R.D. Sharma solutions for Mathematics [English] Class 9 chapter 4 - Algebraic Identities [Latest edition]

Evaluate the following using identities:

`(2x+ 1/x)^2`

Evaluate the following using identities:

(2x + y) (2x − y)

Evaluate the following using identities:

`(a^2b - b^2a)^2`

Evaluate following using identities:

(a - 0.1) (a + 0.1)

Evaluate the following using identities:

(1.5x² − 0.3y²) (1.5x² + 0.3y²)

Evaluate the following using identities:

(399)²

Evaluate the following using identities:

(0.98)²

Evaluate following using identities:

991 ☓ 1009

Evaluate the following using identities:

117 x 83

Simplify the following: 175 x 175 x 2 x 175 x 25 x 25 x 25

Simplify the following:

322 x 322 - 2 x 322 x 22 + 22 x 22

Simplify the following:

0.76 x 0.76 - 2 x 0.76 x 0.24 x 0.24 + 0.24

Simplify the following

`(7.83 + 7.83 - 1.17 xx 1.17)/6.66`

if `x + 1/x = 11`, find the value of `x^2 + 1/x^2`

If \[x - \frac{1}{x} = - 1\]  find the value of  \[x^2 + \frac{1}{x^2}\]

If `x + 1/x = sqrt5`, find the value of `x^2 + 1/x^2` and `x^4 + 1/x^4`

If 9x² + 25y² = 181 and xy = −6, find the value of 3x + 5y

If 2x + 3y = 8 and xy = 2 find the value of `4x^2 + 9y^2`

If 3x - 7y = 10 and xy = -1, find the value of `9x^2 + 49y^2`

Simplify the following products:

`(1/2a - 3b)(1/2a + 3b)(1/4a^2 + 9b^2)`

Simplify the following products:

`(m + n/7)^3 (m - n/7)`

If `x^2 + 1/x^2 = 66`, find the value of `x - 1/x`

if `x^2 + 1/x^2 = 79` Find the value of `x + 1/x`

Simplify the following products:

`(x/2 - 2/5)(2/5 - x/2) - x^2 + 2x`

Simplify the following product:

(x² + x − 2)(x² − x + 2)

Simplify the following products:

`(x^3 - 3x^2 - x)(x^2 - 3x + 1)`

Simplify the following products:

`(2x^4 - 4x^2 + 1)(2x^4 - 4x^2 - 1)`

Prove that a² + b² + c² − ab − bc − ca is always non-negative for all values of a, b and c

Write in the expanded form:

`(a + 2b + c)^2`

Write in the expanded form:

(2a - 3b - c)²

Write the expanded form:

`(-3x + y + z)^2`

Write in the expanded form:

`(m + 2n - 5p)^2`

Write in the expanded form:

`(2 + x - 2y)^2`

Write in the expanded form (a² + b² + c² )²

Write in the expanded form: (ab + bc + ca)²

Write in the expanded form: `(x/y + y/z + z/x)^2`

Write in the expanded form:

`(a/(bc) + b/(ca) + c/(ab))^2`

Write in the expanded form: `(x + 2y + 4z)^2`

Write in the expand form: `(2x - y + z)^2`

Write in the expanded form: (-2x + 3y + 2z)²

If a + b + c = 0 and a² + b² + c² = 16, find the value of ab + bc + ca.

If a² + b² + c² = 16 and ab + bc + ca = 10, find the value of a + b + c.

If a + b + c = 9 and ab + bc + ca = 23, find the value of a² + b² + c².

Find the value of 4x² + y² + 25z² + 4xy − 10yz − 20zx when x = 4, y = 3 and z = 2.

