જો a = Minimum {x^2 + 2x + 3, x in R} અને&nbs | vedclass

Name: PDF Generator App | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

Let $a=\min \left\{x^{2}+2 x+3: x \in Right\}$ and $b=\lim _{	heta ightarrow 0} \frac{1-\cos 	heta}{	heta^{2}}$

Now, $x^{2}+2 x+3=x^{2}+2 x+

Vedclass · Accepted Answer

$\frac{{{4^{n + 1}} - 1}}{{{{3.2}^n}}}$

Vedclass · Answer

Let $a=\min \left\{x^{2}+2 x+3: x \in Right\}$ and $b=\lim _{	heta ightarrow 0} \frac{1-\cos 	heta}{	heta^{2}}$

Now, $x^{2}+2 x+3=x^{2}+2 x+1+2=(x+1)^{2}+2$

$	herefore \min \left\{x^{2}+2 x+3: x \in Right\}=\min \left\{(x+1)^{2}+2: x \in Right\}=2$

Also, $b=\lim _{	heta ightarrow 0} \frac{1-\cos 	heta}{	heta^{2}}$

$=\lim _{	heta ightarrow 0} \frac{\sin 	heta}{2 	heta}=\frac{1}{2}=\left\{\lim _{	heta ightarrow 0} \frac{\sin 	heta}{	heta}=1ight\}$

$\sum_{r=0}^{n} a^{r} \cdot b^{n-r}$

$=\sum_{r=0}^{n} 2^{r}\left(\frac{1}{2}ight)^{n-r}$

$=\quad 2^{-n} \sum_{r=0}^{n} 2^{2 r}$

$=\quad 2^{-n}\left\{1+2^{2}+2^{4}+\ldots . .2^{2 n}ight\}$

$=\frac{4^{n+1}-1}{3.2^{n}}$

જો $a =$ Minimum $\{x^2 + 2x + 3, x \in R\}$ અને $b = \mathop {\lim }\limits_{\theta \to 0} \frac{{1 - \cos \theta }}{{{\theta ^2}}}$ હોય તો $\sum\limits_{r = 0}^n {{a^r}.{b^{n - r}}} $ ની કિમત મેળવો

Similar Questions

${({x^2} - x - 1)^{99}}$ ના સહગુણકનો સરવાળો મેળવો.

$^{10}{C_1}{ + ^{10}}{C_3}{ + ^{10}}{C_5}{ + ^{10}}{C_7}{ + ^{10}}{C_9} = $

$^{15}C_0^2{ - ^{15}}C_1^2{ + ^{15}}C_2^2 - ....{ - ^{15}}C_{15}^2$ = . . .

$(1 + x) (1 + x + x^2) (1 + x + x^2 + x^3) ...... (1 + x + x^2 + ...... + x^{100})$ ના વિસ્તરણમાં બહુપદીનો ઘાતાંક મેળવો

જો ${(1 + x)^n} = {C_0} + {C_1}x + {C_2}{x^2} + .... + {C_n}{x^n}$, તો ${C_0}{C_2} + {C_1}{C_3} + {C_2}{C_4} + {C_{n - 2}}{C_n}$= . . .