Materials

Pre-requisiteObjective
B
is

where
Activity 2
2Pg,
required :
To
knowledge
n(A)verify
: =pthat
Paper,
: and for
Knowledge
different n(B)= two
sets
q.
coloured of and A
sets
B,
pencils. and n(A
2 relations x
B)
=
is pg
required. and
the
total
number of

relations

from
A
Procedure :
1. Let A, = (a,} and B, = (b,, b} be related to each other as below
A, B

>"b,
a,".

2. Let A, = {a,, a,} and B ={b,, b,, b} be related to each other as below:
A, B.

a, ".
>"b,
a,
y"b,

3. Let Ag = a, a, a} and B, = {b4, b, ba) be related to each other as below:

A B.
a, b

a,"

4. Let A = (a,, a,, aq, a} and B, = {b, bo, by ba, b} be related to each other as below :
A4 B,
"b,

"b,

Observations

1. Cartesian product A, x B, = {(a,, b,), (a, b)}.

n(A, xB,) = 2=1 x 2 = n(A,) x n(B,)
.. total number of relations from A, to B, is 2.
(: total number of relations from A, to B, is the total number of subsets of A, x B,).
2. A, x B, ={(a,, b,). (a,, b,). (a,, by). (a,, b,). (ag. b,). (a,. b,)}
n(A, xB,) = 6 = 2 x 3 = n(A,) x n(B,)
.:. total number of relations fromn A, to B, is 20.
3. Ag x B, = {(a, b), (a,, b), (a,, b), (a, b,), (a, b,), (a,, b). (ag, b). (a,. by). (ag, by))
n(A, x B) = 9 = 3 x 3 = n(A) * n(B)
:. total number of relations from A, to B, is 29.

3
 4. Ay xB, ={(a, b,). (a,, b,). (a,, ba). (a,, b4). (a,, bg). (a,, b,), (a, b,), (a, b). (a,, ba), (a,
(a, b,). (a, b,). (ag b,). (ag b), (ag, ba), (a, b,). (a,, b,), (a, b), (a, b4), (a4 b,)
n(A4 x B) =20 = 4 x 5 = n(A) × n(B,)
.:. total number of relations from A, to B, is 220,
Conclusion
For any two non empty sets A and B, n(A x B) =n(A) x n(B) and total number of relations from
to Bis 2n(A x B)

Application: Useful to find total number of relations from set A to another set B.
 Activity 4
Objective:To verify distributive law for three given non-empty sets A, B and C that is
AU(Bnc) =(AUB) n(AuC)
Pre-requisite knowledge : Knowledge of operations on sets and basic knowledge of Venn diagrams.
Materials required : Hardboard with board pins, one white and one pink chart paper, colour pencils, a pair
of scissors and gluestick.
Procedure :
1. Place a white chart paper firmly on a hardboard with the help of board pins.
2. Cut 6rectangular strips of 4 cm by 6 cm from a pink chart paper and paste them on a hardboard in
two consecutive rows such that 3 of them in one row and remaining 3 of them in the next row just
belowthe above three strips, as shown in fig. 4.1.
6cm

4cm

Fig, 4.1
3. Now drawrectangles on each of the strips and write the symbol U in the top left corner on each of
them, as shown in fig. 4.2.
7
 4. Draw three circles and mark them as A, B and C inside each of the rectangles.
5. Shade the different parts of the circles in each of the rectangles, as shown in fig. 4.2.
A BoC Au(Bnc)
U U

B A B
A A

(i) (ii)
AUB AuC (AUB)^(AuC)
U U
A B A B A

(iv) (v) (vi)

Fig. 4.2
Observations
1. In each of the pink strips, Udenotes the universal set represented by the rectangle in each fig
2. Inside the rectangles, circles A, Band Crepresent the subsets of the universal set U.
3. Shaded parts of fig 4.2(), 4.2(i), 4.2(iv) and 4.2(v) represent set A, set Bn C, set A uB
set A uC respectively.
4. Shaded parts of fig 4.2 (ii) and fig 4.2 (vi) represent A u(B nC) and (AuB)o (AuC) respect
Since the two shaded parts are same, so A u(Bo C) =(AuB) (A C)
Conclusion
The activity verifies the following:
For three non empty sets A, B and C: Au(B o C) = (AUB) o (AUC)
 Activity 10
X
Objective: To plot the graph of sinx, sin2x, 2sinx and sin using samne coordinate axes.

Pre-requisite knowiedge : Knowledge of plotting a graph and trigonometric ratios and their properties.
Materials required : Hardboard, board pins, gluestick, ruler, colored pencils and white chart paper.
Procedure:
1. Place achart paper firmly on a hardboard with the help of board pins.
2. On the chart paper, paste a graph paper of convenient size.
3. On the graph paper, draw two perpendicular straight lines XOX and YOY, intersecting each other
at the point O. These perpendicular straight lines are known as coordinate axes.
4. The line XOX is called the x-axis and the line YOY is called the y-axis, while the point O is calle
the origin. X
5. Make a Table 10.1 of ordered pairs for sinx sin2x 2sinx and sin 2
for different values of x, as show
below:
Table 10.1

5T Tt 7T 2T 3T 57 11
Trigonometric 0°
ratios 12 6 4 3 12 2 12 3 4 6 12
0.26 0.50 0.71 0.86 0.97 1 0.97 0.86 0.71 0.50 0.26
sin x
sin2x 0.50 0.86 1 0.86 0.50 0 -0.5 -0.86 -1.0 -0.86 -0.50 0
-
1.42 1.72 1.94 2 1.94 1.72 1.42 1 0.52 0
2sinx 0.52

0.26 0.38 0.50 0.61 0.71 0.79 0.86 0.92 0.97 0.99 11
sin:
n 0 0.13
2

22
 6. Find the ordered pairs (x, sinx), (x, sin2x), x, sin 2 and (x, 2 sinx) to plot the points on a cartesian
plane.
7 NoW, plot the points on the same coordinate axes.
8. Connect the points of plotted ordered pairs by curves by different colour, as shown in the fig 10.1.

