Problem 66 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(\omega(\neq 1)\) is a cube root of unity, and \((1+\omega)^{7}=A+B \omega . \quad 74\) Then \((A, B)\) equals (A) \((-1,1)\) (B) \((0,1)\) (C) \((1,1)\) (D) \((1,0)\)

Step-by-Step Solution

Answer

(A, B) = (1, 1), corresponding to option (C).

Step 1: Understand Roots of Unity

Cube roots of unity are solutions to the equation \(x^3 = 1\). These roots are \(1\), \(\omega\), and \(\omega^2\), where \(\omega = \frac{-1 + i\sqrt{3}}{2}\), and satisfies \(\omega^3 = 1\) and \(1 + \omega + \omega^2 = 0\). Use the property \(\omega^2 = -1 - \omega\) in calculations.

Step 2: Simplify Expression Using Properties

Calculate \((1 + \omega)^2\):\( \,(1 + \omega)^2 = 1 + 2\omega + \omega^2\). Using \(\omega^2 = -1 - \omega\), it follows that \((1 + \omega)^2 = 1 + 2\omega - 1 - \omega = \omega\).

Step 3: Exponentiate the Simpler Expression

Simplify \((1 + \omega)^7\) using \((1+\omega)^2 = \omega\): \((1+\omega)^7 = ((1+\omega)^2)^3 \times (1+\omega) = \omega^3 \times (1+\omega)\). Note that \(\omega^3 = 1\), so \((1+\omega)^7 = 1 \times (1+\omega) = 1 + \omega\).

Step 4: Compare with Given Expression to Solve for A and B

We have \((1+\omega)^7 = 1 + \omega\). Comparing this to \(A + B\omega\), we see \(A = 1\) and \(B = 1\). So, \((A, B)\) equals \((1, 1)\).

Key Concepts

roots of unity

Roots of unity are the solutions to the equation that involves raising a number to a given power ending in the number 1. Specifically, the cube roots of unity are the solutions to the equation \(x^3 = 1\). These roots are both fascinating and vital in complex number theory. The roots include 1, \(\omega\), and \(\omega^2\). Here, \(\omega\) is often represented as \(\frac{-1 + i\sqrt{3}}{2}\).

These cube roots have some fascinating properties:

• \(\omega^3 = 1\), meaning that when \(\omega\) is cubed, it equals 1 again.
• The sum \(1 + \omega + \omega^2 = 0\). This property is very useful in simplifying complex number expressions.
• \(\omega^2 = -1 - \omega\). This relationship allows for simplification when manipulating expressions containing \(\omega\).

Understanding and using these properties can significantly simplify calculations and expressions involving roots of unity.

complex numbers

Complex numbers expand the idea of one-dimensional numbers (like the real numbers) by introducing a second dimension—a real part and an imaginary part. A complex number is generally written as \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. Here, \(i\) is the imaginary unit, defined by the property that \(i^2 = -1\).

Complex numbers are essential in representing quantities which have both magnitude and direction, basically allowing rotation in a plane. They are extended from real numbers and are crucial when dealing with polynomial equations, like finding roots not possible with real numbers alone. In particular:

• The addition of complex numbers is done by adding their real parts and imaginary parts separately.
• When multiplying complex numbers, you apply the distributive property as you would for polynomials, remembering to replace \(i^2\) with \(-1\).
• The conjugate of a complex number \(a + bi\) is \(a - bi\), useful for division of complex numbers.

Learning how to navigate and work with complex numbers allows for solving problems in engineering and physics contexts that involve oscillations and waves, where these numbers naturally arise.

exponentiation of complex numbers

Exponentiation of complex numbers involves raising a complex number to a power. This can get complex quite literally, but understanding Euler's formula and De Moivre's theorem simplifies this process.

Euler's formula is a key principle here: \(e^{ix} = \cos(x) + i\sin(x)\). This relates complex exponentiation to trigonometric functions. De Moivre's theorem, on the other hand, is used for raising complex numbers to integer powers, where \(([\cos(x) + i\sin(x)])^n = \cos(nx) + i\sin(nx)\).

With these in mind:

• For a complex number in polar form \(re^{i\theta}\), the nth power can be elegantly calculated using \(r^n e^{i n \theta}\).
• Exponents of roots of unity simplify over their cyclic nature, like in our cube root example where \(\omega^3 = 1\), capturing a rotational symmetry.
• This symmetry greatly simplifies solving problems regarding powers of complex numbers beautifully aligning them with geometric visualizations.

Working from this basis, one can tackle many different types of problems, whether they involve electrical engineering circuits or predicting waves in physics, thanks to the complex-number powers providing precision and clarity in solutions.