Obtain the polar and exponential representations of z₁, z₂ and z₁z₂, where z1 = 1/2 - i and z₂ = 3+i
हम दिए गए संख्याओं \( z_1 = \frac{1}{2} - i \) और \( z_2 = 3 + i \) का ध्रुवीय (Polar) और घातांक (Exponential) रूप निकालेंगे। ### 1. ध्रुवीय रूप (Polar Representation) ध्रुवीय रूप के लिए, किसी भी जटिल संख्या \( z = a + ib \) को इस रूप में लिखा जाता है: \[ z = r e^{i\theta} \] जहाँ, - \( r = |z| = \sqrt{a^2 + b^2} \) (परिमाण) - \( \theta = \tan^{-1} \left(\frac{b}{a}\right) \) (तर्क) #### ____ ____ ______ ____ ________ _________ ______ _________.
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Find the sum of the 5th roots of unity.
Find the sum of the 5th roots of unity
Which of the following statements are true? Justify your answers. (This means that if you think a statement is false, give a short proof or an example that shows it is false. If it is true, give a short proof for saying so. For instance, to show that "{1, padma, blue) is a set' is true, you need to say that this is true because it is a well-defined collection of 3 objects.)
i) The collection of Venn diagrams is a set.
ii) for any three sets B,A and C .
iii)
iv) , has a root in R only if m is an odd number.
v)
vi) Any system of three linear equations in two variables has no solution
vii) If a matrix has 2 n entries, where n ∈ N , then it is a square matrix.
viii) For n > 4 , the AM of the first n natural numbers is greater than n +1
Find x and y , given that
Is Cramer's Rule applicable for solving the linear system below? If yes, apply it. Otherwise, alter the last equation in the system so that the solution can be obtained by applying the Rule.
Find the polynomial over R of least degree which has i-2 and √3+5i as its roots.
Let x ∈ R such that 0 < x ≤ x ≤...≤x, n ≥2, and
Then show that