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Let u = (log2 x)2 - 6 log2 x + 12, where x is a real number. Then, the equation xu = 256 has
- a)No solution for x
- b)Exactly one solution for x
- c)Exactly two distinct solutions for x
- d)Exactly three distinct solutions for x
Correct answer is option 'B'. Can you explain this answer?
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Let u = (log2 x)2 - 6 log2 x + 12, where x is a real number. Then, the...
Given Equation:
Let u = (log2 x)2 - 6 log2 x + 12
We need to find the number of solutions for the equation xu = 256, where x is a real number.
Solution:
To solve the equation xu = 256, we need to substitute the value of u from the given equation and then solve for x.
Substituting the value of u:
x((log2 x)2 - 6 log2 x + 12) = 256
Simplifying the equation:
(log2 x)2 - 6 log2 x + 12 = 256/x
Observation:
To simplify the equation, let's make a substitution. Consider y = log2 x. The equation becomes:
y2 - 6y + 12 = 256/2^y
y2 - 6y + 12 = 128/2^y
Now, let's solve the equation in terms of y and find the number of solutions.
Quadratic Equation:
The equation y2 - 6y + 12 = 128/2^y is a quadratic equation.
Finding the number of solutions:
To find the number of solutions for the quadratic equation, we need to analyze the discriminant (D) of the equation.
D = b2 - 4ac
= (-6)^2 - 4(1)(12 - 128/2^y)
= 36 - 48 + 128/2^y
= -12 + 128/2^y
Analysis of Discriminant:
To analyze the discriminant, we need to determine the sign of D.
Key Points:
- If D > 0, the quadratic equation has two distinct real solutions.
- If D = 0, the quadratic equation has exactly one real solution.
- If D < 0,="" the="" quadratic="" equation="" has="" no="" real="" />
Determining the sign of D:
Since D = -12 + 128/2^y, we need to analyze the sign of this expression.
-12 is a negative value, and 128/2^y is always positive because 2^y is always positive for real values of y.
Therefore, D = -12 + 128/2^y is negative for all real values of y.
Conclusion:
Since the discriminant is negative, the quadratic equation y2 - 6y + 12 = 128/2^y has no real solutions.
Substituting y back to x:
Since y = log2 x, there are no real values of x that satisfy the equation.
Therefore, the equation xu = 256 has no solution for x. The correct answer is option 'A'.
Let u = (log2 x)2 - 6 log2 x + 12
We need to find the number of solutions for the equation xu = 256, where x is a real number.
Solution:
To solve the equation xu = 256, we need to substitute the value of u from the given equation and then solve for x.
Substituting the value of u:
x((log2 x)2 - 6 log2 x + 12) = 256
Simplifying the equation:
(log2 x)2 - 6 log2 x + 12 = 256/x
Observation:
To simplify the equation, let's make a substitution. Consider y = log2 x. The equation becomes:
y2 - 6y + 12 = 256/2^y
y2 - 6y + 12 = 128/2^y
Now, let's solve the equation in terms of y and find the number of solutions.
Quadratic Equation:
The equation y2 - 6y + 12 = 128/2^y is a quadratic equation.
Finding the number of solutions:
To find the number of solutions for the quadratic equation, we need to analyze the discriminant (D) of the equation.
D = b2 - 4ac
= (-6)^2 - 4(1)(12 - 128/2^y)
= 36 - 48 + 128/2^y
= -12 + 128/2^y
Analysis of Discriminant:
To analyze the discriminant, we need to determine the sign of D.
Key Points:
- If D > 0, the quadratic equation has two distinct real solutions.
- If D = 0, the quadratic equation has exactly one real solution.
- If D < 0,="" the="" quadratic="" equation="" has="" no="" real="" />
Determining the sign of D:
Since D = -12 + 128/2^y, we need to analyze the sign of this expression.
-12 is a negative value, and 128/2^y is always positive because 2^y is always positive for real values of y.
Therefore, D = -12 + 128/2^y is negative for all real values of y.
Conclusion:
Since the discriminant is negative, the quadratic equation y2 - 6y + 12 = 128/2^y has no real solutions.
Substituting y back to x:
Since y = log2 x, there are no real values of x that satisfy the equation.
Therefore, the equation xu = 256 has no solution for x. The correct answer is option 'A'.
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Let u = (log2 x)2 - 6 log2 x + 12, where x is a real number. Then, the equation xu = 256 hasa)No solution for xb)Exactly one solution for xc)Exactly two distinct solutions for xd)Exactly three distinct solutions for xCorrect answer is option 'B'. Can you explain this answer?
Question Description
Let u = (log2 x)2 - 6 log2 x + 12, where x is a real number. Then, the equation xu = 256 hasa)No solution for xb)Exactly one solution for xc)Exactly two distinct solutions for xd)Exactly three distinct solutions for xCorrect answer is option 'B'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about Let u = (log2 x)2 - 6 log2 x + 12, where x is a real number. Then, the equation xu = 256 hasa)No solution for xb)Exactly one solution for xc)Exactly two distinct solutions for xd)Exactly three distinct solutions for xCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let u = (log2 x)2 - 6 log2 x + 12, where x is a real number. Then, the equation xu = 256 hasa)No solution for xb)Exactly one solution for xc)Exactly two distinct solutions for xd)Exactly three distinct solutions for xCorrect answer is option 'B'. Can you explain this answer?.
Let u = (log2 x)2 - 6 log2 x + 12, where x is a real number. Then, the equation xu = 256 hasa)No solution for xb)Exactly one solution for xc)Exactly two distinct solutions for xd)Exactly three distinct solutions for xCorrect answer is option 'B'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about Let u = (log2 x)2 - 6 log2 x + 12, where x is a real number. Then, the equation xu = 256 hasa)No solution for xb)Exactly one solution for xc)Exactly two distinct solutions for xd)Exactly three distinct solutions for xCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let u = (log2 x)2 - 6 log2 x + 12, where x is a real number. Then, the equation xu = 256 hasa)No solution for xb)Exactly one solution for xc)Exactly two distinct solutions for xd)Exactly three distinct solutions for xCorrect answer is option 'B'. Can you explain this answer?.
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Let u = (log2 x)2 - 6 log2 x + 12, where x is a real number. Then, the equation xu = 256 hasa)No solution for xb)Exactly one solution for xc)Exactly two distinct solutions for xd)Exactly three distinct solutions for xCorrect answer is option 'B'. Can you explain this answer?, a detailed solution for Let u = (log2 x)2 - 6 log2 x + 12, where x is a real number. Then, the equation xu = 256 hasa)No solution for xb)Exactly one solution for xc)Exactly two distinct solutions for xd)Exactly three distinct solutions for xCorrect answer is option 'B'. Can you explain this answer? has been provided alongside types of Let u = (log2 x)2 - 6 log2 x + 12, where x is a real number. Then, the equation xu = 256 hasa)No solution for xb)Exactly one solution for xc)Exactly two distinct solutions for xd)Exactly three distinct solutions for xCorrect answer is option 'B'. Can you explain this answer? theory, EduRev gives you an
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