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|(a-b-c, 2a, 2a),(2b, b-c-a,2b),(2c,2c,c...

`|(a-b-c, 2a, 2a),(2b, b-c-a,2b),(2c,2c,c-a-b)| = (a + b + c)^(3)`.

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|[1, a, a^2-b c],[1,b, b^2-c a],[1,c, c^2-a b]|=

The determinant |(b^2-ab,b-c,-ac),(ab-a^2,a-b,b^2-ab),(bc-ac,c-a,ab-a^2)| equals :

a. Minimize z =-3x+4y subject to constraints. x+2yle8 3x+2yle12 xge0, yge0 by graphical method. b. Prove that {:abs((1,a,a^2),(1,b,b^2),(1,c ,c^2)):} = (a - b)(b-c)(c-a)

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)) |:}

If a^2+b^2+c^2=0 and |(b^2+c^2,ab,ac),(ab,c^2+a^2,bc),(ac,bc,a^2+b^2)|=ka^2b^2c^2 , then the value of k is :

If alpha=|(a+b,b+c,c+a),(b+c,c+a,a+b),(c+a,a+b,b+c)|,beta=|(a,b,c),(b,c,a),(c,a,b)| , then

Prove that |(1,a^2,bc),(a,b^2,ca),(1,c^2,ab)|=(a-b)(b-c)(c-a)

Prove that |{:(,1,a,a^(2)),(,1,b,b^(2)),(,1,c,c^(2)):}|=(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)

Using the factor theorem it is found that b + c , c + a and a + b are three factors of the determine : |(-2a,a+b,a+c),(b+a,-2b,b+c),(c+a,c+b,-2c)| . The other factor in the value of the determine is :

SUBHASH PUBLICATION-DETERMINANT-FOUR MARKS QUESTIONS WITH ANSWERS
  1. Prove that |(1,1,1),(a,b,c),(a^3,b^3,c^3)| = (a - b)(b-c)(c-a)(a+b+c).

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  2. Prove that |(x,x^2,yz),(y,y^2,zx),(z,z^2,xy)|= (x-y)(y-z)(z-x)(xy + yz...

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  3. |(x+4, 2x, 2x),(2x, x +4, 2x),(2x, 2x ,x+4)| = (5x + 4)(4 - x)^(2).

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  4. |(y +k, y , y),(y, y + k, y),(y, y, y+k)|= k^(2)(3y + k).

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  5. |(a-b-c, 2a, 2a),(2b, b-c-a,2b),(2c,2c,c-a-b)| = (a + b + c)^(3).

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  6. Prove that |{:(,x+y+2z,x,y),(,z,y+z+2x,y),(,z,x,z+x+2y):}|=2(x+y+z)^(3...

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  7. Prove that |(1,x,x^2),(x^2,1,x),(x,x^2,1)|=(1-x^3)^2.

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  8. |(1+a^(2)-b^(2), 2ab, -2b),(2a, 1 -a^(2)+b^(2),2a),(2b, -2a, 1-a^2-b^2...

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  9. |(a^(2)+1,ab,ac),(ab,b^2+1,bc),(ca,cb,c^2+1)|= 1 + a^2 + b^2 + c^2.

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  10. Prove that |(-a^2,ab,ac),(bc,-b^2,bc),(ca,cb,-c^2)|=4a^(2)b^(2) c^(2).

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  11. |(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,...

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  12. |(1,bc,a(b+c)),(1,ca,b(c+a)),(1,ab,c(a+b))|= 0

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  13. Show that |(1+a,1,1),(1,1+b,1),(1,1,1+c)| = abc(1 + 1/a + 1/b + 1/c) =...

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  14. Evaluate Delta -|(1,a,bc),(1,b,ca),(1,c,ab)|

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  15. Prove that |(b+c,a,a),(b,c+a,b),(c,c,a+b)|= 4ac

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  16. If x,y,z are different and Delta = |(x, x^2, 1+x^3),(y,y^2,1+y^3),(z,z...

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  17. |(a - b, b-c,c-a),(b-c,c-a,a-b),(c-a,a-b,b-c)|=0

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  18. |(0,a,-b),(-a,0,-c),(b,c,0)| = 0

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  19. |(x,a,x+a),(y,b,y+b),(z,c,z+c)| = 0

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  20. Show that |(x,p,q),(p,x,x),(q,q,x)| = (x - p)(x^(2) + px - 2q^2)

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