1. The document contains 10 questions assessing knowledge of complex numbers and quadratic equations.
2. The questions cover topics like solving complex quadratic equations, operations on complex numbers, expressing complex numbers in polar form, and properties of complex functions and arguments.
3. The answers provided solve each question and explain steps like rationalizing expressions, taking conjugates, using trigonometric identities, and applying definitions of absolute value and argument.
![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 />,[object Object]](https://image.slidesharecdn.com/complexnumbers-110816111001-phpapp01/85/Complex-numbers-1-320.jpg)
