Here, → a
Position vector of A = OA = b
→ x
Position vector of P = OP = y
→ x'
Position vector of P' = OP = y'
Now, In ∆AOP, using triangle law,
→ →→
OA + AP = OP
→ → → x a x–a
∴ AP = OP – OA = y – b = y – b
Similarly, In ∆OAP' using triangle law,
→→→
OA + AP' = OP'
→ → → x' a x' – a
∴ AP' = OP' – OA = y' – b = y' – b
Now, AP' = kAP
x' – a x–a
or, y' – b = k y – b
x' – a k(x – a)
or, y' – b = k(y – b)
Now, x' – a = k (x – a)
∴ x' = k (x – a) + a
and y' – b = k (y – b)
∴ y' = k (y – b) + b.
Hence, the image of P(x, y) after enlargement with centre at (a, b) and scale
factor k is P' [k (x – a) + a, k (y – b) + b].
WORKED OUT EXAMPLES
1. Find the image of A (3, – 2) under the enlargement with centre at (1,
2) and scale factor 2.
Solution: Here, the centre of enlargement (a, b) = (1, 2), scale factor (k) = 2.
Now, in enlargement with centre at (a, b) and scale factor k.
P (x, y)→ P' (k (x – a) + a, k (y – b) + b)
A (3, – 2) → A' (2 (3 – 1) + 1, 2 (– 2 – 2) + 2) = A' (5, – 6)
Infinity Optional Mathematics Book - 9 301
2. A (1, 4), B (3, 0) and C (4, 2) are the vertices of ∆ABC. Determine the co-
ordinates of the vertices of the image of ∆ABC after enlargement with
centre at (0, 1) and scale factor – 2. Also show ∆ABC and its image on
the same graph.
Solution: Here, centre of enlargement (a, b) = (0, 1), scale factor (k) = – 2.
Now, In enlargement with centre at (a, b) and scale factor k,
P (x, y) → P' (k (x – a) + a, k (y – b) + b)
A (1, 4) → A' (-2 (1 -0) + 0, – 2 (4 – 1) + 1) = A'(–2, –5)
B (3, 0) → B' (-2 (3 – 0) + 0, – 2 (0 – 1) + 1) = B'(–6, 3)
C (4, 2) → C' (– 2 (4 – 0) + 0, – 2 (2 – 1) + 1) = C'(–8, –1)
Y
C'(–6,3) A(1,4) X
C(4,2)
X'
C'(8,–1) O B(3,0)
A'(–2,–5)
Y'
3. A (1, 3), B (– 3, 5) and C (– 2, – 3) are the vertices of ∆ABC. A' (4, – 3),
B' (12, – 7) and C' (10, 9) are the vertices of ∆A'B'C' which is the image
of ∆ABC after enlargement. Plot both ∆ABC and ∆A'B'C' on the same
graph and find the centre of enlargement and scale factor.
Solution: Y
C'(10,9)
Here, the vertices of ∆ABC are
A (1, 3), B (– 3, 5) and C (– 2, – 3). The B(–3,5)
vertices of the image of ∆ABC are A'
(4, – 3), B' (12, – 7) and C' (10, 9). Here X' A(1,3) X
both ∆ABC and ∆A'B'C' are shown on
the same graph. AA', BB' and CC' P
are joined and they intersect at (2, O
1). This is the centre of enlargement.
Hence the centre of enlargement is P C'(–2,–3) A'(4,–3)
(2, 1).
B'(2,–7)
Again, by actual measurement,
Y'
302 Infinity Optional Mathematics Book - 9
Scale factor = length of A'B' = 12..36 = 2
length of AB
Here, image and object are on the opposite side of the centre of enlargement.
So, scale factor is negative. Scale factor (k) = – 2.
Hence, the centre of enlargement is (2, 1) and scale factor is – 2
Exercise 7.4
Section 'A'
1. Draw the images of the following geometrical figures after enlargement
with centre O and scale factor k.
(a) A (b) A (c) M S
O
O C B D
(d) B
C NR
B A (e) T 3
(g) k = 3 O k = 2
2
S (f) A
UO R O
(k = –2)
C
C P k = 1 Q B
(h) 2
O (k = –1)
C PS
B Q OR
O (k = –2)
(k = –1) A
2. Find the coordinate of the images of A(3,2), B(–7, 6), C(–2, –5) and
D(9, –1) after enlargement E[0, 3] with centre at origin and scale
factor 3.
3. Find the coordinates of the image of the points A(–6, 0), B(0, 5),C(3, 0) and
D(0, –7) after each of the following enlargement.
(i) E[0, 2] (ii) E[0, –3]
Infinity Optional Mathematics Book - 9 303
4. Find the coordinate of the image of the points A(–3, 0), B(7, –5) C(6, 5) and
D (0, 6) after each of the following enlargements.
(i) E[(1, 0), 2] (ii) E [0, 2), 3] (iii) E [(1, 2), – 2]
Section 'B'
5. A (2, 4), B (– 3, 5) and C (– 2, – 3) are the vertices of ∆ABC. Find the coordinates
of the vertices of the image of ∆ABC after enlargement with centre at origin
and scale factor 2. Show ∆ABC and its image on the same graph.
6. A (2, 1), B (4, 5) and C (– 1, 4) are the vertices of ∆ABC. Enlarge the ∆ABC
with centre at (2, – 2) and scale factor – 2. Show ∆ABC and its image on the
same graph.
7. A (2, 1), B (2, 4) and C (5, 3) are the vertices of ∆ABC. Find the image of ∆ABC
under the enlargement with centre at (– 1, 2) and scale factor 3. Present ∆ABC
and its image on the same graph.
8. An enlargement maps point A (3, 2) onto A' (6, 4) and the point B (– 2, 1) onto
B' (– 4, 2). Find the centre and the scale factor of enlargement.
9. An enlargement maps A (2, 3) onto A'(2, 5) and B (6, 4) onto B' (10, 7). Find
the centre and the scale factor of enlargement.
10. A (1, 3), B (1, 5) and C (2, 5) are the vertices of ∆ABC. Also. A'(–4, 4), B'(–4, 8)
and C'(–2, 8) are the, vertices of ∆A'B'C'. Plot ∆ABC' and ∆A'B'C' on the same
graph and find the centre of enlargement and scale factor.
11. P(2, 1), Q(1, 4) and R(6,3)are the vertices of ∆PQR. Also P'(–4, –2), Q'(–2, –8)
and R'(–12, –6) are the vertices of ∆P'Q'R' which is the image of ∆PQR. Plot
∆PQR and ∆P'Q'R' on the same graph and find the centre and scale factor of
the enlargement.
12. A(–2, 2), B(–4, 6) and C(–6, –2) are the vertices of ∆ABC respectively and
∆A'B'C' is the image of ∆ABC under an enlargement where the co-ordinates of
A',B' and C' are (–6, 5), (–10, 13) and (–14, –3) respectively. Find the centre of
enlargement and scale factor using graph paper.
13. M(2, 3), N(6, 4) and S(5, 7) are the vertices of ∆MNS. Enlarge ∆MNS with
centre (3, 1) and scale factor 2. Present ∆MNS and its image on the graphs.
14. A(–2, 1), B(–2, 4), C(4, 4) and D(4, 1) are the vertices of a quadrilateral ABCD.
Enlarge quadrilateral ABCD with centre (0, – 2) and scale factor –1. Plot
quadrilateral ABCD and its image on the same graph.
304 Infinity Optional Mathematics Book - 9
UNIT STATISTICS
8
8.1 Partition Values
Review
The following marks are obtained by 9 students of class 9 in a test of mathematics.
Marks: 32, 59, 24, 46, 75, 66, 35, 81, 53
Let us discuss the above questions based on the above data.
(i) How to arrange the given data in ascending order or in descending order?
(ii) What is the middle value of the given data after arranged in ascending order
or descending order?
(iii) Write the maximum and minimum value.
(iv) What is the difference between the maximum value and the minimum value?
The middle value of the data when they are arranged in ascending or descending
order is known as median. The median divides the given data in two equal parts. If
there are two middle numbers, taking the mean of those numbers we can find the
median. Again, discuss on the following questions in the class room.
(i) What is statistics? (ii) Define frequency.
(iii) What does represent by cumulative frequency?
(iv) What do you mean by average or mean value?
(v) What do you mean by individual series?
(vi) Write one-one example of individual and discrete series.
(vii) What is the range of the above data?
Partition values
The variate value which divide the whole data (arranging in ascending order) into
equal number of parts are known as partition values. The equal parts may two,
four, ten, hundred, etc. The value which divides the whole series into two equal
parts is known as median.
We consider the following 7 observations in ascending order, 19, 26, 38, 43,51,62, 75
(i) Which number divides the whole observation into two equal parts?
(ii) Is 43 divide into two equal pars? Discuss on it.
(iii) How many points or number should be there so that whole observation dividing
into 4 equal parts? Discuss on it.
Infinity Optional Mathematics Book - 9 305
The middle value 43 divides the whole observations into two equal parts. So,
43 is known as the median. 26 divides the lower half 19, 26, 38 into two equal
parts. So, 26 is said to be the first quartile or lower quartile. 62 divides the
second half 51, 62, 75 into two equal parts and is known as the third quartile
or upper quartile.
Quartiles:
The variate values which divide the whole (arranging in ascending order) data into
4 equal parts are known as quartiles. There are three quartiles which are first
quartile (Q1), second quartile (Q2) and third quartile (Q3).
75% 25%
50% 50%
25% 75% 25%
25% 25% 25%
Q1 Q2 Q3
First quartile (Q1) has 25% of the observation below it and 75% of the observation
above it. Second quartile (median) has 50% of the observation below it and 50% of
the observation above it. Upper quartile (Q3) has 75% of the observation below it
and 25% of the observation above it.
The variate value dividing the lower half into two equal parts is known as the first
quartile or lower quartile. The variate value which divides the upper half into two
equal parts is known as third quartile or upper quartile.
Formula for the quartiles:
For Individual series:
If n is the number of observations,
then first quartile (Q1) = value of n+1 th
4
item,
Median or second quartile (Md) or (Q2) = 3(n + 1 ) th
4
item and
Third quartile (Q3) = value of 3 n+1 th
4
item.
If n is odd, the middle value will give the median and if n is even, then the
mean of two middle values will be the median,
306 Infinity Optional Mathematics Book - 9
For Discrete Series:
First quartile (Q1) = value of N+1 th
4
item
Second quartile (Q2) = value of N+1 th
2
item
and third quartile (Q3) = value of 3(N + 1 ) th
4
item.
WORKED OUT EXAMPLES
1. Find the median (2nd quartile) for each of the following set of
observations
(i) 62, 50, 78, 66, 74, 71, 80
(ii) 20, 15, 5, 10, 35, 25, 30, 40
Solution: Here,
i) Arranging the given data (observations) in ascending order, we get
50, 62, 66, 71, 74, 78, 80
No. of observations (n) = 7 n+1 th
2
Median (Md) = The value of item.
= The value of 7+1 th
2
item
= The value of 4th item
= 43
∴ Median (Md) = 43
ii) Arranging the given data in ascending order, we get
5, 10, 15, 20, 25, 30, 35, 40
No. of observations (n) = 8 n+1 th
2
Median (Md) = The value of item
= The value of 8+1 th
2
item
= The value of 9 th
2
item
= The value of 4.5th item.
