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Activity 13
OBJECTIVE MATERIAL REQUIRED
To verify that the graph of a given Cardboard, thick white paper,
inequality, say 5x + 4y – 40 < 0, of the sketch pen, ruler, adhesive.
form ax + by + c < 0, a, b > 0, c < 0
represents only one of the two half
planes. 3. Draw the graph of the linear equation corresponding to the given linear

METHOD OF CONSTRUCTION
1. Take a cardboard of a convenient size and paste a white paper on it.
2. Draw two perpendicular lines X′OX and Y′OY to represent x-axis and
y-axis, respectively. 6

3. Draw the graph of the linear equation corresponding to the given linear
inequality.
4. Mark the two half planes as I and II as shown in the Fig. 13.
DEMONSTRATION
1. Mark some points O(0, 0), A(1, 1), B(3, 2), C(4, 3), D(–1, –1) in half plane I
and points E(4, 7), F(8, 4), G(9, 5), H(7, 5) in half plane II.
2. (i) Put the coordinates of O (0,0) in the left hand side of the inequality.

Value of LHS = 5 (0) + 4 (0) – 40 = – 40 < 0

So, the coordinates of O which lies in half plane I, satisfy the inequality.

(ii) Put the coordinates of the point E (4, 7) in the left hand side of
the inequality.

Value of LHS = 5(4) + 4(7) – 40 = 8 </ 0 and hence the coordinates of the
point E which lie in the half plane II does not satisfy the given inequality.

(iii) Put the coordinates of the point F(8, 4) in the left hand side of the
inequality. Value of LHS = 5(8) + 4(4) – 40 = 16 </ 0

So, the coordinates of the point F which lies in the half plane II do not
satisfy the inequality.

(iv) Put the coordinates of the point C(4, 3) in the left hand side of the
inequality.

Value of LHS = 5(4) + 4(3) – 40 = – 8 < 0

So, the coordinates of C which lies in the half plane I, satisfy the inequality.

(v) Put the coordinates of the point D(–1, –1) in the left hand side of the
inequality.

Value of LHS = 5(–1) + 4 (–) – 40 = – 49 < 0

So, the coordinates of D which lies in the half plane I, satisfy the
inequality.

46 Laboratory Manual
(iv) Similarly points A (1, 1), lies in a half plane I satisfy the inequality. The
points G (9, 5) and H (7, 5) lies in half plane II do not satisfy the inequality.

Thus, all points O, A, B, C, satisfying the linear inequality 5x + 4y – 40 < lie

only in the half plane I and all the points E, F, G, H which do not satisfy the linear
inequality lie in the half plane II.

Thus, the graph of the given inequality represents only one of the two
corresponding half planes.

OBSERVATION
Coordinates of the point A __________ the given inequality (satisfy/does not
satisfy).
Coordinates of G __________ the given inequality.
Coordinates of H __________ the given inequality.
Coordinates of E are __________ the given inequality.
Coordinates of F __________ the given inequality and is in the half plane____.
The graph of the given inequality is only half plane _________.

APPLICATION
This activity may be used to identify the half NOTE
plane which provides the solutions of a
The activity can also be
given inequality. performed for the inequality of
the type ax + by + c > 0.

Mathematics 47
 7
Activity 15
OBJECTIVE MATERIAL REQUIRED
To construct a Pascal's Triangle and to Drawing board, white paper,
write binomial expansion for a given matchsticks, adhesive.
positive integral exponent.

METHOD OF CONSTRUCTION
1. Take a drawing board and paste a white paper on it.
2. Take some matchsticks and arrange them as shown in Fig.15.
 3. Write the numbers as follows:
1 (first row)
1 1 (second row)
1 2 1 (third row)
1 3 3 1 (fourth row), 1 4 6 4 1 (fifth row) and so on (see Fig. 15).
4. To write binomial expansion of (a + b)n, use the numbers given in the
(n + 1)th row.

DEMONSTRATION
1. The above figure looks like a triangle and is referred to as Pascal’s Triangle.
2. Numbers in the second row give the coefficients of the terms of the binomial
expansion of (a + b)1. Numbers in the third row give the coefficients of the
terms of the binomial expansion of (a + b)2, numbers in the fourth row give
coefficients of the terms of binomial expansion of (a + b)3. Numbers in the
fifth row give coefficients of the terms of binomial expansion of
(a + b)4 and so on.

OBSERVATION
1. Numbers in the fifth row are ___________, which are coefficients of the
binomial expansion of __________.
2. Numbers in the seventh row are _____________, which are coefficients
of the binomial expansion of _______.
3. (a + b)3 = ___ a3 + ___a2b + ___ab2 + ___b3
4. (a + b)5 = ___ +___+ ___+ ___ + ___+ ___.
5. (a + b)6 =___a6 +___a5b + ___a4b2 + ___a3b3 + ___a2b4 + ___ab5 + ___b6.
6. (a + b)8 = ___ +___ +___+ ___ + ___+ ___ + ___ + ___+ ___.
7. (a + b)10 =___ + ___ + ___+ ___ + ___+ ___ + ___+___+ ___+ ___+ __.

APPLICATION
The activity can be used to write binomial expansion for (a + b)n, where n is a
positive integer.

Mathematics 51