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### Examination Reforms

Thinking in this direction in the past has given birth to AIEEE, which in due course has become just one more burden to the students. Ideally, there must be just one exam. The students who top that exam may opt for IIT, followed by NIT and thereafter state colleges/other colleges. This looks very difficult to implement but needs to be done keeping in view the long-term efficiency of the system. Here is how to go about it:

• The boards like CBSE, ICSE, State etc. play the role of trainers and not examiners.
• A central body is appointed as examiner.
• This body takes the tests and prepares merit list and a minimum cut-off in each subject and the aggregate.
• This test should be in line with JEE (may be a little easier) having a practical component as well as languages and other papers.
• Students opt for engineering institutes and branches based on their marks in Physics and Mathematics. An institute can ask for more information from the candidate in addition to the marks obtained by him/her in this exam. However, the institute does not take any other written exam. Students opt for medical institues based on their marks in Biology, Chemistry and Physics.
• Students are allotted institutes and branches depending upon their merit and availability.
• The students who do not attain minimum marks in less than or equal to 2 subjects are allowed to take supplementary. If they fail in more than 2 subjects, they are declared fail and they will have to repeat the class.

### $f(x)=x^6+2x^4+x^3+2x+3 $$\mathop {\lim }\limits_{x \to 1} \frac{{{x^n}f(1) - f(x)}}{{x - 1}} = 44$$n=?$

Let $f(x)=x^6+2x^4+x^3+2x+3,x \in R$. Then the natural number n for which $\mathop {\lim }\limits_{x \to 1} \frac{{{x^n}f(1) - f(x)}}{{x - 1}} = 44$ is _ _ _ _ . Solution Since the limit has $\left[ {\frac{0}{0}} \right]$ form, L.H. Rule is applicable. Thus, $\mathop {\lim }\limits_{x \to 1} n{x^{n - 1}}f(1) - f'(x) = 44$ $\therefore nf(1) - f'(1) = 44$ $\therefore n.9 - ({6.1^5} + {8.1^3} + {3.1^2} + 2.1) = 44$ $\Rightarrow 9n - 19 = 44$ $\Rightarrow n=7$