Define \(f: \mathbb{R} \rightarrow \mathbb{R}\) by \[f(x)= \begin{cases}(1-\cos x) \sin \left(\frac{1}{x}\right), & x \neq 0 \\ 0, & x=0\end{cases}\] Then,\(\qquad\) (a) \(f\) is discontinuous. \(\qquad\) (b) \(f\) is continuous but not differentiable. \(\qquad\) \(\qquad\) \(\qquad\) (c) \(f\) is differentiable and its derivative is discontinuous. \(\qquad\) \(\qquad\) \(\qquad\)\(\qquad\)\(\qquad\) \(\qquad\) (d) \(f\) is differentiable and its derivative is continuous.

If two real numbers \(x\) and \(y\) satisfy \((x+5)^{2}+(y-10)^{2}=196\), then the minimum possible value of \(x^{2}+2x+y^{2}-4y\) is

If the maximum and minimum values of \(\sin ^{6} x+\cos ^{6} x,\) as \(x\) takes all real values, are \(a\) and \(b\), respectively, then \(a-b\) equals

The expression \[\sum_{k=0}^{10} 2^{k} \tan \left(2^{k}\right)\] equals

Let \(f: \mathbb{R} \rightarrow[0, \infty)\) be a continuous function such that \[f(x+y)=f(x) f(y)\] for all \(x, y \in \mathbb{R}\). Suppose that \(f\) is differentiable at \(x=1\) and \[\left.\frac{d f(x)}{d x}\right|_{x=1}=2\] Then, the value of \(f(1) \log _{e} f(1)\) is

For \(0 \leq x<2 \pi\), the number of solutions of the equation \[\sin ^{2} x+2 \cos ^{2} x+3 \sin x \cos x=0\] is

Let \[\begin{gathered} p(x)=x^{3}-3 x^{2}+2 x, x \in \mathbb{R}, \\ f_{0}(x)= \begin{cases}\int_{0}^{x} p(t) d t, & x \geq 0, \\ -\int_{x}^{0} p(t) d t, & x<0,\end{cases} \\ f_{1}(x)=e^{f_{0}(x)}, \quad f_{2}(x)=e^{f_{1}(x)}, \quad \ldots \quad, f_{n}(x)=e^{f_{n-1}(x)} \end{gathered}\] How many roots does the equation \(\frac{d f_{n}(x)}{d x}=0\) have in the interval \((-\infty, \infty)?\)