If a,`bgt0` such that ` a^3+b^3=2` , then show that `a+ble 2`.

If a^(3)-3a^(2)+5a-17=0 and b^(3)-3b^(2)+5b+11=0 are such that a+b is a real number, then the value of a+b is

If (x ^(2) -1) is a factor of ax ^(4) + bx ^(3) + cx ^(2) + dx +e, then show that a + c +e =b +d=0

Show that if iz^3+z^2-z+i=0 , then |z|=1

(a) If x + y + z=0, show that x ^(3) + y ^(3) + z ^(3)= 3 xyz. (b) Show that (a-b) ^(3) + (b-c) ^(3) + (c-a)^(3) =3 (a-b) (b-c) (c-a)

Let =|2a_1b_1a_1b_2+a_2b_1a_1b_3+a_3b_1a_1b_2+a_2b_1 2a_2b_2a_2b_3+a_3b_2a_1b_3+a_3b_1a_3b_2+a_2b_3 2a_3b_3| . Expressing as the product of two determinants, show that =0. Hence, show that if a x^2+2h x y+b y^2+2gx+2fy+c=(l x+m y+n)(l^(prime)x+m^(prime)y+n),t h e n|a hgh bfgfc|=0.

If a x^2+b/xgeqc for all positive x where a >0 and b >0, show that 27 a b^2geq4c^3dot

Let z_1, z_2, z_3 be three complex numbers and a ,b ,c be real numbers not all zero, such that a+b+c=0a n da z_1+b z_2+c z_3=0. Show that z_1, z_2,z_3 are collinear.

A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle. Show that the minimum length of the hypotenuse is (a^((2)/(3))+b^((2)/(3)))^((3)/(2)) .

If a^(2) + b^(2) = 7 ab, show that log ((a + b)/(3)) = (1)/(2) (log a + log b ).

Let a=(pi)/(7) , then (a) show that sin^(2)3a-sin^(2)a=sin2asin3a (b) show that co sec a=co sec 2a+co sec4a . (c) Prove that cos a is a root of the equation 8x^(3)+4x^(2)-4x+1=0 .