Algebraic Identities

Page no. 4.11 Ex. 4.2 

Q1.

In the given problem, we have to find expended form

(i) Given

We shall use the identity

By applying in identity we get 

Hence the expended form of

is

(ii) Given

We shall use the identity

Here

By applying in identity we get 

Hence the expended form of is

(iii) Given

We shall use the identity

Here

By applying in identity we get 

Hence the expended form of is.

(iv) Given

We shall use the identity

Here

By applying in identity we get 

Hence the expended form of is

(v) Given

We shall use the identity

Here

By applying in identity we get 

Hence the expended form of is.

(vi) Given

We shall use the identity

Here

By applying in identity we get 

Hence the expended form of is.

(vii) Given

We shall use the identity

Here

By applying in identity we get 

Hence the expended form ofis .

(viii) Given

We shall use the identity

Here

By applying in identity we get 

Hence the expended form of is

(ix) Given

We shall use the identity

Here

By applying in identity we get 

Hence the expended form of is

(x) Given

We shall use the identity

Here

By applying in identity we get 

Hence the expended form of is

(xi) Given

We shall use the identity

Here

By applying in identity we get 

Hence the expended form of is

(xii) Given

We shall use the identity

Here

By applying in identity we get 

Hence the expended form of is.

Q2.

Page no. 4.12 Ex. 4.2 

Q3.

Page no. 4.19 Ex. 4.3

Q1.

In the given problem, we have to find cube of the binomial expressions

(i) Given

We shall use the identity

Here

By applying the identity we get 

Hence cube of the binomial expression is

(ii) Given

We shall use the identity

Here

By applying the identity we get 

Hence cube of the binomial expression of is

(iii) Given

We shall use the identity

Here,

By applying identity we get 

Hence cube of the binomial expression of is

(iv) Given

We shall use the identity

Here

By applying in identity we get 

Hence cube of the binomial expression of is .

Q2.

In the given problem, we have to simplify equation 

(i) Given

We shall use the identity

Here

By applying identity we get 

Hence simplified form of expression is .

(ii) Given

We shall use the identity

Here

By applying identity we get 

By rearranging the variable we get

Hence the simplified value of is

(iii) Given

We shall use the identity

Here

By applying identity we get 

By rearranging the variable we get,

Hence the simplified value of is

(iv) Given

We shall use the identity

Here

By applying the identity we get 

By rearranging the variable we get,

Hence the simplified value of is .

Q3.

In the given problem, we have to find the value of

Given

We shall use the identity

Here putting,

Hence the value of is .

Q4.

In the given problem, we have to find the value of

Given

We shall use the identity

Here putting,

Hence the value of is .

Q5.

In the given problem, we have to find the value of

Given

We shall use the identity

Here putting,

Hence the value of is

Q6.

In the given problem, we have to find the value of

Given

We shall use the identity

Here putting,

Hence the value of is

Q7.

In the given problem, we have to find the value of

Given

We shall use the identity

Here putting,

Hence the value of is .

Q8.

In the given problem, we have to find the value of

Given

We shall use the identity

Here putting,

In order to find we are using identity

Here and

Hence the value of is .

Q9.

In the given problem, we have to find the value of

Given

We shall use the identity

Here putting,

In order to find we are using identity

Here and

Hence the value of is .

Q10.

In the given problem, we have to find the value of

Given,

In order to find we are using identity

Here putting,

Hence the value of is .

Q11.

In the given problem, we have to find the value of

Given,

In order to find we are using identity

Here putting,,

Hence the value of is.

Page no. 4.20 Ex. 4.3

Q12.

Answer :

In the given problem, we have to find the value of

Given

We shall use the identity

Here putting,

In order to find we are using identity

In order to find we are using identity

Here and

Hence the value of is .

Q13.

