RD Sharma 2020 solution class 9 chapter 4 Algebraic Identities Exercise 4.3

Exercise 4.3

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Question 1:

Find the cube of each of the following binomials expressions:

(i) $\frac{1}{x}+\frac{y}{3}$

(ii) $\frac{3}{x}-\frac{2}{x^2}$

(iii) $2x+\frac{3}{x}$

(iv) $4-\frac{1}{3x}$

Answer 1:

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 .

Question 2:

If a + b = 10 and ab = 21, find the value of a³ + b³

Answer 2:

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

Given

We shall use the identity

Here putting,

Hence the value of is .

Question 3:

If a − b = 4 and ab = 21, find the value of a³ −b³

Answer 3:

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

Given

We shall use the identity

Here putting,

Hence the value of is .

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Question 4:

If $x+\frac{1}{x}=5$, find the value of $x^3+\frac{1}{x^3}$

Answer 4:

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

Given

We shall use the identity

Here putting,

Hence the value of is

Question 5:

If $x-\frac{1}{x}=7$, find the value of $x^3-\frac{1}{x^3}$

Answer 5:

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

Given

We shall use the identity

Here putting,

Hence the value of is

Question 6:

If $x-\frac{1}{x}=5$, find the value of $x^3-\frac{1}{x^3}$

Answer 6:

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

Given

We shall use the identity

Here putting,

Hence the value of is .

Question 7:

If $x^2+\frac{1}{x^2}$ = 51, find the value of $x^3-\frac{1}{x^3}$

Answer 7:

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 .

Question 8:

If $x^2+\frac{1}{x^2} =98$, find the value of $x^3+\frac{1}{x^3}$

Answer 8:

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 .

Question 9:

If 2x+3y = 13 and xy = 6, find the value of 8x³ + 27y³

Answer 9:

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 .

Question 10:

If 3x − 2y = 11 and xy = 12, find the value of 27x³ − 8y³

Answer 10:

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.

Question 11:

Evaluate each of the following:

(i) (103)³

(ii) (98)³

(iii) (9.9)³

(iv) (10.4)3

(v) (598)³

(vi) (99)³

Answer 11:

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 .

Question 12:

Evaluate each of the following:

(i) 111³ − 89³

(ii) 46³+34³

(iii) 104³ + 96³

(iv) 93³ − 107³

Answer 12:

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

(i) Given

We can write as

We shall use the identity

Here

${111}^3-{89}^3={\left(100+11\right)}^3-{\left(100-11\right)}^3$

Hence the value of is

(ii) Given

We can write as

We shall use the identity

Here

${46}^3+{34}^3={\left(40+6\right)}^3+{\left(40-6\right)}^3$

Hence the value of is

(iii) Given

We can write as

We shall use the identity

Here

${104}^3+{96}^3={\left(100+4\right)}^3+{\left(100-4\right)}^3$

Hence the value of is

(iv) Given

We can write as

We shall use the identity

Here

Hence the value of is .

Question 13:

If $x+\frac{1}{x}=3$, calculate $x^2+\frac{1}{x^2}, x^3+\frac{1}{x^3}$ and $x^4+\frac{1}{x^4}$

Answer 13:

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.

Question 14:

Find the value of 27x³ + 8y³, if

(i) 3x + 2y = 14 and xy = 8

(ii) 3x + 2y = 20 and xy = $\frac{14}{9}$

Answer 14:

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 .

Question 15:

Find the value of 64x³ − 125z³, if 4x − 5z = 16 and xz = 12.

Answer 15:

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 .

Question 16:

If $x-\frac{1}{x}=3+2\sqrt{2}$, find the value of $x^3-\frac{1}{x^3}$

Answer 16:

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

Given

Cubing on both sides of we get

${\left(x-\frac{1}{x}\right)}^3={\left(3+2\sqrt{2}\right)}^3$

We shall use identity

$$\begin{gathered} 27+16\sqrt{2}+18\sqrt{2}\times 3+18\sqrt{2}\times 2\sqrt{2}=x^3-\frac{1}{x^3}-9-6\sqrt{2} \\ 27+16\sqrt{2}+54\sqrt{2}+72=x^3-\frac{1}{x^3}-9-6\sqrt{2} \end{gathered}$$

 

Hence the value of is .

Question 17:

Simplify each of the following:

(i) (x+3)³ + (x−3)³

(ii) ${\left(\frac{x}{2}+\frac{y}{3}\right)}^3-{\left(\frac{x}{2}-\frac{y}{3}\right)}^3$

(iii) ${\left(x+\frac{2}{x}\right)}^3+{\left(x-\frac{2}{x}\right)}^3$

(iv) (2x − 5y)³ − (2x + 5y)³

Answer 17:

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 .

Question 18:

If $x^4+\frac{1}{x^4}= 194,$find $x^3+\frac{1}{x^3}, x^2+\frac{1}{x^2}$and $x+\frac{1}{x}$

Answer 18:

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.

Question 19:

If $x^4+\frac{1}{x^4}= 119$, find the value of $x^3-\frac{1}{x^3}$

Answer 19:

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

${\left(x-\frac{1}{x}\right)}^2=x^2+\frac{1}{x^2}-2\times x\times \frac{1}{x}$
​

In order to find we are using identity 

Here and 

Hence the value of is .