= 4th item + 5th item
2
= 20 + 25
2
Infinity Optional Mathematics Book - 9 307
= 45 = 22.5
2
∴ Median (Md) = 22.5
2. Obtain the first and third quartiles from the following observations.
102, 145, 126, 115, 136, 120, 148, 155
Solution: Here,
Arranging the given observation in ascending order, we get
102, 115, 120, 126, 136, 145, 148, 155
No. of observation (N) = 8
First quartile (Q1) = The value of n+1 th
4
item
= The value of 8+1 th
4
item
= The value of 9 th
4
item
= The value of 2.25th item.
= 2nditem + 0.25 (3rd item – 2nd item)
= 115 + 0.25 (120 – 115)
= 115 + 0.25 (5)
= 116.25
Again, third quartile (Q3 ) = The value of 3(n + 1) th
4
item.
= The value of (3 × 2.25)th item
= The value of 6.75th item
= 6thitem + 0.75 (7th item – 6t h item)
= 145 + 0.75 (148 – 145)
= 145 + 0.75 × 3
= 147.25
∴ First quartile (Q1) = 116.25 and third quartile (Q3) = 147.25
308 Infinity Optional Mathematics Book - 9
3. Find the median (Second quartile) of the following data.
Marks obtained 25 35 45 55 65 75
No. of students 5 15 10 8 6 2
Solution: Here,
Tabulating the given data in ascending order, we get
Marks obtained No. of students (f) Cumulative frequency (C.f)
25 5 5
35 15 20
45 10 30
55 8 38
65 6 44
75 2 46
Σf = N = 46
Median = value of N+1 th
2
item
= value of 46 + 1 th
2
item
= value of 23.5thitem
In C.f column, 30 is just greater than 23.5 so its corresponding value is 45.
∴ Median (Md) or 2nd quartile (Q2) = 45
4. Find the lower quartile and upper quartile from the following data.
Marks obtained 5 15 25 35 45 55
No. of students 3 7 15 5 8 2
Solution: Here,
Tabulating the given data is ascending order, we get
Marks obtained No. of students (f) Cumulative frequency (C.f)
5 3 3
15
25 7 10
35 15 25
45 5 30
55 8 38
2 40
Σf = N = 40
Infinity Optional Mathematics Book - 9 309
Lower quartile (Q1) = value of N+1 th
4
item
= value of 40 + 1 th = value of 10.25th item.
4
item
In c.f. column 25 is just greater than 10.25 so its corresponding value is 25.
∴ Lower quartile (Q1) = 25 3(N + 1) th
Again, 4
Upper quartile (Q3) = value of item
= value of (3 × 10.25)thitem = value of 30.75th item
In c.f. column, 38 is just greater than 30.75 so its corresponding value is 45.
∴ Upper quartile (Q3) = 45.
Exercise 8.1
Section 'A'
1. (a) Define individual and discrete series. Illustrate them with examples.
(b) What do you understand by quartiles?
(c) Define third quartile. Write the formula to find third quartile in discrete
series.
2. (a) What percentage of values are less than first quartile (Q1).
(b) Define median. How it divides the whole observation?
(c) What percentage of values are more than third quartile (Q3)
Section 'B'
3. (a) Find the median (2nd quartile) from the following data.
(i) 45kg, 60kg, 53kg, 48kg, 51kg, 63 kg
(ii) 8 cm, 16 cm, 28 cm, 60 cm, 30 cm, 60 cm, 8 cm, 12 cm, 8 cm
(iii) 110, 105, 100, 150, 250, 175, 225, 275, 110, 150, 100, 110
(b) Find the first quartile (Q1) from the following data.
(i) Weight (in kg) : 90, 100, 110, 125, 115
(ii) Marks: 20, 30, 60, 45, 110, 90, 80, 118, 115, 120
(iii) Height (in cm) : 22, 26, 14, 30, 18, 17, 35, 41, 12
(c) Find the upper quartile (Q3) from the following data.
(i) 50, 40, 55, 60, 61, 70, 49
(ii) 18, 20, 17, 24, 19, 21, 23
(iii) 63, 38, 47, 59, 24, 50, 75, 81, 8, 110
4. (a) 2x + 1, 3x – 1, 3x + 5, 5x – 7, 51, 63 and 70 are in ascending order. If the
first quartile is 20, what is the value of x.
310 Infinity Optional Mathematics Book - 9
(b) 54, 60, 62, 65, 68, 2x + 30, 72 are in ascending order. If the third quartile
is 70, find the value of x.
5. (a) Out of total of 19 observation arranged in ascending order, the 5th and
(b) 6th observations are 25 and 28 respectively. Find the value of Q1. 23th,
Out of total of 31 observations arranged in ascending order, the
24th and 25th observations are 35, 40 and 45 respectively. What is the
value of Q3?
Section 'C'
6. Find the median from the following data.
(a) Marks obtained 5 6 7 10 9 8
No. of students 2 3 1 2 3 5
(b) Daily Salary (in Rs.) 1000 1500 1700 1800 2000
Employers 6 4 10 9 6
(c) Daily Wages (in Rs.) 50 65 70 75 100 110
No. of workers 579432
7. Calculate lower quartile from the following data.
(a) Weight of children ( in kg) 18 19 20 21 22
No. of children 7 10 15 8 3
(b) Temperature (in °c) 18 17 25 35 37
20
No. of days 12 15 28 25
350
8. Find the third quartile from the following data. 8
30
(a) x 100 150 200 250 300 2
f 10 13 18 25 15 60
7
(b) Marks 25 10 20 15 5
No. of students 3 2 5 4 3
(c) Weight (in kg.) 30 40 20 10 50
Frequency 34532
9. Calculate all the three quartiles from the data given below by making discrete
frequency table.
20, 18, 19, 20, 25, 40, 30, 20, 18, 19, 20, 25, 40, 30, 30, 40, 25, 19, 20
Infinity Optional Mathematics Book - 9 311
Deciles:
Take a long rope. How many places should it be cut so that it is divided into 10
equal parts? Discuss in class.
The variate values dividing the total number of observation into ten equal parts
are known as deciles. There are nine deciles. They are denoted by D1, D2, D3, .......,
D8, D9.
Formula for deciles:
For Individual Series: For Discrete Series:
Dn = N n+1 th Dn =n N+1 th
10 10
item item
where, N = 1, 2, 3, ........ 9 where, n = 1, 2, 3, ........ 9
and n = number of observation. and N = Total sum of the frequency (Σf)
Percentile:
Take a long rope. How many places should it be cut so that it is divided into 100
equal parts? What is the length of each part? Discuss on the above questions.
The variate values dividing the total number of observations (arranged in ascending
order) into hundred equal parts are known as percentiles. There are 99 percentiles.
They are denoted by P1, P2, P3, ......., P98, P99. Which percentiles are also called 1st
quartile, median and 3rd quartiles? Discuss on it.
Formula for percentile: For Discrete Series:
For Individual Series: Pn = n N+1 th
100
item
Pn = N n+1 th where, n = 1, 2, 3, ......., 98, 99
100
item
where, N = 1, 2, 3, ........., 98, 99 N = Total sum of the frequency ( Σf)
n = number of observation.
312 Infinity Optional Mathematics Book - 9
WORKED OUT EXAMPLES
1. Find the 6th decile and 40th percentile of the following data.
62, 50, 78, 66, 74, 71, 80
Solution: Here,
Arranging the given data into ascending order of magnitude, we get
50, 62, 66, 71, 74, 78, 80
Number of observation (n) = 7
By formula,
(i) 6th decile (D6) = value of N n+1 th
100
item.
= value of 6 7+1 th
10
item
= value of (4.8)th item
= 4th item + 0.8 × (5th item – 4thitem)
= 71 + 0.8 × (74 – 71)
= 71 + 2.4
= 73.4
(ii) 40thpercentile (P40) = value of N n+1 th
100
item
= value of 40 7+1 th item
100
= value of 320 th item
100
= value of (3.2)th item
= 3rd item + 0.2 (4th item – 3rd item)
= 66 + 0.2 (71 – 66)
= 66 + 0.2 × 5
= 66 + 1 = 67
∴ 6th decile (D6) = 73.4 and 40th percentile (P40) = 67
Infinity Optional Mathematics Book - 9 313
2. The ages of the 46 students of a secondary school are given below.
Ages (yr.) 10 11 12 13 14 15 16 17
No. of students 2 5 7 9 10 6 4 3
Find 8th decile and 90th percentile.
Solution: Here,
Tabulating the given data in ascending order, we get
Ages No. of students (f) Cumulative frequency (C.f)
10 2 2
11 5 7
12 7 14
13 9 23
14 10 33
15 6 39
16 4 43
17 3 46
Σf = N = 46
By formula, N+1 th
10
(i) 8th decile (D8) = value of n item
= value of 8 46 + 1 th
10
item
= value of 8 × 47 th
10
item
= values of 37.6th item
In c.f column, 39 in just greater than 37.6 so its corresponding value is 15.
∴ D8 = 15. N+1 th
(ii) Again, 90th percentile (P90) = value of n 100
item
= value of 96 46 + 1 th
100
item
= value of 90 × 47 th
100
item
= value of 4230 th
100
item
= value of 42.30th item
In c.f column 43 is just greater than 42.30 so its corresponding value is 16.
H∴e nceP,980t=h 16 (D8) = 15 and 90th percentile (P90) = 16.
decile
314 Infinity Optional Mathematics Book - 9
Exercise 8.2
Section 'A'
1. (a) What do you understand by deciles and percentiles.
(b) Write the formula for 5th decile and 9th decile.
(c) Write the formula for 40th and 80th percentile.
2. (a) 25th percentile also called ............ quartile.
(b) 50th percentile also called ............
(c) 75th percentile also called ............ quartile.
Section 'B'
3. (a) Find the 7th decile from the following data: 16, 19, 21, 26, 24, 32, 31, 28,
34
(b) Find the 6th and 8th decile from the following data.
(i) 32, 24, 59, 46, 75, 35, 66, 53, 81
(ii) 51, 68, 77, 44, 56, 61, 82, 102, 75
4. (a) Find the 30th and 40th percentile from the following data.
(i) 8, 12, 21, 19, 17, 14, 24, 26, 29
(ii) 12, 17, 18, 19, 20, 22, 25, 29, 32, 33, 40, 43, 47, 49, 50, 54, 55, 66, 68
5. (a) Obtain 8th decile and 61st percentile of the following data.
22, 26, 14, 30, 18, 11, 35, 41, 12, 32
(b) Obtain the 3rd decile and 65th percentile of the following data: 6, 8, 5, 10,
4, 15, 16, 3
6. (a) 20, 30, 40, x + 1, 2x – 1, x + 7, 3x + 4, 90, 100 are in ascending order. If
6th decile is 70, find the value of x.
(b) If 12, 17, 2a + 3, 3a + 5, 36, 43 are in ascending order. If its 50th percentile
is 29, find the value of 'a'.
Section 'C'
7. (a) Find the 3rd and 6th deciles from the following data.
Marks obtained 35 45 55 65 75 85
No. of students 7 3 10 5 3 2
(b) Find the 4th and 9th deciles of the given data.
Weight (in kg) 5 10 15 20 25 30
No. of children 3 76257
Infinity Optional Mathematics Book - 9 315
8. (a) Obtain the 36th and 50th percentile from the given data.
Ages (in yrs) 20 40 30 25 35 15
No. of people 3 4 3 2 5 4
(b) Find the 32nd and 80th percentile of the given data.
