SATHEE: Theory of Equations 5 Question 7

Theory of Equations 5 Question 7

7. The largest interval for which $x^{12} - x^{9} + x^{4} - x + 1 > 0$ is

$(1982, 2 M)$

(a) $- 4 < x \leq 0$

(b) $0 < x < 1$

(d) $- \infty < x < \infty$

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Answer:

Correct Answer: 7. (d)

Solution:

Given, $x^{12} - x^{9} + x^{4} - x + 1 > 0$

Here, three cases arises:

Case I When $x \leq 0 \Rightarrow x^{12} > 0, - x^{9} > 0, x^{4} > 0, - x > 0$

$∴ x^{12} - x^{9} + x^{4} - x + 1 > 0, \forall x \leq 0$

Case II When $0 < x \leq 1$

$\begin{aligned} x^{9} < x^{4} and x < 1 \Rightarrow - x^{9} + x^{4} > 0 and 1 - x > 0 \\ ∴ x^{12} - x^{9} + x^{4} - x + 1 > 0, \forall 0 < x \leq 1 \end{aligned}$

Case III When $x > 1 \Rightarrow x^{12} > x^{9}$ and $x^{4} > x$

$∴ x^{12} - x^{9} + x^{4} - x + 1 > 0, \forall x > 1$

From Eqs. (i), (ii) and (iii), the above equation holds for all $x \in R$ .

Theory of Equations 5 Question 7

7. The largest interval for which $x^{12} - x^{9} + x^{4} - x + 1 > 0$ is

Answer:

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Theory of Equations 5 Question 7

7. The largest interval for which x12−x9+x4−x+1>0 is

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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7. The largest interval for which $x^{12} - x^{9} + x^{4} - x + 1 > 0$ is