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Q20 . x^{2}−y^{2}−4xz+4z^{2}
SOLUTION :
On rearranging the terms
= x^{2}−4xz+4z^{2}−y^{2}
= (x)^{2}−2×x×2z+(2z)^{2}−y^{2}
Using the identity x^{2}−2xy+y^{2}=(x−y)^{2}
= (x−2z)^{2}−y^{2}
Using the identity p^{2}−q^{2}= ( p + q )( p – q )
= (x−2z+y)(x−2z−y)
∴ x^{2}−y^{2}−4xz+4z^{2 }= (x−2z+y)(x−2z−y)
Q21 .
SOLUTION :
Splitting the middle term ,
Q22 .
SOLUTION :
Splitting the middle term,
=
Q23 .
SOLUTION :
Splitting the middle term,
Q24 .
SOLUTION :
Splitting the middle term,
Q25 .
SOLUTION :
Splitting the middle term,
Q26 .
SOLUTION :
Splitting the middle term,
Q27 .
SOLUTION :
Splitting the middle term,
Q28 .
SOLUTION :
Using the identity (x−y)^{2}=x^{2}+y^{2}−2xy
Q29 .
SOLUTION :
Splitting the middle term,
Q30 .
SOLUTION :
Splitting the middle term,
Q31 . 9(2a−b)^{2}−4(2a−b)−13
SOLUTION :
Let 2a – b = x
= 9x^{2}−4x−13
Splitting the middle term,
= 9x^{2}−13x+9x−13
= x(9x−13)+1(9x−13)
= (9x−13)(x+1)
Substituting x = 2a – b
= [9(2a−b)−13](2a−b+1)
= (18a−9b−13)(2a−b+1)
∴ 9(2a−b)^{2}−4(2a−b)−13 = (18a−9b−13)(2a−b+1)
Q 32 . 7(x−2y)^{2}−25(x−2y)+12
SOLUTION :
Let x2y = P
= 7P2−25P+12
Splitting the middle term,
= 7P^{2}−21P−4P+12
= 7P(P−3)−4(P−3)
= (P−3)(7P−4)
Substituting P = x – 2y
= (x−2y−3)(7(x−2y)−4)
= (x−2y−3)(7x−14y−4)
∴ 7(x−2y)^{2}−25(x−2y)+12 = (x−2y−3)(7x−14y−4)
Q33 . 2(x+y)2−9(x+y)−5
SOLUTION :
Let x+y = z
= 2z2−9z−5
Splitting the middle term,
= 2z2−10z+z−5
= 2z(z−5)+1(z−5)
= (z−5)(2z+1)
Substituting z = x + y
= (x+y−5)(2(x+y)+1)
= (x+y−5)(2x+2y+1)
∴ 2(x+y)2−9(x+y)−5 = (x+y−5)(2x+2y+1)
Q34 . Give the possible expression for the length & breadth of the rectangle having 35y^{2}−13y−12 as its area.
SOLUTION :
Area is given as 35y^{2}−13y−12
Splitting the middle term,
Area = 35y^{2}+218y−15y−12
= 7y(5y+4)−3(5y+4)
= (5y+4)(7y−3)
We also know that area of rectangle = length ×breadth
∴ Possible length = (5y+4) and breadth=(7y−3)
Or possible length = (7y−3) and breadth= (5y+4)
Q35 . What are the possible expression for the cuboid having volume 3x^{2}−12x.
SOLUTION :
Volume = 3x^{2}−12x
= 3x(x−4)
= 3×x(x−4)
Also volume = Length ×Breadth×Height
∴ Possible expression for dimensions of cuboid are = 3 , x , (x−4)
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