SATHEE: Theory of Equations 1 Question 8

Theory of Equations 1 Question 8

9. If $5, 5 r, 5 r^{2}$ are the lengths of the sides of a triangle, then $r$ cannot be equal to

(a) $\frac{5}{4}$

(b) $\frac{7}{4}$

(d) $\frac{3}{4}$

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Answer:

Correct Answer: 9. (b)

Solution:

Let $a = 5, b = 5 r$ and $c = 5 r^{2}$

We know that, in a triangle sum of 2 sides is always greater than the third side.

$∴ a + b > c; b + c > a$ and $c + a > b$

Now, $a + b > c$

$\Rightarrow 5 + 5 r > 5 r^{2} \Rightarrow 5 r^{2} - 5 r - 5 < 0$

$\Rightarrow r^{2} - r - 1 < 0$

$\Rightarrow r - \frac{1 - \sqrt{5}}{2} r - \frac{1 + \sqrt{5}}{2} < 0$

$[∵$ roots of $a x^{2} + b x + c = 0$ are given by

$\begin{array}{r} x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a} and r^{2} - r - 1 = 0 \\ \Rightarrow r = \frac{1 \pm \sqrt{1 + 4}}{2} = \frac{1 \pm \sqrt{5}}{2}] \end{array}$

$\begin{aligned} \Rightarrow r & \in \frac{1 - \sqrt{5}}{2}, \frac{1 + \sqrt{5}}{2} \\ \frac{1}{\frac{1 - \sqrt{5}}{2}} - \frac{1 + \sqrt{5}}{2} \end{aligned}$

Similarly, $b + c > a$

$\Rightarrow 5 r + 5 r^{2} > 5$

$\Rightarrow r^{2} + r - 1 > 0$

$r - \frac{- 1 - \sqrt{5}}{2} r - \frac{- 1 + \sqrt{5}}{2} > 0$

$∵ r^{2} + r - 1 = 0 \Rightarrow r = \frac{- 1 \pm \sqrt{1 + 4}}{2} = \frac{- 1 \pm \sqrt{5}}{2}$

$\Rightarrow r \in - \infty, \frac{- 1 - \sqrt{5}}{2} \cup \frac{- 1 + \sqrt{5}}{2}, \infty$

$\frac{- 1 - \sqrt{5}}{2} \frac{- 1 + \sqrt{5}}{2}$

$and c + a > b$

$\Rightarrow 5 r^{2} + 5 > 5 r$

$\Rightarrow r^{2} - r + 1 > 0$

$\Rightarrow r^{2} - 2 \cdot \frac{1}{2} r + {\frac{1}{2}}^{2} + 1 - {\frac{1}{2}}^{2} > 0$

$\Rightarrow r - {\frac{1}{2}}^{2} + \frac{3}{4} > 0$

$\Rightarrow r \in R$

From Eqs. (i), (ii) and (iii), we get

$\begin{aligned} r \in \frac{- 1 + \sqrt{5}}{2}, \frac{1 + \sqrt{5}}{2} \\ \underset{- \infty}{⟵} \end{aligned}$

and $\frac{7}{4}$ is the only value that does not satisfy.

Theory of Equations 1 Question 8

9. If $5, 5 r, 5 r^{2}$ are the lengths of the sides of a triangle, then $r$ cannot be equal to

Answer:

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Theory of Equations 1 Question 8

9. If 5,5r,5r2 are the lengths of the sides of a triangle, then r cannot be equal to

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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9. If $5, 5 r, 5 r^{2}$ are the lengths of the sides of a triangle, then $r$ cannot be equal to