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# If  f(x)=|[a,-1,0],[ax,a,-1],[ax^2,ax,a]| , using properties of determinants find the value of f(2x) − f(x). - ISC (Science) Class 12 - Mathematics

ConceptProperties of Determinants

#### Question

If  f(x)=|[a,-1,0],[ax,a,-1],[ax^2,ax,a]|  , using properties of determinants find the value of f(2x) − f(x).

#### Solution

f(x)=|[a,-1,0],[ax,a,-1],[ax^2,ax,a]|

=>f(x)=|[a,-1,0],[ax,a,-1],[ax^2,ax,a]|

Applying C2C2+C1, we get

f(x)=a|[1,0,0],[x,x+a,-1],[x^2,x^2+ax,a]|

=>f(x)=a(a^2+ax+ax+x^2)

=>f(x)=a(a^2+2ax+x^2)

Also,

f(2x)=|[a,-1,0],[2ax,a,-1],[4ax^2,2ax,a]|

f(2x)=a|[1,-1,0],[2x,a,-1],[4x^2,2ax,a]|

Applying C2C2+C1, we get

f(2x)=a|[1,0,0],[2x,2x+a,-1],[4x^2,4x^2+2ax,a]|

⇒f(2x)=a{a(2x+a)+4x^2+2ax}

⇒f(2x)=a(4x^2+a^2+4ax)

∴ f(2x)−f(x)=a(4x^2+a^2+4ax−a^2−2ax−x^2)

=ax(3x+2a)

Is there an error in this question or solution?

#### APPEARS IN

Solution If  f(x)=|[a,-1,0],[ax,a,-1],[ax^2,ax,a]| , using properties of determinants find the value of f(2x) − f(x). Concept: Properties of Determinants.
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