Maths

17. Given, 3x – 2 < 2x + 1

3x – 2x< 2 + 1

x< 3.

So, x (- ∞, 3).

18. $\underset{}{}$$\underset{x?0}{lim}\frac{ax+xcosx}{bsinx}=\underset{x?0}{lim}\frac{x(a+cosx)}{bsinx}$

$=\frac{1}{b}*\frac{(a+cos0)}{1}$

$=\frac{a+1}{b}$

16. Given,

$\frac{2x-1}{3}\ge \frac{3x-2}{4}-\frac{2-x}{5}$

$\frac{(2x-1)}{3}\ge \frac{5(3x-2)-4(2-x)}{4*5}$

20 (2x- 1) ≥3 [15x- 10 - 8 + 4x].

40x-20 ≥3 [19x-18]

40x-20 ≥ 57x – 54.

54 - 20 ≥ 57x- 40x

34 ≥17x

$\Rightarrow \frac{34}{17}\ge x.$

2 ≥x

So, x (- ∞, 2].

14. Given, 37 – (3x + 5) ≥9x –8 (x –3)

37– 3x–5 ≥9x –8x+ 24.

37 – 24–5 ≥9x –8x+ 3x

8 ≥4x.

$\Rightarrow \frac{8}{4}\ge x$

2 ≥x

So, x∈ (–∞, 2]

13. Given 2 (2x + 3) –10 < 6 (x 2)

4x + 6 –10 < 6x 12.

4x –4 < 6x 12

12 –4 < 6x 4x

8 < 2x

$\Rightarrow \frac{8}{2}<\frac{2x}{2}\Rightarrow 4<x$

So, x∈ (4, –∞)

12. Given, $\frac{1}{2}$ $\left(\frac{3x}{5}+4\right)\ge \frac{1}{3}(x-6)$

$\frac{6}{2}\left(\frac{3x}{5}+4\right)\ge \frac{6}{3}(x-6)$

$\Rightarrow 3\left(\frac{3x}{5}+4\right)\ge 2(x-6)$

$\Rightarrow \frac{9}{5}x+12\ge 2x-12$

$\Rightarrow \frac{9}{5}x-2x\ge -12-12.$

$\Rightarrow \frac{9x-10x}{5}\ge -24.$

$\Rightarrow \frac{-x*5\ge }{5}-24*5.$

x ≥–120.

(–1) x (–x)≤ ( –120)* (–1)

x ≤ 120.

So, x ∈ (–∞, 120].

73. Given, f (x) = $\frac{x}{si{n}^{n}x}$

So, f?(x) = $\frac{\left(si{n}^{x}x\frac{dx}{dx}\right)-\left(x\frac{d}{dx}si{n}^{x}x\right)}{{\left(si{n}^{x}x\right)}^2}$

= $\frac{si{n}^{x}x-x·n{(sinx)}^{x-1}\frac{d}{dx}sinx}{{(sinx)}^{2x}}$

$=\frac{{(sinx)}^n-x·x{(sinx)}^{n-1}·cosx}{{(sinx)}^{2n}}$

$=\frac{{(sinx)}^{n-1}[sinx-nx·cosx]}{{(sinx)}^{2n}}$

$=\frac{sinx-n·x·cosx}{{(sinx)}^{2n-(n-1)}}$

$=\frac{sinx-nxcosx}{{(sinx)}^{n+1}}$

11. Given, $\frac{3\left(x-2\right)5}{5}\le \frac{5\left(2-x\right)}{3}$

=> 3 x 3 (x – 2) ≤ 5 x 5 (2 – x)

=> 9x – 18 ≤ 50 – 25x

=> 9x + 25x ≤ 50 + 18

=> 34x ≤ 68

=> $x=\frac{68}{34}$

=> x ≤ 2

So,  $x\in (-\infty ,2]$