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Prove that the following. [[b+c,a,a],[b,...

Prove that the following. `[[b+c,a,a],[b,c+a,b],[c,c,a+b]]`=4ab

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Prove that the following. [[a,a^2,a^3],[b,b^2,b^3],[c,c^2,c^3]] = abc(a-b)(b-c)(c-a)

Prove that the following. [[b^2+c^2,ab,ac],[ab,c^2+a^2,bc],[ca,cb,a^2+b^2]]=4a^2b^2c^2

Prove the following: [[b+c,a+b,a],[c+a,b+c,b],[a+b,c+a,c]] = a^3+b^3+c^3-3abc

Prove that the following. [[1,1,1],[b+c,c+a,c+a],[b^2+c^2,c^2+a^2,a^2+b^2]] =(b-c)(c-a)(a-b)

Prove the following: [[-a^2,ab,ac],[ab,-b^2,bc],[ac,bc,-c^2]]=4a^2b^2c^2

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Prove the following : [[1,bc,a(b+c)],[1,ca,b(c+a)],[1,ab,c(a+b)]] =0

Prove the following: [[(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)

ARIHANT PUBLICATION-DETERMINANTS -PART -1 QUESTION FOR PRACTICE
  1. IF a,b and c are real numbers and Delta=|{:(b+c,c+a,a+b),(c+a,a+b,b+c)...

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  2. Using the properties of determinants prove that |{:(a+b+2c,a,b),(c,b...

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  3. Find the value of x if |(2x,3),(1,x)|=|(3,4),(-1,2)|

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  4. Prove that |[x+y, x, x],[5x+4y, 4x, 2x], [10x+8y, 8x, 3x]|=x^3

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

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  6. Prove that abs[[b+c,c+a,a+b],[q+r,r+p,p+q],[y+z,z+x,x+y]]=2abs[[a,b,c]...

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  7. Show that Delta =Delta1 where Delta=|{:(Ax,x^2,1),(By,y^2,1),(Cz,z^2...

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  8. IFDelta=|{:(1,x,x^2),(1,y,y^2),(1,z,z^2):}| and Delta1=|{:(1,1,1),(yz,...

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  9. Using properties of determinants, prove that |{:(y^2z^2,yz,y+z),(z^2x^...

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  10. If f(x)=|{:(a,-1,0),(ax,a,-1),(ax^2,ax,a):}| then using properties of...

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  11. Find the maximum value of Delta=|{:(1,1,1),(1,1+sin theta,1),(1,1,1+co...

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  12. Prove that the following. [[b+c,a,a],[b,c+a,b],[c,c,a+b]]=4ab

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  13. If a, b and c are all positive real, then prove that minimum value of ...

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  14. Prove that |{:(x^2+1,xy,xz),(xy,y^2+1,yz),(xz,yz,z^2+1):}|=1+x^2+y^2+z...

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  15. Answer any three questions Using properties of determinants, prove t...

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  16. Show that |{:((b+c)^2,a^2,a^2),(b^2,(c+a)^2,b^2),(c^2,c^2,(a+b)^2):}...

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  17. Prove that |{:(x,y,z),(x^2,y^2,z^2),(yz,zx,xy):}|=|{:(1,1,1),(x^2,y^2,...

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  18. Prove that abs((a,b,c),(a^2,b^2,c^2),(bc,ca,ab))=(a-b)(b-c)(c-a)(ab+bc...

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  19. If a, b and c are all positive real, then prove that minimum value of ...

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  20. Prove the following: [[a+b+c,-c,-b],[-c,a+b+c,-a],[-b,-a,a+b+c]] =...

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