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Show that abs((a,b,c),(a-b,b-c,c-a),(b+c...

Show that `abs((a,b,c),(a-b,b-c,c-a),(b+c,c+a,a+b))` = `a^3+b^3+c^3`-3abc.

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|{:(a,b,c),(a-b,b-c,c-a),(b+c,c+a,a+b):}|=a^3+b^3+c^3-3abc

Using properties of determinants , prove that, |{:(a,b,c),(b,c,a),(c,a,b):}|=-(a^3+b^3+c^3-3abc) and hence show that, |{:(2bc-a^2," "c^2," "b^2),(" "c^2,2ca-b^2," "a^2),(" "b^2," "a^2,2ab-c^2):}|=(a^3+b^3+c^3-3abc)^2

|{:(b+c,a-c,a-b),(b-c,c+a,b-a),(c-b,c-a,a+b):}|=8abc

Show that: abs(((b+c)^2,a^2,bc) , ((c+a)^2,b^2,ca) , ((a+b)^2,c^2,ab))=(a^2+b^2+c^2)(a+b+c)(b-c)(c-a)(a-b)

|{:(b+c,a-b,a),(c+a,b-c,b),(a+b,c-a,c):}|=3abc-a^3-b^3-c^3

Show that |[a+b+2c,a,b],[c,b+c+2a,b],[c,a,c+a+2b]|=2(a+b+c)^3 .

Prove that, |(2a, a-b-c,2a),(2b,2b,b-c-a),(c-a-b,2c,2c)|=(a + b+c)^3

Show that: |[3a,-a+b,-a+c],[-b+a,3b,-b+c],[-c+a,-c+b,3c]|=3(a+b+c)(a b+b c+c a)dot

Prove that |[b+c ,a-b, a] ,[c+a, b-c ,b],[ a+b, c-a, c]|=3abc-a^3-b^3-c^3dot

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)

PATHFINDER-DETERMINATES-QUESTION BANK
  1. Without expanding,show that abs((x+1,x+2,x+a),(x+2,x+3,x+b),(x+3,x+4,x...

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  2. Solve fox x: abs((x,3),(5,2x))=abs((x,-4),(5,3))

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  3. Evaluateabs((1,omega,omega^2),(omega^2,1,omega),(omega^2,omega,1)) wer...

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  4. Using determainant find the area of the triangle formed by joining the...

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  5. If x+y+z=0 then show that abs((1,1,1),(x,y,z),(x^3,y^3,z^3)) = 0.

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  6. Evaluate the following determinantabs((x+lamda,x,x),(x,x+lamda,x),(x,x...

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  7. Without expanding prove that,abs((1,ab,1/a+1/b),(1,bc,1/b+1/c),(1,ca,1...

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  8. Show that abs((a,b,c),(a-b,b-c,c-a),(b+c,c+a,a+b)) = a^3+b^3+c^3-3abc.

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  9. If a,b c(all positive)are the p th,q thand r th terms repectively of a...

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  10. If p+-a, q+- b, r+-c and abs((p,b,c),(a,q,c),(a,b,r)) = 0 show that p...

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  11. find the value of :- (101)^2

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  12. Given that x =-9 is a root of abs((x,3,7),(2,x,2),(7,6,x))= 0 .Find th...

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  13. Show that abs(((a-x)^2,(a-y)^2,(a-z)^2) , ((b-x)^2,(b-y)^2,(b-z)^2) ...

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  14. Find the ratio in which the line joining the pts (-3,4,8) and (5,-6,4)...

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  15. Show that: abs(((b+c)^2,a^2,bc) , ((c+a)^2,b^2,ca) , ((a+b)^2,c^2,ab)...

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  16. If A,B,C be the angles of a triangle, then prove that ,abs((-1,cosC,co...

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  17. int(sinx)/(1+cosx)dx= ?

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  18. If x,y,z are all distinct and if abs((x,x^2,1+x^3),(y,y^2,1+y^3),(z,z^...

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  19. Prove that,abs((a^2+1,ab,ac),(ab,b^2+1,bc),(ca,cb,c^2+1))=1+a^2+b^2+c^...

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  20. Prove that,abs((-a^2,ab,ac),(ba,-b^2,bc),(ca,cb,-c^2))=4a^2b^2c^2.

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