cosix+isinix=?
eix
e-ix
ex
e-x
Explanation for the correct option:
Given, cosix+isinix
=coshx+i×isinhx ; ∵cosix=coshx,sinix=isinhx
=coshx–sinhx ; ∵i2=1
=ex+e-x2–ex-e-x2 ; ∵cosh(x)=ex+e-x2,sinh(x)=ex-e-x2
=[ex+e-x–ex+e-x]2
=2e-x2
=e-x
Hence, Option ‘D’ is Correct.
Solve the following quadratic equations :
(i)x2−(3√2+2i)x+6√2i=0
(ii)x2−(5−i)x+(18+i)=0
(iii)(2+i)x2−(5−i)x+2(1−i)=0
(iv)x2−(2+i)x−(1−7i)=0
(v)ix2−4x−4i=0
(vi)x2+4ix−4=0
(vii)2x2+√15ix−i=0
(viii)x2−x+(1+i)=0
(ix)ix2−x+12i=0
(x)x2−(3√2−2i)x−√2i=0
(xi)x2−(√2+i)x+√2i=0
(xii)2x2−(3+7i)x+(9i−3)=0
Find the real values of x and y, if
(i) (x+i y)(2−3i)=4+i(ii) (3x−2i y)(2+i)2=10(1+i)(iii) (1+i)x−2i3+i+(2−3i)y+i3−i=i(iv) (1+i)(x+i y)=2−5i
Prove that (x+1)(x+1-i)(x-1+i)(x-1-i)=x4+4