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Wiley s Mathematics for IIT JEE Main and Advanced Algebra Vol 1 Maestro Series Dr. G S N Murti Dr. U M Swamy PDF

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������������������������������������������������������������������������� ������������������������ ������� �������������������������������������������������������������������������� ��������������� Note to the Students The IIT-JEE is one of the hardest exams to crack for students, for a very simple reason – concepts cannot be learned by rote, they have to be absorbed, and IIT believes in strong concepts. Each question in the IIT-JEE entrance exam is meant to push the analytical ability of the student to its limit. That is why the questions are called brainteasers! Students find Mathematics the most difficult part of IIT-JEE. We understand that it is difficult to get students to love mathematics, but one can get students to love succeeding at mathematics. In order to accomplish this goal, the book has been written in clear, concise, and inviting writing style. It can be used as a self-study text as theory is well supplemented with examples and solved examples. Wher- ever required, figures have been provided for clear understanding. If you take full advantage of the unique features and elements of this textbook, we believe that your experience will be fulfilling and enjoyable. Let’s walk through some of the special book features that will help you in your efforts to crack IIT-JEE. To crack mathematics paper for IIT-JEE the five things to remember are: 1. Understanding the concepts 2. Proper applications of concepts 3. Practice 4. Speed 5. Accuracy About the Cover Picture The Mandelbrot set is a mathematical set of points in the complex plane, the boundary of which forms a fractal. It is the set of complex values of c for which the orbit of 0 under iteration of the complex quadratic polynomial zn+1 = zn 2 + c remains bounded. The Mandelbrot set is named after Benoît Mandelbrot, who studied and popularized it. Special attention has been paid to present an engaging, clear, precise narrative in the layout that is easy to use and designed to reduce math anxiety students may have. CLEAR, CONCISE, AND INVITING WRITING Every new topic or concept starts with de- fining the concept for students. Related ex- amples to aid the understanding follow the definition. DEFINITIONS 4.1 | Quadratic Expressions and Equations In this section, we discuss quadratic expressions and equations and their roots. Also, we derive various properties of the roots of quadratic equations and their relationships with the coefficients. DEFINITION 4.1 A polynomial of the form ax bx c 2 + + , where a, b and c are real or complex numbers and a ¹ 0, is called a quadratic expression in the variable x. In other words, a polynomial f (x) of degree two over the set of complex numbers is called a quadratic expression. We often write f x ax bx c ( ) º + + 2 to denote a quadratic expression and this is known as the standard form. In this case, a and b are called the coefficients of x2 and x, respectively, and c is called the constant term. The term ax2 is called the quadratic term and bx is called the linear term. DEFINITION 4.2 If f x ax bx c ( ) º + + 2 is a quadratic expression and a is a complex number, then we write f (a) for a b c a a 2 + + . If f (a) = 0, then a is called a zero of the quadratic expression f (x). (1) Let f (x) º x2 - 5x - 6. Then f (x) is a quadratic expres- sion and 6 and –1 are zeros of f (x). (2) Let f (x) º x2 + 1. Then f (x) is a quadratic expression and i and –i are zeros of f (x). (3) Let f x x ix ( ) º - + 2 1 2 be a quadratic expression. In this case i and −i/2 are zeros of f (x). (4) The expression x2 + x is a quadratic expression and 0 and –1 are zeros of x2 + x. Examples DEFINITION 4.3 If f (x) is a quadratic expression, then f (x) = 0 is called a quadratic equation. If a