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y (x) = 2^x – x²

y? (x) = 2x log 2 – 2x

M = 3

N = 2

M + N = 5

Start with

(1) $\overline{E}:\frac{6!}{2!}=360$  

(2)    $\overline{GE}:\frac{5!}{2!},$ $\overline{GN}:\frac{5!}{2!}$  

(3) GTE : 4!, GTN: 4!, GTT : 4!

(4) GTWENTY = 1

⇒ 360 + 60 + 60 + 24 + 24 + 24 + 1 = 553

${\left(1+x\right)}^{11}={}^{11}{C}_0+{}^{11}{C}_{1}x+{}^{11}{C}_2{x}^2+.....{+}^{11}{C}_{11}{x}^{11}$

$=\frac{2^{12}-2-24}{12}$

$=\frac{2^{12}-26}{12}=\frac{4070}{12}=\frac{2035}{6}=\frac{m}{n}$

m + n = 2035 + 6 = 2041

${\displaystyle \int \frac{\left(sinx-cosx\right)si{n}^{2}x}{tanx\left(si{n}^{3}x+co{s}^{3}x\right)}dx}$

${\displaystyle \int \frac{\left(sinx-cosx\right)sinxcosx}{si{n}^{3}x+co{s}^{3}x}dx}$ , put sin³x + cos³x = t(3 sin²x×cosx – 3cos²xsinx) dx = dt

-> $\frac{1}{3}{\displaystyle \int \frac{dt}{t}}$

$=\frac{lnt}{3}+c$

$=\frac{ln\left|si{n}^{3}x+co{s}^{3}x\right|}{3}+c$

$f\left(x\right)=\frac{\left(2^x+2^{-x}\right)tanx\sqrt[]{ta{n}^{-1}\left(2x^2-3x+1\right)}}{{\left(7x^2-3x+1\right)}^3}$

$f\left(x\right)=\left(2^x+2^{-x}\right).tanx.\sqrt[]{ta{n}^{-1}\left(2x^2-3x+1\right)}.{\left(7x^2-3x+1\right)}^{-3}$

$f\text{'}\left(x\right)=\left(2^x+2^{-x}\right).se{c}^{2}x.\sqrt[]{ta{n}^{-1}\left(2x^2-3x+1\right)}.{\left(7x^2-3x+1\right)}^{-3}+tanx.\left(Q\left(x\right)\right)$

$f\text{'}\left(0\right)=2.1\sqrt[]{\frac{\pi }{4}}.1$

$=\sqrt[]{\pi }$

$Area=\frac{\pi {\left(13\right)}^2}{2}-\left[\frac{1}{2}\times 25\times 5+{\displaystyle \underset{12}{\overset{13}{\int }}\sqrt[]{\left(169-y^2\right)}}\cdot dy\right]$

$=\frac{169\pi }{2}-\left[\frac{125}{2}+{\left[\frac{y}{2}\sqrt[]{169-y^2}+\frac{169}{2}si{n}^{-1}\frac{y}{13}\right]}_{12}^{13}\right]$

$=\frac{169}{2}\pi -\frac{125}{2}-\left[\frac{169}{2}\times \frac{\pi }{2}-6\times 5-\frac{169}{2}si{n}^{-1}\frac{12}{13}\right]$

$=\frac{169\pi }{4}-\frac{65}{2}+\frac{169}{2}si{n}^{-1}\frac{12}{13}$

${\displaystyle \underset{-\frac{\pi }{2}}{\overset{\frac{\pi }{2}}{\int }}\left(\frac{x^{2}cosx}{1+{\pi }^2}+\frac{1+si{n}^{2}x}{1+e^{{\left(sinx\right)}^{2023}}}\right)dx}=\frac{\pi }{4}\left(\pi +\alpha \right)-2$

${\displaystyle \underset{0}{\overset{\frac{\pi }{2}}{\int }}\left\{\left(\frac{x^{2}cosx}{1+{\pi }^x}+\frac{1+si{n}^{2}x}{1+e^{{\left(sinx\right)}^{2023}}}\right)+\left(\frac{x^{2}cosx}{1+{\pi }^{-x}}+\frac{1+si{n}^{2}x}{1+e^{-{\left(sinx\right)}^{2023}}}\right)\right\}dx}$

$=\frac{\pi }{4}\left(\pi +\alpha \right)-2$

${\displaystyle \underset{0}{\overset{\frac{\pi }{2}}{\int }}\left(x^{2}cosx+1+si{n}^{2}x\right)dx}=\frac{\pi }{4}\left(\pi +\alpha \right)-2$

${\displaystyle \underset{0}{\overset{\frac{\pi }{2}}{\int }}{x}^{2}cosxdx}+{\displaystyle \underset{0}{\overset{\frac{\pi }{2}}{\int }}\left(1+si{n}^{2}x\right)dx}=\frac{\pi }{4}\left(\pi +\alpha \right)-2$ .(1)

Let $I_1={\displaystyle \underset{0}{\overset{\frac{\pi }{2}}{\int }}\left(1+si{n}^{2}x\right)dx}$

$I_1={\displaystyle \underset{0}{\overset{\frac{\pi }{2}}{\int }}1\cdot dx}+{\displaystyle \underset{0}{\overset{\frac{\pi }{2}}{\int }}\left(\frac{1-cos2x}{2}\right)dx}$

$I_1=\frac{\pi }{2}+\frac{1}{2}\left[\frac{\pi }{2}+0\right]$

$I_1=\frac{3\pi }{4}$

Let $I_2={\displaystyle \underset{0}{\overset{\frac{\pi }{2}}{\int }}{x}^{2}cosxdx}$

$I_2={\left[x^2\left(sinx\right)-{\displaystyle \int 2x}{\displaystyle \int cosxdx}\right]}_0^{\frac{\pi }{2}}$