2.00 +
functions1.75+
1.50
1.25 >2sinx
Trigonometric
1.00
0.75
’sin
0.50
> sinx
0.25

of X
7n 47 57 11n
Values -0.25 + 3 122 N23 4 6 3 3

-0.50
-0.75
>sin 2x
-1.00+
Angles (in radians)
Fig. 10.1

Observations

1. From the fig 10.1, we observe that graphs of sinx and 2sinx are of same shape but the maximum
height of the graph of 2sinx is double the maximum height of the graph sin x.
2. The maximum height of the graph of sin2x is 1 unit at x = 4
3. The maximum height of the graph of 2sinx is 2 units at x = 2
X
4. The maximum height of the graph of sin 2 is 1 unit. It is at x = T
5. At x =0 and n, sin x =0
At x = 0 and , sin 2x = 0
X
At x = 0 and 2n, sin = 0
2
Conclusion
We have explored the different sine curves using the same coordinate axes.
Application: Useful in tracing of trigonometric curves and in the field of physics.
 Activity 11
integral powers.
Objective: To interpret geometrically the meaning of i=-1 and its
number iota, ) and its proper
Pre-requisite knowledge : Knowledge of co-ordinate geometry, complex
compasses, adhesive, nails, thread,
Materials required : Cardboard, chart paper, sketch pen, ruler,
Y
Procedure :
1. Take a cardboard of a suitable size and paste a chart paper
on it.
iP,
2. Draw two mutually perpendicular lines X'0X and YOY
representing the coordinate axes. 1=
-1=j P
3. Take athread of one unit length and fix one end of the thread X<
P, P,
at O and the other end at P along OX, as shown in the figure.
4. Loose the other end of the thread at P and rotate the thread
f=-i P.
through the angles of 90°, 180°, 270° and 360° and mark the
free end of the thread in different cases as P, P, P: and P4
respectively, as shown in the figure.
Fig. 11.1
Observations
In the argand plane,
1. P denotes the complex numbers 1, i4, 8, j12, ¡16 ¡4n
2. P, denotes the complex numbers i, i5, ¡9, j13, ¡17 j4n+1
3. P, denotes the complex numbers -1, i2, i, ¡10, i14, ¡18 j4n+2
4. P, denotes the complex numbers i i, i7, j11, ¡15, j4n+3
5. OP = 1, OP, = i, OP, = -1, OP, = , OP, =1
 6. Each time, rotation of OP by 90° clockwise is equivalent to multiplying by i therefore, i is referred to
as the multiplying factor for a rotation of 90°.
7. On rotation of OP through 90°, we get OP, =1 x /= i.
8. On rotation of OP through 180° (2 right angles), we get OP, = 1 x ixj= i? =-1
9. On rotation of OP through 270° (3 right angles), we get OP, = 1x i xix j= j3 = j
10. On rotation of OP through 360° (4 right angles), we get OP, = 1 * ixj xjxji= j# z 1
11. On rotation of OP through n-right angles, we get OP, =1 x i xixi x i x Xn times = n

Conclusion

We can interpret the meaning of i=-1 and its integral powers geometrically as explained above.
Application: Useful while solving problems of geometry in complex analysis.
 Activity 12
Objective : To obtain a quadratic function with the
Pre-requisite knowledge : Knowledge of quadratic help of linear functions
Materials required : Plywood sheet, pieces of wiresequations and conceptsgraphicaly.
of graph of linear
Procedure : and graph paper. equation,
1. Consider the quadratic function f(x) = (x2 -
2. Find the 5x + 6).
ordered pairs (x. fx)) for different values of x. The
tabular form as shown below: values may be presented in the
X -1 2 -2 3 -3 4 -4 5
f(x) 2 12 0 20 5
30 2 42
3. On aplywood sheet, paste a 56
12
graph paper of convenient size. And plot of the following points on
graph paper as shown in figure 12.1.
55|
sfo)
=
(25x+6)

50

40

35

30

25

20

15

10

y=x2
y=x3
3D 45

Fig. 12.1
4. In the figure, the curve
intersects the
y-axis at adistance of y=-2 from O x-axis
in the
at point x = 2 from O in the
positive direction and me
negative direction. These
5. The points A and
Brepresents the straight line given by y (= f(x)) =points
(x
are B and Arespectivey
y-axis at (2, 0) and (0, -2) - 2) intersecting the x a
6. Now, the curve respectively.
intersects the x-axis at point x = 3 from O in the
at adistance of y = -3 from O
in the negative direction as positive direction and mneets y-a
are D and C
respectively. shown in the figure 12.1. These po
26
 7. The points C and Drepresents the straight line given by y= (x-3) intersecting the x-axis and y-axis
at and (3, 0) (0, 3) respectively.
8. Points B and D represent points of intersection for f(x) =x- 5x + 6.
Observations

1. The given line by the linear function y = (x- 2) intersects the x-axis at the point 2 whose coordinates
are (2, 0) and (0, 2).
2. The given line by the linear function y =(x - 3) intersects the x-axis at the point 3 whose
coordinates are (3, 0) and (0, 3).
3. The curve passing through B and D is given by the function y = f(x) = x² - 5x + 6, which is
parabolic function.
Conclusion
From the above activity, we can conclude that the quadratic function f(x) =x- 5x + 6, has two
zeroes which represent the linear functions. And linear functions represent the graph of straight lines.
Application: This activity is useful in understanding the zeroes and the shape of graph of a quadratic
polynomial.