If `(b^2-4a c)^2(1+4a^2)<64 a^2, a<0` , then maximum value of quadratic expression `a x^2+b x+c` is always less than a. 0 b. 2 c. -1 d. -2

If alpha,beta are the roots of a x^2+b x+c=0,(a!=0) and alpha+delta,beta+delta are the roots of A x^2+B x+C=0,(A!=0) for some constant delta then prove that (b^2-4a c)/(a^2)=(B^2-4A C)/(A^2)

If alpha,beta are the roots of the equation a x^2+b x+c=0, then the value of aalpha^2+c//aalpha+b+(abeta^2+c)//(abeta+b) is (b(b^2-2a c))/(4a) b. (b^2-4a c)/(2a) c. (b(b^2-2a c))/(a^2c) d. none of these

If a^(2)+b^(2)+c^(2)=1 where, a,b, cin R , then the maximum value of (4a-3b)^(2) + (5b-4c)^(2)+(3c-5a)^(2) is

Simplify : (2a+3b+4c) ( 4a^(2) + 9b^(2) + 16c^(2) -6ab - 12bc - 8ca )

If a ,b ,a n dc are in H.P., then th value of ((a c+a b-b c)(a b+b c-a c))/((a b c)^2) is ((a+c)(3a-c))/(4a^2c^2) b. 2/(b c)-1/(b^2) c. 2/(b c)-1/(a^2) d. ((a-c)(3a+c))/(4a^2c^2)

Find the product (2a + b+c) (4a ^(2) + b ^(2) + c ^(2) - 2ab -bc -2ca)

If |(a^2,b^2,c^2),((a+b)^2 ,(b+1)^2,(c+1)^2),((a-1)^2 ,(b-1)^2,(c-1)^2)| =k(a-b)(b-c)(c-a) then the value of k is a. 4 b. -2 c.-4 d. 2

Prove that [[1+a^2+a^4, 1+a b+a^2b^2 ,1+a c+a^2c^2],[ 1+a b+a^2b^2, 1+b^2+b^4, 1+b c+b^2c^2],[ 1+a c+a^2c^2, 1+b c+b^2c^2, 1+c^2+c^2]]=(a-b)^2(b-c)^2(c-a)^2

Sove that |(1,a,a^(2)),(1,b,b^(2)),(1,c,c^(2))|^(2)=|(1+a^(2)+a^(4),1+ab+a^(2)b^(2),1+ac+a^(2)c^(2)),(1+ab+a^(2)b^(2),1+b^(2)+b^(4),1+bc+b^(2)c^(2)),(1+ac+a^(2)c^(2),1+ba+b^(2)c^(2),1+c^(2)+c^(4))| and hence show RHS determinent is =(a-b)^(2)(b-c)^(2)(c-a)^(2)