Height (in cm) 10 15 20 25 30 35 40
No. of plants 10 6 15 8 4 5 3
9. (a) The daily wages of 79 workers are given below. Find 5th decile and 90th
percentile.
Wages (in Rs.) 400 500 625 700 900 1000
No. of workers 15 20 6 25 9 4
(b) Find the 4th decile and 80th percentile from the following data.
Fine (in Rs.) 5 10 15 20 25
No. of students 2 8 10 9 6
8.2 Measure of Variability or Dispersion
The marks obtained by 6 students of each two groups of class 9 in an examination
are given below.
Group A 25 26 27 27 28 29
Group B 0 10 18 27 27 80
Study the above given data and discuss on the following questions.
(i) What are the mean marks of groups A and B?
(ii) What are the median marks of groups A and B?
(iii) Do the mean and median represent all the characteristics of the
statistics? If not why?
(iv) Which group obtained marks is more dispersed from the central value?
(v) Does the group A and B have same average marks but different
variability ?
The various measure of central tendency gives us an idea of the concentration of
observation about the central part of the distribution. It cannot explain how the
values of data are scattered? Thus, dispersion is the scatterness of the items from
their central value. The measure of scatterness of item from the central value is
known as measure of dispersion (measure of variability).
The purpose to measure the dispersion is to find the homogeneity and heterogeneity
of the given data.
316 Infinity Optional Mathematics Book - 9
The various measure of dispersion are
(i) Range
(ii) Quartile deviation or semi interquartile range
(iii) Mean deviation or average deviation
(iv) Standard deviation
Quartile deviation:
Find the quartiles of the following data: 9, 7, 5, 15, 13, 11, 17. What does quartiles
represent? Discuss in classroom.
The given data is ascending order
5, 7, 9, 11, 13, 15, 17
It is an individual series and the number of observation (n) = 7
Now, first quartile (Q1) = value of n+1 th
4
item
= value of 7+1 th
4
item
= value of 2nditem
∴ First quartile (Q1) = 7 n+1 th
Second quartile (Q2) = value of 2
item
= value of 7+1 th
2
item
= values of 4th item
∴ Second quartile (Q2) = 11 3 n+1 th
Third quartile (Q3) = value of 4
item
= value of 3 7+1 th
4
item
= value of 6th item
∴ Third quartile (Q3) = 15
From the above results, discuss on the following questions.
(i) What is the average value of quartiles?
(ii) What is the relative difference between the quartiles?
The measure of dispersion depending upon the lower and upper quartile is known
as the "quartile deviation". The difference between the upper quartile (Q3) and the
lower quartile (Q1) is known as the interquartile range. Half of the interquartile
range is known as the "semi-interquartile range" or quartile deviation. To find the
quartile deviation we need first and third quartile.
The quartile divide the whole observation into 4 equal parts. So, there are 3
quartiles. The first quratile or lower quartile (Q1). The second quartile or median is
(Q2) or (Md). The third quartile or upper quartile is (Q3).
Infinity Optional Mathematics Book - 9 317
Interquartile range = Q3 – Q1
∴ Quartile deviation or semi-inter quartile range = Q3 – Q1
2
From above example, Q3 – Q1
2
Quartile deviation (Q.D) = = 15 – 7 = 8 = 4
2 2
The relative measure based on lower and upper quartiles known as the coefficient
of quartile deviation. It is given by
Coefficient of Q. D. = Q3 – Q1
Q3 + Q1
∴ Coefficient of Q. D. = 15 – 7 = 8 = 0.3636 = 36.36%
15 + 7 22
WORKED OUT EXAMPLES
1. Find the quartile deviation and its coefficient from the following data
24, 27, 31, 37, 45, 48, 56
Solution: Here,
Arranging the given data in ascending order.
24, 27, 31, 37, 45, 48, 56
No. of observation (n) = 7
First quartile (Q1) = The values of n+1 th
4
item
= The value of 7+1 th
4
item
= The values of 2th item
∴ First quartile (Q1) = 27
Third quartile (Q3) = The value of 3 n+1 th
4
item
= The value of (3 × 2)th item
= The value of 6th item
∴ Third quartile (Q3) = 48
Again, by formula,
Quartile deviation (Q.D) = Q3 – Q1 = 48 – 27 = 21 = 10.5
2 2 2
Q3 – Q1
and coefficient of Q.D. = Q3 + Q1 = 48 – 2277 = 21 = 0.28 = 28%
48 + 75
318 Infinity Optional Mathematics Book - 9
2. Find the quartile deviation and its coefficient of the following data.
Marks obtained 3 5 7 9 11 13 15 17 19
No. of students 2 10 12 15 20 13 12 10 4
Solution: Here, Tabulating the given data is ascending order, we get
Ages No. of students (f) Cumulative frequency (C.f)
32 2
5 10 12
7 12 24
9 15 39
11 20 59
13 13 72
15 12 84
17 10 94
19 4 98
Σf = N = 98
Now, N+1 th
4
First quartile (Q1) = The value of item
= The value of 98 + 1 th
4
item
= The value of 24.75th item
In c.f. column, 39 is just greater than 24.75 so its corresponding value is 9.
∴ First quartile (Q1) = 9
Again, N+1 th
4
Third quartile (Q3) = The value of 3 item
= The value of 3 × 24.75th item
= The value of 74.25th item.
In c.f column, 84 is just greater than 74.25 so its corresponding value is 15.
∴ Third quartile (Q3) = 15
By formula,
Quartile deviation (Q.D) = Q3 – Q1 = 15 – 9 = 6 = 3
2 2 2
Q3 – Q1
and coefficient of Q.D. = Q3 + Q1 = 15 – 9 = 6 = 0.25 = 25%
15 + 9 24
Infinity Optional Mathematics Book - 9 319
Exercise 8.3
Section 'A'
1. (a) What is dispersion ? Illustrate it with example.
(b) Define interquartile range. Write the formula to find interquartile range.
2. (a) Define semi-interquartile range.
(b) Write the various measure of dispersion.
(c) Write the formula to calculate the quadrile deviation and its coefficient.
3. (a) If the lower quartile and upper quartile of the data are 40 and 60
respectively, find the interquartile range.
(b) If the lower quartile and upper quartile of the data are 20 and 30
respectively, find the quartile deviation.
Section 'B'
4. (a) In a data, first quartile is 30 and the third quartile is 55. Find the
quartile deviation and its coefficient.
(b) In a data, quartile deviation and the first quartiles are 5 and 20
respectively. Find the third quatrile and coefficient of quartile deviation.
5
(c) In a data, the coefficient of quartile deviation is 12 and its upper quartile
is 50 find its lower quartile and inter quartile range. 7
12
5. (a) In a data, the quartile deviation and its coefficient are 14 and
(b) respectively. Find Q1 and Q3. of quartile deviation is 1 and the value
In a certain data, the coefficient 4
of upper quartile is 15 then find the lower quartile and quartile deviation.
6. (a) The third quartile and interquartile range of a data are 51 and 21
respectively. Find the coefficient of the quartile deviation.
(b) The third quartile of a data is 70. If the coefficient of quartile deviation
is 41, find the first quartile and interquartile range.
(c) In a data, the value of first quartile is 'y' and the quartile deviation is
also 'y'. Find the third quartile and the coefficient of quartile deviation.
Section 'C'
7. Find the quartile deviation and its coefficient from the following data.
(i) Price (Rs.) : 13, 27, 6, 14, 13, 19, 8
(ii) Weight (in kg.) : 7, 15, 10, 13, 17, 18, 20
320 Infinity Optional Mathematics Book - 9
(iii) Height (inches) : 4, 3, 10, 7, 8, 15, 12, 19, 24
(iv) Daily expenditure (in Rs.): 140, 123, 132, 130, 112, 118, 138, 135
8. Find the quartile deviation and its coefficient of the following data.
(i) Weight (in Kg.) 40 45 50 55 60 64
No. of people 36954 2
(ii) X 10 25 20 40 30 45
f3 2 5 4 3 2
(iii) Age (in yrs.) 20 25 30 35 40 50 45
No. of members 6 11 7 4 3 1 1
(iv) Wages (in Rs.) 50 60 75 90 82 91
No. of workers 10 12 8 3 5 1
9. Prepare a discrete frequency distribution table and find the quartile deviation
and its coefficient.
11, 12, 19, 11, 14, 12, 15, 13, 14, 12, 16,
16, 17, 15, 13, 14, 17 13, 15, 13, 15,
Mean Deviation
Can you find the mean of the following data?
3, 6, 6, 7, 8, 11, 15, 16
3 + 6 + 6 +7 + 8 + 11 + 15 + 16 72
Mean (X) = 8 = 8 = 9
The distance of each value from the mean. (No minus sign)
Value Distance From (9)
36
63
63
72
81
11 2
15 6
16 7
which looks like this.
Infinity Optional Mathematics Book - 9 321
Mean (X)
1 2
2
3
3 6
6 7
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Find the mean of those distances and the mean in called mean deviation.
Mean deviation = 6 + 3 + 3 + 2 + 1 + 2 + 6 + 7 = 30 = 3.75
8 8
So, the mean = 9, and the mean deviation = 3.75.
Its tell us on average, how far all values are from the middle.
In that example the values are on average, 3.75 away from the middle.
As we know that range depends on the largest and smallest value of the distribution
and quartile deviation depends on 50% of the total distribution, They are not based
on all the observation and they do not measure the scatterness of the items from
the average value. Thus, they are not consider as good measure of dispersion. But
mean deviation measures the variation of each observation of the total distribution
from the average.
Mean deviation is defined as the average of the absolute values of the deviation
(differences) of each item from mean, median or mode. It is also known as average
deviation. Mean deviation calculated from mean is called mean deviation from mean
or simply mean deviation. Similarly, mean deviation is calculated from median is
known as mean deviation from median or simply mean deviation. Mean deviation
is denoted by M.D.
Mean deviation is absolute measure. So to compare two or more series having
different units, the relative measure corresponding to mean deviation is used,
which is called coefficient of mean deviation.
Calculation of mean deviation and its coefficient
(a) For Individual Series
If X and Md be the arithmetic mean and the median of the set of observations,
then
(i) Mean deviation from mean = Σ|X – X| = Σ|D|
Where, |D| = |X – X| n n
(ii) Mean deviation from median = Σ|X – Md| = Σ|D|
n n
322 Infinity Optional Mathematics Book - 9
where, |D| = |X – Md|
n being the number of observations.
(b) For Discrete Series
(i) Mean deviation from mean = Σf|X – X| = Σf|D|
where, |D| = |X – X| N N
N = Total sum of frequencies = Σf, X = Mean, f = frequency
(ii) Mean deviation from median = Σf|X – Md| = Σf|D|
N N
where, |D| = |X – Md|
Md = Median
(c) Coefficient of mean deviation M.D from mean
mean
(i) Coefficient of mean deviation from mean =
(ii) Coefficient of mean deviation from median = M.D from median
median
WORKED OUT EXAMPLES
1. Calculate the mean deviation from mean of the following data. Also,
find its coefficient.
90, 100, 125, 115. 110, 150
Solution: Here,
Tabulating the given data in ascending order, we get
No. of observation (n) = 6
X X– X=D |D|
90 – 25 25
100 – 15 15
110 – 5 5
115 0 0
125 10 10
150 35 35
∑X = 690 ∑|D| = 90
By formula, mean (X) = ΣX = 690 = 115
n 6
Σ|D| 90
Now, mean deviation from mean = n = 6 = 15
and, coefficient of M.D. = Meandeviation from Mean = 15 = 0.13
X 115
Infinity Optional Mathematics Book - 9 323
2. Find the mean deviation from median of the following data. Also, find
the coefficient of mean deviation.
