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Page No 5.13:

Question 1:

Factorize each of the following expressions:

 p3 + 27
 

Answer:

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.

Page No 5.13:

Question 2:

y3 + 125

Answer:

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.

Page No 5.13:

Question 3:

1 − 27a3

Answer:

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.

Page No 5.13:

Question 4:

8x3y3 + 27a3

Answer:

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.

Page No 5.13:

Question 5:

64a3b3

Answer:

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.

Page No 5.13:

Question 6:

x3216-8y3

Answer:

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.

Page No 5.13:

Question 7:

10x4y − 10xy4

Answer:

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.

Page No 5.13:

Question 8:

54x6y + 2x3y4

Answer:

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 5.13:

Question 9:

32a3 + 108b3

Answer:

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:

Question 10:

(a − 2b)3 − 512b3

Answer:

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.

Page No 5.14:

Question 11:

(a + b)3 − 8(ab)3

Answer:

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.

Page No 5.14:

Question 12:

(x + 2)3 + (x − 2)3

Answer:

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.

Page No 5.14:

Question 13:

8x2y3 − x5

Answer:

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.

Page No 5.14:

Question 14:

1029 − 3x3

Answer:

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.

Page No 5.14:

Question 15:

x6 + y6

Answer:

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.

Page No 5.14:

Question 16:

x3y3 + 1

Answer:

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.

Page No 5.14:

Question 17:

x4y4xy

Answer:

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.

Page No 5.14:

Question 18:

a12+ b12

Answer:

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.

Page No 5.14:

Question 19:

x3 + 6x2 + 12x + 16

Answer:

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.

Page No 5.14:

Question 20:

a3 + b3 + a + b

Answer:

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.

Page No 5.14:

Question 21:

a3-1a3 -2a+2a

Answer:

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.

Page No 5.14:

Question 22:

a3 + 3a2b + 3ab2 + b3 − 8

Answer:

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.

Page No 5.14:

Question 23:

8a3 b3 − 4ax + 2bx

Answer:

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.

Page No 5.14:

Question 24:

Simplify:

(i) 173×173×173+127×127×127173×173-173×127+127×127

(ii) 155×155×155-55×55×55155×155+155×55+55×55

(iii) 1.2×1.2×1.2-0.2×0.2×0.21.2×1.2+1.2×0.2+0.2×0.2
 

Answer:

(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:

Question 1:

Factorize:

64a3 + 125b3 + 240a2b + 300ab2
 

Answer:

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.

Page No 5.17:

Question 2:

125x3 − 27y3 − 225x2y + 135xy2

Answer:

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 5.17:

Question 3:

827x3+1+43x2+2x

Answer:

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.

Page No 5.17:

Question 4:

8x3 + 27y3 + 36x2y + 54xy2

Answer:

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.

Page No 5.17:

Question 5:

a3 − 3a2b + 3ab2 b3 + 8

Answer:

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.

Page No 5.17:

Question 6:

x3 + 8y3 + 6x2y + 12xy2

Answer:

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.

Page No 5.17:

Question 7:

8x3 + y3 + 12x2y + 6xy2

Answer:

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.

Page No 5.17:

Question 8:

8a3 + 27b3 + 36a2b + 54ab2

Answer:

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.

Page No 5.17:

Question 9:

8a3 − 27b3 − 36a2+ 54ab2

Answer:

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.

Page No 5.17:

Question 10:

x3 − 12x(x − 4) − 64

Answer:

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.

Page No 5.17:

Question 11:

a3x3 − 3a2bx2 + 3ab2xb3

Answer:

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:

Question 1:

Factorize each of the following expressions:

a3 + 8b3 + 64c3 − 24abc
 

Answer:

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.

Page No 5.22:

Question 2:

x3 − 8y3 + 27z3 + 18xyz

Answer:

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.

Page No 5.22:

Question 3:

27x3y3z3 − 9xyz

Answer:

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 5.22:

Question 4:

127x3-y3+125z3+5xyz

Answer:

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 .

Page No 5.22:

Question 5:

8x3 +27y3 − 216z3 + 108xyz

Answer:

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 .

Page No 5.22:

Question 6:

125 + 8x3 − 27y3 + 90xy

Answer:

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 .

Page No 5.22:

Question 7:

(3x − 2y)3 + (2y − 4z)3 + (4z − 3x)3

Answer:

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.

