If $y+\frac{\mathrm{d}}{\mathrm{d} x}(x y)=x(\sin x+\log x)$ then

If the pair of straight lines $x y-x+y-1=0$ and the line $x+\mathrm{k} y-3=0$ are concurrent, then the value of $k$ is equal to

If the function

$$ f(x)=\left\{\begin{array}{cc} x+a \sqrt{2} \sin x & \text { if } 0 \leq x \leq \frac{\pi}{4} \\ 2 x \cot x+b & \text { if } \frac{\pi}{4} < x \leq \frac{\pi}{2} \\ a \cos 2 x-b \sin x & \text { if } \frac{\pi}{2} < x \leq \pi \end{array}\right. $$

is continuous in $[0, \pi]$ then $a-b=$

Which of the following statements has the truth value T ?

A: cube roots of unity are in Geometric progression and their sum is 1

B: $4+7>10$ iff $2+8<10$

C: $\exists x \in \mathbb{N}$ such that $x^2-3 x+2=0$ and $\exists \mathrm{n} \in \mathbb{N}$ such that n is an odd number

D: $3+\mathrm{i}$ is a complex number or $\sqrt{2}+\sqrt{3}=\sqrt{5}$