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Indicate in the co-ordinate plane all the points whose coordinates( X, y) are such that expressions (2cos t 1/2 cos x cos y) cos x cos y 1 cos x -cos y cos 2 t is positive for any value of t and depict the region formed by these points?
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Indicate in the co-ordinate plane all the points whose coordinates( X,...
**Solution:**
To determine the region formed by the points whose coordinates satisfy the given expression, we need to analyze each term separately and then combine the conditions to find the common region.
**Analyzing the first term: (2cos t)**
Since cosine function oscillates between -1 and 1, the value of 2cos t can vary from -2 to 2. For the given expression to be positive, the value of 2cos t must be greater than 0. Therefore, -2 < 2cos="" t="" />< />
**Analyzing the second term: (1/2 cos x cos y)**
Since cosine function oscillates between -1 and 1, the value of 1/2 cos x cos y can vary from -1/2 to 1/2. For the given expression to be positive, the value of 1/2 cos x cos y must be greater than 0. Therefore, -1/2 < 1/2="" cos="" x="" cos="" y="" />< />
**Analyzing the third term: (cos x cos y)**
Since cosine function oscillates between -1 and 1, the value of cos x cos y can vary from -1 to 1. For the given expression to be positive, the value of cos x cos y must be greater than 0. Therefore, -1 < cos="" x="" cos="" y="" />< />
**Analyzing the fourth term: (1)**
The value of 1 is always positive, so it does not contribute any conditions to the given expression.
**Analyzing the fifth term: (cos x - cos y)**
Since cosine function oscillates between -1 and 1, the value of cos x - cos y can vary from -2 to 2. For the given expression to be positive, the value of cos x - cos y must be greater than 0. Therefore, -2 < cos="" x="" -="" cos="" y="" />< />
**Analyzing the sixth term: (cos 2t)**
Since cosine function oscillates between -1 and 1, the value of cos 2t can vary from -1 to 1. For the given expression to be positive, the value of cos 2t must be greater than 0. Therefore, -1 < cos="" 2t="" />< />
**Combining the conditions:**
By analyzing each term separately, we have obtained the following conditions for the given expression to be positive:
- -2 < 2cos="" t="" />< />
- -1/2 < 1/2="" cos="" x="" cos="" y="" />< />
- -1 < cos="" x="" cos="" y="" />< />
- -2 < cos="" x="" -="" cos="" y="" />< />
- -1 < cos="" 2t="" />< />
To find the common region satisfying all these conditions, we need to intersect the intervals obtained from each condition.
The common region will be the intersection of the intervals (-2, 2), (-1/2, 1/2), (-1, 1), (-2, 2), and (-1, 1), which is (-1/2, 1/2).
Therefore, the points whose coordinates (X, y) satisfy the given expression and form the region in the coordinate plane are the points lying between the lines X = -1/2 and X = 1/2.
To determine the region formed by the points whose coordinates satisfy the given expression, we need to analyze each term separately and then combine the conditions to find the common region.
**Analyzing the first term: (2cos t)**
Since cosine function oscillates between -1 and 1, the value of 2cos t can vary from -2 to 2. For the given expression to be positive, the value of 2cos t must be greater than 0. Therefore, -2 < 2cos="" t="" />< />
**Analyzing the second term: (1/2 cos x cos y)**
Since cosine function oscillates between -1 and 1, the value of 1/2 cos x cos y can vary from -1/2 to 1/2. For the given expression to be positive, the value of 1/2 cos x cos y must be greater than 0. Therefore, -1/2 < 1/2="" cos="" x="" cos="" y="" />< />
**Analyzing the third term: (cos x cos y)**
Since cosine function oscillates between -1 and 1, the value of cos x cos y can vary from -1 to 1. For the given expression to be positive, the value of cos x cos y must be greater than 0. Therefore, -1 < cos="" x="" cos="" y="" />< />
**Analyzing the fourth term: (1)**
The value of 1 is always positive, so it does not contribute any conditions to the given expression.
**Analyzing the fifth term: (cos x - cos y)**
Since cosine function oscillates between -1 and 1, the value of cos x - cos y can vary from -2 to 2. For the given expression to be positive, the value of cos x - cos y must be greater than 0. Therefore, -2 < cos="" x="" -="" cos="" y="" />< />
**Analyzing the sixth term: (cos 2t)**
Since cosine function oscillates between -1 and 1, the value of cos 2t can vary from -1 to 1. For the given expression to be positive, the value of cos 2t must be greater than 0. Therefore, -1 < cos="" 2t="" />< />
**Combining the conditions:**
By analyzing each term separately, we have obtained the following conditions for the given expression to be positive:
- -2 < 2cos="" t="" />< />
- -1/2 < 1/2="" cos="" x="" cos="" y="" />< />
- -1 < cos="" x="" cos="" y="" />< />
- -2 < cos="" x="" -="" cos="" y="" />< />
- -1 < cos="" 2t="" />< />
To find the common region satisfying all these conditions, we need to intersect the intervals obtained from each condition.
The common region will be the intersection of the intervals (-2, 2), (-1/2, 1/2), (-1, 1), (-2, 2), and (-1, 1), which is (-1/2, 1/2).
Therefore, the points whose coordinates (X, y) satisfy the given expression and form the region in the coordinate plane are the points lying between the lines X = -1/2 and X = 1/2.
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Indicate in the co-ordinate plane all the points whose coordinates( X, y) are such that expressions (2cos t 1/2 cos x cos y) cos x cos y 1 cos x -cos y cos 2 t is positive for any value of t and depict the region formed by these points?
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Indicate in the co-ordinate plane all the points whose coordinates( X, y) are such that expressions (2cos t 1/2 cos x cos y) cos x cos y 1 cos x -cos y cos 2 t is positive for any value of t and depict the region formed by these points? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Indicate in the co-ordinate plane all the points whose coordinates( X, y) are such that expressions (2cos t 1/2 cos x cos y) cos x cos y 1 cos x -cos y cos 2 t is positive for any value of t and depict the region formed by these points? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Indicate in the co-ordinate plane all the points whose coordinates( X, y) are such that expressions (2cos t 1/2 cos x cos y) cos x cos y 1 cos x -cos y cos 2 t is positive for any value of t and depict the region formed by these points?.
Indicate in the co-ordinate plane all the points whose coordinates( X, y) are such that expressions (2cos t 1/2 cos x cos y) cos x cos y 1 cos x -cos y cos 2 t is positive for any value of t and depict the region formed by these points? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Indicate in the co-ordinate plane all the points whose coordinates( X, y) are such that expressions (2cos t 1/2 cos x cos y) cos x cos y 1 cos x -cos y cos 2 t is positive for any value of t and depict the region formed by these points? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Indicate in the co-ordinate plane all the points whose coordinates( X, y) are such that expressions (2cos t 1/2 cos x cos y) cos x cos y 1 cos x -cos y cos 2 t is positive for any value of t and depict the region formed by these points?.
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