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32. Using Cofactors of elements of third column, evaluate Δ=|1xyz1yzx1zxy|

0 25 Views | Posted 4 months ago
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  • 1 Answer

  • V

    Answered by

    Vishal Baghel | Contributor-Level 10

    4 months ago

    Given,  Δ=|1xyz1yzx1zxy|

    Co-factor of elements of third column

    ∴ Δ = a13A13 + a23A23 + a33A33

     = yz (z-y) + zx [- (z-x)] + xy (y-x)

     = y z2-y2z-z2x+ zx2 + xy2-x2y.

     = yz2-y2z+ (xy2-xz2) + (zx2-x2y)

     = yz (z-y) + x (y2 – z2) -x2 (y-z)

     = -yz (y-z) + x (y + z) (y-z) -x2 (y-z)

    = (y-z) [-yz + x (y + z) -x2]

     = (y-z) [-yz + xy + xz-x2]

     = (y-z) [-y (z-x) + x (z-x)]

     = (y-z) (z-x) (x -y)

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A = [ 1 0 0 0 α β 0 β α ] then α is (if α, β Î I)
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alok kumar singh

|2A| = 27

8|A| = 27

Now |A| = α2–β2 = 24

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->α + β = 8 and

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->α = 5 and β = 3

If A = [ 2 1 − 1 2 ] , B = [ 1 0 1 1 ] , C = ABAT and X = AC2AT then |X| is equal to
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alok kumar singh

|A| = 3

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->|C| = |ABAT| = |A|B|A7| = |A|2|B|

= 9

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Matrix A of order 3 × 3 is such that |A| = 2 if n = | a d j ( a d j ( a d j . . . . ( a ) ) ) | ︸ 2 0 2 4   t i m e s  then remainder when n is divided by 9 is
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alok kumar singh

|A| = 2

a d j ( a d j ( a d j . . . . ( a ) ) ) ? 2 0 2 4   t i m e s = | A | ( n − 1 ) 2 0 2 4            

= | A | ( n − 1 ) 2 0 2 4                                             

= 2 2 2 0 2 4                                          

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7

If ∫ 0 2 ( 2 x − 2 x − x 2 ) d x = ∫ 0 1 ( 1 − 1 − y 2 − y 2 2 ) d y + ∫ 1 2 ( 2 − y 2 2 ) d y + I     t h e n     I     e q u a l s  
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Vishal Baghel

∫ 0 2 2 x d x − ∫ 0 2 2 x − x 2 d x = ∫ 0 1 d y − ∫ 0 1 1 − y 2 d y − ∫ 0 2 y 2 2 d y + ∫ 1 2 2 d y + I

⇒ 8 3 − ∫ 0 1 1 − t 2 d t = 1 − 8 6 + 2 + I

I = 1 − ∫ 0 1 1 − t 2 d t

If  ∫ 0 2 ( 2 x − 2 x − x 2 ) d x = ∫ 0 1 ( 1 − 1 − y 2 − y 2 2 ) d y + ∫ 1 2 ( 2 − y 2 2 ) d y + I     t h e n     I     e q u a l s

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