(i). 103
107 = (100 + 3) (100 + 7)
= (100)
2 + (3 + 7) 100 + (3) (7)
[By using the identity,
, where
x = 100, a = 3 and b = 7]
= 10000 + 1000 + 21
= 11021
(ii). 95
96 = (100 - 5) (100 - 4)
= (100)
2 + (- 5 - 4) 100 + (- 5) (- 4)
[By using the identity,
, where
x = 100, a = - 5 and b = - 4]
= 10000 - 900 + 20
= 9120
(iii). 104
96 = (100 + 4) (100 - 4)
= (100)
2 - (4)
2 
= 10000 - 16
= 9984
Concept Insight: The key is to use the algebraic identity (x+a) (x+b) = x2+(a+b)x+ab or (x+a) (x-a) = x2 - a2 for such questions. Write each of the numeral as 100 ± k , or any other suitable number whose square can be easily computed.
|
| Q: 3 |
Factorise the following using appropriate identities:
|
| |
| Answer |
|
Concept Insight: Use the appropriate square identity. If the polynomial has only two terms, reduce each term to the perfect square and use the algebraic identity  . When the polynomial has three terms and the term having
unit power of each variable has negative sign use the square identity  else use  .
|
| Q: 4 |
Expand each of the following, using suitable identities:
|
| |
| Answer |
|
Concept Insight: Use the algebraic identity 
. Do consider the sign of terms while multiplying and squaring.
|
| Q: 5 |
 |
| |
| Answer |
|
Concept Insight: Use the algebraic identity
in the reverse order. Write each term as per the terms of the standard identity. Do consider the sign of terms involved.
|
| Q: 6 |
 |
| |
| Answer |
|
Concept Insight: Since the expressions involves cube so cubic identity will be used. If the terms of the given polynomial are separated by positive sign use the identity  or if negative signs are used then use  . Carefully apply the mathematical operations.
| Q: 7 |
Evaluate the following using suitable identities:
(i). (99)3 (ii). (102)3 (iii). (998)3
|
| |
| Answer |
We know that
(i) (99)3 = (100 - 1)3
= (100)3 - (1)3 - 3(100) (1) (100 - 1)
= 1000000 - 1 - 300(99)
= 1000000 - 1 - 29700
= 970299
(ii) (102)3 = (100 + 2)3
= (100)3 + (2)3 + 3(100) (2) (100 + 2)
= 1000000 + 8 + 600 (102)
= 1000000 + 8 + 61200
= 1061208
(iii) (998)3 = (1000 - 2)3
= (1000)3 - (2)3 - 3(1000) (2) (1000 - 2)
= 1000000000 - 8 - 6000(998)
= 1000000000 - 8 - 5988000
= 1000000000 - 5988008
= 994011992
Concept Insight: Use the cubic identity  and  . Write the numerical term as something added or
subtracted from 10,100, 1000 or higher powers of 10 as it's easy to compute higher powers of 10. Carefully apply the mathematical operations.
|
| Q: 8 |
 |
| |
| Answer |
Concept Insight: Since all the polynomial given here has degree 3 so cubic identities would be used here. Now if all the terms of the given polynomial are positive then use identity  while if any two terms has negative sign reduce each of
the term of the polynomial as per the standard cubic identity  .
| Q: 9 |
 |
| |
| Answer |
|
Concept Insight: When the two terms of the polynomial are separated by positive sign use the identity and when by negative sign use .
Carefully take the common term out.
|
| Q: 10 |
 |
| |
| Answer |
|
Concept Insight: Reduce the terms of the polynomial to perfect cube and then if the two terms of the polynomial are separated by positive sign use the identity  and when by negative sign use  .
|
| Q: 11 |
| Factorise: 27x3 + y3 + z3 -9xyz |
| |
| Answer |
We Know that
Concept Insight: Reduce each terms of the polynomial as per the left hand side of the standard identity,  .
|
| Q: 12 |
 |
| |
| Answer |
We know that
Concept Insight: Since the left hand side of the identity resembles the left hand side of identity, , so this identity
will be applicable here. Now the right hand side of the above identity can be written into many forms we need to look at what is required to proved, Accordingly apply mathematical simplifications and square identities to get the desired result.
|
| Q: 13 |
 |
| |
| Answer |
|
Concept Insight: Use the result that  for x + y + z = 0.
|
| Q: 14 |
Without actually calculating the cubes, find the value of each of the following:
|
| |
| Answer |
|
Concept Insight: Use the result since x + y + z = 0. Also consider the
sign of the term. Carefully do the computation.
|
| Q: 15 |
Give possible expressions for the length and breadth of each of the following rectangle, in which their areas are given:
|
| |
| Answer |
We know that, Area = length breadth
Concept Insight: For such questions factorise the expression, given for the area of rectangle by splitting the middle term. One of its factors will be its length and the other will be its breadth.
|
| Q: 16 |
What are the possible expressions for the dimensions of the cuboids whose volumes are given below?
|
| |
| Answer |
We know that, Volume of cuboid = length breadth height
(i). 
So, the possible solutions is
Length = 3, breadth = x, height = x - 4
Concept Insight: For such questions factorise the expression, given for the volume of the cuboid by taking the common term out if it has two terms and by splitting the middle term if the polynomial has three terms. Three factors obtained will be its length breadth and height.
|
| |
|
|
|
|
|
|
|
|
|
|
|
|
|