is a zero of f (x), then a is called a root or a solution of the quadratic equation f (x) = 0. In other words, if f x ax bx c a ( ) , , º + + ¹ 2 0 then a complex number a is said to be a root or a solution of f (x) = 0, if aa 2 + ba + c = 0. The zeros of the quadratic expression f (x) are same as the roots or solutions of the quadratic equation f (x) = 0. Note that a is a zero of f (x) if and only if x − a is a factor of f (x). Examples (1) 0 and –i are the roots of x ix 2 + = 0. (2) 2 is the only root of x x 2 4 4 0 - + = . (3) i and –i are the roots of x2 1 0 + = . (4) i is the only root of x ix 2 2 1 0 - - = . Each chapter starts with an opening vignette, defini- tion of the topic, and contents of the chapter that give you an overview of the chapter to help you see the big picture. CHAPTER OPENER < 0 > 0 = 0 Quadratic Equations A polynomial equation of the second degree having the general form ax2 + bx + c = 0 is called a quadratic equation. Here x represents a variable, and a, b, and c, constants, with a ¹ 0. The constants a, b, and c are called, respectively, the quadratic coefficient, the linear coefficient and the constant term or the free term. The term “quadratic” comes from quadratus, which is the Latin word for “square”. Quadratic equations can be solved by factoring, completing the square, graphing, Newton’s method, and using the quadratic formula (explained in the chapter). Contents 4.1 Quadratic Expressions and Equations Worked-Out Problems Summary Exercises Answers Quadratic Equations 4 A. PEDAGOGY Examples pose a specific problem using concepts already presented and then work through the solution. These serve to enhance the students' understanding of the subject matter. EXAMPLES Example 4.1 Find the quadratic equation whose roots are 2 and –i. Solution: The required quadratic expression is ( )[ ( )] ( )( ) ( ) x x i x x i x i x i - - - = - + = + - - 2 2 2 2 2 Hence the equation is x i x i 2 2 2 0 + - - = ( ) . Example 4.2 Find the quadratic equation whose roots are 1 + i and 1 – i and in which the coefficient of x2 is 3. Solution: The required quadratic expression is 3 1 1 3 1 1 3 1 1 3 6 2 2 [ ( )]( ( )) [( ) )][( ) ] [( ) ] x i x i x i x i x x - + - - = - - - + = - + = - x + 6 Hence the equation is 3x2 - 6x + 6 = 0. Example 4.3 If a and b are roots of the quadratic equation ax bx 2 + + c = 0 and z is any complex number, then find the quadratic equation whose roots are z z a b and . Solution: We have a b ab + = - = b a c a and The equation whose roots are z z a b and is 0 2 2 2 2 = - - = - + + ´ = + - + + = + ( )( ) ( ) [ ( )] x z x z x z z x z z x z x z x z b a a b a b a b a b ab æ èç ö ø÷ + x z c a 2 that is, ax zbx z c 2 2 0 + + = Example 4.4 If a and b are the roots of a quadratic equation ax bx c 2 0 + + = , then find the quadratic equation whose roots are a b + + z z and , where z is any given complex number. Solution: We have Therefore, the required equation is 0 2 = - + ´ - + = + - + - + + + + = a x z x z ax a z z x a z z ax [ ( )] [ ( )] [ ( ) ( )] ( )( ) a b a b a b 2 2 2 + - æ èç ö ø÷ + - + æ èç ö ø÷ a b a z x a c a b a z z Relevant theorems are provided along with proofs to emphasize conceptual un- derstanding rather than rote learning. THEOREMS THEOREM 4.5 If a, b and c are real numbers and a ¹ 0, then ( )/ 4 4 2 ac b a - is the maximum or minimum value of quadratic equation of f x ax bx c ( ) º + + 2 according as a a < > 0 0 or , respectively. PROOF We have f x ax bx c a x b a x c a a x b a ac b a ( ) º + + º + + æ èç ö ø÷ º + æ èç ö ø÷ + - é ë ê ê 2 2 2 2 2 2 4 4 ù û ú ú º + æ èç ö ø÷ + - a x b a ac b a 2 4 4 2 2 If a < 0, then f x ac b a f b a ( ) £ - = - æ èç ö ø÷ 4 4 2 2 for all x Î� Hence ( )/ 4 4 2 ac b a - is the maximum value of f x ( ). If a > 0, then f b a ac b a f x - æ èç ö ø÷ = - £ 2 4 4 2 ( ) for all x Î� Hence ( )/ 4 4 2 ac b a - is the minimum value of f x ( ). ■ Some important formulae and con- cepts that do not require exhaustive