tan A + tan B +
tan C – tan A . tan B . tan C)
Quadrant of coordinate plane: Sector or area when divided
2. cos (A + B + C) = cos A . cos B . cos C (1 – tan B . tan
by the vertical and horizontal real lines to the coordinate C – tan C . tan A – tan A . tan B)
plane. There are four quadrants. First quadrant covers from 3. tan(A + B + C) =
0o to 90o, second quadrant covers from 90o to 180o, third tan A + tan B + tan C – tan A . tan B . tan C
quadrant covers from 180o to 270o and four quadrant covers
from 270o to 360o. 1 – tan B . tan C – tan C . tan A – tan A. tan B
CAST rule: Rule or method from which identify the sign 4. cot(A + B + C) =
cot B . cot C + cot C . cot A + cot A . cot B – 1
(+ or –) when convert the trigonometric ratios of any angle
greater than 90o into standard angles or angles greater than cot A . cot B . cot C – cot A – cot B – cot C
90o. In CAST, C means "cos and sec" which are positive
336
GLOSSARY - 6 (VECTOR)
Vector: Quantity that moves in specific way or physical Negative vector: Vector with the same magnitude but
quantity with magnitude and direction and is denoted by
a , b and AB etc. opposite in direction. The negative vector of a is – a
and the negative vector of AB is BA . So, AB = BA or
Finite direction: Fixed or specific way or direction.
AB = – AB .
Arrow: Symbol which is used for representing direction.
Equal vectors: Vectors with the same magnitude and in the
Magnitude: Length of displacement. same direction. If two vectors are equal their corresponding
components are separately equal and vice versa.
Scalar: Quantity having only magnitude.
Like vectors: Vectors with same direction.
Displacement: Movement of an object in specific
direction. Parallel vector: Vectors with same or exactly opposite in
direction. If ma = b or a = – mb, then a // b.
Directed lines segment: Line segment having initial and
terminal points in fixed direction. Collinear vectors: Overlapping or Coincident vectors to
each other.
Initial point: Starting point of movement or displacement.
Perpendicular or orthogonal vector: Two vectors
Terminal point: Ending or stopping point of movement intersected at right angle.
or displacement.
Localized vector: Vector passing through the given
Component of vector: Quantity of vector along point and parallel to the given vector.
horizontal or vertical.
Co-initial vectors: Vectors having the same initial point.
x-component: Horizontal displacement (left or right) of
any point. Addition of vectors: Adding the corresponding
components of the same ordered vectors and it is the
y-component: Vertical displacement (up or down) of any resultant vector.
point.
Subtraction of vectors: Subtracting the corresponding
Modulus of any point: A pair of vertical bars that components of like vectors.
changes the number into non-negative value.
Resultant vector: Vector which represents the sum of
Absolute value: Value obtained from modules. two vectors.
Magnitude of vector: Non-negative length of vector. Triangle law of vector addition: Resultant vector of a
The magnitude of vector a with initial point as origin is | side of triangle which represents the sum of other two
sides of the triangle in non-cyclic order or direction.
a | = x2 + y2 and initial point which is not origin,
Parallelogram law of vector addition: Resultant Vector
| a | = (x2 – x1)2 + (y2 – y1)2. of the diagonal of a parallelogram which represents the
sum of co-initial vectors of the adjacent sides of the
Direction of vector: Angle made by vector with x-axis. parallelogram.
The direction of vector with initial point as origin is
θ = tan-1 y and initial point other than origin, Polygon law of vector addition: Vector of a side of
x polygon which represents the sum of the vectors of its
y2 – y1 remaining sides in non-cyclic order. The sum of vectors
θ = tan-1 x2 – x1 . of all sides of the polygon in cyclic order is always zero.
Column vector: Vector represented by only column in
the form of x or x2 – x1 . Commutative property of vector addition: If a and b
y y2 – y1 are two vectors then a + b = b + a .
Row vector: Vector represented by only row in the form Associative property in vector addition: If a , b and c
are three vectors then ( a + b) + c = a + ( b + c ).
of [x y] or [x2 – x1 y2 – y1].
Positive vector: Vector which represents the position of
point in the reference of the origin. The position vector of Distributive property of vector addition: If a and b
are two vectors and m and n are scalars then m( a + b) =
(x, y) in the reference of the origin is x . m a + m b and (m + n) a = m a + n a .
y
Null or zero vector: Vector with magnitude Zero.
Unit or identity vector: Vector with magnitude 1. The Scalar Multiplication of Vector: If a = x and m are a
y
a
unit vector of a is a = . vector and a scalar respectively then m a = mx .
|a| my
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GLOSSARY - 7 (TRANSFORMATION)
Transformation: Changing or moving the position of an Reflection on line x = a or x – a = 0: When any point
object under specific rule. (x, y) is reflected on the line x = a or x – a = 0, the image
of (x, y) becomes (2a – x, y) i.e., (x', y') = (2a – x, y), then
Clockwise direction: Direction from where the hand of a the equations for the reflection on the line x = a are
clock moves. x' = –1.x + 0. y + 2a and y' = 0. x + 1.y.
Anti-clock direction: Direction opposite to clockwise Reflection on line y = b or y – b = 0: When any point
direction. (x, y) is reflected on the line y = b or y – b = 0, the image of
(x, y) becomes (x, 2b – y). i.e.(x', y') =(x, 2b – y), then the
Arbitrary constant: Symbol holding a place for an equations for the reflection on the line y = b are x' = 1.x + 0.y
unspecified constant (or numerical value or fixed quantity). and y' = 0.x + (–1).y + 2b.
Object: Anything like as point, line, plane, solid, etc. Rotation: Isometric transformation in which an object
moves about a fixed point through certain angle. The rotation
Image: Result obtained by transforming an object. of P about the centre O through the angle θ is denoted by,
P R[O; θ] P'.