22, 36, 38, 29, 34, 25, 26, 30
Solution: Here,
Arranging the given data in ascending order, we get
22, 25, 26, 29, 30, 34, 36, 38
No. of observation (n) = 8 n+1 th
2
Median (Md) = value of item
= value of 8+1 th
2
item
= value of 4.5th item
29 + 30 59
∴ Median (Md) = 2 = 2 = 29.5
Tabulating the given data in ascending order, we get
X X – Md = D |D|
22 – 7.5 7.5
25 – 4.5 4.5
26 – 3.5 3.5
29 – 0.5 0.5
30 0.5 0.5
34 4.5 4.5
36 6.5 6.5
38 8.5 8.5
Σ|D| = 36
By formula, Σ|D| 36
N 8
Mean deviation from median = = = 4.5
and the coefficient of M.D. = M.D. from median = 4.5 = 0.15
Median 29.5
3. Find the mean deviation from mean of the following data. Also, find
the coefficient of mean deviation.
Age (years) 24 36 50 58 78 83
No. of students 5 6 7 9 3 1
Solution: Here,
Tabulating the given data in ascending order.
324 Infinity Optional Mathematics Book - 9
Age (x) No. of students fx X – X = D |X–X= |D| f|D|
(f)
24 120 – 25.19 25.19 125.95
36 5 216 –13.19 13.19 79.14
50 6 350 0.81 0.81 5.67
58 7 522 8.81 8.81 79.29
78 9 234 28.81 28.81 86.43
83 3 83 33.81 33.81 33.81
1 Σfx = 1525
1525 Σf|D|= 410.29
Σf = N = 31
Now, by formula,
M ean(Xde)v=iatΣNifoxn=fro1m5312m5 e=a4n9=.1Σ9f|ND| =
410.29 = 13.23
31
and coefficient of M.D. = Mean deviation from mean = 13.23 = 0.26
X 49.19
4. Calculate the mean deviation from median of the following frequency
table. Also, find its coefficient.
Height ( in cm) 10 20 30 40 50 60
No. of plants 2 3 9 21 11 5
Solution: Here,
Tabulating the given data in ascending order, we get
Age (x) No, of student (f) cf X – Md = D |D| f|D|
10 2 2 – 30 30 60
20 3 5 – 20 20 60
30 9 14 – 10 10 90
40 21 35 0 00
50 11 46 10 10 110
60 5 51 20 20 100
∑f = 51 ∑f|D| = 420
By formula, N+1 th 51 + 1 th
2 2
Median (Md) = value of item= value of item
= value of 26th item
In c.f. column 35 is just greater than 26 and its corresponding value is 40.
∴ Median (Md) = 40
CNooewff,iMcieenatnodfeMvi.aDti=onMfr.Dom. frmoMmeddimaned=iaΣnf|N=D8|4.20=34=52100.2=085.23
Infinity Optional Mathematics Book - 9 325
Exercise 8.4
Section 'A'
1. (a) What do you mean by mean deviation or average deviation?
(b) Define coefficient of mean deviation.
2. (a) Write the formula to find the mean deviation from mean, in individual
series.
(b) Write the formula to find the coefficient of mean deviation from median.
Section 'B'
3. (a) An individual series has Σx = 120, N = 10 and Σ|D| = 34. Find the mean
deviation and the coefficient of M.D.
(b) An individual series has Σx = 70, Σf = 7 and Σ|X – X| = 34. Find the
mean deviation and the coefficient of mean deviation.
4. (a) In an individual series, median (Md) = 250, N = 7 and Σ|X – Md| = 600.
Find the mean deviation from median and its coefficient.
(b) In an individual series, median (Md) = 140, N = 5 and Σ|X – Md| = 90.
Find the mean deviation and its coefficient.
5. (a) In a discrete series, Σfx = 750, Σf = 25 and Σf|X – X| = 140. Find the
mean deviation from mean and its coefficient.
(b) In a discrete series, median (Md) = 40, Σf = 52 and Σf|X – Md| = 420.
Find the mean deviation from median and its coefficient.
Section 'C'
6. Find the mean deviation from the mean for each of the following set of
observations. Also, find the coefficient of mean deviation from mean.
(i) 7, 10, 2, 3, 4, 5, 11
(ii) 400, 100, 200, 300, 350, 250, 150
(iii) 40, 42, 30, 37, 41, 58, 45, 50, 55, 32
(iv) 10, 5, 15, 25, 45, 50, 30, 20, 35, 40, 55.
7. Calculate the mean deviation from the median for each of the following set of
observations. Also, find the coefficient of M.D. from median.
(i) 105, 100, 125, 130, 140
(ii) 10, 20, 40, 30, 60, 70, 50
(iii) 100, 250, 120, 170, 140, 205
(iv) 32, 34, 20, 36, 27, 24, 28, 23
8. Calculate the mean deviation from mean and its coefficient from the data
given below.
326 Infinity Optional Mathematics Book - 9
(i) Age (in yrs.) 6 8 10 12 14
No. of students 4 2 5 3 1
(ii) Marks 40 20 60 10 50 30
No. of students 10 15 3 5 12 15
(iii) Weight (in Kg.) 35 60 50 45 55 70 75 65
No. of people 86545776
(iv) Price (Rs.) 20 25 30 34 40
Frequency 5 8 12 10 5
9. Find the mean deviation from median and its coefficient from the data given
below.
(i) Marks obtained 6 8 10 12 14 16 18 20
No. of students 1 14 25 27 18 9 4 2
20 25 30 35 40
(ii) Weight (in Kg.) 5 8 12 10 5
No. of students
(iii) X 2 4 6 8 10 12 14 16
f 22453211
(iv) Income (in Rs.) 200 250 300 350 400
No. of persons 5 9 20 10 6
10. Construct a discrete frequency distribution table from the following data and
find the mean deviation from (i) mean and (ii) median.
(i) 15, 20, 18, 20, 18, 22, 24, 28, 12, 18, 22, 24, 15, 20, 18, 20, 22, 15, 28
(ii) 12, 8, 4, 8, 6, 4, 4, 8, 10, 4, 16, 6, 8, 6, 4, 16, 4, 10, 12, 8, 6, 16, 10, 16, 14.
Infinity Optional Mathematics Book - 9 327
Standard Deviation
Let us discuss on the followign questions.
(i) Why we should take absolute value of X - A = D to calculate the mean deviation?
(ii) What would be the result in the square of negative number?
(iii) What are the advantages and disadvantages of mean deviation?
Standard deviation is defined as the positive square root of the mean of squares of
the deviations of the given observations taken from mean. It is also known as root
mean square deviation. It is denoted by the Greek letter s (read as sigma). It was
first introduced by Karl Pearson in 1823.
In mean deviation, we use absolute value of the deviation of the items and ignore the
negative sign. Neglectiong the negative sign is the draw back of the mean deviation.
Standard deviation is the best measure of dispersion because
(i) Its value is based on all the observations.
(ii) The deviation of each item is taken from mean.
(iii) All algebraic sign are also considered.
In standard deviation, greater the value of S.D. the greater the dispersion or
variability and greater will be the magnitude of the variation of the value from
mean. In other words, a small standard deviation means a high degree of uniformity
of the observation as well as homogeneity of the series.
Calculation of standard deviation
(a) Individual series:
(i) Actual mean method : In this method, the deviation is taken from actual
mean
Standard deviation (S.D.) = (s) = Σ(X – X)2 = Σd2
where, d = X – X n n
n = total number of observation
X = Actual mean
(ii) Direct method:
In this method, the actual given data are used.
Standard deviation (S.D) = (s) = ΣX2 – ΣX 2 where X = given data.
n n
(iii) Assumed mean method (short cut method)
In this method, we assume the central value of the given data as an
assumed mean.
328 Infinity Optional Mathematics Book - 9
Standard deviation (S.D.) = (s) = Σd2 – Σd 2
n n
where, d = X – A, and A = assumed mean.
For individual series, n = total no. of observation.
(b) Discrete Series:
(i) Actual mean method:
Standard deviation (S.D.) = (s) = Σf(X – X)2 = Σfd2
where, d = X – X N N
(ii) Direct method
Standard deviation (S.D.) = (s) = Σfx2 – Σfx 2
N N
(iii) Assumed mean method (shortcut method)
Standard deviation (S.D.) = (s) = Σfd2 – Σfd 2
N N
where, d = X – A, A = assumed mean,
In discrete series, N = Sum of the frequency (Sf )
Coefficient of standard deviation
The relative measure of dispersion based on standard deviation is known as
coefficient of standard deviation.
∴ Coefficient of S.D. = Standard deviation (s) = s
Mean X
Variance and coefficient of variation
The square of the standard deviation is known as variance.
∴ Variance = s2
The value of coefficient of standard deviation is so small, due to that it is
multiplied by 100 and converted into percentage then it is called coefficient of
variation. It is denoted by C.V.
∴ Coefficient of variation (C.V) = s × 100%
X
Infinity Optional Mathematics Book - 9 329
WORKED OUT EXAMPLES
1. Calculate the standard deviation and its coefficient of the following
data :
4, 8, 20, 16, 12, 24, 28 by
(i) Actual mean method (ii) Direct method (iii) Short cut method
Solution: Here
Tabulating the given data in ascending order, we get
(i) By actual mean method:
X X– X=D d2
4 – 12 144
8 –8 64
12 – 4 16
16 0 0
20 4 16
24 8 64
28 12 144
∑X = 112 ∑d2 = 144
No. of observation (n) = 7
Now, mean (X) = ΣX = 112 = 16
By formula, N 7
Standard deviation (s) = Σd2 = 448 = 64
∴ S.D. (σ) = 8 n 7
(ii) By Direct method : Here,
X X2
4 16
8 64
12 144
16 256
20 400
24 576
28 784
∑X = 112 ∑X2 = 2,240
n=7
330 Infinity Optional Mathematics Book - 9
By formula, standard deviation (σ) = ΣX2 – ΣX 2 = 2240 – 112 2
n n 7 7
= 320 – 256 = 64
∴ S.D. (σ) = 8
(iii) By short cut method:
Let the assumed mean (A) = 16
X d=X–A d2
4 – 12 144
8 –8 64
12 – 4 16
16 0 0
20 4 16
24 8 64
28 12 144
X d=X–A d2
∑d = 0 ∑d2 = 448
n=7
By formula, S.D. (σ) = Σd2 – Σd 2 = 448 – 0 2= 64 – 0 = 8
∴ S.D. (σ) = 8 n n 7 7
2. Find the standard deviation from the following data. Also, find the
coefficient of S.D. and coefficient of variance.
