MCQ
$\sum\limits_{k = 0}^{10} {^{20}{C_k} = } $
- ✓${2^{19}} + \frac{1}{2}{\,^{20}}{C_{10}}$
- B${2^{19}}$
- C$^{20}{C_{10}}$
- DNone of these
We know that, ${(1 + x)^n} = {\,^n}{C_0} + {\,^n}{C_1}{x^1} + {\,^n}{C_2}{x^2} + .... + {\,^n}{C_n}.{x^n}$
Put $x = 1$; ${2^n} = {\,^n}{C_0} + {\,^n}{C_1} + {\,^n}{C_2} + ..... + {\,^n}{C_n}$
Put $n = 20$; ${2^{20}} = {\,^{20}}{C_0} + {\,^{20}}{C_1} + {\,^{20}}{C_2} + ...... + {\,^{20}}{C_{20}}$
${2^{20}} + \,{\,^{20}}{C_{10}} = 2\,[{\,^{20}}{C_0} + {\,^{20}}{C_1} + ...... + {\,^{20}}{C_{10}}]$
${[^{20}}{C_0} + {\,^{20}}{C_1} + ...... + {\,^{20}}{C_{10}}] = {2^{19}} + \frac{1}{2}{\,^{20}}{C_{10}}$
$\sum\limits_{k = 0}^{10} {^{20}{C_k}} = {2^{19}} + \frac{1}{2}{\,^{20}}{C_{10}}$.
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$f(0)=1 \text { and } \int_0^{\frac{\pi}{3}} f( t ) dt =0$
Then which of the following statements is (are) $TRUE$?
$(A)$ The equation $f( x )-3 \cos 3 x =0$ has at least one solution in $\left(0, \frac{\pi}{3}\right)$
$(B)$ The equation $f( x )-3 \sin 3 x =-\frac{6}{\pi}$ has at least one solution in $\left(0, \frac{\pi}{3}\right)$
$(C)$ $\lim _{x \rightarrow 0} \frac{x \int_0^x f(t) d t}{1- e ^{x^2}}=-1$
$(D)$ $\lim _{ x \rightarrow 0} \frac{\sin x \int_0^{ x } f( t ) dt }{ x ^2}=-1$