Identity transformation: Transformation in which an object
and its image are same. It lies on the axis of transformation. Center of rotation: Fixed point of a rotation from which an
object rotates.
Invariant point: Point in which the identity transformation
is performed. Angle of rotation: Angle of turning around the centre of rotation.
Inverse transformation: Transformation in which the Radius of rotation: Distance of the object or image from
image is made back and invertible of the object. the centre of rotation.
Isometric transformation: Transformation in which an Quarter turn or rotation of ± 90o: Rotation of an object
object and its object are congruent and preserves the distance. through ± 90o.i.e, in anti-clockwise or clockwise direction.
Non-isometric transformation: Transformation in which Positive Quarter turn: Rotation of an object about the
an object is not congruent to its image and it preserves origin O through 90o in anti- clockwise direction (+ 90o),
similarity. which is equivalent to the rotation through – 270o.
Congruent: Objects that are same shape and size. i.e., R[O; 90o ] = R[O; – 270o].
Similar: Objects that are same shape but not size. Negative Quarter turn: Rotation of an object about the
origin O through 90o in clockwise direction (- 90o), which is
Translation vector: Vector by which an object is equivalent to the rotation through + 270o.
transformed. The image of a point p(x, y) under the i.e., R[O ; – 90o] = R[O, + 270o].
a
translation vector T = b is P'(x + a, y + b) and if (x', y') is Half turn: Rotation of an object about the origin O through
translation a x'– x 180o in anti-clockwise or clockwise direction (± 180o).
the image of (x, y), then the vector b = y' – y . i.e., R[O; +180o ] = R[O, –180o ]
Reflection: Transformation in which the image of an object Full turn or complete rotation: Rotation of 360o in anti-
clockwise or clockwise direction.
P forms equidistant from the fixed line and is denoted by i.e., R[O; + 360o ] = R[O; – 360o]
α(P) = P' or R(P) = P'. Positive quarter turn in coordinate plane: The image of
a point (x, y) under positive quarter turn about the origin is
Reflecting axis or axis of reflection: Fixed line in which (– y, x). i.e., (x, y) R[O; + 90o] (– y, x) = (x', y').
performs reflection of any object.
The equation for positive quarter turn about the origin are
Reflection on x-axis or y = 0: When any point (x, y) x ' = 0.x + (– 1).y and y' = 1.x + 0. y.
is reflected on x – axis or the line y = 0, the image of
(x, y) becomes (x, – y), i.e, (x', y') = (x, – y), then the Negative quarter turn in coordinate plane: The image of
equations for reflection about x-axis are x' = 1.x + 0.y and a point (x, y) under negative quarter turn of rotation through
y' = 0.x + (–1).y. – 90° or +270o about the origin is (y,– x).
Reflection on y-axis or x = 0: When any point (x, y) is i.e., (x, y) R R[O; – 90o] (y, – x) = (x', y').
reflected on y-axis or the line x = 0, the image of (x, y)
becomes (– x, y), i,e. (x', y') = (-x, y), then equations for The equations for negative quarter turn about the origin are
the reflection on y-axis are x' = –1.x + 0.y and y' = 0.x + 1.y. x' = 0.x + 1.y and y' = – 1 .x + 0.y.
Reflection on line y = x or x – y = 0: When any point Half turn in coordinate plane: The image of a point (x, y)
(x, y) is reflected on the line y = x or x – y = 0, the image of under half turn about origin O(0, 0) is (– x, – y).
(x, y) becomes (y, x), i.e., (x', y') = (y, x), then the equations i.e., (x, y) R[O; –180o] (–x, –y) = (x', y').
for the reflection about the line y = x are x' = 0.x + 1.y and
y' = 1.x + 0.y. The equations for half turn about the origin are x' = –1.x +
0.y and y' = 0.x + (–1).y.
Reflection on line y = – x or x + y = 0: When any point
(x, y) is reflected on the line y = – x or x + y = 0, the image
of (x, y) becomes (– y, – x), i.e., (x', y') = (– y, – x), then
the equations for the reflection about the line y = – x are
x' = 0.x + (–1).y and y' = –1.x + 0.y.
338
Positive quarter turn about (a, b): The coordinates of the Scalar Factor: Constant ratio of the length of object and its
image or the distance from the centre of enlargement to the
image of the point P(x, y) under the rotation of + 90° or object and its image, generally denoted by k.
– 270° about the centre C(a, b) is P'(–y + a + b, x – a + b). Centre of enlargement: Point of intersection when
producing the lines joining the points of the object and its
i.e. P(x, y) R[(a, b), + 90°/ – 270°] P'(–y + a + b, x – a + b) image under enlargement, generally denoted by O or C.
Negative quarter turn about (a, b): The coordinates of the Enlarging with centre O(0, 0) and scale factor k: P(x, y)
image of the point P(x, y) under the rotation of – 90° about from the centre O(0, 0) by the scale factor k, denoted by
the centre C(a, b) is (y + a – b, – x + a + b). E[O; k] and computed as P(x, y) E[O; k] P'(kx, ky).
i.e. P(x, y) R[(a, b), – 90°/ + 270°] P'(y + a – b, – x + a + b)
Enlargement with centre C(a, b) and scale factor k:
Half turn about (a, b): The coordinates of the image of the Method of enlarging an object P(x, y) from the centre C(a,
point P(x, y) under the rotation of ± 180° about the centre b) by the scale factor k, denoted by E[(a, b); k] and computed
C(a, b) is (– x + 2a, – y + 2b). as, P(x, y) E[(a, b), k] [kx – a(k – 1), ky – b(k – 1)].
i.e. P(x, y) R[(a, b), ± 180°] P'(– x + 2a, – y + 2b)
Inversion: Casting the image of the upward object into
Enlargement: Non-isometric transformation in which the downward.
image of an object becomes larger than the given object.
Reduction: Non-isometric transformation in which the
image of an object becomes smaller than the given object.
