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Let △=|(A x,x^2, 1),(B y, y^2, 1),(C z, ...

Let `△=|(A x,x^2, 1),(B y, y^2, 1),(C z, z^2, 1)|` and`△_1`=`|(A,B, C),( x, y, z),( yz, zx, xy)|,` then show that `△_1=△`.

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Given, `Delta=|(A x , x^{2}, 1)( B y, y^{2} , 1) ( C z ,z^{2} ,1)|`

On taking `x, y` and `z` common from `R_{1}, R_{2}`, and `R_{3}` respectively, we get

`Delta=x y z``| (A ,x , 1 / x) , (B ,y ,1 / y ), (C , z , 1 / z) |`

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Let Delta_(1)=|{:(Ax,x^(2),1),(By,y^(2),1),(Cz,z^(2),1):}| and Delta_(2)=|{:(A,B,C),(x,y,z),(yz,zx,xy):}| , then :

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RD SHARMA-DETERMINANTS-Solved Examples And Exercises
  1. Which of the following is not correct in a given determinant of A , wh...

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  2. Let |(x,2,x),(x^2,x,6),(x,x,6)|=a x^4+b x^3+c x^2+dx+edot Then, the va...

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  3. Let △=|(A x,x^2, 1),(B y, y^2, 1),(C z, z^2, 1)| and△1=|(A,B, C),( x, ...

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  4. If Delta1=|[a, b, c],[ x, y, z],[ p, q, r]| and Delta2=|[q,-b, y],[-p,...

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  5. Without expanding show that: =|cos e c^2thetacot^2theta1cot^2thetacos ...

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  6. Find the value of the determinant =|2 3 4 5 6 8 6x9x 12 x|dot

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  7. Without expanding evaluate the determinant |[41,1,5],[79,7,9],[29,5,3]...

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  8. Show that matrix |(1,a,b+c),(1,b,c+a),(1,c, a+b)|=0

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

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

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

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

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

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

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

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  18. 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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  19. 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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  20. If Delta1=|(1, 1, 1),(a, b, c),( a^2,b^2,c^2)|,Delta2=|(1,bc, a),(1,ca...

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