The modulus and amplitude of $\frac{1 + 2i}{1 - (1 - i)^2}$ are

The angle between the lines whose direction cosines satisfy the equations $l + m + n = 0$ and $l^2 = m^2 + n^2$ is

Efficiency of a heat engine whose sink is at a temperature of $300 ~K$ is $40 \%$. To increase the efficiency to $60 \%$,keeping the sink temperature constant,the source temperature must be increased by (in $~K$)

An organic compound $X$ on treatment with pyridinium chlorochromate in dichloromethane gives compound $Y$. Compound $Y$ reacts with $I_2$ and alkali to form triiodomethane. The compound $X$ is

$A$ father with $8$ children takes them $3$ at a time to the Zoological gardens,as often as he can without taking the same $3$ children together more than once. The number of times he will go to the garden is

Which of the following metals becomes passive towards concentrated $HNO_3$?