GLOSSARY - 8 (STATISTICS)
Partition Values: The points which divide the data (iii) Third Quartile (Q3), which divides the data in
into equal parts. They are (i) Median, (ii) Quartiles, 3
(iii) Deciles and (iv) Percentiles. the ratio 3:1. i.e., the value of 4 th or 75% of the
Central tendency: Central part of the distribution of distribution and is computed by the formula,
data such as mean, median, mode, etc.
Q3 = The values of 3(N + 1)th item
4
Deciles: The points that can be divided into ten equal
Average: Middle value of the data.
parts. There are nine deciles in the data. Deciles are
Mean: Average value of the data, calculated by,
represented as Di, where i = 1 to 9. Deciles are also
Mean (X) = Σx or ΣNfx. calculated as quartiles and the formula is given as:
N
Median: Middle item of the observation is calculated i(N + 1)th
Di = 10 item, where i = 1 to 9 for individual and
N+ 1 th
by, Medium (Md) = The values of 2
item. discrete data.
Mode: Most repeated item of the data in the individual Percentiles: The points that can be divided into hundred
data and item that lies highest frequency in the discrete equal parts. There are ninety nine percentiles in the
data. data. Percentiles are represented as Pi, where i = 1 to 99.
Percentiles are also calculated as quartiles or deciles and
Quartiles: The items which divides the given data
into four equal items or observation. There are three the formula is given as:
quartiles. They are :
Pi = i(N + 1)ththitem, where i = 1 to 99 for individual and
(i) First Quartile (Q1), which divides the data in the 100
ratio 1:3. i.e., the value of 1 th or 25% of the discrete data.
4
Measure of variability: Value of dispersion or measure
distribution and is computed by the formula, of variation of items of the data.
Q1 = The value of N+ 1 th Dispersion: The extent to which a distribution is
4 stretched or squeezed in statistics. It is also called
item. variability, scatter, or spread.
(ii) Second Quartile (Q2), which divides the data in the
2
ratio 2:4 = 1: 2. i.e., the value of 4 th or 50% of
the distribution and is computed by the formula, Deviation: The absolute difference between the element
of central tendency and a given point in statistics.
2(N + 1) N+ 1 th Inter quartile range: Difference between third quartile
4 2 (Q3) and first quartile (Q1). i.e., Q3 – Q1.
Q2 = The value of = item
= Median (Md)
339
Semi-inter quartile range: Half of the inter quartile Standard Deviation in Individual Data: Let X1, X2, X3,
……., Xn are the variants then the SD is calculated by
range, using anyone of following formulae;
i.e., Q3 – Q1.
2
Quartile deviation: Measure of dispersion based on (a) Direct Method:
first quartile (Q1) and third quartile (Q3) and is computed SD/σ = ΣX2 – ΣX 2
Q3 – Q1. N N
2
by, QD = (b) Actual Mean Method:
Coefficient of quartile deviation: Absolute or relative SD/σ = Σ(X – X)2 Σd2
N i.e., N
measure of dispersion for comparative studies of
variability of two distributions with different units of (c) Assumed Mean Method:
i(sMcDom):pAutveedraagseQQo33f+–thQQe12.absolute
measurement and value SD/σ = Σd2 – Σd 2
Mean Deviation N N
taken from the deviation (differences) of each item from where, d = X – A, A = Assumed Mean.
the central value (mean, median or mode). It is also Standard Deviation in Discrete Data: Consider the
discrete data as;
called average deviation.
Mean deviation from mean: While calculating the Class Interval X1 X2 X3 ….. Xn
mean deviation, if the deviations are taken from mean
Frequency (f) f1 f2 f3 ….. fn
Mean deviation from median: While calculating the The SD is calculated by using any one of the following
mean deviation if the deviations taken from the median. formulae;
Mean Deviation in Individual Data: Let X1, X2, X3, (a) Direct Method:
………, Xn be the variate values and X and Md be its
arithmetic mean and median respectively. Then the mean ΣfX2 – ΣfX 2
N N
deviation is computed from the following formulae; SD/σ =
(a) MD from mean = Σ |X – X| (b) Actual Mean Method:
N
Σ |X – Md|
(b) MD from median = N Σf(X – X)2 Σfd2
N i.e., N
Mean Deviation in Discrete Data: Consider the SD/σ =
discrete data as; (c) Assumed Mean Method:
Class Interval X1 X2 X3 ….. Xn SD/σ = Σfd2 – Σfd 2
N N
Frequency (f) f1 f2 f3 ….. fn
where, d = X – A, A = Assumed Mean.
Let X and Md be the arithmetic mean and median of the
Coefficient of Standard Deviation: The relative
data respectively, then the mean deviation is computed
measure of dispersion based on the standard deviation.
from the following formulae;
The coefficient of SD is calculated by the formula;
Σf |X – X|
(a) MD from mean = N SD σx.
Coefficient of SD = mean =
Σf |X – Md|,
(b) MD from median = N Variance: The square of the standard deviation. So, it is
where Σf = N. defined as follows;
Coefficient of Mean Deviation: The relative measure Variance (Var.) = Square of SD = σ2
or absolute measure which compares two or more data
having different units by corresponding mean deviations. Then its computational formulae are given below:
The coefficient of mean deviation is computed from the
following formulae; (a) Var.(X) = Σ(X – X)2 = ΣX2 – ΣX 2
N N N
(a) Coefficient of MD from mean = MD from Mean for individual series.
X
(b) Var.(X) = Σf(X – X)2 = ΣfX2 – ΣfX 2
N N N
(b) Coefficient of MD from median = MD from Median, for discrete series.
