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CONTENTS, , No. :, , Chapter, , Page, No., , 1, , SETS, , 2, , RELATIONS AND FUNCTIONS, , 12, , 3, , TRIGONOMETRIC FUNCTIONS, , 19, , 4, , PRINCIPLES OF MATHEMATICAL INDUCTION, , 23, , 5, , COMPLEX NUMBERS AND QUADRATIC, EQUATIONS, , 26, , 6, , LINEAR INEQUALITIES, , 30, , 7, , PERMUTATIONS AND COMBINATIONS, , 33, , 8, , BINOMIAL THEOREM, , 37, , 9, , SEQUENCE AND SERIES, , 40, , 10, , STRAIGHT LINES, , 43, , 11, , CONIC SECTIONS, , 46, , 12, , INTRODUCTION TO 3D GEOMETRY, , 50, , 13, , LIMITS AND DERIVATIVES, , 54, , 14, , MATHEMATICAL REASONING, , 59, , 15, , STATISTICS, , 61, , 16, , PROBABILITY, , 64, , VIJAYABHERI DISTRICT PANCHAYATH & MAM MALAPPURAM, , 5, , 4

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1, SETS, KEY NOTES, Sets, , , , A set is a well defined collection of objects, Eg : Collection of even integers, Collection of all natural numbers less than 100, Objects or members in a set are called its elements, N, : Set of all natural numbers, Z, : Set of all integers, Q, : Set of all rational numbers, R, : Set of all real numbers, 𝑍 + : Set of all positive integers, 𝑅+ : Set of all positive real numbers, , Representation of sets,  Two methods, Roster form/Tabular form, Described by listing the elements, separated by commas, enclosed with, the braces, Eg : If A is the set of all vowels in, English Alphabet ,, 𝐴 = {𝑎, 𝑒, 𝑖, 𝑜, 𝑢}, , , , , , , , Set builder form/Rule method, In this method we write down a, property or rule which represents all, the elements of the set., Eg : 𝐴 = {𝑥 ∶ 𝑥 is vowels in English, alphabet}, , Subsets, The set A is said to be a subset of a set B if every element of A is also an element, of B, we write it as 𝐴 ⊂ 𝐵, 𝐴 ⊂ 𝐵 if 𝑥 ∈ 𝐴 ⟹ 𝑥 ∈ 𝐵, Eg : {𝑎, 𝑏} is a subset of {𝑎, 𝑏, 𝑐, 𝑑}, 𝜙 is a subset of every set, 𝜙 ⊂ 𝐴, Every set is a subset of itself, 𝐴 ⊂ 𝐴, If 𝐴 ⊂ 𝐵 and 𝐴 ≠ 𝐵, then A is called the proper subset of B and B is called the, superset of A., Eg : {1, 2} is a proper subset of {1, 2, 3}, In a finite set having ‘n’ elements ,, Number of subsets, : 2𝑛, Number of proper subsets : 2𝑛 − 1, , VIJAYABHERI DISTRICT PANCHAYATH & MAM MALAPPURAM, , 5

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LINEAR INEQUALITIES, , 6, , KEY NOTES,  Two real numbers or two algebraic expressions related by the symbol <, >, ≤, ≥, , ≠ form an inequality.,  An inequality is said to be linear, if the variable(s) occur in first degree only and, there is no terms involving the product of the variables., Eg : 𝑎𝑥 + 𝑏 ≤ 0, 𝑎𝑥 + 𝑏𝑦 + 𝑐 > 10 , 𝑎𝑥 ≤ 4,  A linear inequality which have only two variables is called linear inequality in two, variables., Eg: 3𝑥 + 11𝑦 ≤ 0, 4𝑦 + 2𝑥 > 0,  The solution region of a system of inequality is the region which satisfies all the, given inequality in the system simultaneously., QUESTIONS AND ANSWERS, 1., , Solve the following system of linear, inequalities graphically., 𝑥 + 2𝑦 ≤ 8, 2𝑥 + 𝑦 ≤ 8, 𝑥 ≥ 0, 𝑦 ≥ 0, , Ans :, 𝑥 + 2𝑦 = 8, 8, 0, 𝑥, 0, 4, 𝑦, , 𝑥, 𝑦, , 2𝑥 + 𝑦 = 8, 4, 0, 0, 8, , The shaded region in the figure gives the, solution of the system of linear inequalities., 2., , Solve the following system of inequalities, graphically., 5𝑥 + 4𝑦 ≤ 40, 𝑥≥2,𝑦≥3, , VIJAYABHERI DISTRICT PANCHAYATH & MAM MALAPPURAM, , 30

