|[(m+n)^(2), l^(2), mn], [(n+l)^(2), m^(...

`|[(m+n)^(2), l^(2), mn], [(n+l)^(2), m^(2), ln], [(l+m)^(2), n^(2), lm]| =(l^(2) +m^(2) +n^(2))(l-m)(m-n)(n-l)(l+m+n)`

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Apply `C_(1) to (C_(1) + C_(2) -2C_(3)) "and take"(l^(2) + m^(2) + n^(2)) "common from "C_(1)`

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The point of intersection of tangents drawn to the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 at the points where it is intersected by the line lx+my+n=0, is (A) ((-a^(2)l)/(n),(b^(2)m)/(n))(B)((-a^(2)l)/(m),(b^(2)n)/(m))(C)((a^(2)l)/(m),(-b^(2)n)/(m))(D)((a^(2)l)/(m),(b^(2)n)/(m))

Show that the matris [[l_(1),m_(1),n_(1)],[l_(2),m_(2),n_(2)],[l_(3),m_(3),n_(3)]] is orthogonal, if l_(1)^(2) + m_(1)^(2) + n_(1)^(2) = Sigmal_(1)^(2) = 1 = Sigma l_(2)^(2) = Sigma_(3) ^(2) and l_(1) l_(2) + m_(1)m_(2) + n_(1) n_(2) = Sigma l_(1)l_(2) =0 = Sigma l_(2)l_(3) = Sigma l_(3) l_(1).

Given sin alpha+sin beta=l,cos alpha+cos beta=mtan((alpha)/(2))tan((beta)/(2))=n,(A)(l^(2)+m^(2))(1-n)=2l(1+n)(B)(l^(2)+m^(2))(1+n)=2l(1-n)(C)(l^(2)+m^(2))(1-n)=2m(1+n)(D)(l^(2)+m^(2))(1+n)=2m(1-n)

If (l_(1), m_(1), n_(1)) , (l_(2), m_(2), n_(2)) are D.C's of two lines, then (l_(1)m_(2)-l_(2)m_(1))^2+(m_(1)n_(2)-n_(1)m_(2))^2+(n_(1)l_(2)-n_(2)l_(1))^2+(l_(1)l_(2)+m_(1)m_(2)+n_(1)n_(2))^2=

If the direction cosines of two lines are (l_(1), m_(1), n_(1)) and (l_(2), m_(2), n_(2)) and the angle between them is theta then l_(1)^(2)+m_(1)^(2)+n_(1)^(2)=1=l_(2)^(2)+m_(2)^(2)+n_(2)^(2) and costheta = l_(1)l_(2)+m_(1)m_(2)+n_(1)n_(2) If the angle between the lines is 60^(@) then the value of l_(1)(l_(1)+l_(2))+m_(1)(m_(1)+m_(2))+n_(1)(n_(1)+n_(2)) is

RS AGGARWAL-DETERMINANTS-Exercise 6B

|[a^(2), b^(2), c^(2)], [(a+1)^(2), (b+1)^(2), (c+1)^(2)], [(a-1)^(2),...
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|((x-2)^2,(x-1)^2,x^2),((x-1)^2,x^2,(x+1)^2),(x^2,(x+1)^2,(x+2)^2)|=-8...
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|[(m+n)^(2), l^(2), mn], [(n+l)^(2), m^(2), ln], [(l+m)^(2), n^(2), lm...
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Prove that |[a^2, a^2-(b-c)^2, bc],[b^2, b^2-(c-a)^2, ca],[c^2, c^2-(...
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Using properties of determinants, prove that: |[b^2+c^2,a^2,a^2],[b^2,...
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|[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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Prove that |[a,b-c,c+b],[a+c,b,c-a],[a-b,a+b,c]|=(a+b+c)(a^2+b^2+c^2)
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If a,b,c are non-zero real numbers then D=|[b^2 c^2, bc, b+c] , [c^2a^...
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Using properties of determinants, show the following: |(a+C)^2a b c a...
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The determinat Delta=|(b^2-ab,b-c,bc-ac),(ab-a^2,a-b,b^2-ab),(bc-ac,c-...
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Show that | (-a(b^2 + c^2 - a^2), 2b^3, 2c^3), (2a^3, -b(c^2 + a...
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|[x-3, x-4, x-alpha], [x-2, x-3, x-beta], [x-1, x-2, x-gamma]| =0,"whe...
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|[(a+1)(a+2), a+2, 1], [(a+2)(a+3), a+3, 1], [(a+3)(a+4), a+4, 1]| =-2
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Prove that |(1,a^2+bc,a^3),(1,b^2+ca,b^3),(1,c^2+ca,c^3)|=-(a-b)(b-c)(...
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Prove that |[1,a,bc] , [1,b,ca], [1,c,ab]|=|[1,a,a^2] , [1,b,b^2] , [1...
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|[1,bc,b+c],[1,ca,c+a],[1,ab,a+b]|=|[1,a,a^2],[1,b,b^2],[1,c,c^2]|
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Show that x = 2 is a root of the equation |[x, -6, -1], [2, -3x, x-3],...
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|[1,x,x^3],[1,b,b^3],[1,c,c^3]|=0; b!=c
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|[x+a, b, c], [a, x+b, c], [b, b, x+c]|=0
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One root of the equation |(3x-8, 3, 3),(3,3x-8, 3),(3,3,3x-8)|=0 si (A...
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