Class 11th

  $\left|\frac{1}{2}-2i+1\right|=\alpha \left(\frac{1}{2}-2i\right)+\beta \left(1+i\right)$  

$\sqrt[]{\frac{9}{4}+4}=\alpha \left(\frac{1}{2}-2i\right)+\beta \left(1+i\right)$

$\frac{5}{2}=\alpha \left(\frac{1}{2}\right)+\beta +i\left(-2\alpha +\beta \right)$             

$\frac{\alpha }{2}+\beta =\frac{5}{2}$      .(1)

 –2α + β = 0                    …(2)

Solving (1) and (2)

$\frac{\alpha }{2}+2\alpha =\frac{5}{2}$

$\frac{5}{2}\alpha =\frac{5}{2}$            

a = 1

b = 2

-> a + b = 3

Variance = $\frac{\sum x^2}{n}-{\left(\overline{x}\right)}^2$  

$\frac{60^2+60^2+44^2+58^2+68^2+{\alpha }^2+{\beta }^2+56^2}{8}={\left(58\right)}^2=66.2$            

$\frac{7200+1936+3364+4624+3136+{\alpha }^2+{\beta }^2}{8}=3364=66.2$             

$2532.5+\frac{{\alpha }^2+{\beta }^2}{8}-3364=66.2$            

α² + β² = 897.7 * 8

= 7181.6

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

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

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

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

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

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

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

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.

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

12x =

 $3x=\frac{\pi }{4}$

$cos3x=\frac{1}{\sqrt[]{2}}$

$4co{s}^{3}x-3cosx=\frac{1}{\sqrt[]{2}}$

$4\sqrt[]{2}co{s}^{3}x-3\sqrt[]{2}cosx-1=0$            

$x=\frac{\pi }{12}$ is the solution of above equation.

Statement 1 is true

$f\left(x\right)=4\sqrt[]{2}{x}^3-3\sqrt[]{2}x-1$

$f\text{'}\left(x\right)=12\sqrt[]{2}{x}^2-3\sqrt[]{2}=0$

$x=\pm \frac{1}{2}$

$f\left(-\frac{1}{2}\right)=-\frac{1}{\sqrt[]{2}}+\frac{3}{\sqrt[]{2}}-1=\sqrt[]{2}-1>0$            

f(0) = – 1 < 0

one root lies in $\left(-\frac{1}{2},0\right)$ , one root is $cos\frac{\pi }{12}$  which is positive. As the coefficients are real, therefore all the roots must be real.

Statement 2 is false.