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If 1=|(1, 1, 1),(x^2,y^2,z^2),(x, y ,z)|...

If `_1=|(1, 1, 1),(x^2,y^2,z^2),(x, y ,z)|a n d_2=|(1 ,1, 1),(y z, z x,x y), (x ,y, z)|,` without expanding prove that `_1=_2`

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D_1=D_2

`darrx darry darrz`

`D_2=|(1, 1, 1),(yz,zx,xy),(x, y ,z)|`

`C_1rarrC_1x , C_2rarrC_2y , C_3rarrC_3z`

...
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RD SHARMA-DETERMINANTS-Solved Examples And Exercises
  1. Show that matrix |(1,a,b+c),(1,b,c+a),(1,c, a+b)|=0

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  2. Without expanding evaluate the determinant |(x^2+x^(-1))^2(a^2-a^(-1))...

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  3. If 1=|(1, 1, 1),(x^2,y^2,z^2),(x, y ,z)|a n d2=|(1 ,1, 1),(y z, z x,x ...

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  4. Without expanding show that |[b^2c^2,b c, b+c],[c^2a^2,c a ,c+a ],[a^2...

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  5. Without expanding evaluate the determinant |[sinalpha,cosalpha,sin(alp...

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  6. If A+B+C=pi, then the value of |[sin(A+B+C),sin(A+C),cosC],[-sinB,0...

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  7. If Ar=|[1,r,2^r],[2,n,n^2],[n,(n(n+1))/2, 2^(n+1)]|, the value of sum(...

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  8. If the determinant |[a,b,2aalpha+3b],[b,c,2balpha+3c],[2aalpha+3b,2bal...

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  9. If a , b , c are distinct, then the value of x satisfying |(0,x^2-a, x...

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  10. Using the factor theorem it is found that a+b , b+c and c+a are thr...

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  11. Let |(x^2+3x,x-1,x+3),(x+1,-2x x-4),(x-3,x+4 ,3x)|=a x^4+b x^3+c x^2+e...

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  12. If Dk=|[1,n,n],[2k,n^2+n+2,n^2+n],[2k-1,n^2,n^2+n+2]| and sum(k=1)^...

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  13. If Delta1=|(1, 1, 1),(a, b, c),( a^2,b^2,c^2)|,Delta2=|(1,bc, a),(1,ca...

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  14. The value of the determinant |(a^2,a,1),(cosn x,cos(n+1)x,cos(n+2)x),(...

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  15. Evaluate: (i) |(5 ,4),(-2, 3)| (ii) |(sintheta,costheta),(-costheta,s...

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  16. Evaluate: (i) |(x^2+x y+y^2,x+y ),(x^2-x y+y^2,x-y)| (ii) |(1,(log)b ...

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  17. Evaluate D =|[2, 3,-2],[ 1, 2, 3],[-2, 1,-3]| by expanding it along th...

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  18. Evaluate the determinant D=|2 3-2 1 2 3-2 1-3| by expanding it alon...

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  19. Evaluate D =|[2, 3,-2],[ 1, 2, 3],[-2, 1,-3]| by using Sarrus diagram.

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  20. Evaluate =|-1 6-2 2 1 1 4 1-3| by two methods.

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