If a circle with centre at $(-1,1)$ touches the line $x+2 y+4=0$ then the co-ordinates of the point of contact are

A doctor assumes that patient has one of three diseases $\mathrm{d} 1, \mathrm{~d} 2$ or d 3 . Before any test he assumes an equal probability for each disease. He carries out a test that will be positive with probability 0.7 if the patient has disease $\mathrm{d} 1,0.5$ if the patient has disease d 2 and 0.8 if the patient has disease d3. Given that the outcome of the test was positive then probability that patient has disease d2 is

A particle P starts from the point $\mathrm{Z}_0=1+2 \mathrm{i}$ where $\mathrm{i}=\sqrt{-1}$. It moves first horizontally away from the origin by 5 units and then vertically upwards parallel to positive Y -axis by 3 units to reach a point $Z_1$. From $Z_1$ the particle moves $\sqrt{2}$ units in the direction of vector $\hat{\mathrm{i}}+\hat{\mathrm{j}}$ and then it moves through an angle $\frac{\pi}{2}$ in anticlockwise direction on a circle with centre at origin to reach at point $Z_2$, then $Z_2=$

$$ \mathop {\lim }\limits_{x \to 0} \frac{63^x-9^x-7^x+1}{\sqrt{2}-\sqrt{1+\cos x}}=\ldots \ldots $$