Page No 5.22:

Question 8:

(2x − 3y)3 + (4z − 2x)3 + (3y − 4z)3

Answer:

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 .

Page No 5.22:

Question 9:

x2+y+z33 + x3-2y3+z3 + -5x6-y34z33

Answer:

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

Page No 5.22:

Question 10:

(a − 3b)3 + (3bc)3 + (ca)3

Answer:

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.

Page No 5.22:

Question 11:

22a3+33b3+c3-36abc

Answer:

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:

Question 12:

33a3-b3-55c3-315abc

Answer:

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:

Question 13:

8x3 − 125y3 + 180xy + 216

Answer:

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 .

Page No 5.23:

Question 14:

22a3+162b3+c3-12abc

Answer:

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:

Question 15:

Find the value of x3 + y3 − 12xy + 64, when x + y =−4

Answer:

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

Page No 5.23:

Question 16:

Multiply:
(i) x2 + y2 + z2xy + xz + yz by x + yz

(ii) x2 + 4y2 + z3 + 2xy + xz − 2yz by x − 2yz

(iii) x2 + 4y2 + 2xy − 3x + 6y + 9 by x − 2y + 3

(iv) 9x2 + 25y2 + 15xy + 12x − 20y + 16 by 3x − 5y + 4

Answer:

(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:

Question 1:

Factorize: x4 + x2 + 25.

Answer:

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.

Page No 5.24:

Question 2:

Factorize : x2 − 1 − 2aa2

Answer:

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.

Page No 5.24:

Question 3:

If a + b + c = 0, then write the value of a3 + b3 + c3.

Answer:

Recall the formula

When, we have

 

Page No 5.24:

Question 4:

If a2 + b2 + c2 = 20 and a + b + c = 0, find ab + bc + ca.

Answer:

Recall the formula

Given that

Then we have

Page No 5.24:

Question 5:

If a + b + c = 9 and ab + bc + ca = 40, find a2 + b2 +c2.

Answer:

Recall the formula

Given that

,

Then we have

Page No 5.24:

Question 6:

If a2 + b2 + c2 = 250 and ab + bc + ca = 3, find a + b + c.

Answer:

Recall the formula

Given that

,

Then we have

Page No 5.24:

Question 7:

Write the value of 253 − 753 + 503.

Answer:

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:

Question 8:

Write the value of 483 − 303 − 183.

Answer:

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:

Question 9:

Write the value of 123+133-563.

Answer:

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:

Question 10:

Write the value of 303 + 203 − 503.

Answer:

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:

Question 1:

Mark the correct alternative in each of the following:

The factors of a2 − 1 − 2xx2 are

(a) (a − x + 1) (a − x − 1)

(b) (a + x − 1) (a − x + 1)

(c) (a + x +1) (a − x + 1)

(d) none of these

Answer:

The given expression to be factorized is

Take commonfrom the last three terms and then we have

So, the correct choice is (c).

Page No 5.24:

Question 2:

The factors of x4 + x2 + 25 are

(a) (x2 + 3x + 5) (x2 − 3x + 5)

(b) (x2 + 3x + 5) (x2 + 3x − 5)

(c) (x2 + x +5) (x2x + 5)

(d) none of these

Answer:

The given expression to be factorized is

This can be written in the form

So, the correct choice is (a).

Page No 5.24:

Question 3:

The factors of x2 + 4y2 + 4y − 4xy − 2x − 8 are

(a) (x − 2y −4) (x − 2y + 2)

(b) (xy + 2) (x − 4y − 4)

(c) (x + 2y − 4) (x + 2y + 2)

(d) none of these

Answer:

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).

Page No 5.24:

Question 4:

The factors of x3x2yxy2 + y3 are

(a)(x + y) (x2xy + y2)

(b) (x + y) (x2 + xy + y2)

(c) (x + y)2 (xy)

(d) (x − y)2 (x + y)

Answer:

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:

Question 5:

The factors of x3 − 1 + y3 + 3xy are

(a) (x − 1 + y) (x2 + 1 + y2 + x + yxy)

(b) (x + y + 1) (x2 + y2 + 1 −xyx y)

(c) (x − 1 + y) (x2 − 1 − y2 + x + y + xy)

(d) 3(x + y −1) (x2 + y2 − 1)

Answer:

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).