Size 6 9 12 15 18
Frequency 7 12 19 10 3
by (i) Actual mean method (ii) Direct method
(iii) Assumed mean method
Solution: Here,
Tabulating the given data in ascending order, we get
(i) Actual mean method:
Size (X) Frequency (f) fx X – X = d d2 fd2
6 7 42 – 5.41 29.26 204.87
69.69
9 12 108 – 2.41 5.808 7.08
193.2
12 19 228 0.59 0.34 118.62
Σfd2 = 593.46
15 10 150 3.59 12.88
18 3 54 6.59 43.42
Σf = N = 51 Σfx = 582
Now,
Infinity Optional Mathematics Book - 9 331
Mean (X) = Σfx = 582 = 11.41
N 52
Σd2 593.46
Standard deviation (s) = n = 51 = 11.63 = 3.41
∴ S.D (σ) = 3.41
(ii) Direct method:
Size (x) Frequency (f) fx x2 fx2
6 7 42 36 252
9 12 108 81 972
12 19 228 144 2736
15 10 150 225 2250
18 3 54 324 972
Σf = N = 51 Σfx = 582 Σfx2 = 7182
By formula, Σfx2 – Σfx 2 = 7182 – 582 2
S.D (σ) = N N 51 51
= 140.82 – 130.22 = 10.6 = 3.25
(iii) Assumed mean method
Let, assumed mean (A) = 12
Size (X) Frequency (f) X – A = d fd d2 fd2
6 7 – 6 – 42 36 252
9 12 – 3 – 36 9 108
12 19 0 00 0
15 10 3 30 9 90
18 3 6 18 36 108
Σf = N = 51 Σfx = 30 Σfd2 = 558
By formula,
Standard deviation (σ) = Σfd2 – Σfd 2 = 558 – –30 2
N N 51 51
= 10.94 – 0.346 = 10.594 = 3.25
Again,
s 3.25
Coefficient of standard deviation = X = 11.41 = 0.284
and Coefficient of variance (C.V) = s × 100% = 0.284 × 100% = 28.48%
X
Note:
Sd
(i) In individual series actual mean (X) = A + S N
(ii) In discrete series, actual mean (X) = A + Sfd where, A = assumed mean.
N
332 Infinity Optional Mathematics Book - 9
Exercise 8.5
Section 'A'
1. (a) Define standard deviation.
(b) Write the formula to find the standard deviation of discrete series by
actual mean method and assumed mean method.
2. (a) What do you mean by coefficient of standard deviation?
(b) What is coefficient of variation?
(c) Why coefficient of standard deviation is multiplied by 100?
3. (a) In an individual series, X = 18, Σd2 = 112, Σx = 126 and d = X – X then
find the standard deviation and its coefficient.
(b) In an individual series Σx = 114, X = 19 and Σ(X – X)2 = 232, find
standard deviation and its coefficient.
(c) In an individual series Σd2 = 84, Σd = –20, n = 10, find standard deviation.
4. (a) In a discrete series, Σfd = –20, Σfd2 = 5400, N = 35, d = X – A and A = 40
find S.D and coefficient of S.D..
(b) In a discrete series N = 20, Σfx = 570, Σfx2 = 19100 and X = 28.50 then
find the coefficient of variation.
Section 'B'
5. (a) Find the standard deviation and its coefficient from the following data.
Also, find the coefficient of variation.
(i) Marks: 5, 10, 15, 20, 25
(ii) Weight (Kg.) : 60, 50, 80, 40, 90, 95, 70
(iii) Size : 27, 24, 31, 48, 45, 37, 56
(iv) Rainfall (mm) : 34, 23, 46, 37, 40, 28, 32, 35, 44, 50
(v) Weight (Kg.) : 40, 50, 60, 70, 80, 90, 100, 110, 120, 130
(vi) Temperature (0°C) : 12, 6, 7, 3, 15, 10, 18, 5
Section 'C'
6. Find the standard deviation, its coefficient and coefficient of variation.
(i) X 61 64 67 70 73
f 5 18 42 27 8
Infinity Optional Mathematics Book - 9 333
(ii) Height (in cm) 10 15 20 25 30 35
No. of plants 356431
(iii) Marks 40 44 50 55 60 65
No. of students 2 5 6 4 3 3
(iv) Size 15 25 35 45 55
Frequency 5 8 13 15 12
(v) Age (in years) 10 20 25 30 35 40
No. of Boys 1 5 10 12 8 4
(vi) Marks obtained 35 45 50 55 60 65 70 75
No. of students 8 4 5 5 6 6 6 7
7. Construct the frequency distribution table and find the variance and its
coefficient
(i) Mark obtained by 36 students are :
25, 20, 10, 15, 10, 20, 15, 20, 40, 35, 25, 25, 30, 35, 30, 25, 25, 10, 30, 15,
20, 15, 30, 20, 20, 30, 20, 25, 25, 35, 25, 25, 15, 25, 25, 20, 20
(ii) Wages per hour (Rs.)
55, 65, 35, 55, 45, 75, 55, 65, 75, 55, 45, 65, 35, 75, 45, 55, 65, 75, 55, 65
8. Collect the marks of opt. mathematics obtained by all the students of grade 9
in first terminal examination of your school. Show them in discrete frequency
table and find the standard deviation, its coefficient and coefficient of variation.
334 Infinity Optional Mathematics Book - 9
ANSWER SHEET
Exercise 1.1
3. (a) – 3, 5 (b) 4, 3 (c) 1, – 7 (d) 7, 1
(e) (h) 2, 1
4. (a) 6, 1 (f) 1, 1 (g) 2, – 1
(c)
(d) (2, 2), (8, 8) (b) (4, 8) (6, 12)
(2, 2), (2, 8), (2, 10), (2, 12), (4, 8), (4, 12), (6, 12), (8, 8)
(2, 2), (2, 4), (6, 3), (8, 2), (8, 8)
Exercise 1.2
8. (a) A × B = {(0, 0), (0, 2), (0, 4), (1, 0), (1, 2), (1, 4), (2, 0), (2, 2), (2, 4)}
B × A = {(0, 0), (0, 1), (0, 2), (2, 0), (2, 1), (2, 2), (4, 0), (4, 1), (4, 2)}
(b) A × B = {(2, 3), (2, – 3), (3, 3), (3, – 3), (4, 3), (4, – 3)}
B × A = {(3, 2), (3, 3), (3, 4), (– 3, 2), (– 3, 3), (– 3, 4)}
(c) A × B = {(0, – 2), (0, 3), (1, – 2), (1, 3), (2, – 2), (2, 3)}
B × A = {(– 2, 0), (– 2, 1), (– 2, 2), (3, 0), (3, 1), (3, 2)}
Exercise 1.3
6. (a) {(2, 1), (4, 2)} (b) { } (c) {(2, 2), (3, 3)} (d) {(4, 2)}
7. (a) {(1, 2)} (b) {(1, 2), (1, 3), (2, 3)}
(c) { } (d) {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2)}
(e) {(3, 2) (3, 3) (f) {(2, 1), (2, 2), (3, 1), (3, 3)}
8. (a) R1 = {(2, 4), (3, 6), (4, 8)} (b) R2 = {(2, 4), (2, 6), (2, 8), (2, 10). (3, 6), (4, 4), (4, 8)}
9. {(1, 3), (1, 7), (2, 3), (2, 7), (3, 1), (3, 2), (3, 7)}
Exercise 1.4
6. (a) {(2, 2), (3, 3)}, {2, 3}, {2, 3} (b) {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)}, {1, 2, 3}, {2, 3, 4}
(c) {(1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (3, 2)}, {1, 2, 3}, {2, 3, 4} (d) {(2, 4), (3, 3)}, {2, 3}, {4, 3}
Exercise 1.5
Consult your teacher
Exercise 1.6
Consult your teacher
Exercise 1.7 (b) – 4, 3, 20 (c) 1 (d) 2
2. (a) – 17, – 5, 3
Infinity Optional Mathematics Book - 9 335
3. (a) { – 6, – 1, 4} (b) {– 2, 1, 6, 13}
4. (a) { 1, 2, 3} (b) {– 2, 2, – 3, 3, – 4, 4}
5. (a) (1 –3) (b) (3, 7) (c) (2, 2) (d) (3, 7)
6. (a) 4x + 17, 37 (b) 2x – 5, – 9
x
8. (a) – 2, 9, – 6, 10 (b) – 1, 1, – 5, –11
7. (a) h + 3, x + h + 3, h
1 –1 x 1 (b) 2, 1, 2x + 1
9. (a) 2, 2 , 2 – 2
Exercise 1.8 –3
(b) – 1 10. (a) 2 (b) 2, 4
9. (a) – 5
Exercise 1.9 x2 2x2
5 5
1. (a) 9x2 – 9x + 10 (b) 7x3 + 3x2 – 6x + 3 (c) 3x3 – 1 (d) x3 + + 19x + 1
+ x – 3 6 4
2. (a) – 2x2 + 10x – 18 (b) 2x2 + 6x – 6 (c) 6x3 – 6x2 + 10x – 13 (d) 5x4 – 6x3 + 15x + 3
3. (a) (i) 5y3 + 4y2 + 4y + 11 (ii) – y3 – 4y2 + 12y – 1 (b) (i) 14x2 – x + 8 (ii) 10x2 – 4x + 18
3 3
4. (a) – x2 + 15x – 11 (b) 2x4 – 5x3 + 4x2 + 6 + x
5. (a) x2 – 9 (b) x3 – 3x – 2 (c) x4 + x2 + 1 (d) 5x4 + 18x3 – 16x2 – 10x + 3
7. (a) 5x3 + x2 – 5x + 15 (b) 3x3 – 20x2 + 10x – 6
Exercise 1.10
Consult your teacher
Exercise 1.11
2. (a) 3, 5, 7, 9 (b) 1, 5, 9, 13 (c) 4, 7, 12, 19 (d) 2, 7, 14, 23
(e)
3. (a) 2, – 3 , 34, – 45 (f) 2, 54, 68, 7 (d) 1, 2, 3, 5
2 16 (d) n2+2n–1, 254
