SATHEE: Theory of Equations 5 Question 6

Theory of Equations 5 Question 6

6. If $a + b + c = 0$ , then the quadratic equation $3 a x^{2} + 2 b x + c = 0$ has

$(1983, 1 M)$

(a) at least one root in $(0, 1)$

(b) one root in $(2, 3)$ and the other in $(- 2, - 1)$

(d) None of the above

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Answer:

Correct Answer: 6. (a)

Solution:

$f (x) = a x^{3} + b x^{2} + c x + d$

$∴$

$f (0) = d and f (1) = a + b + c + d = d$

$[∵ a + b + c = 0]$

$∴ f (0) = f (1)$

$f$ is continuous in the closed interval $[0, 1]$ and $f$ is derivable in the open interval $(0, 1)$ .

Also,

$f (0) = f (1) .$

$∴$ By Rolle’s theorem, $f^{'} (α) = 0$ for $0 < α < 1$

Now,

$\begin{aligned} f^{'} (x) = 3 a x^{2} + 2 b x + c \\ f^{'} (α) = 3 a α^{2} + 2 b α + c = 0 \end{aligned}$

$\Rightarrow$

$∴$ Eq. (i) has exist atleast one root in the interval $(0, 1)$ . Thus, $f^{'} (x)$ must have root in the interval $(0, 1)$ or $3 a x^{2} + 2 b x + c = 0$ has root $\in (0, 1)$ .

Theory of Equations 5 Question 6

6. If $a + b + c = 0$ , then the quadratic equation $3 a x^{2} + 2 b x + c = 0$ has

Answer:

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Theory of Equations 5 Question 6

6. If a+b+c=0, then the quadratic equation 3ax2+2bx+c=0 has

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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6. If $a + b + c = 0$ , then the quadratic equation $3 a x^{2} + 2 b x + c = 0$ has