Page No 5.25:

Question 6:

The factors of 8a3 + b3 − 6ab + 1 are

(a) (2a + b − 1) (4a2 + b2 + 1 − 3ab − 2a)

(b) (2ab + 1) (4a2 + b2 − 4ab + 1 − 2a + b)

(c) (2a + b + 1) (4a2 + b2 + 1 −2abb − 2a)

(d) (2a − 1 + b) (4a2 + 1 − 4ab − 2ab)

Answer:

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).

Page No 5.25:

Question 7:

(x + y)3 − (x − y)3 can be factorized as

(a) 2y (3x2 + y2)

(b) 2x (3x2 + y2)

(c) 2y (3y2 + x2)

(d) 2x (x2+ 3y2

Answer:

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).

Page No 5.25:

Question 8:

The factors of x3 − 7x + 6 are

(a) x (x − 6) (x − 1)

(b) (x2 − 6) (x − 1)

(c) (x + 1) (x + 2) (x + 3)

(d) (x − 1) (x + 3) (x − 2) 

Answer:

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).

Page No 5.25:

Question 9:

The expression (ab)3 + (b c)3 + (ca)3 can be factorized as

(a) (ab) (b c) (ca)

(b) 3(ab) (bc) (ca)

(c) −3(ab) (bc) (ca)

(d) (a + b + c) (a2 + b2 + c2abbcca)

Answer:

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).

Page No 5.25:

Question 10:

The expression x4 + 4 can be factorized as

(a) (x2 + 2x + 2) (x2 − 2x + 2)

(b) (x2 + 2x + 2) (x2 + 2x + 2)

(c) (x2 − 2x − 2) (x2 − 2x + 2)

(d) (x2 + 2) (x2 − 2)

Answer:

The given expression to be factorized is

This can be written in the form 

So, the correct choice is (a).

Page No 5.25:

Question 11:

If 3x = a + b + c, then the value of (xa)3 + (xb)3 + (xc)3 − 3(xa) (xb) (xc) is

(a) a + b + c

(b) (ab) (bc) (ca)

(c) 0

(d) none of these

Answer:

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).

Page No 5.25:

Question 12:

If (x + y)3 − (xy)3 − 6y(x2y2) = ky2, then k =

(a) 1

(b) 2

(c) 4

(d) 8

Answer:

The given equation is

Recall the formula 

Using the above formula, we have

, provided.

So, the correct choice is (d).

Page No 5.25:

Question 13:

If x3 − 3x2 + 3x − 7 = (x + 1) (ax2 + bx + c), then a + b + c =

(a) 4

(b) 12

(c) −10

(d) 3

Answer:

The given equation is

x3 − 3x2 + 3x − 7 = (x + 1) (ax2 + bx + c)

This can be written as

x3-3x2+3x-7=x+1ax2+bx+cx3-3x2+3x-7=ax3+bx2+cx+ax2+bx+cx3-3x2+3x-7=ax3+a+bx2+b+cx+c
 

Comparing the coefficients on both sides of the equation.

We get,


c = -7 .......(4)

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 = -10

So, the correct choice is (c).

Page No 5.25:

Question 14:

The value of (2.3)3-0.027(2.3)2+0.69+0.09

(a) 2

(b) 3

(c) 2.327

(d) 2.273

Answer:

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 5.26:

Question 15:

The value of (0.013)3+(0.007)3(0.013)2-0.013×0.007+(0.007)2 is

(a) 0.006

(b) 0.02

(c) 0.0091

(d) 0.00185

Answer:

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).



Page No 5.9:

Question 1:

Factorize:

1. x3 + x - 3x2 - 3

Answer:

The given expression to be factorized is

Take common x from the first two terms and -3 from the last two terms. That is

Finally, take common x2 + 1from the two terms. That is

We cannot further factorize the expression. 

So, the required factorization is.

Page No 5.9:

Question 2:

Factorize:

2. a(a+b)3 − 3a2b (a + b)
 

Answer:

The given expression to be factorized is

Take common from the two terms. That is

Expand the term within the second braces.

We cannot further factorize the expression. 

So, the required factorization of is.

Page No 5.9:

Question 3:

Factorize:

3. xx3-y3 + 3xy (x-y)

Answer:

The given expression to be factorized is

We know that

The given expression then becomes

Take common from the two terms. That is

We cannot further factorize the expression. 

So, the required factorization of is.

Page No 5.9:

Question 4:

Factorize:

a2x2 + (ax2 + 1)x + a

Answer:

The given expression to be factorized is

Simplify the middle term. That is

Take common from the first two terms and 1 from the last two terms. That is

Finally, take commonfrom the two terms. That is

We cannot further factorize the expression. 