Answer :

In the given problem, we have to find the value of numbers

(i) Given

In order to find we are using identity

We can write as

Hence where

The value of is

(ii) Given

In order to find we are using identity

We can write as

Hence where

The value of is

(iii) Given

In order to find we are using identity

We can write as

Hence where

The value of is

(iv) Given

In order to find we are using identity

We can write as

Hence where

The value of is

(v) Given

In order to find we are using identity

We can write as

Hence where

The value of is

(vi) Given

In order to find we are using identity

We can write as

Hence where

The value of is .

Q14.

Answer :

In the given problem, we have to find the value of numbers

(i) Given

We can write as

We shall use the identity

Here

Hence the value of is

(ii) Given

We can write as

We shall use the identity

Here

Hence the value of is

(iii) Given

We can write as

We shall use the identity

Here

Hence the value of is

(iv) Given

We can write as

We shall use the identity

Here

Hence the value of is .

Q15.

Answer :

In the given problem, we have to find the value of

Given

We shall use the identity

Here putting,

Again squaring on both sides we get,

We shall use the identity

Again cubing on both sides we get,

We shall use identity

Hence the value of is respectively.

Q16.

Answer :

In the given problem, we have to find the value of

Given

By adding and subtracting in left hand side of we get,

Again by adding and subtracting in left hand side of we get,

Now cubing on both sides of we get

we shall use identity

Hence the value of is respectively.

Q17.

Answer :

In the given problem, we have to find the value of

(i) Given

On cubing both sides we get,

We shall use identity

Hence the value of is

(ii) Given

On cubing both sides we get,

We shall use identity

Hence the value of is .

Q18.

Answer :

From given problem we have to find the value of

Given

On cubing both sides of we get

We shall use identity

Hence the value of is .

Q19.

Answer :

In the given problem, we have to find the value of

Given

Cubing on both sides of we get

We shall use identity

Hence the value of is .

Factorization Of Algebraic Expressions

Page no. 5.13 Ex. 5.2 

Q1.

The given expression to be factorized is

This can be written in the form

Recall the formula for sum of two cubes

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization of is.

Q2.

The given expression to be factorized is

This can be written in the form

Recall the formula for sum of two cubes

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization of is.

Q3.

The given expression to be factorized is

This can be written in the form

Recall the formula for difference of two cubes

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization of is.

Q4.

The given expression to be factorized is

This can be written in the form

Recall the formula for sum of two cubes

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization of is.

Q5.

The given expression to be factorized is

This can be written in the form

Recall the formula for difference of two cubes

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization of is.

Q6.

The given expression to be factorized is

This can be written in the form

Recall the formula for difference of two cubes

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization of is.

Q7.

The given expression to be factorized is

Take common from the two terms,. Then we have

This can be written in the form

Recall the formula for difference of two cubes

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization of is.

Q8.

The given expression to be factorized is

Take common from the two terms,. Then we have

This can be written in the form

Recall the formula for sum of two cubes

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization of is.

Q9.

The given expression to be factorized is

Take common from the two terms,. Then we have

This can be written in the form

Recall the formula for sum of two cubes

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization of is.

Page no. 5.14 Ex. 5.2 

Q10.

The given expression to be factorized is

This can be written in the form

Recall the formula for difference of two cubes

Using the above formula, we have 

We cannot further factorize the expression. 

So, the required factorization of is.

Q11.

The given expression to be factorized is

This can be written in the form

Recall the formula for difference of two cubes

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization of is.

Q12.

The given expression to be factorized is

Recall the formula for sum of two cubes

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization of is.

Q13.

The given expression to be factorized is

Take common. Then we have

This can be written as

Recall the formula for difference of two cubes

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization of is.

Q14.

The given expression to be factorized is

Take common 3. Then we have from the above expression,

This can be written as

Recall the formula for difference of two cubes

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization of is.

Q15.

The given expression to be factorized is

This can be written as

Recall the formula for sum of two cubes

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization of is.

Q16.

The given expression to be factorized is

This can be written as

Recall the formula for sum of two cubes

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization of is.

