Theory of Equations

Some simple problems... more to follow...

Please scroll to the bottom of the post ONLY after you solve all the questions.

(1) If 2+i√3 is a root of x² + px + q = 0 and p,q are real, then

(p, q) = (__, __).

(2) Solve for x:

| x² + 4x + 3 | + 2x + 5 = 0 where |.| denotes the modulus function.

(3) Solve for x:
sin x + cos x = x + (1/x) where x is expressed in radians. Consider only the positive values of 'x'.

(4) If x² + ax + b = 0 and x² + bx + a = 0 have a common root, find the numeric value of (a+b) if a and b are distinct.

(5) Find the number of solutions of the equation:




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All the above questions are previous years IIT-JEE questions! Surprised, aren't you?

Jst fr fn!

Don't you think that I was influenced by today's assembly.

In an attempt to keep the blog alive though I have changed the blog's purpose.

SUDOKU MASTERS 2008

National Sudoku Championship

REGISTRATION OPENS ON SAT 8 MARCH 2008

ONLINE QUALIFYING TEST ON SUN 16 MARCH 2008

FINALS IN BANGALORE ON SUN 30 MARCH 2008

SELECTION OF TEAM REPRESENTING INDIA AT
THIRD WORLD SUDOKU CHAMPIONSHIP (WSC)
TO BE HELD IN HOLIDAY INN RESORT GOA,
14-17 APRIL 2008

Website
http://sudokumasters.in/

Interesting Indeed...

Go to

http://puzzlecodebreaking.50webs.com/rules.htm

to begin.

“ Take nothing for granted. Use every resource you have.....Google, page title, page source, hidden links etc. No information is left accidental. If you think anything doesn't/shouldn't belong to the page, then think again. Many a times, the key to the next step is embedded in the page So, don't trust any piece of information. Besides, your brain should be with you through out the battle. ”

Good luck! Enjoy!

The Appointees

As the Appointment season is on, I have decided to put some appointments on our blog to maintain its standard. Here are my proposals.

POSTS

President :: Sajal Jain
Chief Technical Head :: Ananth Govind Rajan
Director of Innovation :: Shikhar Srivastav
Legal Head :: Shashwat Gangwar
Editor/Co-ordinator :: Mayank Gupta
Member :: Dipanshu Malhotra

DUTIES

The President has to look after all the duties performed by his Cabinet

It shall be the duty of the Technical Head, Legal Head and the Co-ordinator to judge the level of a question. ( 1/2/3/4/5). In case of a dispute, The President shall intervene.
Exciting new concepts shall be proposed by the Director of Innovation.
Technical Head should accomplish the task of maintaining the quality of the blog.
Legal Head has to look after the scores.
Coordinator has to make sure that there is active participation on the blog.(Polls shall also be maintained by him). It is his duty to invite more members to the blog.
The member...hmmm...has to enjoy.

These are just some proposals. Any dispute regarding the same shall be reported as a comment.
If you have some new proposals, please please please post them.

Divisors & Remainders

Two numbers, when divided by a divisor give 4375 and 2989 as the remainder. Their sum, when divided by the same divisor gives 2361 as the remainder. What is the divisor?

If q1, q2 and q3 be the three quotients in the respective divisions, the value of (q1 + q2) - q3 may be:

(a) -2
(b) -1
(c) -0
(d) +0
(e) +1
(f) +2

The question may have more than one possible answer.

One more for you

The value of is

[A].

[B].

[C].

[D]. This question gives us a general trick/method which can be used whenever we have some coefficients in the angle term of cos...x,2x,3x,....,(n+1)x The answer has been posted as a comment.

Two good syllabus questions

1

Quadratic Equations

If α, β, γ, δ are the four positive roots of the equation x⁴ -12x³+ax²+bx +81=0, then find “a” and “b”.

2

Permutations and Combinations

Find all possible arrangements of the word "COMBINATORICS" such that all the vowels are in alphabetical order and consonants are also in alphabetical order.

Note: Consonants and vowels to be considered separately.

Note: These are not puzzles but JEE-level questions

What's probable & what's not?

!!Please comment on the correctness of this question!!

In a lucky draw competition, a contestant had to draw 3 tickets from a pile of well-shuffled tickets. If the numbers on these tickets were in arithmetic progression, he would be adjudged the winner of the contest. If the numbers were not in arithmetic progression, he would return them to the pile. The tickets were numbered from 1 to 201. What is the minimum number of contestants to ensure that at least one of them wins the contest? If 100 people took part in the lucky draw, do you think we could take for granted that at-least one contestant would win the prize?

The Consecutive Numbers

A Math teacher, in order to keep two of her students occupied thought of two consecutive numbers in the range 1 to 10, and told one of the students Alex one of the numbers and the other student, Sam the other.

Sam and Alex then had the following conversation:

Alex: I don't know your number.
Sam: I don't know your number, either.
Alex: Now I know!

How many such sets of numbers can you find? Can you find all the sets which could have been thought up by the Math teacher?