So, the required factorization of is.

Page No 5.9:

Question 5:

Factorize:
x2 + y − xy − x

Answer:

The given expression to be factorized is

Rearrange the given expression as

Take common x from the first two terms and -1 from the last two terms. That is

Finally, take commonfrom the two terms. That is

We cannot further factorize the expression. 

So, the required factorization of is.

Page No 5.9:

Question 6:

Factorize:

x3 − 2x2y + 3xy2 − 6y3

Answer:

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

We cannot further factorize the expression. 

So, the required factorization of is.

Page No 5.9:

Question 7:

Factorize:

6ab − b2 + 12ac − 2bc

Answer:

The given expression to be factorized is

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

Finally, take commonfrom the two terms. That is

We cannot further factorize the expression. 

So, the required factorization of is.

Page No 5.9:

Question 8:

Factorize:

x2+1x2 - 4 x + 1x+6

Answer:

The given expression to be factorized is

We have

Use the above result in the original expression to get

Substituting in the above , we get

Put.

We cannot further factorize the expression. 

So, the required factorization of is.

Page No 5.9:

Question 9:

Factorize:

x(x-2)(x-4) + 4x -8

Answer:

The given expression to be factorized is

Take common 4 from the last two terms. That is

Again take commonfrom the two terms of the above expression.

                                   

We cannot further factorize the expression. 

So, the required factorization of is.

Page No 5.9:

Question 10:

Factorize:

(x+2) (x2+25) − 10x2 − 20x

Answer:

The given expression to be factorized is

Take common from the last two terms. That is

Again take commonfrom the two terms of the above expression. Then

We cannot further factorize the expression. 

So, the required factorization of is.

Page No 5.9:

Question 11:

Factorize:

2a2 + 26ab+3b2
 

Answer:

The given expression to be factorized is

This can be written in the form

We cannot further factorize the expression. 

So, the required factorization of is.

Page No 5.9:

Question 12:

Factorize:

(a − b + c)2 + (b − c + a)2 + 2(a − b + c) (b − c + a)

Answer:

The given expression to be factorized is

This can be written as

We cannot further factorize the expression. 

So, the required factorization of is.

Page No 5.9:

Question 13:

Factorize:

a2 + b2 + 2(ab + bc + ca)

Answer:

The given expression to be factorized is

This can be written as

Take commonfrom the last two terms.

Finally, take common from the two terms of the above expression.

We cannot further factorize the expression. 

So, the required factorization of is.

Page No 5.9:

Question 14:

Factorize:

4(x − y)2 − 12(x − y) (x + y) + 9(x + y)2

Answer:

The given expression to be factorized is

Substituting andin the above expression, we get

=

This can be arrange in the form

Putand.

Take common -1 from the expression within the braces.

We cannot further factorize the expression. 

So, the required factorization of is.

Page No 5.9:

Question 15:

Factorize:
a2 + b2 + 2bc − c2

Answer:

The given expression to be factorized is

This can be arrange in the form

Substitutingin the above expression, we get.

Put.

We cannot further factorize the expression. 

So, the required factorization of is.

Page No 5.9:

Question 16:

Factorize:

a2 + 2ab +b2c2

Answer:

The given expression to be factorized is

This can be arrange in the form

Substitutingin the above expression, we get.

Put.

We cannot further factorize the expression. 

So, the required factorization of is.

Page No 5.9:

Question 17:

Factorize:

a2 + 4b2 − 4ab − 4c2

Answer:

The given expression to be factorized is

This can be arrange in the form

Substitute.

Put.

We cannot further factorize the expression. 

So, the required factorization of is.

Page No 5.9:

Question 18:

Factorize:

xy
9yx9

Answer:

The given expression to be factorized is

This can be written in the form

Take commonfrom the two terms of the above expression

We cannot further factorize the expression. 

So, the required factorization of is

Page No 5.9:

Question 19:

Factorize:

x4 + x2y2 + y4

Answer:

The given expression to be factorized is

Add and subtract the termin the given expression.

Substitutingin the above expression, we get

Putin the above expression,

We cannot further factorize the expression. 

So, the required factorization of is.

Page No 5.9:

Question 20:

Factorize:

x
2y2 − 4xz + 4z2

Answer:

The given expression to be factorized is

Rearrange the terms as

Substitutingin the avove expression, 

Put.

