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Consider the quadratic equation (c - 5)x...

Consider the quadratic equation `(c - 5)x^(2) - 2cx + (c - 4) = 0, c ne 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

A

11

B

18

C

10

D

12

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To solve the quadratic equation \((c - 5)x^2 - 2cx + (c - 4) = 0\) and find the integral values of \(c\) such that one root lies in the interval (0, 2) and the other root lies in the interval (2, 3), we can follow these steps: ### Step 1: Identify the roots of the quadratic equation The roots of the quadratic equation can be found using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \(a = c - 5\), \(b = -2c\), and \(c = c - 4\). ### Step 2: Calculate the discriminant The discriminant \(\Delta\) is given by: \[ \Delta = b^2 - 4ac = (-2c)^2 - 4(c - 5)(c - 4) \] Calculating this gives: \[ \Delta = 4c^2 - 4[(c^2 - 9c + 20)] = 4c^2 - 4c^2 + 36c - 80 = 36c - 80 \] ### Step 3: Roots conditions For the roots to be real, the discriminant must be non-negative: \[ 36c - 80 \geq 0 \implies c \geq \frac{80}{36} = \frac{20}{9} \approx 2.22 \] ### Step 4: Evaluate the roots at specific points Let the roots be \(r_1\) and \(r_2\). We need to ensure: - One root \(r_1\) is in (0, 2) - The other root \(r_2\) is in (2, 3) Using Vieta's formulas, we know: \[ r_1 + r_2 = \frac{2c}{c - 5} \quad \text{and} \quad r_1 r_2 = \frac{c - 4}{c - 5} \] ### Step 5: Set up inequalities for the roots 1. For \(r_1\) to be in (0, 2): - \(r_1 > 0\) implies \(c - 4 > 0\) or \(c > 4\) - \(r_1 < 2\) gives \(\frac{2c}{c - 5} < 4\) which simplifies to \(c < 24\) 2. For \(r_2\) to be in (2, 3): - \(r_2 > 2\) gives \(\frac{2c}{c - 5} > 4\) which simplifies to \(c > 20\) - \(r_2 < 3\) gives \(\frac{2c}{c - 5} < 6\) which simplifies to \(c < 30\) ### Step 6: Combine the inequalities From the above conditions, we have: - From \(r_1\): \(4 < c < 24\) - From \(r_2\): \(20 < c < 30\) ### Step 7: Find the intersection of the intervals The intersection of the intervals \( (4, 24) \) and \( (20, 30) \) gives: \[ 20 < c < 24 \] ### Step 8: Determine integral values of \(c\) The integral values of \(c\) in the interval \( (20, 24) \) are: \[ c = 21, 22, 23 \] Thus, there are \(3\) integral values. ### Final Answer The number of elements in set \(S\) is \(3\).
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