Q17.

The given expression to be factorized is

Take common. Then we have from the above expression,

This can be written as

Recall the formula for difference of two cubes

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization of is.

Q18.

The given expression to be factorized is

This can be written as

Recall the formula for difference of two cubes

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization of is.

Q19.

The given expression to be factorized is

This can be written as

Take common x² from first two terms, 2x from the next two terms andfrom the last two terms. Then we have,

Finally, take common. Then we get,

We cannot further factorize the expression. 

So, the required factorization of is.

Q20.

The given expression to be factorized is

This can be written as

Recall the formula for sum of two cubes

Using the above formula, we have

Take common. Then we have

We cannot further factorize the expression. 

So, the required factorization of is.

Q21.

The given expression to be factorized is

This can be written as

Recall the formula for sum of two cubes

Using the above formula and taking common from the last two terms, we get

Take common. Then we have, 

We cannot further factorize the expression. 

So, the required factorization of is.

Q22.

The given expression to be factorized is

Recall the well known formula

The given expression can be written as

Recall the formula for difference of two cubes

Using the above formula and taking common –2 from the last two terms, we get 

We cannot further factorize the expression. 

So, the required factorization of is.

Q23.

The given expression to be factorized is

The given expression can be written as

Recall the formula for difference of two cubes

Using the above formula and taking common from the last two terms, we get 

Take common. Then we have,

We cannot further factorize the expression. 

So, the required factorization of is.

Q24.

(i) The given expression is

Assumeand. Then the given expression can be rewritten as

Recall the formula for sum of two cubes

Using the above formula, the expression becomes

Note that both and b are positive. So, neithernor any factor of it can be zero.

Therefore we can cancel the termfrom both numerator and denominator. Then the expression becomes

(ii) The given expression is

Assumeand. Then the given expression can be rewritten as

Recall the formula for difference of two cubes

Using the above formula, the expression becomes

Note that both, b is positive and unequal. So, neithernor any factor of it can be zero.

Therefore we can cancel the termfrom both numerator and denominator. Then the expression becomes

(iii) The given expression is

Assumeand. Then the given expression can be rewritten as

Recall the formula for difference of two cubes

Using the above formula, the expression becomes

Note that both, b is positive and unequal. So, neithernor any factor of it can be zero.

Therefore we can cancel the termfrom both numerator and denominator. Then the expression becomes

Page no. 5.17 Ex. 5.3

Q1.

The given expression to be factorized is

This can be written in the form

Take common from the last two terms,

This can be written in the following form

Recall the formula for the cube of the sum of two numbers

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization of is.

Q2.

The given expression to be factorized is

This can be written in the form

Take common from the last two terms,. Then we get

This can be written in the following form

Recall the formula for the cube of the difference of two numbers

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization is of is.

Q3.

The given expression to be factorized is

This can be written in the form

Take common from the last two terms,. Then we get

This can be written in the following form

Recall the formula for the cube of the sum of two numbers

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization is of is.

Q4.

The given expression to be factorized is

This can be written in the form

Take common from the last two terms. Then we get

This can be written in the following form

Recall the formula for the cube of the sum of two numbers

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization is of is.

Q5.

The given expression to be factorized is

This can be written in the form

Take common from the third and fourth terms. Then we get

This can be written in the following form

Recall the formula for the cube of the difference of two numbers

Using the above formula, we have

This can be written in the following form

Recall the formula for the sum of two cubes

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization is ofis

Q6.

The given expression to be factorized is

This can be written in the form

Take common from the last two terms. Then we get

This can be written in the following form

Recall the formula for the cube of the sum of two numbers

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization is of is.

Q7.

The given expression to be factorized is

This can be written in the form

Take common 6xy from the last two terms,. Then we get

This can be written in the following form

Recall the formula for the cube of the sum of two numbers

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization is of is.

Q8.