X
Coefficient of Variation: The relative measure of
Standard Deviation (SD/σ): The absolute dispersion dispersion based on the SD in the form of percentage.
which is the positive square root of the average of square Then, Coefficient of Variation,
of the deviation taken from the mean, also called root-
mean square deviation. CV = σ × 100%
x
340
HOME ASSESSMENT - 1 (ALGEBRA)
1. Find the values of x and y:
(a) (2x – 3, 4 – 3y) = (5, y) (b) (x2 – 5, 4) = (4, 2x + y) (c) (2x + y, x + 1) = (2, y + 2)
2. If P = {1, 2, 4} and Q = {2, 3}, find the following Cartesian products and show them in tree
diagram, mapping diagram, table and graph.
(a) P × Q (b) Q × P (c) P × P (d) Q2
3. If A = {1, 4, 6} and B = {1, 2, 3}, find A × B. Also, write down the set of ordered pairs of the
following relations in the set builder form and table.
(a) f = "is less than" (b) g = "is square root of" (c) h = "is the half of"
4. If P = {0, 1, 3, 5, 7} and Q = {2, 4, 6}, find the following relations from P to Q.
(a) 2x – y = 4 (b) y = 2x2 (c) 2y = 3x
5. Find the domain, range and inverse of the following relations:
(a) R1 = {(1, 2), (2, 3), (3, 4), (4, 5)} (b) R2 = {(1, 1), (2, 8), (3, 27)}
6. Find the set of ordered pairs of the following relation for the given conditions.
(a) R1 = {(x, y) : y = 2x – 3, Domain = {–1, 0, 4}
(b) R2 = {(x, y) : y = 3x + 1, y ∈{– 2, 1, 4}
7. State whether the following relations are the functions. Give reasons.
(a) f = {(2, 1), (4, 2), (5, – 2), (3, – 2)} (b) g = {(3, 2), (4, – 1), (2, 2), (4, 1)}
(c) h (d) (e) Y (f) Y
a Y
1
b2
c O X OX
OX
8. It is given that A = {1, 2, 3} and B = {6, 12, 18}. If the function f:A → B is defined by "is the
thrice of", then represent the function f in the following methods:
(a) Roster form (b) Set-builder Method (c) Formula
(d) Mapping diagram (e) Lattice diagram
9. If f(x) = 2x2 + x – 2, find the values of f(0), f(2) and f(–2).
10. Find the range, domain and co-domain of the following functions.
(a) f = {(4, 5), (6, 8), (8, 11)} (b) g = {(x, y) : y = x2 – 1 and 0 < x ≤ 3}
(c) (d) (e) Y (f)
h -1 k
1
1 2 x 0 246 1
4 8 y – 2 2 14 34 X2 4
9
6 12 O
16
11. Find the set of ordered pairs of the following functions:
(a) f(x) = 7 – 2x for D = {2, 4, 6} (b) g(x) = (2x – 3) for x ∈ {0, 1, 2}
(c) h(x) = {(x, y): y = 2x2 – 3 and R = {5, 29} (d) k(x) = {(x, y): 2y = x + 2 for 1 ≤ y < 4}
341
12. (a) If f = {x, 2x2 – 1} is a function, find the image of 4.
(b) If f : x → 3x – 1 is a function, find the pre-image of –2.
2
2x + 1
(c) Which element in the range has the pre-image 7 under the function f(x) = x–4 .
13. (a) If f(x + 3) = f(x) + f(3), prove that f(0) = 0 and f(–3) = –f(3).
(b) If g(x + 4) = g(x) + g(6), prove that g(2) = 0 and g(–2) = –g(6).
14. If f(1) = 4, f(4) = 7, f(7) = 10, find the value of f(22).
15. (a) If the function f:x → 2x + 1 is defined from A = {2, 3, 4} to the set B = {5, 6, 7, 8, 9}, find the
range of the function f. Is the function an one-one and onto? Give reason.
(b) Is the function g(x) = 3x2 – 2 into function? Give reason.
16. If f(x) = ax + b, f(1) = 2 and f(–1) = 4, find the values of a and b.
17. Find the degree of the following polynomials:
(a) 3x3y + 3x2y3 + 4xy5 – xy + y – 2 (b) 3x2y2z3 + 5x3yz4 + 4x4y2z
18. Classify the following polynomials based on the degree.
(a) 2x (b) 3x2yz + 2x – 3 (c) 3x2 + y
19. Add the next three numbers in the following sequences:
(a) 3, 7, 11, 15, .... (b) 7, 1, –5, –11, ... (c) 6, 12, 24, 48, .... (d) 20, 23, 28, 35, ...
20. Differentiate the types of the following sequences with reasons.
(a) 5, 8, 11, 14. (b) 14, 9, 4, – 1, ... (c) 18, 19, 21, 24. (d) 4, 12, 36, 108, ....
21. Find the first five terms of the sequence whose nth terms as follows.
(a) tn = 3n – 2 (b) an = (n – 1) (2n – 1) (c) tn = 2n2 – 5 (d) un = 2n + 3
22. Find the general term of the following sequences and find the 15th term and 50th term.
(a) 3, 5, 7, 9, .... (b) 2, –1, – 4, –7, .... (c) –1, 3, 9, 17, .... (d) 3, 0, – 5, – 12, ...
23. Add the next two figures in the given figures and find the number of dots in 10th figure and 15th figure.
(a) (b) (c)
,, , , ..... , , , , ...... ,, , , .....
