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Without expanding, prove that |(a+b x,...

Without expanding, prove that `|(a+b x, c+dx, p+q x), (a x+b, c x+d, p x+q),( u, v, w)|=(1-x^2)|(a, c, p),( b, d, q),( u, v, w)|` .

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Without expanding, prove that |a b c x y z p q r|=|x y z p q r a b c|=|y b q x a p z c r|

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Using the property of determinants and without expanding, prove that: |[b+c, q+r, y+z],[ c+a, r+p, z+x],[ a+b, p+q, x+y]|=2|[a, p, x],[ b, q ,y],[ c, r, z]|

Prove that: |{:(a, b, c), (x, y, z), (p, q, r):}|=|{:(y, b, q), (x, a, p), (z, c, r):}|

Using properties of determinants, prove that |(b+c,q+r,y+z),(c+a,r+p,z+x),(c+b,p+q,x+y)|=2|(a,p,x),(b,q,y),(c,r,z)|

Without, prove that : |{:(1+b,b+c,c+a),(p+q,q+r,r+p),(x+y,y+z,z+x),:}|=2|{:(a,b,c),(p,q,r),(x,y,z):}|

Using the property of determinants and without expanding, prove that: |b+c q+r y+z c+a r+p z+x a+b p+q x+y|=2|a p x b q y c r z|

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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]| .

Prove that [a b c][u v w]=|[a*u, b*u, c*u], [a*v, b*v, c*v], [a*w, b*w, c*w]|

RD SHARMA ENGLISH-DETERMINANTS-All Questions
  1. Prove that: |[0,a,-b],[-a,0,-c],[ b, c,0]|=0 .

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  2. Without expanding or evaluating show that |(0,b-a, c-a),( a-b,0,c-b)...

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  3. Without expanding, prove that |(a+b x, c+dx, p+q x), (a x+b, c x+d, ...

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  4. Prove that: |-a^2 ab ac ba -b^2 bc ...

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  5. Prove that: |(1, 1, 1),( 1, 1+x,1) ,(1, 1, 1+y)|=x y .

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  6. Evaluate: |(1,a ,a^2), (1,b,b^2) ,(1,c,c^2)| .

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  7. Show that : |[x, y, z ],[x^2,y^2,z^2],[x^3,y^3,z^3]|=x y z(x-y)(y-z)(z...

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  8. Prove that: |[alpha,beta,gamma],[alpha^2,beta^2,gamma^2],[beta+gamma,g...

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  9. In a triangleABC, if |(1,1,1),(1+sinA,1+sinB,1+sinC),(sinA+sin^2A,sinB...

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  10. In a A B C ,if|1 1 1 1+cosA 1+cosB 1+cosC cos^2A+A cos^2B+cosB cos^2...

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  11. Show that |[a ,b ,c],[ a^2,b^2,c^2],[bc, ca, ab]|=|[1, 1, 1],[a^2,b^2,...

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  12. If x!=y!=z and |x x^2 1+x^3 y y^2 1+y^3 z z^2 1+z^3|=0 , then prove th...

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  13. For any scalar p prove that =|[x,x^2, 1+p x^3],[y, y^2, 1+p y^3],[z, z...

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  14. Using properties of determinants, show that |1 a a^2 -b c 1 b ...

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  15. Prove that: |a^2+2a2a+1 1 2a+1a+2 1 3 3 1|=(a-1)^3 .

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  16. Let a ,\ b and c denote the sides B C ,\ C A and A B respectively of ...

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  17. If f(x)=|[a-1, 0,a] , [x,a-1,a], [x^2a, x, a]| , using properties of ...

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  18. Show that: x p q p x q q q xx-p)(x^2+p x-2q^2) .

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  19. If m in N and mgeq2 prove that: |1 1 1\ ^m C1\ ^(m+1)C1\ ^(m+2)C1\ ^m...

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  20. Evaluate: =|10 ! 11 ! 12 ! 11 ! 12 ! 13 ! 12 ! 13 ! 14 !|

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