1 1 1 43
2, 6, 18, 54 (b) 1, 3, 9, 1 (c) – 1, 0, 2 ,
27
4. (a) 4n – 2, 58 (b) 13 – 5n, –62 (c) n2 + 3, 228
(e) 23nn – 2 , 4313 (f ) (– 1)n +1 2n , 30
+ 1 2n + 33
3
Exercise 1.12
5. (a) 45 (b) 141 (c) 34
(d) 122 (e) 642209 1(0f) 156 8
7
6. (a) ∑ (2n – 1) (b) ∑ (5n – 20) (c) ∑ (– 1)n . 2n (d) ∑ 3n
n=1 n=1 n=1
n=1
5 (– 1)n + 1 . 2n – 1 5 (5n – 1 + 1)
(e) ∑ (f) ∑
n= 1 4n + 1 n= 1
336 Infinity Optional Mathematics Book - 9
Exercise 2.1
5. (a) 31, 91, 217, 811, 2143 , 1 ; 0 (b) 21, 18, 118, 312, 510, 712; 0
729
(d) 54, 87, 1110, 1143, 1167, 2190; 1
(c) 21, 13, 14, 15,16, 1 ; 0 (f) 0, 1, 0, 12, 0, 13; 0
7
(e) 2, 143, 296, 1469, 2754, 109 ; 3
36
Exercise 2.2
Consult your teacher
Exercise 2.3
2. (a) 0 (b) 0 (c) 6 (d) ∞
(e) 16 (f) 10
3. (a) 8, 12, 14, 15, ...... ; 16 (b) 6, 9, 21 , 45 , ...... ; 12
2 4
Exercise 2.4
2. (a) 8, 7.97 (b) 5,5.002 3. (a) 0.04 (b) 2.1198
(d) – 8
4. (a) 13 (b) 23,0
5. (a) & (b) Consult your teacher
6. (a) 3 (b) 5 (c) 6
Exercise 2.5 (b) 2 (c) – 2 (d) 4 (e) 4
4. (a) 3
Exercise 3.1
5. (a) 2 3 4 (b) 1 2 3 3 4 5
3 4 5 2 4 6 (c) 5 6 7
(f) 2 3 4
1 – 1 – 3 – 1 1 – 1
(d) 4 2 0 (e) – 2 4 – 8 5 6 9
Exercise 3.2 (d) – 2, 6
5. (a) 1 0 1 9 16 (e) –1, 2, –3, – 2
3 2 (b) 9 16 25
6. (a) –1, 2 (b) 3, 73 (c) 5, – 7
7. (a) 4 (b) – 1, 6
8. (a) 1, 0 (b) 3, –3 (c) 1 , 3
2 2 2
9. (a) 6, 3, – 2, 0 (b) – 4, 2, 7, 7 (c) 3, 2 (d) – 1, 3
Infinity Optional Mathematics Book - 9 337
Exercise 3.3
5. (a) 3 (b) (4 8 – 1) (c) 0 5 –1
1
10 7 – 11 6
(e) – 2
4 1 12 4 0 – 2
–2 12 5
(d) – 4 (f) 12 – 5 16
7 – 16 13 6
6. (a) – 8 (b) (4 1 5) (c) 8 –3
2 1 –3
(d)
2 0 –2 –6 –1 –5 1 1
00 0
(e) – 2 2 (f) 9 12 0
34 8 – 15 7
8. (a) 19 – 18 (b) 17 – 2 26
– 11 –3 5 6 14
9. (a) 1, –23 (b) 2, – 2, – 6
10. (a) 2, – 5
(b) – 5, 2, 0, – 3
11. (a) (i) –6 3 0
(b) (i) 14 (ii) – 7 (iii) 0
–9
11 – 13
13 15
–9 – 7 (ii) 6 3 (iii) – 13 – 15
0 –5 97
13. (a) – 8, 130, 14 (b) – 1, 1, – 10
14. (a) – 9 18
15. (a) 20 5 (b) –1 1 3
2 (b) –7 2 4
2 4,1 4 2 3,1
3 53 4 52
Exercise 3.4
6. (a) – 1, – 6 (b) – 2, – 5, – 4, 1
Exercise 3.5
3. (a) ( 5 ) 2 10 12 (c) 1 1 (d) 4 – 25
(b) 3 15 18 0 1 7 –4
0 00
4. (a) 0 –2 (b) – 5 – 7 (c) 10 6 (d) – 5 –6
8 10 13 15 6 10 9 10
6. (a) 5 , 3 (b) 1, 2 (c) 9, – 16 (d) – 2
7 (c) 1, – 2 (b) 1 1
7. (a) ± 1 (b) 12 9.(a) (4 5) 18 – 4
8. (a) 29 21 (b) 14 21
21 18 44 54
338 Infinity Optional Mathematics Book - 9
10. (a) 2 (b) 3 9 (c) 2
0 – 4 – 14 –2
11. (a) 3, 8 (b) 3, 2
13. (a) 4 , 0, 5 (b) 2, 1 , 3 (c) 131, 104, – 2
3 4 2 2
Exercise 3.6
4. (a) 3, 2 (b) 1 , 0
2
Exercise 4.1
1. (a) d = (x2 – x1)2 + (y2 – y1)2 (b) Show to your teacher (c) x2 + y2
2. (i) MN = 5 units (ii) MN = 5 units
3. (a) 5 units (b) (i) 10 units, (ii) 26 units
5. (a) k = 1 (b) a = 1 6. (a) (i) x = 4 (ii) y + 3 = 0 (iii) y = 5 (iv) x = 3
7. (a) k = 3 (b) a = 5
8. (a) x2 + y2 = 25 (b) x2 + y2 + 4x – 6x + 9 = 0
9. (a) x2 – 4y + 4 = 0 (b) y2 – 6x + 9 = 0
(c) x + y = z (d) x2 + y2 + h2 + k2 – 2xh – 2yk = r2
10. (a) (i) x2 + y2 = 25, (ii) 3x2 + 3y2 – 50x + 75 = 0
(b) x2 + y2 + 20 = 0
11. ( a) 3x2 + 3y2 – 28x – 2y + 55 = 0 (b) 8x2 + 8y2 + 6x – 36y + 27 = 0
(c) 3x2 + 4y2 + 16x – 40y + 116 = 0 (d) 5x2 + 5y2 + 68x – 94y – 47 = 0
12. (a) 5x2 + 5y2 – 127 – 46y + 77 = 0 (b) x2 + y2 + 5x + 4 = 0
(c) x = 1
Exercise 4.2
3. (a) b = 15 (b) (–1, –4)
4. (a) (i) (0, 0) (ii) (1, 3) (iii) (0, 4) (b) (i) (–5, 4) (ii) (25, –22) (iii) (2, 12)
(c) (i) 3, 34 , internal (ii) (–36, –37), external
7
5. (a) (i) (2, 2) (ii) (0, 2 2 ) (b) x = –1, y = –7 (c) (–1, 1), (1, –1), (3, –4)
6. (a) 1:1 (b) 2:3 7. (a) (4, –2) (b) (4, 2)
8. (a) m1 : m2 = – 5:8, x = –1 (b) m1 : m2 = 1 : 1, – 23, 0 (c) m1 : m2 = 5 : 1, 0, – 13
9. (a) (i) (2, –3) and (0, – 3) (ii) (–1, 0) and (–4, 2) 3
(b) (–1, 2) and (–8, 5) (c) (0, 5), –1, 7 and 1, 13
2 2
Infinity Optional Mathematics Book - 9 339
10. (c) x = 6, y = 3 (d) (i) (–2, 9) (ii) (8, 8) 12. – 27, – 20
7
11. (a) 5 units, (11, 10) (b) 2 units, (4, 2)
14. (a) (3, 0), (1, 4), (–1, –2) (b) 53 units
Exercise 4.3
1. (c) y = mx + b (d) m = d – b 2. (a) y – 6 = 0 (b) x – 6 = 0
c – a
3. (a) (i) 3 (ii) ∞ (iii) –1
4. (a) (i) 3 , (ii) –1 (iv) – 1 (b) 0°, (ii) 30°, (iii) 135°, (iv) 120°
5. (a) 3 x – 7 = 0 3
6. (a) y – 4 = 0 1
(b) (i) 1, (ii) = 1 (c) – 3
(b) x – y + 4 = 0
(b) x + 7 = 0
7. (a) (i) 120° (ii) 45° (c) b = 4 (d) x = 5
8. (a) r = 4 (b) m = 9
9. (a) (i) x – y + 5 = 0 (ii) 3 x + y + 1 = 0 (iii) 3 x + y = 7
(b) (i) x – 3 y = 0 (ii) 3 x – y = 0 (iii) x + y = 0 (iv) x – 3y = 0
(c) y = 3
10. (a) (i) 0, 1, – 3 , (ii) 0, 1, – 1 (b) 0, –1, ∞ 1, 0, ∞
12. (a) x – y + 3 = 0 (b) 3 x + y = 4
13. (a) x – y + 6 = 0, x + y = 6 (b) x – y = 4, x + y + 4 = 0 (c) x – y = 0, x + y = 0
Exercise 4.4
2. (i) 2x + y = 10, (ii) 2x – y = 8, (iii) 7x – 3y + 21 = 0 3. 3x – 4y = 12
4. (a) (i) x + y = 5, (ii) x + y + 7 = 0 (iii) x – y = 5 (iv) x – y = 11 (b) (i) x + 2y = 7, (ii) 3x + y = 0
5. (a) 4x = 3y = 24 (b) 3x + 2y + 24 = 0
6. (a) 3x + y = 9 (b) 16x + 35y + 140 = 0
7. (a) x + y = 3, 2x + y = 4 (b) x + 2y = 4 (c) x + y = 2, 2x + y = 3
8. (a) 3x + 4y – 24 = 0 or 3x + 4y + 24 = 0 (b) 3x + 2y + 12 = 0, 3x – 2y = 12
Exercise 4.5 (ii) x + 3 y = 2 (iii) 3 x – y + 4 2 = 0 (iv) 7x – 7y + 5 2 = 0
2. (i) 3 x + y = 8 (b) 3 x – y + 4 3 = 0 4. (i) 3 x – y + 2 3 = 0 (ii) 3 x + y + 6 = 0
(v) x + 3 y = 2 (b) x + 3 y = 4 6. 3 x +y = 4
3. ( a) x – 3 y + 6 = 0
5. (a) x – 3 y + 2 = 0
340 Infinity Optional Mathematics Book - 9
Exercise 4.6
1. (a) y = mx + b, x + y = 1, xcosα + ysinα = p (b) –4, –3 (c) 2, 3
2. (c) a b
cosα = 53, sinα = 4 and p = 2
5
3. (i) y = –x + 2, m = –1, b = 2 (ii) y = – 2 x + 6, m = –2 2 , b = 6
(iii) y = 78x + 185, m = 78, b = 185 (iv) y = 23x – 5 m = 3 , b=– 5
2, 2 2
4. (i) x – y = 1, a = 3, b = –4 3 ((iiiv)) 1–x8x12+5 +2y51y=2 = 1, a =– 125, b= 12
(iii) 3 4 1, a = 18, b= 25
x + y = 1, a = 2, b = 2
2 23
5. (i) 23x – 21y = 2, p = 2 α = 330° 11
(ii) – 2x + 2y = 2 2 , p = 2 2 α = 135°
(iii) – 12x + 12y = 2, p = 2, α = 135° (iv) – 1 x + 3 y = 4, p = 4, α = 150°
2
2
5
6. a = 3, b = –5, m = – 3
7. (a) y= 3 x – 4, x + y = 1, 23x – y = 2 (b) – 23x + 12y = 2 p = 2, α = 210
4 –4 2
8. (a) – m 1x + 31 1y = b + 1 (b) b b2x + a b2 y = ab
m2 + m2 + m2 a2 + a2 + a2 + b2
9. (a) A = 27sq. units (b) A = 10sq. units (c) h = ± 24
Exercise 4.7
2. (a) y – k = g(x – h) (b) x – y = 0
3. (a) (i) x + y = 5 (ii) 2x – 3y + 8 = 0
(b) (i) 3 x – y – 2(1 + 3 ) = 0 (ii) x + y = 0 (iii) x + 3 y = 1 + 2 3 (iv) x – 3y + 9 = 0
(c) x – y + 1 = 0
4. (i) 2x + y = 1 (ii) 3x – 5y + 18 = 0
(iii) ax – by – ab = 0 (iv) ax – by – 2bx + (b2 + 2ab – a2) = 0
5. (a) (i) 3x + 5y = 0 (ii) 3x + 2y – 6 = 0 (iii) bx + ay = 3ab (iv) 3x + y = 7