24. Write down the expanded form and find the values of the following sigma notations:
6 64
∑ (a) ∑ ∑(b) (2n2 – n + 1) (c) 2n2 – 8
(2n – 3)
n=2 n=3 n=1 n + 1
25. Find the 10th term of the sequence obtained from the following sigma notation.
7
6 ∑(b) n 5
∑ (a) n+1
(3n + 1) ∑(c) 4
n=2 4n – 3
n=3 n=4
26. Find the sum of first five terms of the sequences whose nth terms are given below.
2n – 1 4 – n2
(a) tn = 2n – 7 (b) tn = n+1 (c) an = 2n – 3
27. Find the 10th of the following series whose sums of the first n terms are given below.
(a) Sn = 3n2 + 1 (b) Sn = 5n2 + 4 (c) Sn = (2n + 7)2
342
ANSWERS
1. (a) 4, 1 (b) ± 3, –2 or 10 (c) 1, 0
2. Show to your teacher.
3. {(1, 1), (1, 2), (1, 3), (4, 1), (4, 2), (4, 3), (6, 1), (6, 2), (6, 3)}
(a) {(1, 2), (1, 3)}, {(x, y) : x < y}
(b) {(1, 1)}, {(x, y) : x = y}
(c) {(1, 2)}, {(x, y) : x = 1 y}; Table: show to your teacher.
2
4. (a) {(3, 2), (5, 6)} (b) {(1, 2)} (c) { }
5. (a) D = {(1, 2, 3, 4)}, R = {2, 3, 4, 5}, R1-1 = {(2, 1), (3, 2), (4, 3), (5, 4)}
(b) D = {(1, 2, 3)}, R = {1, 8, 27}, R2-1 = {(1, 1), (8, 2), (27, 3)}
6. (a) {(–1, –5), (0, –3), (4, 5)} (b) {(–1, –2), (0, 1), (1, 4)}
7. (a) Yes (b) No (c) No (d) Yes (e) No (f) No
8. Show to your teacher.
9. f(x) = – 2, f(2) = 8, f(–2) = 4
10. (a) D = {4, 6, 8}, R = {5, 8, 11}, C = {5, 8, 11} (b) D = {1, 2, 3}, R = {0, 3, 8}, C = {0, 3, 8}
(c) D = {1, 4, 6}, R = {2, 8, 12}, C = {2, 8, 12, 16} (d) D = {0, 2, 4}, R = C = {–2, 2, 14, 34}
(e) D = {1, 3, 5}, R = C = {0, 1, 2, 4} (f) D = {–1, 1, 2}, R = {1, 4}, C = {1, 4, 9}
11. (a) {(2, 3), (4, –1), (6, –5)} (b) {(0, – 3), (1, – 1), (2, 1)}
(c) {(2, 5), (– 2, 5), (– 4, 29), (4, 29)} (d) {(0, 1), (2, 2), (4, 3)}
12. (a) 31 (b) –1 (c) 5
14. 25 15. {5, 7, 9}, No 16. –1, 3 17. (a) 6 (b) 8
18. (a) Monomial (b) Trinomial (c) Binomial
19. (a) 19, 23, 27 (b) –17, –23, –29 (c) 96, 192, 384 (d) 44, 55, 68
20. (a) Finite, ascending, linear or arithmetic (b) Infinite, descending, linear
(c) Finite, ascending, quadratic (d) Infinite, ascending, geometric
21. (a) 1, 4, 7, 10,13 (b) 0, 3, 10, 21, 36 (c) –3, 3, 13, 27, 45 (d) 16, 32, 64, 128, 256
22. (a) 2n + 1, 31, 101 (b) 5 – 3n, – 40, – 145 (c) n2 + n – 3, 237, 2547 (d) 4 – n2, –221, –2496
23. Fig. show to your teacher. (a) 22, 32 (b) 19, 29 (c) 100, 225
24. (a) 25 (b) 158 (c) 43
10
25. (a) 37 (b) 13 (c) 4
14 41
26. (a) –5 (b) 113 (c) – 151
27. (a) 65 15
20
(b) 95 (c) 80
PROJECT WORK
Study the following two problems.
(i) You deposit Re. 1 in the money-collecting box on the first day, Rs. 2 on the second day, Rs. 4 on the
third day, Rs. 7 on the fourth day and so on. How much money do you deposit on the 30th day? Find it.
(ii) You deposit Re. 1 in the money-collecting box on the first day, Rs. 2 on the second day, Rs. 4 on the
third day, Rs. 8 on the fourth day and so on. How much money do you deposit on the 30th day? Find
it. In which case can you deposit the amount on the 30th day? Decide your conclusion.
343
HOME ASSESSMENT - 2 (LIMIT)
1. (a) What is the limit of a sequence?
(b) Define an infinite series.
(c) Write the definition of the limit of a function.
2. (a) What is the ninth term of the sequence 0.1, 0.01, 0.001, ...... ?
(b) Write down the eighth term of the sequence 3, 130, 1030, ...... .
3. Find the general term of the following sequences and also find their 25th term:
(a) 210, 2100, 20100, ................. (b) 0.7, 0.07, 0.007, ............
4. (a) What is the approximately last term of the sequence 7.9, 7.99, 7.999, ......... when its number
of terms is increased?
(b) What are the nearest whole numbers of each term of the sequence 2.1, 2.01, 2.001, ...........?
5. (a) What is the limit of the sequence 2.9, 2.99, 2.999, 2.9999, ............?
(b) Write down the sequence having the general term 5n + 2 and convert into decimal form. Also,
n
find its limit when n is countable infinitely large number.
6. (a) Draw a line segment MN of the length 8 cm. Bisect it 8 times towards N. Write down the
sequence formed by the eight midpoints. When it is bisected 40 times, what are the 40th
midpoint and its limit?
(b) A man cuts half of a stick. He cuts again and again half in each cut. At what times does he
finish to cut the stick? What amount of stick is left at last?
7. (a) Find the sequences of the sides, perimeters and areas of the squares formed by joining the
midpoints of the sides of the square ABCD having the length 8 cm.
(b) The given square of length 4 cm side is bisected by its 4 cm 4 cm
diagonal as shown in the shaded portion alongside.
Write down the sequence formed by the area of the Fig (ii) Fig (iii)
shaded portion. Also, find its 40th term and 100th term
and find the limit of the sum of all shaded portion of Fig (i)
the last one.
8. (a) Find the first five terms of the sequence having the nth term 2 + 1n. Also, find the limiting value
n2
of the nth term of the sequence when n is countable infinitely greater.