Simplify (a + b + c)² + (a - b + c)²

Simplify: `(a + b + c)^2 - (a - b + c)^2` 

Simplify (a + b + c)² + (a - b + c)² + (a + b - c)²

Simplify (2x + p - c)² - (2x - p + c)²

Simplify `(x^2 + y^2 - z)^2 - (x^2 - y^2 + z^2)^2`

Simplify the expression: 

`(x + y + z)^2 + (x + y/2 + 2/3)^2 - (x/2 + y/3 + z/4)^2`

Simplify the following expressions:

`(x + y - 2z)^2 - x^2 - y^2 - 3z^2 +4xy`

Simplify the following expressions:

`(x^2 - x + 1)^2 - (x^2 + x + 1)^2`

Find the cube of the following binomials expression :

\[\frac{1}{x} + \frac{y}{3}\]

Find the cube of the following binomials expression :

\[\frac{3}{x} - \frac{2}{x^2}\]

Find the cube of the following binomials expression :

\[2x + \frac{3}{x}\]

Find the cube of the following binomials expression :

\[4 - \frac{1}{3x}\]

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

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

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

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

If  \[x^2 + \frac{1}{x^2}\], find the value of \[x^3 - \frac{1}{x^3}\]

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

Evaluate of the following:

(103)³

Evaluate the following:

 (98)³

Evaluate of the following:

 (9.9)³

Evaluate of the following: 

`(10.4)^3`

Evaluate of the following: 

 (598)³

Evaluate of the following: 

(99)³

Evaluate of the following:

 111³ − 89³

Evaluate of the following:

 46³+34³

Evaluate of the following:

104³ + 96³

Evaluate of the following:

93³ − 107³

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

Find the value of 27x³ + 8y³, if 3x + 2y = 14 and xy = 8

Find the value of 27x³ + 8y³, if  3x + 2y = 20 and xy = \[\frac{14}{9}\]

If `x - 1/x = 3 + 2sqrt2`, find the value of `x^3 - 1/x^3`

Simplify of the following:

(x+3)³ + (x−3)³

Simplify of the following:

Simplify of the following:

\[\left( x + \frac{2}{x} \right)^3 + \left( x - \frac{2}{x} \right)^3\]

Simplify of the following:

(2x − 5y)³ − (2x + 5y)³

If `x^4 + 1/x^4 = 194, "find"  x^3 + 1/x^3`

If `x^4 + 1/x^4 = 194, "find"  x^2 + 1/x^2`

If `x^4 + 1/x^4 = 194, "find"  x+ 1/x`

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

Find the following product:

(3x + 2y) (9x² − 6xy + 4y²)

Find the following product:

(4x − 5y) (16x² + 20xy + 25y²)

Find the following product:

 (7p⁴ + q) (49p⁸ − 7p⁴q + q²)

Find the following product:

\[\left( \frac{x}{2} + 2y \right) \left( \frac{x^2}{4} - xy + 4 y^2 \right)\]

Find the following product:

\[\left( \frac{3}{x} - \frac{5}{y} \right) \left( \frac{9}{x^2} + \frac{25}{y^2} + \frac{15}{xy} \right)\]

Find the following product:

\[\left( 3 + \frac{5}{x} \right) \left( 9 - \frac{15}{x} + \frac{25}{x^2} \right)\]

Find the following product:

Find the following product:

Find the following product:

Find the following product:

Find the following product:

Find the following product:

If x = 3 and y = − 1, find the values of the following using in identify:

 (9y² − 4x²) (81y⁴ +36x²y² + 16x⁴)

If x = 3 and y = − 1, find the values of the following using in identify:

\[\left( \frac{3}{x} - \frac{x}{3} \right) \left( \frac{x^2}{9} + \frac{9}{x^2} + 1 \right)\]

If x = 3 and y = − 1, find the values of the following using in identify:

\[\left( \frac{x}{7} + \frac{y}{3} \right) \left( \frac{x^2}{49} + \frac{y^2}{9} - \frac{xy}{21} \right)\]

If x = 3 and y = − 1, find the values of the following using in identify:

\[\left( \frac{x}{y} - \frac{y}{3} \right) \frac{x^2}{16} + \frac{xy}{12} + \frac{y^2}{9}\]

If x = 3 and y = − 1, find the values of the following using in identify:

\[\left( \frac{5}{x} + 5x \right)\] \[\left( \frac{25}{x^2} - 25 + 25 x^2 \right)\]

If x = −2 and y = 1, by using an identity find the value of the following

If x = −2 and y = 1, by using an identity find the value of the following

If x = −2 and y = 1, by using an identity find the value of the following

Find the following product:

