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Using the properties of determinants, prove that following `|(a-b-c,2a,2a),(2b,b-c-a,2b),(2c,2c,c-a-b)|=(a+b+c)^3`

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By using properties of determinants , show that : (i) {:|( a-b-c, 2a, 2a),( 2b, b-c-a, 2b ) , ( 2c,2c, c-a-b) |:} = ( a+b+c)^(3) (ii) {:|( x+y+2z, x,y ),( z, y+z+2x, y ),( z,x,z+x+2y )|:} =2( x+y+z)^(3)

Prove that, |(2a, a-b-c,2a),(2b,2b,b-c-a),(c-a-b,2c,2c)|=(a + b+c)^3

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Using properties of determinant prove that |[a+b+c,-c,-b],[-c, a+b+c,-a],[-b,-a, a+b+c]|=2(a+b)(b+c)(c+a)

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By using properties of determinants , show that : (i) {:|( 1,a,a^(2)),( 1,b,b^(2)),( 1,c,c^(2))|:}=(a-b)(b-c) (c-a) (ii) {:|( 1,1,1),( a,b,c) ,(a^(3) , b^(3), c^(3))|:} =( a-b) (b-c)( c-a) (a+b+c)

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CENGAGE PUBLICATION-DETERMINANTS-All Questions
  1. Prove that |a ,b+c, a^2,b, c+a, b^2,c, a+b, c^2|=-(a+b+c)xx(a-b)(b-c)...

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  2. If a >0 and discriminant of a x^2+2b x+c is negative, then |[a,b,ax+b]...

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  3. Using the properties of determinants, prove that following |(a-b-c,2a,...

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  4. If a1,a2,a3,.....an.... are in G.P. then the determinant Delta=|[log a...

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  5. Prove that,abs((a^2+1,ab,ac),(ab,b^2+1,bc),(ca,cb,c^2+1))=1+a^2+b^2+c^...

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  6. Let vec ar=xr hat i+yr hat j+zr hat k ,r=1,2,3 be three mutually perp...

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  7. The number of distinct roots of abs((sinx, cosx, cosx),(cosx, sinx, co...

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  8. Let a ,b ,c be real numbers with a^2+b^2+c^2=1. Show that the eq...

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  9. If lines p x+q y+r=0,q x+r y+p=0 and r x+p y+q=0 are concurrent, the...

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  10. If plambda^4+qlambda^3+rlambda^2+slambda+t=|[lambda^2+3lambda, lambda-...

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  11. Find the value of lambdaif2x^2+7x y+3y^2+8x+14 y+lambda=0 represent a ...

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  12. If x ,y ,z are different from zero and "Delta"=|[a, b-y ,c-z],[ a-x, b...

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  13. If A , B ,C are angles of a triangle, then the value of |[e^(2i A),e^(...

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  14. If a^2+b^2+c^2=1, then prove that |[a^2 +(b^2+c^2)cosθ, ab(1−cosθ), ...

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  15. For the equation |[1,x,x^2],[x^2, 1,x],[x,x^2, 1]|=0, a) There are ex...

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  16. Let Deltar=|[r-1,n,6],[(r-1)^2,2n^2,4n-2],[(r-1)^3,3n^3,3n^2-3n]|. Sho...

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  17. Let m be a positive integer and Deltar=|[2r-1,.^mCr,1],[m^2-1,2^m,m+1]...

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  18. a^(−1)+b^(−1)+c^(−1)=0 such that |[1+a,1,1,],[1,1+b,1,],[1,1,1+c,]| =△...

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  19. Find the area of a triangle whose vertices are A(3,2),B(11 ,8) and C(8...

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  20. If Dk=|[1, n, n];[ 2k, n^2+n+1, n^2+n];[ 2k-1, n^2, n^2+n+1]|a n...

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