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QUESTIONS AND ANSWERS, 1., , Find the number of 4 letter, words, with or without, meaning, which can be formed, out of the letters of the word, ROSE, where the repetition of, the letters is not allowed., , Ans :, , 1, 2, 3, 4, 5, 5 ways 2 ways, Total 2 digit numbers = 5 × 2, = 10, , Ans :, 4., , 4 ways, , 3, , 2, , 1, , Total Words(without repetition), = 4×3×2×1, = 24, 2., , How many 3-digit numbers can be, formed from the digits 1, 2, 3, 4 and, 5 assuming that, (a) repetition of the digits is allowed?, (b) repetition of the digits is not, allowed?, Ans :, (a) 1, 2, 3, 4, 5, , Given 4 flags of different colors,, how many different signals can, be generated, if a signal requires, the use of 2 flags one below the, other?, , 5 chances, , 5 chances, , 5 chances, , Total 3 digit numbers = 5 × 5 × 5, = 125, (b), , Ans :, , 4 chances, 5, , 3 chances, Total number of Signals = 4 × 3, = 12, 3., , How many 2 digit even numbers, can be formed from the digits 1,, 2, 3, 4, 5 if the digits can be, repeated?, , 4, , 3, , Total 3 digit numbers = 5 × 4 × 3, = 60, , 5., , How many 5-digit telephone, numbers can be constructed using, the digits 0 to 9 if each number, starts with 67 and no digit appears, more than once?, Ans :, 0 → 9 , total 10 digits, , VIJAYABHERI DISTRICT PANCHAYATH & MAM MALAPPURAM, , 34

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2𝑛(2𝑛−1)(2𝑛−2), , 6, , 𝑛(𝑛−1)(𝑛−2), 4(2𝑛−1), , 7, , 𝑛−2, , 1 way, , 1 way, , 8, , 7, , Total numbers starting with 67, = 1×1×8×7×6, = 336, 6. (a) If 𝑛𝐶8 = 𝑛𝐶9 , find 𝑛𝐶17, (b) In how many ways can 3 boys, and 2 girls be selected from 5, boys and 6 girls., Ans :, (a) 𝑛𝐶8 = 𝑛𝐶9 ⟹ 𝑛 = 8 + 9 = 17, 𝑛, 𝐶17 = 17𝐶17 = 1, (b) 3 boys from 5 boys, selection, = 5𝐶3, 2 girls from 6 girls ,, Selection = 6𝐶2, Total Selection of 3 boys and 2, girls = 5𝐶3 × 6𝐶2, 5×4×3, , 6×5, , = 1×2×3 × 1×2, = 150, , 7., (a) Find 𝑛 if 2𝑛𝐶3 ∶ 𝑛𝐶3 = 12 ∶ 1, (b) How many chords can be drawn, through 21 points on a circle?, Ans :, 2𝑛, , 𝐶3, 𝑛𝐶, 3, , =, , 12, , 1, 2𝑛(2𝑛−1)(2𝑛−2)⁄, 1×2×3, 𝑛(𝑛−1)(𝑛−2)⁄, 1×2×3, , =, , 12, 1, , 6, , =, =, , 12, 1, 12, 1, , 8𝑛 − 4 = 12𝑛 − 24, 4𝑛 = 20, 𝑛=5, (b) The number of chords = 21𝐶2, 21×20, = 1×2 = 210, 8. (a) Find the value of 'r' such that 10𝐶𝑟 is, maximum, (b) What is the number of ways of choosing, 4 cards from a pack of 52 playing cards?, In how many of these, (i) four cards are of the same suit,, (ii) four cards belong to four different, suits,, (iii) are face cards,, (iv) two are red cards and two are black, cards, Ans :, (a) 𝑟 = 5, (b) Total selections = 52𝐶4, (i) 4 Diamonds or 4 Hearts or 4 spades or 4, clubs, Total selection = 13𝐶4 + 13𝐶4 + 13𝐶4 + 13𝐶4, = 4 × 13𝐶4, = 2860, (ii) 1 Diamonds and 1 Hearts and 1 spades, and 1 clubs, Total selection = 13𝐶1 × 13𝐶1 × 13𝐶1 × 13𝐶1, 4, , = ( 13𝐶1 ), = 134, (iii) Total face cards = 12, ∴ Total selection of 4 face cards = 12𝐶4, = 495, (iv) Total Red = 26, Total Black = 26, ∴ 2 Red and 2 Black , total, selection = 26𝐶2 × 26𝐶2, = 105625, , VIJAYABHERI DISTRICT PANCHAYATH & MAM MALAPPURAM, , 35