(b) What is the sum of the even number of the terms of the infinite series having the nth term
5 + (– 1)n ? Find it.
5
x2 – 24.
9. (a) The given function is f(x) = x – Does f(x) denote a fixed real number at x = 2? For this,
evaluate the following:
(i) What are the values of f(x) at x = 1.9, 1.99, 1.999 and 1.9999?
(ii) What are the values of f(x) at x = 2.1, 2.01, 2.001 and 2.0001?
(iii) Can the values of f(x) that are computed in (i) and (ii), express in whole numbers by
rounding off?
344
4x – 2, x > 1
x–1
(b) Find the limit of the function h(x) = x2 – 1, x < 1 at x = 1
x–1
10. (a) Write the notation of the limit as x approaches a of a function f(n) = 3xx2+–aa.
(b) Write the mathematical sentences of the following limit notation.
(i) x → – b (ii) lim – f(x) (iii) lim 2x2 – 1 = 1
x→2 →1 4 4
x
11. Find the value of the following functions for the given limiting conditions by using tables:
(a) lim 3x + 1 (b) x lim 2 x2 – 24 (c) lim x2 – 5x – 6
x→2 →– x+ x→6 x–6
(d) lim f(x), where f(x) = x2 – 1, x > 1
x→2 x – 1 at x = 1
x2 – 2x + 1, x < 1
x–1
ANSWERS
2. (a) 0.000000001 (b) 3 3. (a) 210, 5 × 10– 26 (b) 118, 7 × 10– 25
10000000
4. (a) 8 (b) 2, 2, 2, .........
5. (a) 3 (b) 17, 122, 137, 242, ...............; 7.6, 5.67, 5.5, ......., 5
6. (a) 4, 6, 7, 7.5, 7.75, ...; 8 × 1, 8 × 3, 8 × 7, 8 × 15, ... = 4, 6, 7, 7.5, ...........; 8 × (240 – 1) = 7.999999 ...
2 4 8 16 240
(b) Countable infinite times, approximately zero
7. (a) 8, 4 2 , 4, 2 2 , .........; 32, 16 2 , 16, 8 2 , .........; 64, 32, 16, 8, .........
(b) 8, 12, 14, 15, .........; 1.4901 × 10 – 8 cm2, 1.2622 × 10 – 29 cm2; 16 cm2
8. (a) 2 + 1, 2 + 12, 2 + 1 , 2 + 14, 2 + 15; 0 (b) 5n
1 4 9 3 16 25
9. (a) No, (i) and (ii) show to you teacher. (iii) Yes, 4 (b) 2
10. (a) lim a f(x) = lim a 3x2 – a (b) Show to your teacher.
x→ x→ 2a
11. (a) 7 (b) – 4 (c) 7 (d) 0
PROJECT WORK
Draw a square of the length 6 cm. Draw a diagonal in the square and colour a triangle formed by diagonal.
Draw a median in the non-coloured triangle from the vertex of the square and colour the half triangle.
Similarly, draw the medians in the successive non-coloured triangle 4 times. Write the sequence formed
by the area of the coloured triangles. Also, find the area of all coloured triangles when the medians drawn
at countable infinite times. What is the limit of the infinite series?
345
HOME ASSESSMENT - 3 (MATRIX)
1. Write down the order of the following matrices:
(a) 2 (b) [–2 4] (c) [2 3 0] (d) 1 2 (e) 1 23
1 4 3 4 –1 0
2. Find the values of a11, a13, a23, a32 and a33 from the following matrices:
(a) [2 –1 3] (b) 4 –3 8 (c) 4 5 –3 2
57 2 –1 3 1 4
7 1 8 –2
3. Construct 2 × 2 and 2 × 3 matrices from the following elements of general form aij = 3i2 – j3.
4. Identify the types of the following matrices:
(a) 1 4 (b) [7, –2 1] (c) 0 00
4 3 0 00
100 (e) 20 (f) 200
(d) 0 1 0 02 420
001 1 –1 3
5. Find the values of x, y and z from the following cases:
(a) 2x + 1 3y – 1 = 3y – 1 5 (b) x – 2y 4y – 3 = –3 5
2z + x 3 4 3 6x – 2z 2y – z 4 3
6. (a) For what values of a, b, c and d, the matrices 3a + b 3c – d = 3 6 are equal?
a– b a+b+d 5 2d –1
(b) For what values of p and q the matrix 2p – q 2q – 4 is an identity matrix?
0 1
(c) If 3m – n 3l + n is a scalar matrix, find the values of l, m and n.
2l – 4 3
(d) For what values of w, x, y and z, the matrix 2w – 4 w + 2x is a null matrix?
3y – z – 1 2y – z
7. If P = 4 5 ,Q= –2 1 and R = 2 –3 , find the followings matrices:
2 –1 3 –5 0 4
(a) P + Q (b) Q – R (c) P + Q + R (d) P – Q + R
8. (a) If A + B = 4 5 and B = 1 3 , find the matrix A.
–1 0 –2 –1
(b) If M = 2 –1 and N = –2 4 , find the matrix L such that L + N = M.
57 3 –6
(c) If P = 2 –1 and Q = 3 –2 , find the matrix R such that P – Q + R is a null matrix.
–2 4 1 –4
(d) If A = –3 2 and B = 7 3 , find the matrix C such that A – B – C is a 2 × 2 identity matrix.
4 5 2 –1
9. Solve the following:
(a) 2 3 – A= 31 + I, where I is a 2 × 2 unit matrix.
5 7 –2 –3
(b) 2B – 44 = O, where O is a 2 × 2 null matrix.
2 –6
346
(c) 2A – B = 1 3 and A + 2B = 3 –1 .
0 5 –5 0
(d) 2P + Q = 4 1 and P – Q = 22 .
