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Show that: |[b+c,c+a ,a+b],[ q+r, r+p, p...

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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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]|

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|

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):}|

If A=|[a, b, c],[ x, y, z],[ p, q, r]| and B=|[q, -b, y],[ -p, a, -x],[ r,-c, z]| , without expanding or evaluating A and B , show that A+B=0 .

If Delta_1=|[a, b, c],[ x, y, z],[ p, q, r]| and Delta_2=|[q,-b, y],[-p, a,-x],[ r,-c, z]|, without expanding or evaluating Delta_1 and Delta_2 , show that Delta_1+Delta_2=0

Prove that |[a x-b y-c z, a y+b x, c x+a z], [a y+b x, b y-c z-a x, b z+c y],[c x+a z, b z+c y, c z-a x-b y]|=(x^2+y^2+z^2)(a^2+b^2+c^2)(a x+b y+c z)dot

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

If p+q+r=0=a+b+c , then the value of the determinant |[p a, q b, r c],[ q c ,r a, p b],[ r b, p c, q a]|i s (a) 0 (b) p a+q b+r c (c) 1 (d) none of these

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|

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|

RD SHARMA ENGLISH-DETERMINANTS-All Questions
  1. Show that: |[1 ,1+p,1+p+q],[2, 3+2p,1+3p+2q],[3, 6+3p,1+6p+3q]|=1 .

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  2. Show that: |[a, a+b ,a+b+c],[2a,3a+2b,4a+3b+2c],[3a,6a+3b, 10 a+6b+3c]...

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  3. Show that: |[b+c,c+a ,a+b],[ q+r, r+p, p+q],[ y+z ,z+x,x+y]|=2|[a, b, ...

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  4. Prove that: determinant \begin{vmatrix} 1+ a & 1 & 1 \\ 1 & 1+b...

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  5. If a ,\ b ,\ c are the roots of the equation x^3+p x+q=0 , then find...

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  6. Prove that: |(b+c)^2a^2a^2b^2(c+a)^2b^2c^2c^2(a+b)^2|=2a b c(a+b+c)^3

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  7. Show that: |(b+c)^2b a c a a b(c+a)^2c b a c b c(a+b)^2|=2a b c(a+b+c)...

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  8. Show that: |1+a^2-b^2 2a b-2b2a b1-a^2+b^2 2a2b-2a1-a^2-b^2|=(1+a^2+b^...

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  9. Show that: |b^2+c^2a b a c b a c^2+a^2b cc a c b a^2+b^2|=4a^2\ b^2\ c...

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  10. Prove that: |a b a x+b y b c b x+c y a x+b y b x+c y0|=(b^2-a c)(a x^2...

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

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  12. If a ,\ b ,\ c are roots of the equation x^3+p x+q=0 , prove that d...

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  13. Find dy/dx if y=cos^2x

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  14. If a+b+c!=0 and , \begin{vmatrix} a & b & c \\ b & c & a \\ ...

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  15. If a , b , c are real numbers, prove that \begin{vmatrix} a & b & ...

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  16. Show that: |a b-cc+b a+c b c-a a-bb+a c|=(a+b+c)(a^2+b^2+c^2)dot

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  17. Using properties of determinants. Prove that |[3a,-a+b,-a+c], [-b+a,3b...

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  18. Using properties of determinants, solve for x:|[a+x, a-x, a-x], [a-x...

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  19. Using properties of determinants, solve the following for x: |[x-2,...

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  20. If a,b,c are all distinct and |[a,a^3,a^4-1],[b,b^3,b^4-1],[c,c^3,c^4-...

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