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Show that |{:(1+a^(2)-b^(2),,2ab,,-2...

Show that
`|{:(1+a^(2)-b^(2),,2ab,,-2b),(2ab,,1-a^(2)+b^(2),,2a),(2b,,-2a,,1-a^(2)-b^(2)):}| = (1+a^(2) +b^(2))^(3)`

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a) Prove that int_(a)^(b)f(x)dx=int_(a)^(b)f(a+b-x)dx" and evaluate "int_(pi//6)^(pi//3)(dx)/(1+sqrt(tanx)) b) Prove that |{:(1+a^(2)-b^(2), 2ab, -2b), (2ab, 1-a^(2)+b^(2), 2a), (2, -2a, 1-a^(2)-b^(2)):}|=(1+a^(2)+b^(2))^(3)

|(1+a^(2)-b^(2), 2ab, -2b),(2a, 1 -a^(2)+b^(2),2a),(2b, -2a, 1-a^2-b^2)|=(1 + a^2 + b^2)^(3) .

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)

|(b^(2)c^(2),bc,b+c),(c^(2)a^(2),ca,c+a),(a^(2)+b^(2),ab,a+b)|=

(cos2A)/(a^(2))-(cos2B)/(b^(2))=1/(a^(2))-1/(b^(2))

(cos2A)/(a^(2))-(cos2B)/(b^(2))=1/(a^(2))-1/(b^(2))

Prove that abs{:(a^(2) + 1, ab , ac),(ab, b^(2) + 1, bc),(ca, cb, c^(2) +1):}=1 + a^(2) + b^(2) +c^(2)

Without expanding the determinant, prove that {:|( a, a ^(2), bc ),( b ,b ^(2) , ca),( c, c ^(2) , ab ) |:} ={:|( 1, a^(2) , a^(3) ),( 1,b^(2) , b^(3) ),( 1, c^(2),c^(3)) |:}

Prove that |{:(,1,a,a^(2)),(,1,b,b^(2)),(,1,c,c^(2)):}|=(a-b)(b-c)(c-a)

OSWAAL PUBLICATION-DETERMINANTS-TOPIC -1 DETERMINANTS, MINORS & COFACTORS (LONG ANSWER TYPE QUESTIONS-I)
  1. 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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  2. |[1+a^2-b^2,2ab,-2b],[2ab,1-a^2+b^2,2a],[2b,-2a,1-a^2-b^2]|=(1+a^2+b^2...

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  3. Show that |{:(1+a^(2)-b^(2),,2ab,,-2b),(2ab,,1-a^(2)+b^(2),,2a),(2...

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  4. Prove that |(1,a,a^2),(1,b,b^2),(1,c,c^2)|=(a-b)(b-c)(c-a)

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  5. |[b+c, a,a] , [b,c+a,b] , [c,c,a+b]|=4abc

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  6. Using properties of determinants, prove that |a+x y z x a+y z x y a+z...

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  7. Using Properties of determinants, prove that {:|(x+lamda,2x,2x),(2...

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

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

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  10. Using Properties of determinants, prove that: {:|(x^2+1,xy,yz),(xy,y...

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

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  12. Using Properties of determinants, prove that {:|(x+y,x,x),(5x+4y,4x,...

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  13. Using Properties of determinants, prove that: {:|(b+c,c+a,a+b),(q+r,...

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  14. IF a+b+c ne 0 and {:|(a,b,c),(b,c,a),(c,a,b)|=0, then using properties...

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  15. Prove that |[1,a,a^3],[1,b,b^3],[1,c,c^3]|=(a-b)(b-c)(a+b+c)

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  16. Prove that: (i) |{:(,1,1,1),(,a,b,c),(,a^(3),b^(3),c^(3)):}|=(a-b)(b...

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

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  18. Find the equation of the line joining A( 1,3) and B (0,0) using det...

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  19. Using properties of determinants, prove the following: |xx+y x+2y\...

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  20. Using properties of determinants, prove the following: |alphabetag...

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