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DIRECTIONS for the question: Solve the following question and mark the best possible option.
Which of the following cannot be the value of 4sin x cos x (1 + sin x cos x)?
- a)-1
- b)0
- c)3
- d)4
Correct answer is option 'D'. Can you explain this answer?
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DIRECTIONSfor the question:Solve the following question and mark the b...
Question Analysis:
We are given the expression 4sinxcos x(1 - sin x cos x) and we need to determine which of the given options cannot be the value of this expression.
Solution:
Let's simplify the expression step by step and analyze each option.
Simplifying the Expression:
We can use the trigonometric identity sin2x = 1 - cos2x to simplify the expression.
4sinxcos x(1 - sin x cos x)
= 4sinxcos x - 4sin2x cos2x
Now, let's simplify the expression further.
Option Analysis:
We will analyze each option and determine if it can be the value of the expression.
a) -1:
If the expression is equal to -1, then we can equate the expression to -1 and solve for x.
4sinxcos x - 4sin2x cos2x = -1
However, when we solve this equation, we find that there is no real solution for x. Therefore, option a) cannot be the value of the expression.
b) 0:
If the expression is equal to 0, then we can equate the expression to 0 and solve for x.
4sinxcos x - 4sin2x cos2x = 0
By factoring out a common factor of 4sinx cosx, we get:
4sinx cosx (1 - sinx cosx) = 0
This equation is satisfied when either sinx cosx = 0 or 1 - sinx cosx = 0.
If sinx cosx = 0, then either sinx = 0 or cosx = 0. This implies that x can be any multiple of π/2.
If 1 - sinx cosx = 0, then sinx cosx = 1. However, there is no real solution for x in this case.
Therefore, option b) can be the value of the expression.
c) 3:
If the expression is equal to 3, then we can equate the expression to 3 and solve for x.
4sinxcos x - 4sin2x cos2x = 3
However, when we solve this equation, we find that there is no real solution for x. Therefore, option c) cannot be the value of the expression.
d) 4:
If the expression is equal to 4, then we can equate the expression to 4 and solve for x.
4sinxcos x - 4sin2x cos2x = 4
However, when we solve this equation, we find that there is no real solution for x. Therefore, option d) cannot be the value of the expression.
Conclusion:
From our analysis, we can conclude that the expression 4sinxcos x(1 - sin x cos x) cannot have a value of 4. Therefore, the correct answer is option D.
We are given the expression 4sinxcos x(1 - sin x cos x) and we need to determine which of the given options cannot be the value of this expression.
Solution:
Let's simplify the expression step by step and analyze each option.
Simplifying the Expression:
We can use the trigonometric identity sin2x = 1 - cos2x to simplify the expression.
4sinxcos x(1 - sin x cos x)
= 4sinxcos x - 4sin2x cos2x
Now, let's simplify the expression further.
Option Analysis:
We will analyze each option and determine if it can be the value of the expression.
a) -1:
If the expression is equal to -1, then we can equate the expression to -1 and solve for x.
4sinxcos x - 4sin2x cos2x = -1
However, when we solve this equation, we find that there is no real solution for x. Therefore, option a) cannot be the value of the expression.
b) 0:
If the expression is equal to 0, then we can equate the expression to 0 and solve for x.
4sinxcos x - 4sin2x cos2x = 0
By factoring out a common factor of 4sinx cosx, we get:
4sinx cosx (1 - sinx cosx) = 0
This equation is satisfied when either sinx cosx = 0 or 1 - sinx cosx = 0.
If sinx cosx = 0, then either sinx = 0 or cosx = 0. This implies that x can be any multiple of π/2.
If 1 - sinx cosx = 0, then sinx cosx = 1. However, there is no real solution for x in this case.
Therefore, option b) can be the value of the expression.
c) 3:
If the expression is equal to 3, then we can equate the expression to 3 and solve for x.
4sinxcos x - 4sin2x cos2x = 3
However, when we solve this equation, we find that there is no real solution for x. Therefore, option c) cannot be the value of the expression.
d) 4:
If the expression is equal to 4, then we can equate the expression to 4 and solve for x.
4sinxcos x - 4sin2x cos2x = 4
However, when we solve this equation, we find that there is no real solution for x. Therefore, option d) cannot be the value of the expression.
Conclusion:
From our analysis, we can conclude that the expression 4sinxcos x(1 - sin x cos x) cannot have a value of 4. Therefore, the correct answer is option D.
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Community Answer
DIRECTIONSfor the question:Solve the following question and mark the b...
We know that sin 2q = 2sin q cos q.
So, 4sin x cos x (1 + sin x cos x) = 2sin 2x(1 + ½ sin 2x) = sin 2x (2 + sin 2x).
So, 4sin x cos x (1 + sin x cos x) = 2sin 2x(1 + ½ sin 2x) = sin 2x (2 + sin 2x).
We know that the minimum and maximum values of sin q are –1 and 1 respectively.
Substituting these values in sin 2x (2 + sin 2x), the minimum value of the expression is –1(2 – 1) = –1 and the maximum value of the expression is 1(2 + 1) = 3.
Since –1 ≤ 4sin x cos x (1 + sin x cos x) ≤ 3, the best answer is option 4.
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