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the value of the determinant |{:((a(1...

the value of the determinant
`|{:((a_(1)-b_(1))^(2),,(a_(1)-b_(2))^(2),,(a_(1)-b_(3))^(2),,(a_(1)-b_(4))^(2)),((a_(2)-b_(1))^(2),,(a_(2)-b_(2))^(2) ,,(a_(2)-b_(3))^(2),,(a_(3)-b_(4))^(2)),((a_(3)-b_(1))^(2),,(a_(3)-b_(2))^(2),,(a_(3)-b_(3))^(2),,(a_(3)-b_(4))^(2)),((a_(4)-b_(1))^(2),,(a_(4)-b_(2))^(2),,(a_(4)-b_(3))^(2),,(a_(4)-b_(4))^(2)):}|` is

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det[[2a_(1)b_(1),a_(1)b_(2)+a_(2)b_(1),a_(1)b_(3)+a_(3)b_(1)a_(1)b_(2)+a_(2)b_(1),2a_(2)b_(2),a_(2)b_(3)+a_(3)b_(2)a_(1)b_(3)+a_(3)b_(1),a_(3)b_(2)+a_(2)b_(3),2a_(3)b_(3)]]=

Show that if x_(1),x_(2),x_(3) ne 0 |{:(x_(1) +a_(1)b_(1),,a_(1)b_(2),,a_(1)b_(3)),(a_(2)b_(1),,x_(2)+a_(2)b_(2),,a_(2)b_(3)),(a_(3)b_(1),,a_(3)b_(2),,x_(3)+a_(3)b_(3)):}| =x_(1)x_(2)x_(3) (1+(a_(1)b_(1))/(x_(1))+(a_(2)b_(2))/(x_(2))+(a_(3)b_(3))/(x_(3)))

prove that (1)/((a-a_(1))^(2)),(1)/(a-a_(1)),(1)/(a_(1))(1)/((a-a_(2))^(2)),(1)/(a-a_(2)),(1)/(a_(2))(1)/((a-a_(3))^(2)),(1)/(a-a_(3)),(1)/(a_(3))]|=(-a^(2)(a_(1)-a_(2))(a_(2)-a_(3))(a_(3)-a_(1)))/(a_(1)a_(2)a_(3)(a-a_(1))^(2)(a-a_(2))^(2)(a-a_(3))^(2))

Suppose a_(1),a_(2),a_(3) are in A.P. and b_(1),b_(2),b_(3) are in H.P. and let /_\=|(a_(1)-b_(1),a_(1)-b_(2),a_(1)-b_(3)),(a_(2)-b_(1),a_(2)-b_(2),a_(2)-b_(3)),(a_(3)-b_(1),a_(3)-b_(2),a_(3)-b_(3))| then

The determinant |(b_(1)+c_(1),c_(1)+a_(1),a_(1)+b_(1)),(b_(2)+c_(2),c_(2)+a_(2),a_(2)+b_(2)),(b_(3)+c_(3),c_(3)+a_(3),a_(3)+b_(3))|

The value of the determinant Delta = |(1 + a_(1) b_(1),1 + a_(1) b_(2),1 + a_(1) b_(3)),(1 + a_(2) b_(1),1 + a_(2) b_(2),1 + a_(2) b_(3)),(1 + a_(3) b_(1) ,1 + a_(3) b_(2),1 + a_(3) b_(3))| , is

If f(x)=(a_(1)x+b_(1))^(2)+(a_(2)x+b_(2))^(2)+...+(a_(n)x+b_(n))^(2), then prove that (a_(1)b_(1)+a_(2)b_(2)+...+a_(n)b_(n))^(2)<=(a_(1)^(2)+a_(2)^(2)+...+a_(n)^(2))(b_(1)^(2)+b_(2)^(2)+...+b_(n)^(2))

(a_(1))/(a_(1)+a_(2))+(a_(3))/(a_(3)+a_(4))=(2a_(2))/(a_(2)+a_(3))

Suppose a_(1),a_(2),a_(3) are in A.P. and b_(1),b_(2),b_(3) are in H.P. and let Delta=|(a_(1)-b_(1),a_(1)-b_(2),a_(1)-b_(3)),(a_(2)-b_(1),a_(2)-b_(2),a_(2)-b_(3)),(a_(3)-b_(1),a_(3)-b_(2),a_(3)-b_(3))| then prove that

ARIHANT MATHS-DETERMINANTS -Exercise (Questions Asked In Previous 13 Years Exam)
  1. the value of the determinant |{:((a(1)-b(1))^(2),,(a(1)-b(2))^(2),,...

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  2. If a^(2)+b^(2)+c^(2)=-2 and f(x)= |{:(1+a^(2)x,(1+b^(2))x,(1+c^(2))x...

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  3. The system of equations alphax+y+z=alpha-1, x+alphay+z=alpha-1 ...

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  4. If a1,a2,a3,.....an.... are in G.P. then the determinant Delta=|[logan...

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  5. If D =|{:(1,1,1),(1,1+x,1),(1,1,1+y):}|"for" " "xne0,yne0 then D is

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  6. Consider the system of equations x-2y+3z=-1 -x+y-2z=k x-3y+4z=1 ...

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  7. Let a,b,c, be any real number. Suppose that there are real numbers x,y...

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  8. Let a,b,c be such that b(a+c)ne 0. If |{:(,a,a+1,a-1),(,-b,b+1,b-1),(,...

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  9. If f(theta)=|{:(1,tantheta,1),(-tantheta,1,tantheta),(-1,-tantheta,1):...

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  10. The number of values of k which the linear equations 4x+ky+2z=0 kx...

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  11. If the trivial solution is the only solution of the system of equation...

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  12. The number of values of k, for which the system of equations (k""+"...

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  13. If alpha,beta!=0 , and f(n)""=alpha^n+beta^n and |3 1+f(1)1+f(2)1+f(1)...

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  14. The set of the all values of lamda for which the system of linear equa...

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  15. Which of the following values of alpha satisfying the equation |(1+alp...

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  16. The system of linear equations x+lambday-z=0 lambdax-y-z=0 x+y-l...

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  17. The total number of distinct x in R for which |[x, x^2, 1+x^3] , [2x,...

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  18. Let a,lambda,mu in R, Consider the system of linear equations ax+2y=la...

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  19. If S is the set of distinct values of ' b for which the following syst...

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