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Ncert Solutions Maths class 12th Maths Class 12th Application of Derivatives

31. On which of the following intervals is the function f given by f(x) = x¹⁰⁰ + sin x –1 decreasing ?

(A) (0,1) (B) $(\frac{π}{2}, π)$ (C) $(0, \frac{π}{2})$ (D) None of these

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Answered by

Vishal Baghel | Contributor-Level 10

4 months ago

We have, f (x) = x 100 + sin x - 1.

So, f (x) = 100x⁹⁹ + cosx.

(A) When x (0,1). We get.

x>0

$\Rightarrow$ x⁹⁹> 0

$\Rightarrow$ 100 x⁹⁹> 0.

Now, 0 radian = 0 degree

and 1 radian = $\frac{18 0^{°} \times 1}{π} = \frac{18 0^{°}}{(\frac{22}{7})} = 5 7^{°} .$

So, cosx> 0 for x∈ (0,1) radian = (0,57)

∴f (x) > 0 for x∈ (0,1).

(B) When x ∈ $(\frac{π}{2}, π)$ we get,

So, x> 1

$\Rightarrow$ x⁹⁹> 1

$\Rightarrow$ 100x⁹⁹> 100. ${\begin{matrix} ? (\frac{π}{2}, π) = (1.57, 3.14) & which is great than 1 \end{matrix}}$

And cosx is negative between -1 and 0.

So, f (x) = 100x⁹⁹ + cosx> 100 - 1 = 99 > 0.

∴f(x) is strictly increasing on $(\frac{π}{2}, π)$

(c) When x ∈ $(0, \frac{π}{2}),$ we get,

$\Rightarrow$ x> 0

$\Rightarrow$ x⁹⁹> 0

$\Rightarrow$ 100x⁹⁹> 0

and cosx> 0. (firstquadrant).

I e, f(x) > 0.

∴f(x) is strictly increasing on $(0, \frac{π}{2})$

Hence, option (D) is correct.

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y (x) = ∫? (2t² - 15t + 10)dt
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2a² - 15a + 10 = -3
2a² - 15a + 13 = 0 (The provided solution has 2a²-15a+7=0, suggesting a different problem or a typo)
Following the image: 2a² - 15a + 7 = 0
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b = ∫? (2t² - 15t + 10)dt = [2t³/3 - 15t²/2 + 10t] from 0 to 7.
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Let F₁(A,B,C) = $(A \land \sim B) \lor$ $[\sim C \land (A \lor B)] \lor \sim A a n d F_{2} (A, B) = (A \lor B) \lor (B \to \sim A)$ be two logical expressions. Then:

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By truth table

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From option let it be isosceles where AB = AC then

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So $A B = \sqrt[]{h^{2} + x^{2}} = \sqrt[]{\frac{9 r^{2}}{4} + \frac{3 r^{2}}{4}} = \sqrt[]{3} r$ .

Hence $Δ$ be equilateral having each side of length $\sqrt[]{3} r .$

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31. On which of the following intervals is the function f given by f(x) = x¹⁰⁰ + sin x –1 decreasing ?

(A) (0,1) (B) $(\frac{π}{2}, π)$ (C) $(0, \frac{π}{2})$ (D) None of these

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31. On which of the following intervals is the function f given by f(x) = x100 + sin x –1 decreasing ? (A) (0,1) (B) (π2,π) (C) (0,π2) (D) None of these

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31. On which of the following intervals is the function f given by f(x) = x¹⁰⁰ + sin x –1 decreasing ?

(A) (0,1) (B) $(\frac{π}{2}, π)$ (C) $(0, \frac{π}{2})$ (D) None of these