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If a , b , c are all positive and are pth ,qth and rth terms of a G.P., then show that `=|(loga, p,1),(logb, q,1),(logc ,r,1)|=0`

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To show that the determinant \[ D = \begin{vmatrix} \log a & p & 1 \\ \log b & q & 1 \\ \log c & r & 1 \end{vmatrix} = 0 ...
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RD SHARMA ENGLISH-DETERMINANTS-All Questions
  1. Solve : |[x-2,2x-3,3x], [-4x-4,2x-9,3x-16], [x-8,2x-27,3x-64]| = 0

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  2. If x+y+z=0 , prove that |[xa, yb,zc],[yc, za, xb], [zb, xc, ya]|=x y ...

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  3. If a , b , c are all positive and are pth ,qth and rth terms of a G.P....

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  4. Prove that: |[bc-a^2,ca-b^2,ab-c^2],[ca-b^2,ab-c^2,bc-a^2],[ab-c^2,bc-...

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  5. |[bc-a^2,ca-b^2,ab-c^2],[ca-b^2,ab-c^2,bc-a^2],[ab-c^2,bc-a^2,ca-b^2]|...

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  6. Prove that: |[(b+c)^2,a^2,a^2],[b^2,(c+a)^2,b^2],[c^2,c^2,(a+b)^2]|=2a...

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  7. Show that |[b^2+c^2,ab,ac],[ba,c^2+a^2,bc],[ca,cb,a^2+b^2]|=4a^2b^2c^2

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  8. Prove that: |[a, b, ax+by],[ b, c, bx+cy], [ax+by, bx+cy,0]|=(b^2-a c)...

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  9. Without expanding the determinant, show that (a+b+c) is a factor of th...

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  10. If m in N and mgeq2, prove that: |1 1 1m(C1)m+1(C1)m+2(C1)m(C2)m+1(C2...

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  11. Evaluate: =|(10 !, 11 !, 12 !), (11 !, 12 !, 13 !), (12 !, 13 !, 14 !)...

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  12. Show that: |b+c c+a a+b q+r r+p p+q y+z z+x x+y|=2|a b c p q r x y z|

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  13. Prove that |[1+a,1, 1], [1,1+b,1], [1, 1, 1+c]|=a b c(1+1/a+1/b+1/c)=a...

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  14. If a , b , c , are roots of the equation x^3+p x+q=0, prove that |[a...

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  15. If a+b+c!=0 and |a b c b c a c a b|=0 , then prove that a=b=cdot

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  16. Let a , band c detnote the sides BC,CA andAB respectively of triangl...

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  17. Prove that |[a^2+2a,2a+1,1],[2a+1,a+2,1],[3,3,1]|=(a-1)^3

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  18. Using properties of determinant show that: |[1 , a , bc] , [1 , b , ca...

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  19. Using properties of determinants, show that triangle ABC is isosceles,...

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  20. In a triangleABC, if |[1,1,1][1+sinA,1+sinB,1+sinC],[sinA+sin^2A, sinB...

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