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9. Determine the number of 5 card, combinations out of a deck of 52 cards if, there is exactly one ace in each, combination., Ans :, 1 Ace and 4 Non Ace, total selection, = 4𝐶1 × 48𝐶4, = 778320, , 10. A bag contains 5 black and 6 red balls., Determine the number of ways in which, 2 black and 3 red balls can be selected., Ans :, Black: 5 , Red: 6, 2 Black and 3 Red, Total selection, = 5𝐶2 × 6𝐶3, = 10 × 20, = 200, , PRACTICE PROBLEMS, 1. (a) How many 3-digit numbers are there, with no digit repeated?, (b) In how many ways can 3 vowels and 2, consonants be selected from the letters, of the word INVOLUTE, , 2. Find the number of ways of selecting, 9 balls from 6 red balls, 5 white balls, and 5 blue balls if each selection, consists of 3 balls of each color., 3., , A group consists of 4 girls and 7 boys., In how many ways can of 5 members be, selected if the team has team, (i) no girl?, (ii) at least one boy and one girl?, (iii) at least 3 girls?, , , , VIJAYABHERI DISTRICT PANCHAYATH & MAM MALAPPURAM, , 36

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⟹ 3 × 𝑟 3 = 81, ⟹ 𝑟 3 = 27, ⟹𝑟=3, ∴ 𝑎2 = 3 × 3 = 9, 𝑎3 = 9 × 3 = 27, ∴ we can insert 9 and 27 between 3, and 81 so that the resulting sequence is, a G.P, 8., , present at the end of 2nd hour 4th hour, and 𝑛𝑡ℎ hour. ?, Ans :, Here 𝑎1 = 30 and 𝑟 = 2, At the end of 2nd hour,, number of bacteria = 𝑎3 = 𝑎𝑟 2, = 30 × 𝑟 2 = 120, At the end of 4th hour,, number of bacteria = 𝑎5 = 𝑎𝑟 4, = 30 × 𝑟 4 = 480, At the end of 𝑛𝑡ℎ hour,, number of bacteria = 𝑎𝑛+1 = 𝑎𝑟 𝑛, = 30 × 2𝑛, , The number of bacteria in a certain, culture, doubles in every hour. If there, were 30 bacteria present in the culture, originally, how many bacteria will be, , PRACTICE PROBLEMS, 1., 2., 3., 4., , In a G.P, the third term is 24 and 6th, term is 192. Find the 10th term, Find the sum of 𝑛 terms of the G.P, 2 4, 1, 3 , 9,….., , 5., , 6., , Which term of the G.P √3, 3, 3√3, …., is 729 ?, The 𝑛𝑡ℎ term of the sequence, 5,, 5, −5/2 , 5/4,……is 1024. Find 𝑛, , The sum of first 3 terms of a G.P is 16, and the sum of next three terms is 128., Find the first term, common ratio and, sum of 𝑛 terms., The sum of first three terms of a G.P is, 39, and their product is 1. Find the, 10, common ratio and the terms, , , , VIJAYABHERI DISTRICT PANCHAYATH & MAM MALAPPURAM, , 42

