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For any scalar p prove that =|xx^2 1+p x...

For any scalar `p` prove that `=|xx^2 1+p x^3y y^2 1+p y^3z z^2 1+p z^3|=(1+p x y z)(x-y)(y-z)(z-x)` .

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MODERN PUBLICATION-DETERMINANTS-Exercise 4(b) (LONG ANSWER TYPE QUESTIONS (II))
  1. Prove the following : |{:(2ab,a^(2),b^(2)),(a^(2),b^(2),2ab),(b^(2),...

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  2. Prove the following : |(1,x,x^(2)-yz),(1,y,y^(2)-zx),(1,z,z^(2)-xy)|=...

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  3. For any scalar p prove that =|xx^2 1+p x^3y y^2 1+p y^3z z^2 1+p z^3|=...

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  4. |(x+y+z,-z,-y),(-z,x+y+z,-x),(-y,-x,x+y+z)|=2(x+y)(y+z)(z+x)

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  5. Prove: |2y y-z-x2y2z2z z-x-y x-y-z2x2x|=(x+y+z)^3

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

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  7. Show that: |3a-a+b-a+c-b+a3b-b+c-c+a-c+b3c|=3(a+b+c)(a b+b c+c a)dot

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  8. Using properties of determinants. Find the value of 'x' |(4-x,4+x,4+...

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  9. Solve: |a+x a-x a-x a-x a+x a-x a-x a-x a+x|=0

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  10. Prove that |[x, sintheta, costheta],[-sintheta, -x, 1],[costheta, 1, ...

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  11. Prove that |{:(1+a,1,1),(1,1+b,1),(1,1,1+c):}| =abc (1+(1)/(a)+(1)/...

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

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  14. |[(b+c)^2,ab,ca],[ab,(a+c)^2,bc],[ac,bc,(a+b)^2]|=2abc(a+b+c)^3

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  15. Show that Delta=|(y+z)^2x y z xx y(x+z)^2y z x z y z(x+y)^2|=2x y z(x+...

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  16. If a, b, c are positive and unequal, show that value of the determinan...

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  17. Using properties of determinants, prove that |a a+b a+b+c2a3a+2b4a...

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  18. Using properties of determinants. Prove that |1 1+p1+p+q2 3+2p4+3p+2q3...

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  19. Q. |(x+y,x,x),(15x+4y,4x,2x),(10x +8y,8x,3x)|=x^3

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  20. Prove |[-bc, b^2+bc, c^2+bc] , [a^2+ac, -ac, c^2+ac] , [a^2+ab, b^2+ab...

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