Continuity and ...

If $$y=1-\cos { \theta  } ,x=1-\sin { \theta  } $$, then $$\cfrac { dy }{ dx } $$ at $$\theta =\cfrac { \pi  }{ 4 } $$ is

Consider the parametric equation $$x = \cfrac {a(1 - t^{2})}{1 + t^{2}}, y = \cfrac {2at}{1 + t^{2}}$$.

Consider the following statements.
$$1$$. The function is discontinuous at $$x=3$$.
$$2$$. The function is not differentiable at $$x=0$$.
Which of the statements is$$/$$ are correct?

If $$f(x)=\sqrt{25-x^2}$$, then what is $$\displaystyle \lim_{ x\rightarrow 1 } \dfrac{f(x)-f(1)}{x-1}$$ equal to?

If $$x=\sec { \theta  } -\cos { \theta  } $$ and $$y=\sec ^{ n }{ \theta  } -\cos ^{ n }{ \theta  } $$, then $${ \left( \dfrac { dy }{ dx }  \right)  }^{ 2 }$$ is

If $$x = a \cos^3 \theta$$ and $$y = a\sin^3 \theta$$, then $$1 + \left( \dfrac{dy}{dx} \right )^2$$ is

The function $$f(x) = \left\{\begin{matrix}x^{2}/a, & 0\leq x < 1\\ a, & 1\leq x < \sqrt {2}\\ \dfrac {2b^{2} - 4b}{x^{2}}, & \sqrt {2} \leq x < \infty\end{matrix}\right.$$ is continuous for $$0\leq x < \infty$$, then the most suitable values of $$a$$ and $$b$$ are

If $$2^x + 2^y = 2^{x + y}$$, then $$\dfrac{dy}{dx}$$ is equal to

If $$s=\sec ^{ -1 }{ \left( \cfrac { 1 }{ 2{ x }^{ 2 }-1 }  \right)  } $$ and $$t=\sqrt { 1-{ x }^{ 2 } } $$, then $$\cfrac { ds }{ dt } $$ at $$x=\cfrac { 1 }{ 2 }$$ is

If $$x = \sin t$$ and $$y = \tan t$$, then $$\dfrac{dy}{dx}$$ is equal to