The given expression to be factorized is

This can be written in the form

Take common from the last two terms,. Then we get

This can be written in the following form

Recall the formula for the cube of the sum of two numbers

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization is of is.

Q9.

The given expression to be factorized is

This can be written in the form

Take common – 18ab from the last two terms,. Then we get

This can be written in the following form

Recall the formula for the cube of the difference of two numbers

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization is ofis.

Q10.

The given expression to be factorized is

This can be written in the form

Take common – 12x from the last two terms,. Then we get

This can be written in the following form

Recall the formula for the cube of the difference of two numbers

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization is of is.

Q11.

The given expression to be factorized is

This can be written in the form

Take common from the last two terms,. Then we get

This can be written in the following form

Recall the formula for the cube of the difference of two numbers

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization is of is.

Page no. 5.22 Ex. 5.4

Q1.

The given expression to be factorized is

This can be written in the form

Recall the formula

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization of is

Q2.

The given expression to be factorized is

This can be written in the form

Recall the formula

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization is ofis

Q3.

The given expression to be factorized is

This can be written in the form

Recall the formula

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization is of is

Q4.

The given expression to be factorized is

This can be written in the form

Recall the formula

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization is ofis

Q5.

The given expression to be factorized is

This can be written in the form

Recall the formula

Using the above formula, we have 

We cannot further factorize the expression. 

So, the required factorization is ofis

Q6.

The given expression to be factorized is

This can be written in the form

Recall the formula 

Using the above formula, we have 

We cannot further factorize the expression. 

So, the required factorization is ofis

Q7.

The given expression to be factorized is

Let, and. Then the given expression becomes

Note that

Recall the formula

When, this becomes

So, we have the new formula

, 

when.

Using the above formula, the given expression can be written as

Put, and. Then we have

We cannot further factorize the expression. 

So, the required factorization is ofis

Q8.

The given expression to be factorized is

Let, and. Then the given expression becomes

Note that

Recall the formula

When, this becomes

So, we have the new formula

, when.

Using the above formula, the given expression can be written as

Put, and. Then we have

We cannot further factorize the expression. 

So, the required factorization is ofis

Q9.

The given expression to be factorized is 

Let, and. Then the given expression becomes

Note that 

Recall the formula 

When, this becomes 

So, we have the new formula

, when.

Using the above formula, the given expression can be written as

Put, and.

Then we have 

We cannot further factorize the expression. 

So, the required factorization is of is

Q10.

The given expression to be factorized is

Let, and. Then the given expression becomes

Note that 

Recall the formula

When, this becomes 

So, we have the new formula

, when.

Using the above formula, the given expression can be written as

Put, and. Then we have

We cannot further factorize the expression. 

So, the required factorization is ofis

Q11.

The given expression to be factorized is

This can be written in the form

Recall the formula

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization is of is .

Page no. 5.23 Ex. 5.4

Q12.

The given expression to be factorized is

This can be written in the form

Recall the formula

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization is of is

Q13.

The given expression to be factorized is

This can be written in the form

Recall the formula

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization is ofis .

Q14.

The given expression to be factorized is

This can be written in the form

Recall the formula

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization is of is

Q15.

The given expression is

It is given that

The given expression can be written in the form

Recall the formula

Using the above formula, we have

Q16.

(i) The given expression is

We have to multiply the above expression by.

The required product is

Recall the formula

Using the above formula, we have

(ii) The given expression is

We have to multiply the above expression by.

The required product is

Recall the formula

Using the above formula, we have

(iii) The given expression is

We have to multiply the above expression by.

The required product is

Recall the formula

Using the above formula, we have

(iii) The given expression is

We have to multiply the above expression by.

The required product is

Recall the formula

Using the above formula, we have

Page no. 5.24 Formative Assessment_VSA

Q1.

The given expression to be factorized is

This can be written in the form

We cannot further factorize the expression. 

So, the required factorization is.

Q2.

