Let $\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$ be defined as
$f(x+y)+f(x-y)=2 f(x) f(y), f\left(\frac{1}{2}\right)=-1 .$ Then, the value of $\sum_{\mathrm{k}=1}^{20} \frac{1}{\sin (\mathrm{k}) \sin (\mathrm{k}+\mathrm{f}(\mathrm{k}))}$ is equal to:
Let $\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$ be defined as
$f(x+y)+f(x-y)=2 f(x) f(y), f\left(\frac{1}{2}\right)=-1 .$ Then, the value of $\sum_{\mathrm{k}=1}^{20} \frac{1}{\sin (\mathrm{k}) \sin (\mathrm{k}+\mathrm{f}(\mathrm{k}))}$ is equal to:
- [JEE MAIN 2021]
- A
$\operatorname{cosec}^{2}(1) \operatorname{cosec}(21) \sin (20)$
- B
$\sec ^{2}(1) \sec (21) \cos (20)$
- C
$\operatorname{cosec}^{2}(21) \cos (20) \cos (2)$
- D
$\sec ^{2}(21) \sin (20) \sin (2)$
Similar Questions
The function $f\left( x \right) = \left| {\sin \,4x} \right| + \left| {\cos \,2x} \right|$, is a periodic function with period
The function $f\left( x \right) = \left| {\sin \,4x} \right| + \left| {\cos \,2x} \right|$, is a periodic function with period
- [JEE MAIN 2014]
$f : R \to R$ is defined as
$f(x) = \left\{ {\begin{array}{*{20}{c}}
{{x^2} + 2mx - 1\,,}&{x \leq 0}\\
{mx - 1\,\,\,\,\,\,\,\,\,\,\,\,\,,}&{x > 0}
\end{array}} \right.$
If $f (x)$ is one-one then the set of values of $'m'$ is
$f : R \to R$ is defined as
$f(x) = \left\{ {\begin{array}{*{20}{c}}
{{x^2} + 2mx - 1\,,}&{x \leq 0}\\
{mx - 1\,\,\,\,\,\,\,\,\,\,\,\,\,,}&{x > 0}
\end{array}} \right.$
If $f (x)$ is one-one then the set of values of $'m'$ is
Show that the function $f: R_* \rightarrow R_*$ defined by $f(x)=\frac{1}{x}$ is one-one and onto, where $R_*$ is the set of all non-zero real numbers. Is the result true, if the domain $R_*$ is replaced by $N$ with co-domain being same as $R _*$ ?
Show that the function $f: R_* \rightarrow R_*$ defined by $f(x)=\frac{1}{x}$ is one-one and onto, where $R_*$ is the set of all non-zero real numbers. Is the result true, if the domain $R_*$ is replaced by $N$ with co-domain being same as $R _*$ ?
Let $R =\{ a , b , c , d , e \}$ and $S =\{1,2,3,4\}$. Total number of onto function $f: R \rightarrow S$ such that $f(a) \neq$ 1 , is equal to $.............$.
Let $R =\{ a , b , c , d , e \}$ and $S =\{1,2,3,4\}$. Total number of onto function $f: R \rightarrow S$ such that $f(a) \neq$ 1 , is equal to $.............$.
- [JEE MAIN 2023]
The domain of the function $f(x){ = ^{16 - x}}{\kern 1pt} {C_{2x - 1}}{ + ^{20 - 3x}}{\kern 1pt} {P_{4x - 5}}$, where the symbols have their usual meanings, is the set
The domain of the function $f(x){ = ^{16 - x}}{\kern 1pt} {C_{2x - 1}}{ + ^{20 - 3x}}{\kern 1pt} {P_{4x - 5}}$, where the symbols have their usual meanings, is the set