Iit Foundation

IIT Jee Foundation Vidyalankar April 13, 2009

Abstract Welcome to students who are entering into the arena of the most coveted entrance exam in India for Engg/Higher Education.

These study material is meant for those students who are just beginning their ad-

venture towards preparing IIT Jee exam.

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Contents I

Motivation

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1

What to do and what not to!

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Books to refer when we are starting with Mathematics

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II 3

Let's start!

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Warm up problems

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3.1

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.2

Numerical problems

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Part I

Motivation 1

What to do and what not to!

Following is the list of personal experiences while preparing students for IITJee. • When you start preparation make sure that you have total control of your situation during these two years in

your own hands.

• Every reason is an excuse. • Mistakes are just ne, rather they have their own importance but repeating mistakes is just not ne. • 1st year of your college is the time when you can explore Jee the best, so do as much as you can when you

are in your XI class

• Right attitude for Jee should be there since the beginning to the end, attitude decides whether your are in or

out.

• Smart work is more important along with hard work. • Make sure you clear all your doubts however silly they are, because a small re can destroy the whole city. • Referring lot of books is not required, referring right book at the right time is the best you can do. • Remember you are preparing for Jee not cleared Jee, so never become overcondent. • Consistency of prepartion in highs and lows during these two years is a must. There will times when you

would be able to solve everything-dont become overcondent & at times you won't be able to do a single prdoblem don't lose condence.

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Books to refer when we are starting with Mathematics

Right books at the right time is a must for best preparation. Good start is half done! Following is the list of books to refer in the beginning. 1. Higher Algebra - Hall & Knight 2. Plane Trigonometry (part I)- S. L. Loney 3. Calculus and Analytical Geometry (6ed)- Thomas Finney - Best book to start with in the rst year. Basic skills are sharpened with this book. Topics need to concentrate on are - Ratio, Proportion, Miscellaneous theorems and examples (few topics from this chapter) & Theory of equations.

Higher-Algebra

- Whole of trigonometry is best covered here, is a good reference book as all books refer this book for problems and theory. So its the mother book for trigonometry. But the topic is better dealt is other books.

Plane-Trigonometry

- Good book when you start your coordinate geometry understanding few month down the line of your prepartion. Its a very good introduction to coordinate geometry leading to calculus that becomes the basis for mathematics & physics. So faster you reach a maturity in calculus better for your JEE endeavor. You can read this book at your own pace.

Thomas-Finney

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On reading books you are refering at times you wont understand things at the beginning. So don't hung yourself to it for long.

Just leave that and start working on something else.

topic again.

After some time say a day or two come to the same

You will see that you are better understanding that topic this time.

Same with problem solving, try some

problem for some time. If you are not able to solve it then go to the next and come back again to that problem- and to your surprice it would solve!

Part II

Let's start! 3

Warm up problems

3.1

Theory

1. What is the dierence between an expression & an equation ? 2. What is a polynomial equation in x? (a) A polynomial in x is an expression of the form an xn + an−1 xn−1 + · · · + a0 3. What are coecients in the above polynomial? (a) ai 's are all coecients in the polynomials. And specially called as ak is the polynomial of xk 4. What do we mean by roots of a polynomial equation or expression ? (a) Roots of an equation are those value of x for which the equation is true. Like x2 − 1 = 0 is true or is said to be satised for x = 1or x = −1. Hence these are said to be roots of this equation. And for an expression like f (x) = x2 − 1 those values of x for which the expression vanishes means it becomes zero i.e. f (α) becomes zero is called as roots of an expression. 5. What is degree of a polynomial? (a) The highest power taken by x in the polynomial is called as degree of the polynomial. For e.g. degree of f (x) = x3 + 2x + 1 is 3. 6. What are the parts we get on dividing a polynomial by a lower degree term like x − α? (a) like on dividing 7 by 2 we can write 7 = 2x3 + 1.i.e. Divident = divisor x Quotient + Remainder . Similarly in polynomials we can do a similar thing. Any polynomial f (x) when divided by D(x) can be expressed as f (x) = Q(x)D(x) + R(x) where Q(x) is the Quotient & R(x) is the remainder. i. Remainder R(x) will always be of one degree less than the Divisor D(x). So if we divide a polynomial by x − α then the remainder would be or one degree less that is of degree zero. That means it would be a constant number. 7. Remainder theorem : Any polynomial in x, f (x) = an xn + an−1 xn−1 + · · · + a0 when divided by x − α then (a) f (x) = (x − α)φ(x) + C , where C is a constant. (b) Remainder is f (α) (c) If f (α) = 0 then i. α is the root of this polynomial f (x) ii. x − α is the factor of this polynomial f (x). i.e. f (x) = (x − α)φ(x) + 0

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8. Note few nice things (a) f (0) = a0 = constant term in the polynomial (b) f (1) = an + an−1 + · · · + a0 =sum of all the coecients of the polynomials (c) f (−1) = a0 − a1 + a2 − a3 + · · · 9. Absolute value function or Modulus of a function (

(a) |x| =

x −x

x>0 where x ∈ R x≤0

10. (Cyclic) Symmetry in JEE Mathematics. (a) Symmetry is seen in many topics from trigo to algebra. So what do we exactly mean by symmetry. Any expression in say a, b, c is said to be symmetric if the expression doesn't change on changing a → b, b → a. For. e.g a + b + c becomes b + a + c. Similarly a3 + b3 + c3 − 3abc becomes b3 + a3 + c3 − 3bac. (b) So if an expression is symmetric then the factors also will be symmetric! thats the gain! e.g. i.

a3 + b3 + c3 − 3abc = (a + b + c)(a2 + b2 + c2 − ab − bc − ac) = (a + b + c)(

(a − b)2 + (b − c)2 + (c − a)2 ) 2

(c) Advantage of knowing symmetry in problem solving i. If a term of a particular type occurs in a symmetric function, then all the (cyclic) terms of that type also occur ii. The sum, dierence, product & quotient of two symmetric functions is also symmetric. 3.2

Numerical problems

1. What is the dierence between x2 − x & x2 = x ? 2. What are the roots of the equation x2 − x = 0 or f (x) = x2 − x 3. Synthetic division: what is the quotient and remainder on (a) dividing x3 − 3x2 + x + 1 by x + 2 (b) dividing x3 − 2 by x + 2 4. Is (bc − ad)(ca − bd)(ab − cd) symmetric with respect to a, b, c, d 5. Show that the expressions are cyclic with regard to a, b, c, d taken in this order (a) (a − b + c − d)2 (b) (a − b)(c − d) + (b − c)(d − a)

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