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Without expending, prove that : (i) |{...

Without expending, prove that :
(i) `|{:(b^(2)c^(2),bc,b+c),(c^(2)a^(2),ca,c+a),(a^(2)b^(2),ab,a+b):}|=0`
(ii)`|{:(x,y,z),(x^(2),y^(2),z^(2)),(yz,zx,xy):}|=|{:(1,1,1),(x^(2),y^(2),z^(2)),(x^(3),y^(3),z^(3)):}|`
(iii) `|{:(1,2x,x^(2)-yz),(1,y,y^(2)-zx),(1,z,z^(2)-xy):}|`
`("Taking 2, 3 and "2/3"common from "C_(1),C_(2)" and "C_(3)" repectively")`
`=4xx49 ["from eq.(1)"]`
=198.
(iv) `|{:(sinx,cosx,sin(x+alpha)),(siny,cosy,sin(y+alpha)),(sinz,cosz,sin(z+alpha)):}|=0`

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NAGEEN PRAKASHAN-DETERMINANTS-Exercise 4b
  1. Find the values of the following determinants : (i) |{:(12,3,4),(16,...

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  2. Prove that : |{:(a+b,b+c,c+a),(c,a,b),(1,1,1):}|

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  3. Without expending, prove that : (i) |{:(b^(2)c^(2),bc,b+c),(c^(2)a^(...

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  4. Prove that : |{:(x+a,x+2a,x+3a),(x+2a,x+3a,x+4a),(x+4a,x+5a,x+6a):}|=0

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  5. Find the value of : |{:(1,x,yz),(1,y,zx),(1,z,xy):}|

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  6. The value of Det[[x,x^2,x^3],[y,y^2,y^3],[z,z^2,z^3]] will be

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  7. Prove that : |{:(1,x,x^(3)),(1,y,y^(3)),(1,z,z^(3)):}|

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

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

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

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

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

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

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

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

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

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

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  20. 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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