3 7 0 –1
10. (a) If P = 2 –1 , Q = 3 2y + 1 and R = 5 6 4 , find the values of x, y and z such that
x 3 0 1 –2 2z +
P + Q = R.
(b) If X = 2x –1 ,Y= x–4 2y – 1 and Z = 2 –1 , find the values of x, y and z such
2y 2z – 1 0 5 1 –4
that X – Z = Y.
11. If L = –2 1 ,M= 1 3 and N = 2 –2 , prove that following relations:
3 0 –4 2 –4 5
(a) (L – M) + N = L – (M – N) (b) (M + N) – L = M + (N – L)
12. (a) If P = 3 –1 and Q = 57 , verify the commutative property of matrix addition.
2 4 –9 –5
(b) If A = 4 8 ,B= –8 –10 and C = –5 –6 , show the associative property of matrix addition.
7 9 –12 –15 25
13. (a) If the additive inverse of the matrix 2a – 5b – 1 is 6 –10 , find the values of a, b, c and d.
4c – 6d 8 –11
(b) If the sum of matrices 2x – y 2y and 2 4 is an identity, find the values of x, y and z.
3y + z 8 3 –8
(c) If A = x 2 ,B= 2x – 1 3y – 1 , C = 2y + 1 z –1 , D = 3 7 and A + C = B + D, find
–3 –1 3z + 2 0 –4 2 3 1
the values of x, y and z.
14. Find the transpose of the following matrices: 32 1
–3 –1 0
(a) 2 (b) [2 3 4] (c) 1 –2 4 (d) 4 13
4 3 2 –1
15. If A = 2 –4 ,B= –2 3 and C = –3 1 , prove the following relations:
37 5 –4 3 2
(a) A = (AT)T (b) (A + B)T = AT + BT (c) (A – C)T = AT – CT
(d) (A + B + C)T = AT + BT + CT (e) (A – B – C)T = AT – BT – CT
16. If X = 3 4 ,Y= –1 5 and Z = 02 , find;
–3 7 2 –4 –2 –5
(a) XT + Y (b) XT – (Y + Z)T
17. (a) If P = 3 –2 , prove that P + PT is a symmetric matrix.
–1 5
(b) If A = –3 7 and B = 2 4 , prove that (A + B) + (A + B)T is a symmetric matrix.
2 9 –2 –2
18. (a) If P = sinθ cosθ and P+ PT = I, find the values of the angles θ and α.
– cosθ – sinα
(b) If A = cos2θ 1 and A + AT = 1 0 , find the measure of the angles θ and β.
tanβ –2 0 –4
347
19. (a) Find the values of p, q and r if P = 2p – 3 3 ,Q= 5 2r and PT = Q.
2q – r 3r 3 5r + 4
(b) If A = 2x - 3y -2y + z and B = –1 6 , find the values of x, y and z such that A = BT.
3x + 3y 5 7 5
20. (a) If 2 2 x +3 –2 1 = y –3 , find the values of x, y and z.
–1 0 2 1 z 3
(b) If P = 2 1 ,Q= y z and 3P – 2Q = 2P, find the values of x, y and z.
3x –2 2 –1
21. (a) If 2M + 3N = –1 2 and M – 2N = 3 –6 then find the matrices M and N.
18 15 –5 –10
(b) If A = 1 –3 and B = 2 –1 then find a matrix C such that 2A + 3C = 4B.
0 4 0 3
22. Find the product of the following matrices:
(a) [–1 3] –2 (b) [2 4] –1 –3
1 2 0
(c) 12 24 (d) –1 4 2 –1 0
–4 3 –2 1 2 3 424
23. (a) If P = –1 –3 and Q = 2 4 , show that PQ ≠ QP.
4 1 0 1
(b) If M = 2 3 and N = 4 –3 then show that MN = NM.
6 4 –6 2
24. If A = 1 2 ,B= 2 –3 and C = –4 2 then show the following properties:
–2 5 1 0 1 3
(a) A(B + C) = AB + AC (b) (AB)C = A(BC)
25. (a) If A = 4 2 , prove that A2 + 6I = 5A, where I is a 2 × 2 unit matrix.
–1 1
(b) If P = 2 5 and P + 2Q = P2 + 2I, find the matrix Q, where I is an identity matrix.
–2 1
26. (a) If 1 2 x = 8 , find the values of x and y.
–2 –1 y –7
(b) If [a b] 2 0 = [–3 1], find the values of a and b.
–1 1
27. (a) If x y –2 = 4 , find the values of x and y.
y x 3 –1
(b) If P = 2 –1 , Q = 0 1 and PR = Q find the value of the matrix R.
–2 3 4 –3
28. (a) Which matrix pre-multiplies to the matrix 2 4 and gives 2 4 ?
–1 3 –3 –1
(b) Which matrix post multiplies to 2 4 then gets a matrix –8 ?
–2 1 –7
29. If P = 1 –3 and Q = –2 0 then prove that:
2 0 3 4
(a) (PQ)T = QTPT (b) (2A – 3B)T = 2AT – 3BT.
