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

`|(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)`.

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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)

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)

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

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

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)) |:}

|(a^(2)+1,ab,ac),(ab,b^2+1,bc),(ca,cb,c^2+1)|= 1 + a^2 + b^2 + c^2 .

Prove that |{:(a-b-c, 2a, 2a), (2b, b-c-a, 2b), (2c, 2c, c-a-b):}|=(a+b+c)^(3)

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

|[1, a, a^2-b c],[1,b, b^2-c a],[1,c, c^2-a b]|=

SUBHASH PUBLICATION-DETERMINANT-FOUR MARKS QUESTIONS WITH ANSWERS

Prove that |(1,1,1),(a,b,c),(a^3,b^3,c^3)| = (a - b)(b-c)(c-a)(a+b+c).
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Prove that |(x,x^2,yz),(y,y^2,zx),(z,z^2,xy)|= (x-y)(y-z)(z-x)(xy + yz...
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|(x+4, 2x, 2x),(2x, x +4, 2x),(2x, 2x ,x+4)| = (5x + 4)(4 - x)^(2).
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|(y +k, y , y),(y, y + k, y),(y, y, y+k)|= k^(2)(3y + k).
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|(a-b-c, 2a, 2a),(2b, b-c-a,2b),(2c,2c,c-a-b)| = (a + b + c)^(3).
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Prove that |{:(,x+y+2z,x,y),(,z,y+z+2x,y),(,z,x,z+x+2y):}|=2(x+y+z)^(3...
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Prove that |(1,x,x^2),(x^2,1,x),(x,x^2,1)|=(1-x^3)^2.
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|(1+a^(2)-b^(2), 2ab, -2b),(2a, 1 -a^(2)+b^(2),2a),(2b, -2a, 1-a^2-b^2...
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|(a^(2)+1,ab,ac),(ab,b^2+1,bc),(ca,cb,c^2+1)|= 1 + a^2 + b^2 + c^2.
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Prove that |(-a^2,ab,ac),(bc,-b^2,bc),(ca,cb,-c^2)|=4a^(2)b^(2) c^(2).
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|(b+c, q+r, y+z),(c+a,r+p,z +x),(a+b,p+q,x+y)|=2|(a,p,x),(b,q,y),(c,r,...
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|(1,bc,a(b+c)),(1,ca,b(c+a)),(1,ab,c(a+b))|= 0
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Show that |(1+a,1,1),(1,1+b,1),(1,1,1+c)| = abc(1 + 1/a + 1/b + 1/c) =...
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Evaluate Delta -|(1,a,bc),(1,b,ca),(1,c,ab)|
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Prove that |(b+c,a,a),(b,c+a,b),(c,c,a+b)|= 4ac
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If x,y,z are different and Delta = |(x, x^2, 1+x^3),(y,y^2,1+y^3),(z,z...
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|(a - b, b-c,c-a),(b-c,c-a,a-b),(c-a,a-b,b-c)|=0
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|(0,a,-b),(-a,0,-c),(b,c,0)| = 0
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|(x,a,x+a),(y,b,y+b),(z,c,z+c)| = 0
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Show that |(x,p,q),(p,x,x),(q,q,x)| = (x - p)(x^(2) + px - 2q^2)
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