Solution Of System Of Linear Inequalities In Two Variables

We are given two unit vectors $%5Coverrightarrow%7Ba%7D$ and $%5Coverrightarrow%7Bb%7D$ , and the angle between them is $%5Cfrac%7B%5Cpi%7D%7B4%7D$ .
The formula for the dot product of two vectors $%5Coverrightarrow%7Bu%7D$ and $%5Coverrightarrow%7Bv%7D$ is:
$%5Coverrightarrow%7Bu%7D%20%5Ccdot%20%5Coverrightarrow%7Bv%7D%20%3D%20%7C%5Coverrightarrow%7Bu%7D%7C%20%7C%5Coverrightarrow%7Bv%7D%7C%20%5Ccos%20%5Ctheta.$
Since $%5Coverrightarrow%7Ba%7D$ and $%5Coverrightarrow%7Bb%7D$ are unit vectors, their magnitudes are both 1, so: $%5Coverrightarrow%7Ba%7D%20%5Ccdot%20%5Coverrightarrow%7Bb%7D%20%3D%20%5Ccos%20%5Cfrac%7B%5Cpi%7D%7B4%7D%20%3D%20%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%7D.$ Now, we need to calculate the expression: $(%5Coverrightarrow%7Ba%7D%20%2B%20(%5Coverrightarrow%7Ba%7D%20%5Ccdot%20%5Coverrightarrow%7Bb%7D)%20%5Coverrightarrow%7Bb%7D)%20%5Ccdot%20(%5Coverrightarrow%7Ba%7D%20-%20(%5Coverrightarrow%7Ba%7D%20%5Ccdot%20%5Coverrightarrow%7Bb%7D)%20%5Coverrightarrow%7Bb%7D).$ Substitute $%5Coverrightarrow%7Ba%7D%20%5Ccdot%20%5Coverrightarrow%7Bb%7D%20%3D%20%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%7D$ into the expression: $(%5Coverrightarrow%7Ba%7D%20%2B%20%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%7D%20%5Coverrightarrow%7Bb%7D)%20%5Ccdot%20(%5Coverrightarrow%7Ba%7D%20-%20%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%7D%20%5Coverrightarrow%7Bb%7D).$ Use the distributive property of the dot product: $%5Coverrightarrow%7Ba%7D%20%5Ccdot%20%5Coverrightarrow%7Ba%7D%20-%20%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%7D%20%5Coverrightarrow%7Ba%7D%20%5Ccdot%20%5Coverrightarrow%7Bb%7D%20%2B%20%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%7D%20%5Coverrightarrow%7Bb%7D%20%5Ccdot%20%5Coverrightarrow%7Ba%7D%20-%20%5Cfrac%7B1%7D%7B2%7D%20%5Coverrightarrow%7Bb%7D%20%5Ccdot%20%5Coverrightarrow%7Bb%7D.$ Since $%5Coverrightarrow%7Ba%7D%20%5Ccdot%20%5Coverrightarrow%7Ba%7D%20%3D%201$ and $%5Coverrightarrow%7Bb%7D%20%5Ccdot%20%5Coverrightarrow%7Bb%7D%20%3D%201$ , and $%5Coverrightarrow%7Ba%7D%20%5Ccdot%20%5Coverrightarrow%7Bb%7D%20%3D%20%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%7D$ , the expression becomes: $1%20-%20%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%7D%20%5Ccdot%20%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%7D%20%2B%20%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%7D%20%5Ccdot%20%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%7D%20-%20%5Cfrac%7B1%7D%7B2%7D.$ Simplify: $1%20-%20%5Cfrac%7B1%7D%7B2%7D%20%2B%20%5Cfrac%7B1%7D%7B2%7D%20-%20%5Cfrac%7B1%7D%7B2%7D%20%3D%20%5Cfrac%7B1%7D%7B2%7D.$
Thus, the value of the expression is $%5Cfrac%7B1%7D%7B2%7D$ , and the correct answer is option (D).

Step 1: Express

Z

and

%5Coverline%7BZ%7D

.
We are given that

Z%20%3D%201%20%2B%20i

. The conjugate of

Z

, denoted

%5Coverline%7BZ%7D

, is:

%5Coverline%7BZ%7D%20%3D%201%20-%20i

Step 2: Substitute

Z

and

%5Coverline%7BZ%7D

into the equation.
We are also given the equation

Z%20-%2024%5Coverline%7BZ%7D%20%3D%20%5Clambda%20Z%5E2

. Substituting the values of

Z

and

%5Coverline%7BZ%7D

(1%20%2B%20i)%20-%2024(1%20-%20i)%20%3D%20%5Clambda%20(1%20%2B%20i)%5E2

Step 3: Simplify the equation.
First, simplify the left-hand side:

(1%20%2B%20i)%20-%2024(1%20-%20i)%20%3D%201%20%2B%20i%20-%2024%20%2B%2024i%20%3D%20-23%20%2B%2025i

Now, simplify the right-hand side:

%5Clambda%20(1%20%2B%20i)%5E2%20%3D%20%5Clambda%20(1%20%2B%202i%20-%201)%20%3D%20%5Clambda%20(2i)

Thus, the equation becomes:

-23%20%2B%2025i%20%3D%20%5Clambda%20%5Ccdot%202i

Step 4: Solve for

%5Clambda

.
Equating the real and imaginary parts, we get:

%5Ctext%7BReal%20part%3A%20%7D%20-23%20%3D%200%20%5Cquad%20%5Ctext%7B(This%20is%20already%20satisfied.)%7D

%5Ctext%7BImaginary%20part%3A%20%7D%2025%20%3D%202%5Clambda%20%5Cquad%20%5CRightarrow%20%5Cquad%20%5Clambda%20%3D%20%5Cfrac%7B25%7D%7B2%7D

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Solution Of System Of Linear Inequalities In Two Variables

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Solution Of System Of Linear Inequalities In Two Variables

High-Yield Trend

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Official Solution

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About Solution Of System Of Linear Inequalities In Two Variables - KEAM

Frequently Asked Questions

Why focus on Solution Of System Of Linear Inequalities In Two Variables PYQs?

How to best use this analysis?

Chapter Questions
2 MCQs