348
ANSWERS
1. (a) 2 × 1 (b) 1 × 2 (c) 1 × 3 (d) 2 × 2 (e) 2 × 3
2. (a) 2, 3 (b) 4, 8, 2 (c) 4, – 3, 1, 1, 8 3. (a) 2 –5 (b) 2 – 5 – 24
4. (a) Square (b) Row 11 4 11 4 –15
(c) Zero (d) Identity
(e) Scalar (f) Lower triangular
5. (a) 2, 2, 1 (b) 1, 2, 1
6. (a) 2, – 3, 2, 0
(b) 3 , 2 (c) 2, – 1, – 6 (d) 2, – 1, 1, 2
2
7. (a) 26 (b) –4 4 (c) 4 3 (d) 8 1
5 –6 3 –9 5 –2 –1 8
8. (a) 32 (b) 4 –5 (c) 1 –1 (d) – 11 –1
11 2 13 3 –8 2 5
9. (a) –2 2 (b) 22 (c) 1 1 , 1 –1 (d) 2 1 , 0 –1
79 1 –3 –1 2 –2 –1 1 2 1 3
10. (a) – 2, 3, 0 (b) –2, 1 , 1,
2
13. (a) – 3, – 2, – 2, – 2 (c) – 7, – 3, – 4
(b) – 2, – 2, 3
14. (a) [2 4] 2 13 (d) 3 –3 4
2 –1 1
(b) 3 (c) – 2 2
4 4 –1 1 03
16. (a) 2 2 (b) 4 –3
6 3 –3 16
18. (a) 30o or 150o, 210o, 330o (b) 45o, 135o, 225o, 315°; 135o, 315o
19. (a) 4, – 3, – 2 (b) 1, 1, 9 20. (a) – 3, – 2, 4 (b) 4 , 1 ,12
3
1 –2 –1 2 2 2/3
21. (a) 3 0 , 4 5 (b) 0 4/3 22. (a) [5] (b) [6 – 6]
(c) –2 6 (d) 14 9 16 25. (b) –3 5 26. (a) 2, 3 (b) – 1, 1
– 14 – 13 16 4 12 –2 –4
27. (a) 1, 2 10 28. (a) 1 0 (b) 2
(b) 2 – 1 –1 1 –3
PROJECT WORK
Look a purchased bill including quantity, rate and amount at your house and copy it in full page
of A4 paper. Write the answers of the following questions:
(i) Write a matrix obtained by the values of the records of amount. Name the type of matrix.
(ii) Write a matrix obtained by the values of the records of quantity, rate and amount. Name
the type of matrix.
(iii) How many rows and columns are there in the matrix (ii) ?
(iv) Write the order of matrices obtained in the numbers (i) and (ii).
(v) What are the elements in the second column? Write them.
(vi) What are the elements in the third column? Write them.
(vii) Which column represents the amount of each item?
(viii) Which column represents the rate of goods?
349
HOME ASSESSMENT - 4 (COORDINATE GEOMETRY)
1. Find the coordinates of the point A, B, C and D from the adjoining figure. Y
Also, find the distance between the following points: BA
(a) A and C (b) B and D (c) B and E X' O C X
DE
2. Find the distance of the following points from the origin:
Y'
(a) (p – q, q) and (p, p + q) (b) (psin θ + qcos θ, pcos θ – qsin θ)
(c) (pcos α, psin α)
3. (a) Find the length of the line segment which makes the x-intercept 5 units and y-intercept 12 units.
(b) If P(8, –3) and Q(2, 5) are two end points of the line segment PQ and R(2, 3) and S(–2, 0) are
two end points of the line segment PQ, then prove that PQ = 2RS.
4. (a) If the distance between the points (–a, 4a) and (3a, a) is 20 units, find the value of a.
(b) Find the coordinates of the points on the x-axis which are at a distance of 2 5 units from the point (1, 2).
(c) Find the coordinates of the point on the y-axis which is equidistant from the points (3, 0) and
(7, 4).
5. (a) Prove that the points A(–3, –1), B(3, 1) and C(6, 2) are collinear.
(b) If the points A(p + q, q – p) and B(p – q, p + q) are equidistant from the point C(x, y), prove that qx = py.
6. (a) Show that the points (3, 3), (2, –1) and (–1, 4) are the vertices of an isosceles triangle.
(b) Prove that the triangle with vertices (–5, 5), (5, –5) and (–5 3 , 5 3 ) is an equilateral triangle.
7. (a) Prove that the points (1, –1), (0, 3) and (4, 5) are the vertices of a right-angled triangle.
(b) Prove that the triangle with vertices (2, 1), (4, 2) and (5, 0) forms an isosceles right triangle.
(c) Prove that the points (–3, 1) and (–1, –3) subtend a right-angled triangle at the origin and the
triangle which is formed by these points is also isosceles.
8. (a) Prove that the points (2, 0), (1, 2), (–3, 3) and (–2, 1) are the vertices of a parallelogram.
(b) Show that the quadrilateral with vertices (–4, 0), (–6, 1), (4, 2) and (–2, 1) is a rhombus.
(c) Prove that the points P(2, –1), Q(3, 0), R(0, 3) and S(–1, 2) are the vertices of a rectangle.
(d) Prove that the quadrilateral ABCD with vertices A(2, 1), B(5, 2), C(6, –1) and D (3, –2) is a square.
9. (a) P(1, –1), Q(3, 3), R(5, 1) and S(3, 2) are the corners of a geometric figure. What type of figure is
made from the vertices?
(b) Which shape makes by joining the points (–1, 1), (1, –1), (1, 1) and (2, 2) ?
(c) Prove that the points A(1, –1), B(4, 2), C(–1, 5) and D(6, 2) lie on the circle having center (1, 2).
10. (a) Find the coordinates of a point which is equidistant from the points (3, 4), (0, 5) and (–3, –4).
(b) Find the center and radius of the circumcircle circumscribing the triangle with vertices (0, 0),
(0, 3) and (4, 3).
11. (a) Find the coordinates of a point which divides the line segment joining the points (1, 2) and
(4, 2) internally in the ratio 1:2.
(b) Which point divides the line joining the points (–2, 1) and (0, –1) externally in the ratio 3:1 ?
12. (a) In what ratio does the point (–2, 0) divide the line segment joining the points (–4, –1) and (4, 3)?
(b) In what ratio is the line joining the points (–6, –2) and (4, 3) divided by the y-axis?
13. (a) The point P(a, 0) divides the line segment joining the points A(–2, 1) and B(7, –1). If AP =
2BP, find the value of a.
(b) If A(4, 9) and C(–2, 3) are the opposite vertices of a quadrilateral ABCD, find the midpoint of
the diagonal BD.
350
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