explanation, but their mention is im- portant, are presented in this section. These are marked with a magnifying glass. QUICK LOOK QUICK LOOK 2 Let f x ax bx c ( ) º + + = 2 0 be a quadratic equation and a b and be its roots. Then the following hold good. 1. f (x - z) = 0 is an equation whose roots are a + z and b + z, for any given complex number z. 2. f x z ( / ) = 0 is an equation whose roots are z z a b and for any non-zero complex number z. 3. f x ( ) - = 0 is an equation whose roots are -a and -b. 4. If ab ¹ ¹ 0 0 and c , f(1/x) = 0 is an equation whose roots are 1 1 / / . a b and 5. For any complex numbers z1 and z2 with z1 0 ¹ , f x z z [( )/ ] - = 2 1 0 is an equation whose roots are z z z z 1 2 1 2 a b + + and . Within each chapter the stu- dents would find problems to reinforce and check their understanding. This would help build confidence as one progresses in the chapter. These are marked with a pointed finger. TRY IT OUT At the end of every chapter, a summary is presented that organ- izes the key formulae and theorems in an easy to use layout. The related topics are indi- cated so that one can quickly summarize a chapter. SUMMARY Try it out Verify the following properties: 1. ((a, b) + (c, d)) + (s, t) = (a, b) + ((c, d) + (s, t)) 2. (a, b) + (c, d) = (c, d) + (a, b) 3. (a, b) + (0, 0) = (a, b) 4. (a, b) + (-a, -b) = (0, 0) 5. (a, b) + (c, d) = (s, t) Û (a, b) = (s, t) - (c, d) Û (c, d) = (s, t) - (a, b) DEFINITION 3.2 For any complex numbers (a, b) and (c, d), let us define ( ) ( ) ( ) a b c d ac bd ad bc , , , × = - + This is called the product of (a, b) and (c, d) and the process of taking products is called multiplication. Try it out Verify the following properties for any complex numbers (a, b), (c, d) and (s, t). 1. [( ) ( )] ( ) ( ) [( ) ( )] a b c d s t a b c d s t , , , , , , × × = × × 2. ( ) ( ) ( ) ( ) a b c d c d a b , , , , × = × 3. ( ) [( ) ( )] ( ) ( ) ( ) ( ) a b c d s t a b c d a b s t , , , , , , , × + = × + × 4. ( ) ( ) ( ) a b a b , , , × = 1 0 5. ( ) ( ) ( ) a c d ac ad , , , 0 × = 6. ( ) ( ) ( ) a c ac , , , 0 0 0 × = 7. ( ) ( ) ( ) a c a c , , , 0 0 0 + = + 4.1 Quadratic expressions and equations: If a, b, c are real numbers and a ≠ 0, the expression of the form ax2 + bx + c is called quadratic expression and ax2 + bx + c = 0 is called quadratic equation. 4.2 Let f (x) º ax2 + bx + c be a quadratic expression and a be a real (complex) number. Then we write f (a) for aa2 + ba + c. If f(a) = 0, the a is called a zero of f(x) or a root of the equation f(x) = 0. 4.3 Roots: The roots of the quadratic equation ax2 + bx + c = 0 are - + - - - - b b ac a b b ac a 2 2 4 2 4 2 and 4.4 Discriminant: b2 - 4ac is called the discriminant of the quadratic expression (equation) ax2 + bx + c = 0. 4.5 Sum and product of the roots: If a and b are roots of the equation ax2 + bx + c = 0, then a b a b + = - = b a c a 2 and 4.6 Let ax2 + bx + c = 0 be a quadratic equation and Δ = b2 - 4ac be its discriminant. Then the following hold good. (1) Roots are equal Û Δ = 0 (i.e., b2 = 4ac). (2) Roots are real and distinct Û Δ > 0. (3) Roots are non-real complex (i.e., imaginary) Û Δ > 0. 4.7 Theorem: Two quadratic equations ax2 + bx + c = 0 and ¢ + ¢ + ¢ = a x b x c 2 0 have same roots if and only if the triples (a, b, c) and (a¢, b¢, c¢ ) are proportional and in this case ax bx c a a a x b x c 2 2 + + = ¢ + ¢ + ¢ ¢ ( ) 4.8 Cube roots of unity: Roots of the equation x3 - 1 = 0 are called cube roots of unity and they are 1 1 2 3 2 , - ± i -1 2 3 2 / / ± i are called non-real cube roots of unity. Further each of them is the square of the other and the sum of the two non-real cube roots of unity is equal to -1. If w ≠ 1 is a cube root of unity and n is any positive integer, then 1 + wn + w2n is equal to 3 or 0 according as n is a multiple of 3 or not. 4.9 Maximum and minimum values: If f(x) º ax2 + bx + c and a ≠ 0, then f b a ac b a - æ èç ö ø÷ = - 2 4 4 2 is the maximum or minimum value of f according as a < 0 or a > 0. 