10 Question MCQ (including Assertion)

Assertion (A): A finction must be continuous at a point if it is differentiable at that point.
Reason (R): If the limit of a function exists at a point, it must be differentiable at that point.

The number of points at which the function \(\mathrm{f}(\mathrm{x})=\frac{1}{x-[x]}\) is not continuous is

If \(f(x)=2 x\) and \(g(x)=\frac{x^2}{2}+1\), then which of the following can be a discontinuous function

The set of points where the functions \(f\) given by \(f(x)=|x-3| \cos x\) is differentiable is

The function \(\mathrm{f}(\mathrm{x})=\left\{\begin{array}{ll}\frac{\sin x}{x}+\cos \mathrm{x} & , \text { if } \mathrm{x} \neq 0 \\ k & , \text { if } \mathrm{x}=0\end{array}\right.\) is continuous at \(\mathrm{x}=0\), then the value of k is

Differential coefficient of \(\sec \left(\tan ^{-1} x\right)\) w.r.t. \(x\) is

Assertion (A): The function \(f(x)=\sin \left(x^2\right)\) is a continuous function.
Reason (R): The composition of two continuous functions is a continuous function.

If \(f^{\prime \prime}(x)>0\) over an interval, what does this imply about the rate of change of the function?

Assertion (A): The function \(f(x)=\cos |x|\) is differentiable at \(x=0\).
Reason (R): The cosine function is an even function, so \(\cos (-x)=\cos (x)\).