(b) (i) w = 6
6. (a) x + y = 7, (7, 0) (ii) w = 2 (iii) w = – 1 (iv) w = –1
2
(b) x + y = 3, (0, 3)
7. (a) 7x + y = 11, x + 3y + 7 = 0, 3x – y + 1 = 0 (b) 3x + 4y – 14 = 0, 5 units
(c) x + 8y + 9 = 0, 65 units
8. (a) x – y = 0 (b) x + y + 1 = 0 9. (a) x + 3y = 5 (b) x + y = 0
10. (a) x + y = 6, 4 2 units (b) x + y – 9 = 0 11.(a) 2:1 (b) 1:1
12. (a) 4x – 5y = 0
Infinity Optional Mathematics Book - 9 341
Exercise 4.8
2. (a) (i) 33 units (ii) 35 units (iii) ( 3 – 1) units
5 13
(iv) 2 units (v) m2 + n2 units
5
(b) (i) m = 20 or 6 (ii) 3 or 13 (iii) z = – 5
21
3. (i) 61 units (ii) 3 units (iii) 1 units
41 2 34
(iv) 3 3 –5 3 units 4. 2( 2 – 1) units, 18 ( 2 – 1) units
2
7. (a) 7 sq. units (b) 5 46 sq. units (c) 13 sq. units
2 5
Exercise 4.9
3. (a) (i) 10 sq. units (ii) 26 sq. units (iii) 4 sq. units (iv) 20 sq. units
4. (a) p = 5 (b) k = 4 (c) a + 3b – 7 = 0
6. (a) m = 1 (b) n = 14 or 2
7. (a) (i) 11 sq. units (ii) 72 sq. units (iii) 28 sq. units (iv) 44 sq. units
(b) k = 192 (c) a = 4
2
9. (c) (i) D(2, 3), E(5, 3) (ii) 3 sq. units and 6 sq. units
2
(iii) 300% more (iv) 9 sq. units 10. (a) k = –3 (b) k = 85
2
11. (a) O(7, 2) (b) m = 65
4
12. (12, 2), (13, 6), (10, 13) taking CB and CD as coordinates axes 9 sq units, 9 sq. units, areas are the
2 2
same in both the case
Exercise 5.1
2. (a) 98146" (b) 68718" (c) 504540" (d) 428450"
(e) 1458" (f) 1256085" (g) 775836" (h) 155600"
(i) 820070" (j) 7589"
3. (a) 50.258° (b) 125.944° (c) 35.3° (d) 140.012°
(e) 0.261°
4. (a) 60.526g (b) 8.0706g (c) 181.56g (d) 20.005g
(e) 0.8557g
5. (a) 140° (b) 500g (c) 30g29'18" (d) 83g8'46"
(e) 114g38'89" (f) 166g68'52"
6. (a) 135° (b) 234° (g) 64'75"
(e) 185°13'12" (f) 76°30'32.4"
(c) 73°24'28.3" (d) 115°13'40.8"
(g) 40'57.8"
342 Infinity Optional Mathematics Book - 9
7. 42°34'52" 10. 36° 11. 80g 12. 45° and 90°
13. 50g and 30g
Exercise 5.2
2. (a) c (b) 23 c (c) 76 c (d) 4 c
6
(e) 3 c (f) 1230 c
2
3. (a) 108° (b) 40° (c) 50° (d) 120°
4. (a) 80g (b) 140g (c) 10g (d) 32g
5. (a) 5 : 36 (b) 2 : 7
6. (a) 31 c (b) 117°
180
7. (a) 305° (b) 55 c
36
9. (a) 1000 g (b) 130 c (c) 20g (d) 20g, 40g, 140g
9
c 7 c 2 c 30g, 80g (g) 100g
4 20 5
(e) , , (f)
7 c
20
(h)
10. (a) 60°, 40°, 80° (b) 60°, 51°, 69° (c) 50g, 40g, 110g (d) 81°, 9°
(e) 24°60°, 96° (f) 63°, 72°, 45°
c 2 c 4 C
2 3 5
11. (a) (i) 90°, (ii) 120°, (iii) 144°,
(ii) 45°, 50g 400 g
9
(b) (i) 72°, 80g, (ii) 40°,
(c) 8 (d) 54°, 81°, 108°, 135°, 162° (e) 12°
(f) 10 (g) 12, 6 (h) 18, 6 (i) 8
12. (a) (i) 270° (ii) 150° (iii) 240°
c 5 c (iii) 356 c
2 6
(b) (i) (ii)
255 c
2
(c) (i) (ii) 90° (iii) 180° (d) 105°
Exercise 5.3
1. (a) 4 c (b) 4.6° (c) 171.81°
3 (b) 14 cm (c) 12.83 m
(b) 14.32 cm (c) 35.6 cm
2. (a) 3.67 m
3. (a) 175 cm
Infinity Optional Mathematics Book - 9 343
4. (a) 44 m (b) 41.24 m 5. (a) 37.7 cm (b) 52.38 cm
6. (a) 12 m
7. (a) 11 cm, 25 cm (b) 0.25c (c) 15 cm
(c) 88 cm, 130 cm
(b) 19.56 cm, 47.56 cm
Exercise 5.4
(d) 66 cm, 91.2 cm
2. 32 3. 13 4. b , a2 – b2 5. –259
5 a a
6. x2 + y2 7. 1101 12. 13 14. 6653
x2 – y2
Exercise 5.5
1. (a) 83 (b) 24 (c) 13 (d) 81
(e) 147 (f) 53 (g) 8 (h) 94
(i) –21 (j) 3
4. (a) 169 (b) 23 (c) 45
Exercise 5.6
1. (a) sin2θ – cos2θ (b) sin3θ + cos3θ
(c) 35tan2θ – 31cosθ . tanθ + 6cos2θ (d) 1 – tan4θ
(e) 6sin4θ – sin2θ . cos2θ – 2cos4θ
2. (a) 6sinA – 11cosA (b) sin2θ + 6sinθ
(c) 18sin2A – 4tan2A – 2 (d) (sin2A – cos2A) (sinA – cosA)2
(e) – 4sinA . secA (f) 1 2–csoisnA2A
(h) 1
(g) sec22Ata–ntAan2A (b) cos2θ (cosθ – tanθ) (cosθ + tanθ)
3. (a) (sinA – cosA) (sinA + cosA)
(c) (sec2θ + cos2θ) (secθ + cosθ) (secθ – cosθ) (d) (sinθ – 2) (4sinθ + 3)
(e) (cosθ – sinθ) (cosθ + sinθ) (sin2θ – sinθ . cosθ + cos2θ) (sin2θ + sinθ . cosθ + cos2θ)
(f) (sin2θ – 2sinθ . cotθ + 2cos2θ) (sin2θ + 2sinθ . cotθ + 2cot2θ)
Exercise 5.7
4. (a) – 23 (b) –21 (c) 23 (d) – 23
(e) –21 (f) 21 (g) 23 (h) – 13
344 Infinity Optional Mathematics Book - 9
(i) – 13 (j) 1 (k) 2 (l) 2
(m) – 13 (d) – (2 + 2)
(n) 2 (o) – 1
5. (a) 1 3
(e) 14 (b) 1 (c) –31
8. (a) 0 2
(b) 1 (c) – sin2A (d) 1
(e) – 1 (f) – sec2θ (g) 1 (h) cotA
(i) cotA (j) cosA
10. (a) 2 (b) – 2 3 (c) 3 9– 2 (d) cosecθ
3
(f) 2.
(e) cot2θ
12. (a) 30° (b) 9° (c) 10° (d) 10°
(e) 10° (f) 452°
Exercise 5.8
1. (a) 3 –1 (b) 3 –1 (c) 1 – 3 (d) 3 +1
22 22 22 22
(e) 2 – 3 (f) 2 – 3 (g) 2 ( 3 + 1) (h) 2 ( 3 – 1)
(i) ( 3 – 2) (j) 1 – 3 (k) 3 – 2 (l) 1 – 3
22 22
3.(a) (i) 56 (ii) 33 (b) (i) 31 (ii) 17
65 65 25 2 25 2
(c) (i) 1 (ii) 1 (iii) 1 (iv) 7
7
(d) (i) 1 (ii) 3 (iii) 1 (iv) 0
22
14. (a) sinA. cosB cosC + cosA . sinB . cosC + cosA . cosB sinC – sinA sinB. sinC.
(b) cosA. cosB. cosC – sinA.sinB. cosC – sinA . cosB sinC – cosA. sinB. sinC.
(c) tanA + tanB + tanC – tanA tanB tanC
1 – tanA. tan B. – tanC . tanA – tanB. tanC
(d) sinA cosBcosC + cosA sinB cosC – cosAcosB.sinC + sinA. sinB. sinC
(e) sinA.cosB.cosC – cosA. sinB. cosC + cosA cosB sinC + sinA. sinB.sinC.
Exercise 6.1
4. (a) OH = 4 (b) OL = ––43
7
5. (a) (i) AB = 3 (ii) CD = –52 (iii) EF = 8 (iv) GH = –2
2 12 3
(b) PQ = (–1, –3), –1 (c) RS = (9, 17), 9 ,
–3 17
Infinity Optional Mathematics Book - 9 345
6. (i) OZ = (3, 5) 3 (ii) OW = (5, 0) 5 (iii) OV = (0, 6) 0
5 0 6
(iv) OA = (–4,–3) –4
–3
(v) ON = (5, –3) 5 (vi) OA = (–3, –3) –3
–8 –3
8. OA = (5, 5) OM = (–3, –5) ON = 4
6
OZ = (–4, 6) OR = (–4, 0) OQ = (4, 0)
9. (i) For AB x-comp = 5 y-comp = 4 AB = 5
y-comp = –7 4
(ii) For EF , x-comp = –5 y-comp = 8
y-comp = 0 EF = –5
(iii) For PQ , x-comp = – 3 y-comp = –4 –7
y-comp = –8
(iv) For GH x-comp = –10 AB = DC , DA = CB PQ = –3
8
(v) For ST x-comp = 4
GH = –10
(vi) For UV , x-comp = 0 0
10. AB , DC , DA , CB
ST = 4
Exercise 6.2 –4
UV = 0
–8
4. (i) like (ii) unlike (iii) unlike
5. (i) 14 units, 120° (ii) 4 2 units, 135° (iii) 7 units, ∞ (iv) 85 units, 130.60°
(v) 61 units, 230.19° (vi) 3 5 units, 153.43°
6. (i) 212° (ii) 90° (iii) 45° (iv) 135°
7. (i) 34 units, 323.13° (ii) 13 units, 56.30° (iii) 2 41 units, 308.65° (iv) 3 2 units, 45°
8. (b) –4 or 8 10. (i) 3 , – 4 (ii) 354 , – 3 (iii) 1 ,– 1 (iv) (1,1)
5 5 34 2 2
(v) 2 , – 3 4 , – 5
13 13 (vi) 41 41
11. (i) –7 i – 6 j , – 7 , – 6 (ii) i – 3 j , 110 , – 3
85 85 10
(iii) – 4 i + 3 j , – 4 , 3
5 5
12. (a) 0 (b) 2 or –2
13. (a) ± 2 10 , 64.2° or 245.37° (b) ± 2 30 , 32.57° or 147.42° (c) 8 units 14. (c) Yes
2 or – 2 16.(a)–7,10,4 2 units (b) (2,8) (c) (–1, –11), 2 13 units
15. 2 –2
346 Infinity Optional Mathematics Book - 9
Exercise 6.3
3. (a) 6 (b) –68 (c) a || b
–9
4. (a) 2 (b) 4 5. (a) 8 i + 5 j (b) 2 i + 3 j
10 –5
6. (a) 3 (b) –2
7. (a) 1 –15
–1 , 7 , 2 units, 7 2 units (b) 3, 7 , 109 units, 65 units.
