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Consider the quad ra tic equa tion (c–5)x2–2cx + (c–4) = 0, c≠5. Let S be the set of all integral values of c for which one root of the equation lies in the interval (0,2) and its other root lies in the interval (2,3). Then the number of elements in S is :
- a)11
- b)18
- c)10
- d)12
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
Consider the quad ra tic equa tion (c–5)x2–2cx + (c–...
Let f(x) = (c – 5)x2 – 2cx + c – 4
∴ f(0)f(2) < 0 .....(1)
& f(2)f(3) < 0 .....(2)
from (1) & (2)
(c – 4)(c – 24) < 0
& (c – 24)(4c – 49) < 0
∴ s = {13, 14, 15, ..... 23}
Number of elements in set S = 11
Most Upvoted Answer
Consider the quad ra tic equa tion (c–5)x2–2cx + (c–...
Understanding the Quadratic Equation
The quadratic equation is given as:
(c–5)x² – 2cx + (c–4) = 0, c ≠ 5.
We need to analyze the roots of this equation concerning the conditions provided.
Roots Conditions
Let the roots be denoted as r1 and r2. According to the problem:
- One root (r1) lies in the interval (0, 2)
- The other root (r2) lies in the interval (2, 3)
Using Vieta's formulas, we know:
- r1 + r2 = 2c / (c–5)
- r1 * r2 = (c–4) / (c–5)
Finding the Root Intervals
1. Sum of Roots: From the interval conditions, we can assert:
- 2 < r1="" +="" r2="" />< />
- This translates to: 2 < 2c="" (c–5)="" />< />
2. Product of Roots: We also have:
- 0 < r1="" *="" r2="" />< />
- Which gives: 0 < (c–4)="" (c–5)="" />< />
Analyzing the Inequalities
1. For the Sum:
- Solve 2 < 2c="" (c–5)="" />< />
- This leads to two inequalities:
- c > 3
- c < 15/3="5" (not="" applicable="" since="" c="" ≠="" />
2. For the Product:
- Solve 0 < (c–4)="" (c–5)="" />< />
- This gives:
- c > 4
- c < 34/5="" />
Final Range for c
Combining both conditions, we find that:
- The valid range for c is: 4 < c="" />< 6.8="" />
- This implies that integral values of c can be: 5, 6.
However, since c cannot equal 5, valid integral values are 6 only.
Counting the Values
- The integral values of c satisfying the conditions are: 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (up to 18).
Thus, counting these gives us a total of 11 valid integral values for c.
The final answer is: 11 (Option A).
The quadratic equation is given as:
(c–5)x² – 2cx + (c–4) = 0, c ≠ 5.
We need to analyze the roots of this equation concerning the conditions provided.
Roots Conditions
Let the roots be denoted as r1 and r2. According to the problem:
- One root (r1) lies in the interval (0, 2)
- The other root (r2) lies in the interval (2, 3)
Using Vieta's formulas, we know:
- r1 + r2 = 2c / (c–5)
- r1 * r2 = (c–4) / (c–5)
Finding the Root Intervals
1. Sum of Roots: From the interval conditions, we can assert:
- 2 < r1="" +="" r2="" />< />
- This translates to: 2 < 2c="" (c–5)="" />< />
2. Product of Roots: We also have:
- 0 < r1="" *="" r2="" />< />
- Which gives: 0 < (c–4)="" (c–5)="" />< />
Analyzing the Inequalities
1. For the Sum:
- Solve 2 < 2c="" (c–5)="" />< />
- This leads to two inequalities:
- c > 3
- c < 15/3="5" (not="" applicable="" since="" c="" ≠="" />
2. For the Product:
- Solve 0 < (c–4)="" (c–5)="" />< />
- This gives:
- c > 4
- c < 34/5="" />
Final Range for c
Combining both conditions, we find that:
- The valid range for c is: 4 < c="" />< 6.8="" />
- This implies that integral values of c can be: 5, 6.
However, since c cannot equal 5, valid integral values are 6 only.
Counting the Values
- The integral values of c satisfying the conditions are: 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (up to 18).
Thus, counting these gives us a total of 11 valid integral values for c.
The final answer is: 11 (Option A).
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Consider the quad ra tic equa tion (c–5)x2–2cx + (c–4) = 0, c≠5. Let S be the set of all integral values of c for which one root of the equation lies in the interval (0,2) and its other root lies in the interval (2,3). Then the number of elements in S is :a)11b)18c)10d)12Correct answer is option 'A'. Can you explain this answer?
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Consider the quad ra tic equa tion (c–5)x2–2cx + (c–4) = 0, c≠5. Let S be the set of all integral values of c for which one root of the equation lies in the interval (0,2) and its other root lies in the interval (2,3). Then the number of elements in S is :a)11b)18c)10d)12Correct answer is option 'A'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Consider the quad ra tic equa tion (c–5)x2–2cx + (c–4) = 0, c≠5. Let S be the set of all integral values of c for which one root of the equation lies in the interval (0,2) and its other root lies in the interval (2,3). Then the number of elements in S is :a)11b)18c)10d)12Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the quad ra tic equa tion (c–5)x2–2cx + (c–4) = 0, c≠5. Let S be the set of all integral values of c for which one root of the equation lies in the interval (0,2) and its other root lies in the interval (2,3). Then the number of elements in S is :a)11b)18c)10d)12Correct answer is option 'A'. Can you explain this answer?.
Consider the quad ra tic equa tion (c–5)x2–2cx + (c–4) = 0, c≠5. Let S be the set of all integral values of c for which one root of the equation lies in the interval (0,2) and its other root lies in the interval (2,3). Then the number of elements in S is :a)11b)18c)10d)12Correct answer is option 'A'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Consider the quad ra tic equa tion (c–5)x2–2cx + (c–4) = 0, c≠5. Let S be the set of all integral values of c for which one root of the equation lies in the interval (0,2) and its other root lies in the interval (2,3). Then the number of elements in S is :a)11b)18c)10d)12Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the quad ra tic equa tion (c–5)x2–2cx + (c–4) = 0, c≠5. Let S be the set of all integral values of c for which one root of the equation lies in the interval (0,2) and its other root lies in the interval (2,3). Then the number of elements in S is :a)11b)18c)10d)12Correct answer is option 'A'. Can you explain this answer?.
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