4.10 Theorems (change of sign of ax2 + bx + c): Let f(x) º ax2 + bx + c where a, b, c are real and a ≠ 0. If a and b are real roots of f(x) = 0 and a < b, then (1) (i) f(x) and a (the coefficient of x2) have the same sign for all x < a or x > b. (ii) f(x) and a will have opposite signs for all x such that a < x < b. (2) If f (x) = 0 has imaginary roots, then f(x) and a will have the same sign for all real values of x. 4.11 If f(x) is a quadratic expression and f (p)f (q) < 0 for some real numbers p and q, then the quadratic equation f (x) = 0 has a root in between p and q. SUMMARY B. WORKED-OUT PROBLEMS AND ASSESSMENT – AS PER IIT-JEE PATTERN In-depth solutions are provided to all worked-out problems for students to understand the logic behind and formula used. WORKED-OUT PROBLEMS Mere theory is not enough. It is also important to practice and test what has been proved theoretically. The worked-out problems and exercise at the end of each chapter are in resonance with the IIT-JEE paper pattern. Keeping the IIT-JEE pattern in mind, the worked-out problems and exercises have been divided into: 1. Single Correct Choice Type Questions 2. Multiple Correct Choice Type Questions 3. Matrix-Match Type Questions 4. Comprehension-Type Questions 5. Assertion–Reasoning Type Questions 6. Integer Answer Type Questions 1. If the equations x ax 2 1 0 + + = and x x a 2 0 - - = have a real common root, then the value of a is (A) 0 (B) 1 (C) −1 (D) 2 Solution: Let a be a real common root. Then a a a a 2 2 1 0 0 + + = - - = a a Therefore a a ( ) ( ) ( )( ) a a a + + + = + + = 1 1 0 1 1 0 If a = -1, then the equations are same and also cannot have a real root. Therefore a + ¹ 1 0 and hence a = -1, so that a = 2. Answer: (D) m m m m m m < + - > Þ < - + > 0 4 4 0 0 2 2 0 2 and 3 and 3 ( )( ) This gives m < -2 and so x x x x x 2 5 6 0 2 3 0 2 3 - + < Þ - - < Þ Î ( )( ) ( , ) Answer: (C) 4. If p is prime number and both the roots of the equation x px p 2 444 0 + - = ( ) are integers, then p is equal to (A) 2 (B) 3 (C) 31 (D) 37 Solution: Suppose the roots of x px p 2 444 0 + - = ( ) are integers. Then the discriminant p p p p 2 + = + ´ 4 444 4 444 ( ) { ( )} must be a perfect square. Therefore p divides p + 4 ´ (444). This implies p divides 4 444 2 3 37 4 ´ = ´ ´ ( ) Th f WORKED-OUT PROBLEMS Single Correct Choice Type Questions Multiple correct choice type questions have four choices provided, but one or more of the choices provided may be correct. MULTIPLE CORRECT CHOICE TYPE QUESTIONS Multiple Correct Choice Type Questions 1. Suppose a and b are integers and b ¹ -1. If the quadratic equation x2 + ax + b + 1 = 0 has a positive integer root, then (A) the other root is also a positive integer (B) the other root is an integer (C) a b 2 2 + is a prime number (D) a b 2 2 + has a factor other than 1 and itself Solution: Let a and b be the roots and a be a positive integer. Then a b - + = a and ab = b + 1 b a = - -a implies b is an integer and a b 2 2 2 2 2 2 2 2 2 2 1 1 1 1 + = + + - = + + + = + + ( ) ( ) ( )( ) a b ab a b a b a b Since a2 + > 1 1 and b2 1 1 + > , it follows that a2 + 1 is a factor of a b 2 2 + other than 1 and itself. Answers: (B), (D) Solution: Case 1: Suppose b is even, that is, b m = 2 . Then b ac 2 4 - = 4 4 2 ( ) . m ac k - = Case 2: Suppose b is odd, that is, b m = - 2 1. Then b ac m ac m m ac 2 2 2 4 2 1 4 4 4 1 4 - = - - = + + - ( ) = + - + = + 4 1 4 1 2 ( ) m m ac k Answers: (A), (B) 3. If a and b are roots of the equation x ax b 2 0 + + = , then (A) a = 0, b = 1 (B) a b = = 0 (C) a b = = - 1 1 , (D) a b = = - 1 2 , Solution: If a + b = -a and ab = b, then a = 0 = b or a = 1, b = -2. Answers: (B), (D) These are the regular mul- tiple choice questions with four choices provided. Only one among the four choices will be the correct answer. SINGLE CORRECT CHOICE TYPE QUESTIONS

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