(3x + 2y + 2z) (9x² + 4y² + 4z² − 6xy − 4yz − 6zx)

Find the following product:

(4x − 3y + 2z) (16x² + 9y² + 4z² + 12xy + 6yz − 8zx)

Find the following product:

(2ab − 3b − 2c) (4a² + 9b² +4c² + 6 ab − 6 bc + 4ca)

Find the following product:

(3x − 4y + 5z) (9x² +16y² + 25z² + 12xy −15zx + 20yz)

Evaluate:

25³ − 75³ + 50³

Evaluate:

48³ − 30³ − 18³

If x + \[\frac{1}{x}\] = then find the value of \[x^2 + \frac{1}{x^2}\].

If \[x + \frac{1}{x} = 3\]  then find the value of \[x^6 + \frac{1}{x^6}\].

If \[x - \frac{1}{x} = \frac{1}{2}\],then write the value of \[4 x^2 + \frac{4}{x^2}\]

If \[a^2 + \frac{1}{a^2} = 102\] , find the value of \[a - \frac{1}{a}\].

If a + b + c = 0, then write the value of \[\frac{a^2}{bc} + \frac{b^2}{ca} + \frac{c^2}{ab}\]

Mark the correct alternative in each of the following:

If \[x + \frac{1}{x} = 5\] then \[x^2 + \frac{1}{x^2} = \]

If \[x + \frac{1}{x} = 2\], then \[x^3 + \frac{1}{x^3} =\]

If \[x + \frac{1}{x}\] 4, then \[x^4 + \frac{1}{x^4} =\]

If \[x + \frac{1}{x} = 3\] then \[x^6 + \frac{1}{x^6}\] = 

If \[x^2 + \frac{1}{x^2} = 102\], then \[x - \frac{1}{x}\] = 

If \[x^3 + \frac{1}{x^3} = 110\], then \[x + \frac{1}{x} =\]

If \[x^3 - \frac{1}{x^3} = 14\],then \[x - \frac{1}{x} =\]

If a + b + c = 9 and ab + bc + ca = 23, then a² + b² + c² =

(a − b)³ + (b − c)³ + (c − a)³ =

If \[\frac{a}{b} + \frac{b}{a} = - 1\] then a³ − b³ =

If a − b = −8 and ab  = −12, then a³ − b³ =

If the volume of a cuboid is 3x² − 27, then its possible dimensions are

75 × 75 + 2 × 75 × 25 + 25 × 25 is equal to

(x − y) (x + y) (x² + y²) (x⁴ + y⁴) is equal to ______.

If \[x^4 + \frac{1}{x^4} = 623\] then \[x + \frac{1}{x} =\]

If  \[x^4 + \frac{1}{x^4} = 194,\] then \[x^3 + \frac{1}{x^3} =\]

If \[x - \frac{1}{x} = \frac{15}{4}\], then \[x + \frac{1}{x}\] = 

If  \[3x + \frac{2}{x} = 7\] , then \[\left( 9 x^2 - \frac{4}{x^2} \right) =\]

If a² + b² + c² − ab − bc − ca =0, then

If a + b + c = 0, then \[\frac{a^2}{bc} + \frac{b^2}{ca} + \frac{c^2}{ab} =\]

If a^1/3 + b^1/3 + c^1/3 = 0, then

If a + b + c = 9 and ab + bc + ca =23, then a³ + b³ + c³ − 3abc =

\[\frac{( a^2 - b^2 )^3 + ( b^2 - c^2 )^3 + ( c^2 - a^2 )^3}{(a - b )^3 + (b - c )^3 + (c - a )^3} =\]

The product (a + b) (a − b) (a² − ab + b²) (a² + ab + b²) is equal to

The product (x²−1) (x⁴ + x² + 1) is equal to

If \[\frac{a}{b} + \frac{b}{a} = 1\] then a³ + b³ =

If 49a² − b = \[\left( 7a + \frac{1}{2} \right) \left( 7a - \frac{1}{2} \right)\] then the value of b is