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PRACTICE PROBLEMS, 1., , 2., , 3., , (ii) Is ∆𝐴𝐵𝐶 a right angled triangle?, Justify, , Find the slope of the straight line, passing through the points (2, -1) and, (4,5), 4., , Find the equation of the straight line, whose 𝑥 and 𝑦 intercepts are 2 and −5, respectively., , 5., , Equation of a line is 3𝑥 − 2𝑦 − 6 = 0 ., Find its, (i) Slope, (ii) 𝑥 and 𝑦 intercepts, , 6., , Reduce the equation 2𝑥 + 5𝑦 − 10 = 0, of straight line into intercept form., Hence write its 𝑥 and 𝑦 intercepts, , A line passes through (𝑥1 , 𝑦1 ) and, (ℎ, 𝑘 ). If the slope of the line is 𝑚,, show that 𝑘 − 𝑦1 = 𝑚(ℎ − 𝑥1 )., Without using Pythagorus theorem, show that the points A(4,4), B(3,5) and, C(-1,-1) are the vertices of a right, angled triangle. [Hint : Find slopes of, AB, BC and AC], OR, Consider a triangle ABC with vertices, A(4,4), B(3,5) and C(-1,-1), (i) Find the slopes of the sides AB, BC, and AC, , , , VIJAYABHERI DISTRICT PANCHAYATH & MAM MALAPPURAM, , 45

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PRACTICE PROBLEMS, 1., , Find the equation of the circle with, centre (2,4) and radius is 5, , 4., , Find the equation of the parabola with, focus (6,0) directrix 𝑥 = −6, , 2., , Find the centre and radius of the circle, , 5., , Find the coordinates of the foci, the, vertices , length of major axis and, minor axis ,eccentricity and latus, , 𝑥 2 + 𝑦 2 − 8𝑥 + 12𝑦 − 3 = 0, 3., , 𝑥2, , 𝑦2, , rectum of the ellipse 100 + 400 = 1, , Find the coordinates of focus, axis of, parabola , equation of directrix, length, of latus rectum of the parabola, 𝑥 2 = −16𝑦, , 6., , Find the equation of the ellipse with, length of major axis 20 and foci are, (0, ±5), , , , VIJAYABHERI DISTRICT PANCHAYATH & MAM MALAPPURAM, , 49

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Eg : Find the distance between the points (-1,1,1), (1,2,3), Distance = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2 + (𝑧2 − 𝑧1 )2, 2, , = √(1 − (−1)) + (2 − 1)2 + (3 − 1)2, = √(2)2 + 12 + 22, = √4 + 1 + 4, = √9, =3, QUESTIONS AND ANSWERS, 1., , Fill in the blanks :, (a) The 𝑥-axis and 𝑧-axis taken together determine a plane known as ……., (b) The coordinates of points in the XY plane are of the form ………., (c) If P is a point on YZ plane then its 𝑥 coordinates is ………., , Ans :, (a) XZ-plane, (b) (𝑥, 𝑦, 0), (c) Zero, 2., , Which of the following points lies in the sixth octant, (i) (−4,2, −5), (ii) −4, −2, −5), (iii) (4, −2, −5), (iv) (4,2,5), Ans : (i) or (−4,2, −5), 3., Show that the points 𝑃(−2,3,5), 𝑄(1,2,3) and 𝑅(7,0, −1) are collinear., Ans :, We know that points are said to be collinear if they lie on a line, 𝑃𝑄 = √(1 + 2)2 + (2 − 3)2 + (3 − 5)2, = √9 + 1 + 4, = √14, 𝑄𝑅 = √(7 − 1)2 + (0 − 2)2 + (−1 − 3)2, = √36 + 4 + 16, = √56, = 2√14, 𝑃𝑅 = √(7 + 2)2 + (0 − 3)2 + (−1 − 5)2, = √81 + 9 + 36, = √126, = 3√14, Thus 𝑃𝑄 + 𝑄𝑅 = 𝑃𝑅. Hence P, Q and R are collinear, VIJAYABHERI DISTRICT PANCHAYATH & MAM MALAPPURAM, , 51

