The set of values of $'a'$ for which the inequality ${x^2} - (a + 2)x - (a + 3) < 0$ is satisfied by atleast one positive real $x$ , is

Let $\mathrm{f}(\mathrm{x})$ be a polynomial of degree $3$ such that $\mathrm{f}(\mathrm{k})=-\frac{2}{\mathrm{k}}$ for $\mathrm{k}=2,3,4,5 .$ Then the value of $52-10 \mathrm{f}(10)$ is equal to :

Let $A=\{0,1,2,3,4,5,6,7\} .$ Then the number of bijective functions $f: A \rightarrow A$such that $f(1)+f(2)=3-f(3)$ is equal to $.....$

The range of the function $f(x) = \frac{{x + 2}}{{|x + 2|}}$ is

Let $f : R -\{0,1\} \rightarrow R$ be a function such that $f(x)+f\left(\frac{1}{1-x}\right)=1+x$. Then $f(2)$ is equal to :

Show that none of the operations given above has identity.