We cannot further factorize the expression. 

So, the required factorization ofis.

Page No 5.9:

Question 21:

Factorize:

x2 + 62x+10

Answer:

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.

Finally, take commonfrom the above expression. Then we have

We cannot further factorize the expression.

So, the required factorization is.

Page No 5.9:

Question 22:

Factorize:

x2-22x-30

Answer:

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,

Finally take commonfrom the above expression,

We cannot further factorize the expression. 

So, the required factorization of is.

Page No 5.9:

Question 23:

Factorize:

x2-3x-6

Answer:

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. Then we have

We cannot further factorize the expression. 

So, the required factorization of is.

Page No 5.9:

Question 24:

Factorize:

x2+55x + 30

Answer:

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,

We cannot further factorize the expression.

So, the required factorization of is.

Page No 5.9:

Question 25:

Factorize:

x2+23x-24

Answer:

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,

Finally take commonfrom the above expression,

We cannot further factorize the expression. 

So, the required factorization of is.

Page No 5.9:

Question 26:

Factorize:

2x2-56x+112

Answer:

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,

Finally take commonfrom the above expression,

We cannot further factorize the expression. 

So, the required factorization of is.

Page No 5.9:

Question 27:

Factorize:

x2+1235x+135

Answer:

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,

Finally take commonfrom the above expression,

We cannot further factorize the expression. 

So, the required factorization of is.

Page No 5.9:

Question 28:

Factorize:

21x2-2x+121

Answer:

The given expression to be factorized is

This can be written in the form

We cannot further factorize the expression. 

So, the required factorization of is.

Page No 5.9:

Question 29:

Factorize:

55x2+20x+35

Answer:

The given expression to be factorized is

This can be written in the form

Take commonfrom the first two terms andfrom the last two terms,

Finally take commonfrom the above expression,

We cannot further factorize the expression. 

So, the required factorization of is.

Page No 5.9:

Question 30:

Factorize:

2x2+35x+5

Answer:

The given expression to be factorized is

This can be written in the form

Take commonfrom the first two terms andfrom the last two terms,

Finally take commonfrom the above expression,

We cannot further factorize the expression. 

So, the required factorization of is.

Page No 5.9:

Question 31:

Factorize:

9(2ab)2 − 4(2ab) − 13

Answer:

The given expression to be factorized is

Substitutingin the above expression, we get

This can be written in the form

Take common x from the first two terms and 1 from the last two terms,

Finally take commonfrom the above expression,

Put,

We cannot further factorize the expression. 

So, the required factorization ofis.

Page No 5.9:

Question 32:

Factorize:

7(x − 2y)2 − 25(x − 2y) + 12

Answer:

The given expression to be factorized is

Substitutingin the above expression, we get

This can be written in the form

Take commonfrom the first two terms andfrom the last two terms,

Finally take commonfrom the above expression,

Put in the above expression,

We cannot further factorize the expression. 

So, the required factorization of is.

Page No 5.9:

Question 33:

Factorize:

2(x + y)2 − 9(x + y) − 5

Answer:

The given expression to be factorized is

Substitutingin the above expression, we get

This can be written in the form

Take commonfrom the first two terms andfrom the last two terms,

Finally take commonfrom the above expression,

Put. Then we have

We cannot further factorize the expression. 

So, the required factorization of is.

Page No 5.9:

Question 34:

Give possible expressions for the length and breadth of the rectangle having 35y2 + 13y − 12 as its area.

Answer:

The area of the rectangle is

First we will factorize the above expression. This can be written in the form

Take commonfrom the first two terms andfrom the last two terms,

Finally take commonfrom the above expression,

The area of a rectangle having length a and breadth bis ab.

Here we don’t know the bigger or the smaller factor. So, the two possibilities are

(i) Length isand breadth is

(ii) Length is and breadth is

Page No 5.9:

Question 35:

What are the possible expressions for the dimensions of the cuboid whose volume is 3x2− 12x.

Answer:

The volume of the cuboid is

First we will factorize the above expression. 

Take commonfrom the two terms of the above expression,

The volume of a cuboid having length, breadth b and height is.

Here the word ‘dimensions’ stands for the length, breadth and height of the cuboid. So, the three possibilities are

(i) Length is, breadth is x and height is

(ii) Length is x , breadth isand height is

(iii) Length is, breadth isand height is x

There are many other possibilities also, because we can consider the product of two simple factors as a single factor.



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