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Chapter 2: Chapter 2 · MATHEMATICS-VOLUME 1 · EN medium
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| Since the sides are equal, the given points form an equilateral triangle. Example . Let z z , and z be complex numbers such that z and z Prove that z z z z z z Given that z = z z z z z z z ⇒ z = r Therefore z = r = r z z z z z z z z z = | z z z z z z z z z z = r z z z z z z z and | ) z z z = r z z z z z z z z z z z z O Re Im −+ −− Fig. . Complex Numbers ⇒ z z z z z z = r . (given that z Thus, z z z z z z = r . Example . Show that the equation z = has four solutions. We have, z = z . ⇒ | z = z ⇒ | z = , ⇒ | z = , or | z = . | z = is a solution, | z = zz . Given z = z ⇒ z = z ⇒ z = . It has non-zero solutions. Hence including zero solution, there are four solutions. . . Square roots of a complex number Let the square root of a ib be x iy That is ib = x iy where x y ∈ ib = x iy i xy Equating real and imaginary parts, we get = a and xy | |||||||||
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| . If is a cube root of unity, show that (i) ( (ii) . If z , find the rotation of z by θ radians in the counter clockwise direction about the origin when (i) (ii) (iii) . Complex Numbers EXERCISE . Choose the correct or the most suitable answer from the given four alternatives : . i is ( ) ( ) ( ) − ( ) i . The value of is ( ) + i ( ) i ( ) ( ) . The area of the triangle formed by the complex numbers z iz , , and z iz in the Argand’s diagram is ( ) | z ( ) | z ( ) | z ( ) | z . The conjugate of a complex number is i − . Then, the complex number is ( ) i + ( ) ( ) − ( ) i − . If z , then | z is equal to ( ) ( ) ( ) ( ) . If z is a non zero complex number, such that iz then | z is ( ) ( ) ( ) ( ) . If | , then the greatest value of | z is ( ) ( ) ( ) ( ) . If z , then the least value of | z is ( ) ( ) ( ) ( ) . If | z = , then the value of z is ( ) z ( ) z ( ) ( ) . The solution of the equation | z is ( ) − i ( ) i ( ) ( ) . If | , | , | and | z z z z z z , then the value of | is ( ) ( ) ( ) ( ) . If z is a complex number such that z ∈ \ and z , then | z is |
| ( ) ( ) ( ) ( ) . z z and z are complex numbers such that z and | = then is ( ) ( ) ( ) ( ) . If z is purely imaginary, then | z is ( ) ( ) ( ) ( ) . If z iy is a complex number such that | , then the locus of z is ( ) real axis ( ) imaginary axis ( ) ellipse ( ) circle . The principal argument of i is ( ) − ( ) − ( ) − ( ) − p . The principal argument of (sin is ( ) ( ) ( ) ° ( ) ° . If ( )( )( ni iy , then ( is ( ) ( ) i ( ) x ( ) + n . If is a cubic root of unity and ( B , then ( , ) A B equals ( ) ( , ) ( ) ( , ) − ( ) ( , ) ( ) ( , ) . The principal argument of the complex number is ( ) ( ) p ( ) ( ) p . If α and β are the roots of x , then is ( ) − ( ) − ( ) ( ) . The product of all four values of cos is ( ) − ( ) − ( ) ( ) . If is a cubic root of unity and k , then k is equal to ( ) ( ) − ( ) i ( ) − i . The value of is ( ) cis ( ) cis ( ) − cis ( ) − cis Complex Numbers . If cis , then the number of distinct roots of ( ) ( ) ( ) ( ) |
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