$I_2={\left[x^2\left(sinx\right)-2{\displaystyle \int xsinx}\right]}_0^{\frac{\pi }{2}}$

$I_2={\left[x^{2}sinx-2\left(x\left(-cosx\right)+{\displaystyle \int cosx}\right)\right]}_0^{\frac{\pi }{2}}$

$I_2={\left[x^{2}sinx-2\left(-xcosx+sinx\right)\right]}_0^{\frac{\pi }{2}}$

$I_2=\left(\frac{{\pi }^2}{4}-2\right)$

Put l₁ and l₂ in (1)

$\frac{{\pi }^2}{4}-2+\frac{3\pi }{4}$

$\frac{{\pi }^2}{4}+\frac{3\pi }{4}-2$

$\frac{\pi }{4}\left(\pi +3\right)-2$

α = 3

$f\left(x\right)=\{\begin{array}{cc}\hfill 2+2x,\hfill & \hfill x\in \left(-1,0\right)\hfill \\ \hfill 1-\frac{x}{3},\hfill & \hfill x\in [0,3)\hfill \end{array}$

$g\left(x\right)=\{\begin{array}{cc}\hfill x,\hfill & \hfill x\in [0,1)\hfill \\ \hfill -x,\hfill & \hfill x\in \left(-3,0\right)\hfill \end{array}$   ->g(x) = |x|, x Î (–3, 1)

$f\left(g\left(x\right)\right)=\{\begin{array}{cc}\hfill 2+2\left|x\right|,\hfill & \hfill \left|x\right|\in \left(-1,0\right)\Rightarrow x\in ?\hfill \\ \hfill 1-\frac{\left|x\right|}{3},\hfill & \hfill \left|x\right|\in [0,3)\Rightarrow x\in \left(-3,1\right)\hfill \end{array}$            

$f\left(g\left(x\right)\right)=\{\begin{array}{cc}\hfill 1-\frac{x}{3},\hfill & \hfill x\in [0,1)\hfill \\ \hfill 1+\frac{x}{3},\hfill & \hfill x\in \left(-3,0\right)\hfill \end{array}$

Range of fog(x) is [0, 1]

            Range of fog(x) is [0, 1]

First term = a

Common difference = d

Given: a + 5d = 2        . (1)

Product (P) = (a₁a₅a₄) = a (a + 4d) (a + 3d)

Using (1)

P = (2 – 5d) (2 – d) (2 – 2d)

-> = (2 – 5d) (2 –d) (– 2) + (2 – 5d) (2 – 2d) (– 1) + (– 5) (2 – d) (2 – 2d)

$\frac{dP}{dd}$ = –2 [ (d – 2) (5d – 2) + (d – 1) (5d – 2) + (d – 1) (5d – 2) + 5 (d – 1) (d – 2)]

= –2 [15d² – 34d + 16]

$d=\frac{8}{5}or\frac{2}{3}$

at  $\left(\frac{8}{5}\right)$ , product attains maxima

-> d = 1.6

16cos²θ + 25sin²θ + 40sinθ cosθ = 1

16 + 9sin²θ + 20sin 2θ = 1

$16+9\left(\frac{1-cos2\theta }{2}\right)$            + 20sin 2θ = 1

$\frac{-9}{2}cos2\theta +20sin2\theta =\frac{-39}{2}$            

– 9cos 2θ + 40sin 2θ = – 39

$-9\left(\frac{1-ta{n}^2\theta }{1+ta{n}^2\theta }\right)+40\left(\frac{2tan\theta }{1+ta{n}^2\theta }\right)=-39$            

48tan²θ + 80tanθ + 30 = 0

24tan²θ + 40tanθ + 15 = 0

  $tan\theta =\frac{-40\pm \sqrt[]{{\left(40\right)}^2-15\times 24\times 4}}{2\times 24}$        

  $tan\theta =\frac{-40\pm \sqrt[]{160}}{2\times 24}$           

$=\frac{-10\pm \sqrt[]{10}}{12}$            

-> $tan\theta =\frac{\sqrt[]{10}-10}{12}$ , $tan\theta =\frac{-\sqrt[]{10}-10}{12}$  

So $tan\theta =-\frac{\sqrt[]{10}-10}{12}$  will be rejected as $\theta \in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$  

Option (4) is correct.

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  $\frac{dy}{dx}-\left(\frac{sin2x}{1+co{s}^{2}x}\right)y=\frac{sinx}{1+co{s}^{2}x}$

IF = $e^{-{\displaystyle \int \frac{sin2xdx}{1+co{s}^{2}x}}}$  

$=e^{ln\left(1+co{s}^{2}x\right)}=\left(1+co{s}^{2}x\right)$        

So, y(1 + cos² x) = ${\displaystyle \int \frac{sinx}{\left(1+co{s}^{2}x\right)}\cdot \left(1+co{s}^{2}x\right)dx}$  

y(1 + cos² x) = – cos x + c

$?$      y(0) = 0

0 = – 1 + c

-> c = 1

$y=\frac{1-cosx}{1+co{s}^{2}x}$   

Now, $y\left(\frac{\pi }{2}\right)=1$  

a, ar, ar², ….ar⁶³

a+ar+ar² +….+ar⁶³ = 7 [a + ar² + ar⁴ +.+ar⁶²]

$\frac{a\left(1-r^{64}\right)}{\left(1-r\right)}=7\frac{a\left(1-r^{64}\right)}{\left(1-r^2\right)}$            

1 + r = 7

r = 6

|2A| = 2⁷

8|A| = 2⁷

Now |A| = α²–β² = 2⁴

α² = 16 + β²

α²– β² = 16

(α–β) (α+β) = 16

->α + β = 8 and

α – β = 2

->α = 5 and β = 3