Column I Column II (a) If αis root x^5=1 with α≠0, then the value (p) 0 of α^101+α^102+…+α^230

Column I Column II (a) If αis root x^5=1 with α≠0, then...

Key Concepts

Roots of Unity

Roots of unity are the complex solutions of the equation z^n = 1 for a given positive integer n. They have a symmetric distribution on the unit circle in the complex plane and their properties, such as periodicity and summation identities, are fundamental in simplifying expressions involving powers of these numbers.

Periodicity of Powers

When a complex number is a root of unity, its powers repeat in a fixed cycle governed by its order. This periodicity is key to reducing and summing series that involve high powers, as many terms can cancel or combine in predictable ways.

Modulus of Complex Numbers

The modulus of a complex number, defined as the distance from the origin in the complex plane, is a central concept in many areas of complex analysis. It is particularly important in determining the maximum and minimum magnitudes of complex functions or curves.

Complex Curve Representation

Complex numbers can be used to represent curves in the plane, with their real and imaginary parts corresponding to the x and y coordinates, respectively. This allows for geometric analysis of curves, including the determination of extremal values of the modulus.

Solving Complex Equations

Solving equations that involve complex numbers and their modulus often involves separating the real and imaginary parts and using properties unique to complex arithmetic. Techniques in this area are essential for handling equations that mix polynomial expressions with absolute values.

Determinants with Complex Entries

Determinants that include complex numbers rely on the same algebraic principles as those with real numbers, but with additional nuances due to complex arithmetic. Evaluating such determinants often involves leveraging properties of roots of unity and the behavior of complex conjugates.

Transcript

00:01 In this question, we need to match the elements of column 1 with the elements of column 2.

00:07 In part a, we need to find the value of alpha to the power 1001 plus alpha to the power 1002 plus so on plus alpha to the power 230.

00:32 And it is given that x to the power 5 is equal to 1.

00:40 And alpha is not equals to 0.

00:48 Now since x to the power 5 is equal to 1, therefore we can write alpha to the power 5 is equals to 1.

01:02 Hence alpha to the power 101 plus alpha to the power 102 plus so on plus alpha to the power 230 will be equal to the power 1002 plus so on plus alpha to the power 230 will be equal.

01:18 Equals to alpha to 2 the power 101 into 1 minus alpha to the power 130 upon 1 minus alpha and this will be equals to alpha to the power 101 into 1 minus alpha to the power 5 to the power 26 upon 1 minus alpha and this will be equals to 0.

01:55 Hence the correct match is a with p so this is the first correct match now come to point b in part b we need to find the maximum and minimum value of modulus z where that represents the curve z is equals to 3 upon 2 plus cos theta plus iota sine theta since z represents a complex number and z is equals to x plus iota y therefore we can write the above equation as x plus iota y is equal 2 plus cos theta plus iota sine theta and this expression can also be written as 1 upon x plus iota y is equal to 2 plus cos theta plus iota sine theta upon 3 and this will be equals to x minus iota y upon x square plus y square is equal to 2 plus cos theta plus iota sine theta upon 3 now equate the real parts real parts we get x upon x upon x square plus y square is equal to 2 plus cos theta upon 3.

04:51 Therefore, cost theta is equal to 3x upon x square plus y square minus 2.

05:08 Similarly equate the imaginary parts.

05:18 Parts we get minus y upon x square plus y square is equal to sine theta upon three or we can write sine theta is equal to minus three y upon x square plus y square since sine square t, plus cos squared theta is equal to 1.

06:09 Therefore, from the values of sine theta and cost theta, we can write 3x upon x square plus y square minus 2 to the power 2 plus 3y upon x square plus y square to the power 2 is to 1 and this will be equals to 9 x square upon x square plus y square to the power 2 plus 9 y square upon x square plus y square to the power 2 minus 12 x square upon x square plus y square plus 4 is equal to 1.

07:14 When we further solve it we get x minus 2 plus y square is equal to 1.

07:29 This is a equation of circle whose center is 2 .0 and the radius is 1.

07:56 We can make this circle as shown below.

08:06 In this circle, this point is 1, this point is 3 and this is the center 2 .0.

08:18 Therefore we can conclude that the maximum value of modulus z is equals to 3 and the minimum value of modulus is equal to 1.

08:35 Hence the correct match is b and s q.

08:46 This is the correct answer for option b...