Find the greatest number which divides 2011 and 2623 leaving remainders 9 and 5 respectively.

Find the greatest 4-digit number which is divisible by 15, 24, and 36.

Two tankers contain 825 liters and 675 liters of kerosene oil, respectively. What can be the maximum capacity of an uncalibrated container which can be used to measure the kerosene oil of both the tankers when used an integral number of times?

दो टैंकर क्रमशः 825 लीटर और 675 लीटर केरोसिन धारित करते हैं। एक खाली कंटेनर की अधिकतम क्षमता क्या हो सकती है जिसका इस्तेमाल करके दोनों टैंकरों के केरोसिन को पूरा-पूरा मापा जा सके।

• 75 liters
• 80 liters
• 65 liters
• 60 liters

Let $f(x)$ be a polynomial function satisfying the following conditions: $\lim _{x \rightarrow \infty} \frac{f(x)}{|x|^3}=0$, $\lim _{x \rightarrow \infty}(\sqrt{f(x)}-x)=-1$, and $f(0)=0$ (where $[.]$ denotes the greatest integer function).

Which of the following is/are correct?

A The number of points of discontinuity of $g(x)=[f(x)]$ in $[0,3]$ is 3.

B The number of points of discontinuity of $g(x)=[f(x)]$ in $[0,3]$ is 5.

C The number of points of non-derivability of $h(x)=f(|x|)$ is 4.

D The number of points of non-derivability of $h(x)=|f(|x|)|$ is 3.

Find the greatest and least values of:

$$ \frac{x+2}{2x^{2}+3x+6} \quad \forall x \in \mathbb{R} $$

The sum of three consecutive even natural numbers is 48. Find the greatest of these numbers.