# COMPLEX NUMBER, Sn dey solution

Mathematics (SOURENDRANATH NATH DE) is a very famous book of  West Bengal board  and Students also like this book. So we are here to provide you some help in the solution of sn dey.

2 MARKS

1. Do you think 4 as a complex number? If so why?

2.i  If x, y are real and x+iy=-i(-2+3i), find x and y.

2.ii If x,y are real and x+iy=(5/(-3+4i)), find x and y.

3. If (1+i)(2+i)(3+i)….(n+i)=a+ib , show that , 2 . 5 . 10….(n^2+1)=a^2+n^2

4. Prove that, |frac{x-iy}{-x+iy}|=1

5. if Z=frac{2+i}{-2+i} , find conjugate Z in the form a+ib

6. Express in the form A+iB (A, B are real):
6.i left(1-iright)^3

6.ii frac1{1+i}+frac{1+i}i

6.iii frac{x+iy}{y-ix}“left(x^2+y^2ne0right)

6.iv frac i{2+i}+frac3{1+4i}

6.v frac{sqrt3-isqrt2}{2sqrt3-isqrt2}

6.vi frac1{1-costheta-isintheta}

7. Find the conjugate:
i. sqrt{-5}-2

ii. 2-isqrt3-sqrt2

iii.frac{2-i}{left(1-2iright)^2}

8. Simplify:
i. frac{i+i^2+1^3+1^4}{1+i}

ii.left(1+i^3right)left(1+frac1iright)^2left(i^4+frac1{i^4}right)

iii.left(1+iright)^2+left(frac{1-i}{1+1}right)^2

iv. left(1+iright)^{-2}-left(1-1right)^{-2}

v.left(frac{1+i}{1-i}right)^2+left(frac{1-i}{1+1}right)^2

9. If x, y are real and (x+3i) and (-2+iy) are conjugate of each other , find x and  y.

10. Find the least positive integral value of n so that left(frac{1+i}{1-i}right)^n=1

11. Find the modulus:
i. 2sqrt2-isqrt6

ii.left(sqrt5+isqrt3right)left(-2+iright)

iii.frac{1-i}{3-4i}

iv.frac2{1+costheta+isintheta}

12. Find the arguments of each of the following complex number:
i. 2-2i

ii. -3-3i

iii. (1+i)left(sqrt3+iright)

iv. frac{sqrt3+i}{-1-isqrt3}

13. If a=frac{1+i}{sqrt2} , show that a^6+a^4+a^2+1=0

14. Find the cube root of -1 and 27

15. Factorise a^2+1

16. If 1,w,w^2 denote the cube root of unity, find the roots of left(x+5right)^3+27=0