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Prove that |[1, 1, 1]; [a, b, c]; [ bc+...

Prove that `
|[1, 1, 1];
[a, b, c];
[ bc+a^2, ac+b^2, ab+c^2]|`=`2(a-b)(b-c)(c-a)`

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(b) answer any one question : (i) prove that |[1,a,a^2],[1,b,b^2],[1,c,c^2]| =(a-b)(b-c)(c-a)

prove that , |{:(a,a^2,a^3+bc),(b,b^2,b^3+ca),(c,c^2,c^3+ab):}|=(a-b)(b-c)(c-a)(abc+bc+ca+ab)

Prove that . |[1,a,a^2-bc],[1,b,b^2-ca],[1,c,c^2-ab]|=0

Without expanding the determinant, prove that |{:(1,a,a^2),(1,b,b^2),(1,c,c^2):}|=(a-b)(b-c)(c-a) .

By using properties of determinants , show that : (i) {:|( 1,a,a^(2)),( 1,b,b^(2)),( 1,c,c^(2))|:}=(a-b)(b-c) (c-a) (ii) {:|( 1,1,1),( a,b,c) ,(a^(3) , b^(3), c^(3))|:} =( a-b) (b-c)( c-a) (a+b+c)

Prove |{:(1,a^2+bc,a^3),(1,b^2+ca,b^3),(1,c^2+ab,c^3):}|=-(a-b)(b-c)(c-a)(a^2+b^2+c^2)

Prove that [[1+a^2+a^4, 1+a b+a^2b^2 ,1+a c+a^2c^2],[ 1+a b+a^2b^2, 1+b^2+b^4, 1+b c+b^2c^2],[ 1+a c+a^2c^2, 1+b c+b^2c^2, 1+c^2+c^4]]=(a-b)^2(b-c)^2(c-a)^2

|{:(1,1,1),(a,b,c),(bc,ca,ab):}|=(a-b)(b-c)(c-a)

Prove that {:|( a^(2) , bc, ac+c^(2)),( a^(2) +ab,b^(2) ,ac),( ab,b^(2) +bc,c^(2)) |:} =4a^(2) b^(2) c^(2)

Without expanding the determinant, prove that {:|( a, a ^(2), bc ),( b ,b ^(2) , ca),( c, c ^(2) , ab ) |:} ={:|( 1, a^(2) , a^(3) ),( 1,b^(2) , b^(3) ),( 1, c^(2),c^(3)) |:}

CENGAGE PUBLICATION-DETERMINANTS-All Questions
  1. " if " |{:(a^(2),,b^(2),,c^(2)),((a+1)^(2),,(b+1)^(2),,(c+1)^(2)),((a-...

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  2. Statement 1: "Delta"= |[m y+n z, m q+n r, m b+n c],[k z-m x, k r-m ...

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  3. Prove that |[1, 1, 1]; [a, b, c]; [ bc+a^2, ac+b^2, ab+c^2]|=2(a-b)(b...

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  4. If =|sin theta cos var phi sin theta sin var phi cos theta cos theta c...

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  5. The system of equations -2x+y+z=a, x-2y+z=b, x+y-2z=c, has: (a)no so...

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  6. Show that if x1, x2, x3!=0 abs((x1+a1.b1,a1.b2,a1.b3),(a2.b1,x2+a2.b...

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  7. If determinant |[cos(theta+phi),-sin(theta+phi),cos2phi],[sin theta,co...

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  8. If Deltar =|[2^(r-1) ,2.3^(r-1), 4.5^(r-1)],[alpha,beta,gamma],[2^n-1,...

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  9. The values of k in R for which the system of equations x+k y+3z=0,k x...

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  10. A determinant of second order is made with the elements 0 and 1. Fi...

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  11. If a ,b , c are nonzero real numbers such that |[b c,c a, a b],[ c a, ...

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  12. Prove that the determinant |[x, sintheta,costheta],[-sintheta,-x,1],[...

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  13. Consider the determinant f(x)=|[0,x^2-a, x^3-b], [x^2+a,0,x^2+c], [x^4...

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  14. Find the value of |[1, 2, 4],[-1,3, 0],[4 ,1, 0]|

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  15. Consider the system of the equation k x+y+z=1,x+k y+z=k ,a n dx+y+k z=...

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  16. If x+y+z=0 prove that |[a x, b y, c z],[ c y, a z ,b x],[ b z ,c x ,a ...

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  17. Each question has four choices a, b, c and d, out of which only one is...

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  18. If a ,b ,c in R , then find the number of real roots of the equat...

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  19. Delta = |(a, a^2, 0),(1, 2a+b,(a+b)),(0, 1, 2a+3b)| is divisible by ...

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  20. Without expanding at any stage, prove that the value of each of the...

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