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# Complex numbers

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### Complex numbers

1. 1. ASSESSMENT OF COMPLEX NUMBERS & QUADRATIC EQUATIONS (XI)<br /> Level...1<br /> Q.1 Solve X2 – (3√2 – 2i) x - 6√2i = 0 (3√2, -2i)<br /> Q.2 1+ix-2i3+i + 2-3iy+i3-i = i, find real values of x and y. <br /> Q.3 If z = (1+i1-i ), then z4 is<br /> (i) 1 (ii) -1 (iii) 0 (iv) none of them.<br /> Q.4 If z = 11-I(2+3I) , then |z| is<br /> (i) 1 (ii) 1/√(26) (iii) 5/√(26) (iv) none <br /> Q.5 If x+iy = 3+5i7-6i , then y = <br /> (a) 9/85 (b) -9/85 (c) 53/85 (d) -53/85<br /> Q.6 If z = 11-cosφ-isinφ , then Re(z) is <br /> (a) 0 (b) ½ (c) cot(φ/2) (d) (½)cot(φ/2)<br /> Level......2<br /> Q.1 Find the real values of ϴ for which the complex number 1+icosϴ1-2icosϴ is purely real.<br /> Q.2 If (1 - i) (1 - 2i)(1 - 3i)...........(1 - ni) = (x - yi) , show that 2.5.10...........(1+n2) = x2+y2 .<br /> Q.3 Prove that arg(z) = 2π – arg(z) ,z ≠0<br /> Q.4 Express in polar form: -21+i√3 . <br /> Q.5 If iz3 +z2 – z+ i = 0, then show that |z| = 1. <br /> Q.6 If a+ib = c+ic-i , then a2+b2 = 1 and b/a = 2cc²-1 <br /> Q.7 If x = - 5 +2√(-4) , find the value of x4+9x3+35x2 – x+4. <br /> Q.8 Show that a real x will satisfy equation 1-ix1+ix = a – ib, if a2+b2 = 1 where a, b are real.<br /> Q.9 A variable complex z is such that arg ( z-1z+1 ) = π2 , show that x2+y2 – 1=0<br /> Q.10 Find the values of x and y if x2 – 7x +9yi and y2i+20i – 12 are equal. <br /> Answers of Level—2<br /> 1. Rationalise it with 1+2icosϴ , then equate imaginary part to 0, value will be 2nπ±π2 , n ЄZ.<br /> 2. By taking modulus or conjugate on the both sides. <br /> 3. Let z = r( cosϴ + i sinϴ) , z = r (cosϴ - i sinϴ) = r (cos(2π-ϴ) + i sin(2π-ϴ).<br /> 4. (cos2π/3 +isin2 π/3)<br /> 5. take i outside, make factors as z2 (z – i)+ i (z – i) = 0 ⇨ z = i, z2= -i <br />|z| = |i| =1, |z2| =|-i|=1 ⇨ |z|2 = |z2| =1⇨ |z| = 1.<br />6. By taking conjugate on the both sides and use the given value.<br />7. Divide given poly. By x2 +10x +41=0 as (x+5)2= (4i)2 <br /> x2 +10x +41 x4+9x3 +35x2 – x+4<br /> x4+10x3 +41x2<br /> -x3 - 6x2 – x<br /> -x3 – 10x2 – 41x <br /> 4x2 + 40x + 4<br /> 4x2 + 40x +164<br /> -160 -> answer. <br /> <br /> 8. By C &D then we will get x = 2b1+a2²+b² .<br /> 9. Assume z = x+iy, arg ( z-1z+1 ) = π2 ⇨ arg( x+iy-1x+iy+1 ) = π2 ⇨ arg( x-1+iyx+1+iy x x+1-iyx+1-iy ) = π2 <br /> ⇨ tan-1π2 = 2xyx²+y²-1 .<br /> 10. x =4, 3 and y =5, 4.<br />it can be used in finding principal argument of complex numbers.<br /> ϴ, x>0, y>0<br /> π - ϴ, x<0, y>0 <br /> arg (z) = ϴ - π , x<0, y<0 <br /><ul><li> - ϴ, x>0, y<0
2. 2.