The given expression to be factorized is

Take commonfrom the last three terms and then we have 

We cannot further factorize the expression. 

So, the required factorization is.

Q3.

Recall the formula

When, we have

Q4.

Recall the formula

Given that

Then we have

Q5.

Recall the formula

Given that

,

Then we have

Q6.

Recall the formula

Given that

,

Then we have

Q7.

The given expression is

Let, and. Then the given expression becomes

Note that

Recall the formula

When, this becomes

So, we have the new formula

, when.

Using the above formula, the value of the given expression is

Q8.

The given expression is

Let, and. Then the given expression becomes

Note that

Recall the formula

When, this becomes 

So, we have the new formula

, when.

Using the above formula, the value of the given expression is

Q9.

The given expression is

Let, and. Then the given expression becomes

Note that:

Recall the formula

When, this becomes

So, we have the new formula

, when.

Using the above formula, the value of the given expression is

Q10.

The given expression is

Let, and. Then the given expression becomes

Note that

Recall the formula

When, this becomes

So, we have the new formula

, when.

Using the above formula, the value of the given expression is

Page no. 5.24 Formative Assessment_MCQ

Q1.

The given expression to be factorized is

Take commonfrom the last three terms and then we have

So, the correct choice is (c).

Q2.

The given expression to be factorized is

This can be written in the form

So, the correct choice is (a).

Q3.

The given expression to be factorized is

This can be arrange in the form

Let. Then the above expression becomes

Put.

So, the correct choice is (a).

Q4.

The given expression to be factorized is

Take common from the first two terms and from the last two terms. That is

Finally, take commonfrom the two terms. That is

So, the correct choice is (d).

Page no. 5.25 Formative Assessment_MCQ

Q5.

The given expression to be factorized is

This can be written in the form

Recall the formula

Using the above formula, we have 

So, the correct choice is (a).

Q6.

The given expression to be factorized is

This can be written in the form

Recall the formula

Using the above formula, we have 

So, the correct choice is (c).

Q7.

The given expression to be factorized is

Recall the formula for difference of two cubes

Using the above formula, we have,

So, the correct choice is (a).

Q8.

The given expression to be factorized is

This can be written in the form

Take common x from the first two terms andfrom the last two terms. Then we have

Finally, take commonfrom the above expression,

So, the correct choice is (d).

Q9.

The given expression is

Let, and. Then the given expression becomes

Note that:

Recall the formula

When, this becomes

So, we have the new formula

, when.

Using the above formula, the value of the given expression is

So, the correct choice is (b).

Q10.

The given expression to be factorized is

This can be written in the form 

So, the correct choice is (a).

Q11.

The given expression is

Recall the formula

Using the above formula the given expression becomes

Given that

Therefore the value of the given expression is

So, the correct choice is (c).

Q12.

The given equation is

Recall the formula 

Using the above formula, we have

, provided.

So, the correct choice is (d).

Q13.

The given equation is

This can be written as

Comparing the coefficients on both sides of the equation.

We get,

Putting the value of a from (1) in (2)

We get,

So the value of a, b and c is 1, – 4 and 7 respectively.

Therefore,

a + b + c =1-4 + 7 = 4

So, the correct choice is (a).

Q14.

The given expression is

This can be written in the form

Assumeand. Then the given expression can be rewritten as

Recall the formula for difference of two cubes

Using the above formula, the expression becomes

Note that both a and b are positive, unequal. So, neithernor any factor of it can be zero.

Therefore we can cancel the termfrom both numerator and denominator. Then the expression becomes

So, the correct choice is (a).

Page no. 5.25 Formative Assessment_MCQ

Q15.

The given expression is

Assumeand. Then the given expression can be rewritten as

Recall the formula for sum of two cubes

Using the above formula, the expression becomes

Note that both and b are positive. So, neithernor any factor of it can be zero.

Therefore we can cancel the termfrom both numerator and denominator. Then the expression becomes

So, the correct choice is (b).