Relations and F...

Let $$f_k(x)=\dfrac{1}{k}(sin^kx+cos^kx)$$ for $$k=1,2,3,....$$ Then for all $$x\in R$$, the value of $$f_4(x)-f_6(x)$$ is equal to:-

Let  $$S = \{ 1,2,3 , \ldots , 100 \} .$$  The number of non-empty subsets  $$A$$  of  $$S$$  such that the product of elements in  $$A$$  is even is :-

Let $$a, b, c,$$ $$\epsilon\ R$$. If $$f(x)=ax^2+bx+c$$ is such that $$a+b+c=3$$ and $$f(x+y)=f(x)+f(y)+xy, \forall \, x, y\,\epsilon\, R,$$ the $$\displaystyle \sum_{n=1}^{10}f(n)$$ is equal to.

Let $$f : (-1,1)\to R$$ be a be a function defined by $$f(x) = max \left\{ -|x|, -\sqrt{1-x^2}\right\}$$. If $$K$$ be the set of all points at which f is not differentiable, then $$K$$ has exactly :

For every pair of continuous functions $$f, g : [0, 1]  \rightarrow R$$ such that $$\mathrm{m}\mathrm{a}\mathrm{x} \{f(x) : x \in [0, 1]\} = \mathrm{m}\mathrm{a}\mathrm{x} \{g(x) : x \in [0, 1]\}$$, the correct statement(s) is (are)

Let $$f(x) = \dfrac {ax + b}{cx + d}$$, then $$fof(x) = x$$, provided that

$$A$$ and $$B$$ are two sets having $$3$$ and $$4$$ elements respectively and having $$2$$ elements in common. The number of relations which can be defined from $$A$$ to $$B$$ is:

Let $$R$$ be a relation from a set $$A$$ to a set $$B$$, then:

If $$f(x)=3x+1, g(x)= x^{3}+2,$$ then $$(f+g)(0)-f(0)g(0)=$$

If $$f(x)=x^3-x^2+x+1$$, then the value of $$\dfrac {f(1)+f(-1)}{2}$$ will be