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Show that abs((a,b,c),(a+2x,b+2y,c+2z),(...

Show that `abs((a,b,c),(a+2x,b+2y,c+2z),(x,y,z))` =0

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The correct Answer is:
`Delta =Delta _1`
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If a x_1^2+b y_1^2+c z_1^2=a x_2 ^2+b y_2 ^2+c z_2 ^2=a x_3 ^2+b y_3 ^2+c z_3 ^2=d ,a x_2 x_3+b y_2y_3+c z_2z_3=a x_3x_1+b y_3y_1+c z_3z_1=a x_1x_2+b y_1y_2+c z_1z_2=f, then prove that |(x_1, y_1, z_1), (x_2, y_2, z_2), (x_3,y_3,z_3)|=(d-f){((d+2f))/(a b c)}^(1//2)

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NCERT BANGLISH-DETERMINANTS -Miscellaneous Exercises on Chapter 4
  1. Show that abs((a,b,c),(a+2x,b+2y,c+2z),(x,y,z)) =0

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  2. Prove that the determinant {:|( x, sin theta ,cos theta ),( -sin thet...

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  3. Without expanding the determinant, prove that {:|( a, a ^(2), bc ),( ...

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  4. Evaluate {:|( cos alpha cos beta , cos alpha sin beta , -sin alpha ),...

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  5. If a,b and c are real numbers, and Delta ={:|( b+c,C+a,a+b),( c+a,a...

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  6. Solve the equation {:|( x+a,x,x),(x,x+a,x),(x,x,x+a) |:}=0,ane 0

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  7. Prove that {:|( a^(2) , bc, ac+c^(2)),( a^(2) +ab,b^(2) ,ac),( ab,b^(2...

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  8. If A^(-1) ={:[( 3,-1,1),(-15,6,-5),(5,-2,2) ]:}and B= {:[( 1,2,-2),( -...

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  9. Evaluate {:|( x,y , x+y),( y,x+y,x),( x+y,x,y)|:}

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  10. Evaluate {:|(1,x,y),( 1,x+y,y),( 1,x,x+y)|:}

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  11. Using properties of determinants in Exercises prove that : {:|( alph...

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  12. {:|( x,x^(2) , 1+ px^(3) ),( y,y^(2) , 1+ py^(2)),( z,z^(2) , 1+pz^(2)...

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  13. {:|( 3a,-a+b,-a+c),( -b+a, 3b,-b+c) ,( -c+a,-c+b,3c) |:} = 3( a+b+c) (...

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  14. {:|( 1,1+p,1+p+q),( 2,3+2p,4+3p+2q),( 3,6+3p,10+6p+3q)|:}=1

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  15. {:|( sin alpha , cos alpha ,cos (alpha +delta) ),( sinbeta, cos beta,c...

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  16. Solve the system of equations (2)/( x) + ( 3)/( y) +(10 )/( z) = 4 ...

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  17. If a,b,c are in A.P. then the determinant {:|( x+2,x+3,x+2a),( x+3,...

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  18. If x,y,z are nonzero real number , then the inverse of matrix A= {:[( ...

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  19. Let A={:[( 1,sin theta , 1),( -sin theta , 1, sin theta ),( -1 ,-sin ...

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