If a,b are real and `a^(2)+b^(2)=1,` then show that, the equation `(1-ix)/(1+ix)=a-ib` is satisfied by a real value of x.

If a,b are real and a^2+b^2=1 ,then prove that the equation. (1-ix)/(1+ix)=a-ib is satisfied for all real values of x.

If a,b are real and a^(2)+b^(2)=1 then show that the equation (sqrt(1+x)-isqrt(1-x))/(sqrt(1+x)+isqrt(1-x))=a-ib is satisfied by a real value of x.

If a,b, are real, then show that the roots of the equation (x-a)(x-b) = b^2 are always real.

Show that the equation sintheta=x+1/x is not possible if x is real.

If a,b,c,d,x,y are real and (a+ib)(c+id)=x+iy, show that , (a-ib)(c-id)=x-iy

Show that the equation e^(x-1)+x-2=0 has no real root greater than 1.

Show that a real value of x will satisfy the equation (1-i x)//(1+i x)=a-i b if a^2+b^2=1 ,where a ,b real.

Use cramer's rule to solve the equations: a_2x+b_2y+c_2=0.a_3x+b_3y+c_3=0,a_2b_3-b_2a^3 ne0 and hence , find the condition in the form of a determinant of third order so that the three equations a_ix+b_iy+c_i=0 (i=1,2,3) are satisfied by the same values of x and y .

If x is real , show that the expressions (x^2-ab)/(2x-a-b) has no real value between a and b.

If a,b,x,y are real and 3sqrt(x+iy)=a+ib show that , 3sqrt(x-iy)=a-ib