1 7 10 4
(c) i – j , 7 i + 7 j , 2 units, 98 units
8. i) –12 (ii) –1145 (iii) –23 (iv) –6
11 4
(v) –20 (vi) –20
23 25
10. (a) –4 (b) –4
11. (a) 1 (b) –5 , 5 (c) 2
2 12 –12 16
12. (c) (i) AC (ii) AB (iii) BA (iv) AC (v) AC
13. (a) 1 (b) 3 (c) c – b + a 11
(d) 2 (2 c + a ), 2 ( a – 2 c )
14. (a) –2 ,2 10 units, 251.56°, – 1 , 3 (b) –1 , 17 units, 255.96°, – 1 , – 4
–6 10 10 –4 17 17
10 (ii) 2 26 units 1 (b) (i) –1 (ii) –1
15. (a) (i) 2 (iii) 4 3 3
17. (b) (i) m + n (ii) m + n + p = 2 n
Exercise 7.1
2. (i) A' (2, – 3) 3. (i) A' (1, – 9) 4. (i) A' (4, – 6)
(ii) B' (– 6, – 5)
(iii) C' (– 7, 6) (ii) B' (7, 8) (ii) B' (– 7, – 5)
(iv) D'(8, 4)
(v) P (0, 0) (iii) C' (2, – 6) (iii) C' (– 6, 9)
(vi) E' (5, 0)
(vii) F' (– 6, 0) (iv) D'(– 3, – 5) (iv) D'(3, 2)
(viii) G'(0, – 4)
(ix) H' (0, 7) (v) E' (– 5, 0) (v) E' (0, 6)
(vi) F' (3, 0) (vi) F'(8, 0)
(vii) G'(0, 5) (vii) G'(0, – 5)
(viii) H' (0, – 2) (viii) H' (9, 0)
G' (0, 5) and
(x) I' (– 9, – 6) H (0, – 2) are invariable points
P' (0, 0), E(5, 0)
F'(– 6, 0) are invariable points
Infinity Optional Mathematics Book - 9 347
5. (i) A' (– 7, 0) 6. (i) A' (1, 1) 7. (i) A' (3, 5)
(ii) B' (– 8, 6) (ii) B' (10, 2) (ii) B' (2, 10)
(iii) C' (6, – 5) (iii) C' (– 1, 11)
(iii) C' (3, – 5) (iv) D'(– 3, 6) (iv) D'(– 7, – 2)
(v) E' (10, 0) (v) E' (7, 6)
(iv) D'(2, 1) (vi) F'(– 1, 0) (vi) F'(– 6, 6)
(vii) G'(4, – 7) (vii) G'(0, 11)
8. (i) A' (2, 3), B' (6, – 7), C' (– 8, – 5) (viii) H' (4, 2) (viii) H' (0, 4)
(i) y-axis A' (5, – 2)
(ii) A' (– 2, – 3), B' (– 6, 7), C' (8, 5) (ii) x-axis 11. (i) B' (– 3, – 3)
(iii) y=x (ii) C' (1, – 6)
(iii) A' (– 9, 2), B' (1, 6), C' (– 1, – 8) (iv) y=–x (iii)
(v) x=2
(iv) A' (3, – 6), B' (– 7, – 10), C' (– 5, 4) (vi) y=–3
(v) A' (3, – 2), B' (– 7, – 6), C' (– 5, 8) 10.
(vi) A' ( - 3, 2), B' (7, 6), C' (5, – 8)
9. (i) A (– 7, – 6)
(ii) A (– 3, 5)
(iii) A (– 6, – 7)
(iv) A' (2, – 6)
(v) A' (1, – 3)
(vi) A' (–1, 2)
12. (i) A' (– 6, – 6) 13. (i) A' (– 3, – 2)
(ii) B' (– 3, 4) (ii) B' (1, – 1)
(iii) C' (– 9, 1) (iii) C' (1, 4)
14. A' (– 2, – 3) 15. M' (2, – 1)
B' (– 4, 1) N' (1, – 5)
C' (– 5, – 1) K' (8, – 7)
A" (2, – 3) M" (– 1, 2)
B" (4, 1) N" (– 5, 1)
C" (5, – 1) K" (– 7, 6)
Exercise 7.2
2. (i) A' (– 4, 3) 3. (i) M' (4, 6) 4. (i) A' (6, 7) 5. (i) M' 5, 0)
(ii) N' (6, 7)
(ii) B' (– 4, – 6) (ii) N' (5, – 7) (ii) B' (– 8, – 2) (iii) R' (– 3, – 5)
(iv) S'(7, – 1)
(iii) C' (8, – 6) (iii) O' (– 6, – 2) (iii) C' (– 9, 5) A' (– 3, – 6)
6. (i) B' (5, 7)
(iv) D'(5, 1) (iv) K'(– 5, 1) (iv) D'(1, – 3) (ii) C' (2, – 1)
(iii) D'(– 8, 7)
(v) E' (– 8, 0) (v) S' (6, 0) (v) E' (7, 0) (iv)
(vi) F' (0, – 6) (vi) T' (0, – 7) (vi) F'(0, 5)
(vii) G'(7, 0) (vii) U'(– 8, 0) (vii) G'(– 8, 0)
(viii) H' (0, 5) (viii) V' (0, 9) (viii) H' (0, – 3)
7. A' (– 5, 2), B' (– 3, – 1), C' (– 2, 4) 8. A' (0, 2), B' (– 2, – 1), C' (1, – 2), D' (3, 1)
9. A' (– 3, 2), B' (– 1, 5), C' (3, 4), A" (3, – 2), B" (1, – 5), C" (– 3, – 4)
10. A' (2, – 3), B' (– 1, – 6), C' (– 6, – 1), A" (– 3, – 2), B" (– 6, 1), C" (– 1, 6)
348 Infinity Optional Mathematics Book - 9
Exercise 7.3
2. (i) A' (8, 4) 3. (i) A' (6, 5), B' (– 3, – 3), C' (9, – 4), D' (7, – 6)
(ii) B' (– 4, 5) (ii) A' (3, 7), B' (– 6, – 1), C' (6, – 2), D' (4, – 4)
(iii) C' (– 3, – 2) (iii) A' (11, 0), B' (2, – 8), C' (14, – 9), D' (12, – 11)
(iv) D'(11, 1) (iv) A' (1, – 3), B' (– 8, – 11), C' (4, – 12), D' (2, – 14)
(v) E' (3, 8) 4. T = – 3 , B' (– 5, – 4), C' (4, – 3)
–5
(vi) F' (11, 2) 5. T = 2 , A' (11, 9), B' (1, 9)
7
(vii) G' (3, –1)
(viii) H' (–3, 2)
6. A'(5, 8), B' (2, 6), C' (9, 4)
7. A'(–2, –1), B' (4, 0), C' (3, –6), D' (–4, –6)
8. P' (– 3, – 11), Q' (0, – 4), R' (1, – 7)
9. (i) A' (8, 7), B' (11, 5), C' (14, 10) (ii) A' (– 4, 8), B' (– 1, 6), C' (2, 11)
10. P' (0, – 1), Q' (– 1, 1), R' (– 6, – 9), S' (4, – 5)
Exercise 7.4
2. (A' (9, 6), B' (– 21, 18), C' (– 6, – 15), D' (27, – 3) 6. A' (2, – 8), B' (– 2, – 16), C' (8, – 14)
3. (i) A' (– 12, 0), B' (0, 10), C' (6, 0), D' (0, – 14) 7. A' (8, – 1), B' (8, 8), C' (17, 5)
8. [(0, 0), 2]
(ii) A' (18, 0), B' (0, – 15), C' (– 9, 0), D' (0, 21) 9. [(2, 1), 2]
4. (i) A' (– 7, 0), B' (13, – 10), C' (11, 10), D' (– 1, 12) 10. [(6, 2), 2]
11. [(0, 0), – 2]
(ii) A' (– 9, – 4), B' (21, – 19), C' (18, 11), D' (0, 14) 12. [(2, – 1), 2]
(iii) A' (9, 6), B' (– 11, 16), C' (– 9, – 4), D' (3, – 6) 13. M' (1, 5), N' (9, 7) , S' (7, 13)
14. A' (2, – 5), B' (2, – 8) , C' (– 4, – 5),
5. A' (4, 8), B' (–6, 10), C' (–4, –6)
D'(–4, –5)
Exercise 8.1
3. (a) (i) 52kg (ii) 16 cm (iii) 130
(b) (i) 95 kg (ii) 41.25 (iii) 15.5 cm
(c) (i) 61 (ii) 23 (iii) 78
4. (a) 7 (b) 20 5. (a) 25
6. (a) 8 (b) Rs.1700 (c) Rs. 70 (b) 40
8. (a) 300 (b) 25 (c) 60 kg 7. (a) 11kg (b) 25°C
9. 19, 20, 30
Exercise 8.2
3. (a) D7 = 31 (b) (i) 59, 75 (ii) 75, 82 4. (a) (i) 14, 26 (ii) 22, 29
5. (a) 34.40, 28.84
(b) 4.70, 9.70 6. (a) 63 (b) 10
7. (a) 55, 55
(b) 15 kg, 30 kg 8. (a) 25 yrs, 30 yrs (b) 20cm, 30 cm
9. (a) Rs. 625, Rs. 900
(b) Rs.15, Rs. 20
Infinity Optional Mathematics Book - 9 349
Exercise 8.3
3. (a) 20, 5 (b) 5
4. (a) 12.5, 0.29 (b) 7.5, 0.6 (c) Q1 = 295
5. (a) 10, 38 (b) 9, 3
6. (a) 30, 0.26 (b) 42, 28 (c) 3y, 0.5
7. (i) Rs.5.5, 0.47 (ii) 4 kg, 0.285 (iii) 5.75 inches, 0.51 (iv) Rs. 8.87, 0.069
8. (i) 5, 0.1 (ii) 10, 0.33, (iii) 5, 0.16, (iv) 12.5, 0.2 9. 1.5, 0.103
Exercise 8.4
3. (a) 3.4, 0.28 (b) 4.8, 0.48 (iv) 13.63, 0.45
4. (a) 85.71, 0.34 (b) 18, 0.12 (iv) 4.5, 0.163
5. (a) 5.6, 0.18 (b) 8.07, 0.201 (iv) 4.5, 0.15
6. (i) 2.85, 0.47 (ii) 17.14, 0.42 (iii) 7.2, 0.167 (iv) 41, 0.13
7. (i) 13, 0.104 (ii) 17.14, 0.42 (iii) 44.16, 0.28
8. (i) 2.13, 0.22 (ii) 11.83, 0.35 (iii) 11.63, 0.21
9. (i) 2.24, 0.18 (ii) 4.75, 0.158 (iii) 2.65, 0.1325
10. (i) 3.2, 3.21, 0.16 (ii) 3.45, 3.36, 0.42
Exercise 8.5 (b) 6.21, 0.32 (c) 2.09
(b) 41.89 %
3. (a) 4, 0.22 (ii) 18.65, 0.27, 27% (iii) 10.95, 0.29, 29%
4. (a) 12.4, 0.31 (v) 6.5, 0.18, 18% (vi) 4.87, 0.51, 51 %
5. (i) 7.07, 0.471, 47.1 % (ii) 6.89, 0.336, 33.6% (iii) 7.70, 0.1482, 14.82%
(iv) 7.05, 0.24, 24 % (v) 6.54, 0.225, 22.5% (vi) 19.7, 0.35, 34.8%
6. (i) 2.92, 0.043, 4.3% (ii) 151, 21.18%
(iv) 12.48, 0.32, 32 %
7. (i) 51.69, 31.1%
350 Infinity Optional Mathematics Book - 9
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Infinity Optional mathematics book 9 Final for CTP 2077
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