Class XI Chapter 5 – Complex Numbers and Quadratic Equations Maths ______________________________________________________________________________ On comparing imaginary parts, we obtain 1 Im 0 zz 1 1 Question 13: Find the modulus and argument of the complex number Solution 13: Let 1 3i z 1 3i , then 2 1 2i 1 3i 1 3i 2i 6i 1 5i 6 1 z 2 2 13i 1 3i 1 3 1 9 5 5i 5 5i 1 1 i 10 10 10 2 2 Let i.e., z r cosir sin 1 r cos 2 and 1 r sin 2 On squaring and adding, we obtain 2 2 2 2 2 1 1 r cos sin 2 2 2 1 1 1 r 4 4 2 1 r Conventionally, r 0 2 1 1 1 1 cos and sin 2 2 2 2 1 1 cos andsin 2 2 3 4 4 1 2i 1 3i [As lies in the II quadrant] Therefore, the modulus and argument of the given complex number are respectively. 1 2 and 3 4

Class XI Chapter 5 – Complex Numbers and Quadratic Equations Maths ______________________________________________________________________________ Question 14: Find the real numbers x and y if x iy35i is the conjugate of 6 24i . Solution 14: Let z x iy3 5i 2 z 3x 5xi 3yi 5yi 3x 5xi 3yi 5y 3x 5y i 5x 3y 3 5 5 3 z x y i x y It is given that, z 6 24i 3x 5y i 5x 3y 6 24i Equating real and imaginary parts, we obtain 3x 5y 6 ........ i 5x 3y 24 ........ ii Multiplying equation (i) by 3 and equation (ii) by 5 and then adding them, we obtain 9x15y 18 25x15y 120 34x 102 102 x 3 34 Putting the value of x in equation (i), we obtain 3 3 5y 6 5y 6 9 15 y 3 Thus, the values of x and y are 3 and 3 respectively. Question 15: Find the modulus of 1 i 1 i 1i 1 i . Solution 15: 1 i 1 i 2 2 1i 1i 1 i 1 i 1 i 1 i 2 2 1 2 1 2 i i i i 2 2 1 1

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