SATHEE: Theory of Equations 4 Question 3

Theory of Equations 4 Question 3

3. If both the roots of the quadratic equation $x^{2} - m x + 4 = 0$ are real and distinct and they lie in the interval $[1, 5]$ then $m$ lies in the interval

(2019 Main, 9 Jan II)

(a) $(4, 5)$

(b) $(- 5, - 4)$

(d) $(3, 4)$

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Answer:

Correct Answer: 3. (a)

Solution:

According to given information, we have the following graph

Now, the following conditions should satisfy

(i) $D > 0 \Rightarrow b^{2} - 4 a c > 0$

$\begin{array}{lc} \Rightarrow & m^{2} - 4 \times 1 \times 4 > 0 \\ \Rightarrow & m^{2} - 16 > 0 \\ \Rightarrow & (m - 4) (m + 4) > 0 \\ \Rightarrow & m \in (- \infty, - 4) \cup (4, \infty) \end{array}$

(ii) The vertex of the parabola should lie between $x = 1$ and $x = 5$

$∴ - \frac{b}{2 a} \in (1, 5) \Rightarrow 1 < \frac{m}{2} < 5 \Rightarrow m \in (2, 10)$

(iii) $f (1) > 0 \Rightarrow 1 - m + 4 > 0$

$\Rightarrow m < 5 \Rightarrow m \in (- \infty, 5)$

(iv) $f (5) > 0 \Rightarrow 25 - 5 m + 4 > 0 \Rightarrow 5 m < 29 \Rightarrow m \in - \infty, \frac{29}{5}$

From the values of $m$ obtained in (i), (ii), (iii) and (iv), we get $m \in (4, 5)$ .

Theory of Equations 4 Question 3

3. If both the roots of the quadratic equation $x^{2} - m x + 4 = 0$ are real and distinct and they lie in the interval $[1, 5]$ then $m$ lies in the interval

Answer:

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Theory of Equations 4 Question 3

3. If both the roots of the quadratic equation x2−mx+4=0 are real and distinct and they lie in the interval [1,5] then m lies in the interval

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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3. If both the roots of the quadratic equation $x^{2} - m x + 4 = 0$ are real and distinct and they lie in the interval $[1, 5]$ then $m$ lies in the interval