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Prove that : (i) |{:(a,c,a+c),(a+b,b,a...

Prove that :
(i) `|{:(a,c,a+c),(a+b,b,a),(b,b+c,c):}|=2 abc`
(ii) 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)`

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det[[a^(2),bc,ac+c^(2)a^(2)+ab,b^(2),acab,b^(2)+bc,c^(2)]]=4a^(2)b^(2)c^(2)

Prove that ,,a^(2),bc,ac+c^(2)a^(2)+ab,b^(2),acab,b^(2)+bc,c^(2)]|=4a^(2)b^(2)c^(2)

Prove that : |{:(b^(2)c^(2),bc, b+c),(c^(2)a^(2),ca, c+a),(a^(2)b^(2),ab, a+b):}|=0

Prove that |{:(a^(2)+1,ab,ac),(ab,b^(2)+1,bc),(ac,bc,c^(2)+1):}|=1+a^(2)+b^(2)+c^(2) .

|(1,a^(2)+bc,a^(3)),(1,b^(2)+ac,b^(3)),(1,c^(2)+ab,c^(3))|=-(a-b)(b-c)(c-a)(a^(2)+b^(2)+c^(2))

Prove that |[-a^(2),ab,ac],[ba,-b^(2),bc],[ca,cb,-c^(2)]|=4a^(2)b^(2)c^(2)

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

Prove that: |(a^2+1, ab, ac),(ab, b^2+1, bc),(ac, bc, c^2+1)|=1+a^2+b^2+c^2

Prove that |[(a+b)^(2),ca,bc],[ca,(b+c)^(2),ab],[bc,ab,(c+a)^(2)]|=2abc(a+b+c)^(3)

Prove that: 2a^(2)+2b^(2)+2c^(2)-2ab-2bc-2ac=[(a-b)^(2)+(b-c)^(2)+(c-a)^(2)]

NAGEEN PRAKASHAN-DETERMINANTS-Exercise 4b
  1. Prove that : |{:(1,x,x^(3)),(1,y,y^(3)),(1,z,z^(3)):}|

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

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  3. Prove that : |{:(x+a,b,c),(a,x+b,c),(a,b,x+c):}|=x^(2)(x+a+b+c)

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  4. Prove that : |{:(x-y-z,2x,2x),(2y,y-z-x,2y),(2z,2z,z-x-y):}|

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

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

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

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  8. Prove that : |{:(a^(2),b^(2)+c^(2),bc),(b^(2),c^(2)+a^(2),ca),(c^(2),a...

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  9. Prove that : (i) |{:(a,c,a+c),(a+b,b,a),(b,b+c,c):}|=2 abc (ii) Pr...

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  10. Find the value of |{:(a,b,c),(a^(2),b^(2),c^(2)),(bc,ca,ab):}|

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  11. Prove that : |{:(0,a-b,a-c),(b-a,0,b-c),(c-a,c-b,0):}|=0

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  12. Find the value of x |[x,-6,-1],[2,-3x,x-3],[-3,2x,x+2]|=0

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  13. (i) Solve the equation |{:(x-2,2x-3,3x-4),(x-4,2x-9,3x-16),(x-8,2x-...

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  14. Solve the equation |{:(x+a,c+b,x+c),(x+b,x+c,x+a),(x+c,x+a,x+b):}|=0

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  15. One root of the equation |(3x-8, 3, 3),(3,3x-8, 3),(3,3,3x-8)|=0 si (A...

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  16. If 2 s=a+b+c, then show that : |{:(a^(2),(s-a)^(2),(s-a)^(2)),((s-b)^(...

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  17. If the sides of a DeltaABC and a, b, c and |{:(a^(2),b^(2),c^(2)),((a+...

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  18. If the pth, qth and rth terms of a G.P, are x,y and z repectively, the...

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  19. Prove that : |{:(1,1,1),(x,y,z),(x^(3),y^(3),z^(3)):}|=(x-y)(y-z)(x+y+...

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

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