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7., , Consider the following figure. Find the distance PQ, Z, S, P, 5, 3 O, , Q, , 4, , Y, R, , X, , Ans :, 𝑃(3,4,5), 𝑄(0,4,0), 𝑃𝑄 = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2 + (𝑧2 − 𝑧1 )2, 𝑃𝑄 = √(0 − 3)2 + (4 − 4)2 + (0 − 5)2, = √9 + 0 + 25, = √34, PRACTICE PROBLEMS, 1., , 2., , 3., , 4., , Name the octants in which the following points lie, (i) (−5,4,7), (ii) (−5, −3, −2), (iii) (2, −5, −7), (iv) (7,4, −3), Given three points 𝐴(−4,6,10), 𝐵(2,4,6) and 𝐶(14,0, −2)., (a) Find AB, (b) Prove that the points A, B and C are collinear., Verify the following, (a) (0,7, −10), (1,6, −6) and (4,9, −6) are the vertices of an isosceles triangle., (b) (−1,2,1), (1, −2,5), (4, −7,8) and (2,3, −4) are the vertices of a parallelogram., Find the equation of the set of points P such that its distance from the points 𝐴(3,4, −5) and, 𝐵(−2,1,4) are equal., , , , VIJAYABHERI DISTRICT PANCHAYATH & MAM MALAPPURAM, , 53

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14, MATHEMATICAL REASONING, KEY NOTES, , , , , , , , , , , , , Statement, A statement is a sentence which is either always true or always false, but not both, simultaneously., Simple Statements, A statement is called simple if it cannot be broken down into two or more statements., Compound Statements, A compound statement is the one which is made up of two or more simple statement., Negation of a statement, The denial of a statement is called the negation of the statement. The negation of a, statement p in symbolic form is written as “~p”., The Conditional Statement, If p and q are any two statements, then the compound statement “if p then q” formed, by joining p and q by a connective ‘if-then’ is called a conditional statement or an, implication and is written in symbolically p → q or p ⇒ q,, Converse of a Conditional Statement, The conditional statement “q → p” is called the converse of the conditional statement, “p → q”., Contrapositive of Conditional Statement, The statement “(~q) → (~p) ” is called the contrapositive of the statement p → q., QUESTIONS AND ANSWERS, , 1., , Write the negation of the, following statements, (a) √2 is irrational, (b) √2 is not a complex number, (c) Every natural number is greater, than zero, , Ans :, (a) It is false that √2 is irrational, (b) It is false that √2 is not a, complex number, , (c) It is false that every natural number is, greater than zero, 2., , Write the converse and contrapositive of, the following statements, (a) If a number is divisible by 9, then it is, divisible by 3, (b) If the integer n is odd, then n2 is odd, (c) If a triangle is equilateral ,then it is, isosceles, , VIJAYABHERI DISTRICT PANCHAYATH & MAM MALAPPURAM, , 59

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having no common Factor, , Ans :, (a) Converse : If a number is divisible by, 3, then it is divisible by 9, Contrapositive: If a number is not, divisible by 3, then it is not divisible by 9, (b) Converse: If the integer n2 is odd,, then n is odd, Contrapositive: If the integer n2 is, not odd, then n is not odd, (c) Converse:, If a triangle is, isosceles ,then it is equilateral, Contrapositive: If a triangle is not, isosceles, then it is not equilateral, 3., , 𝑎2, , Squaring , we get 2 = 𝑏2, ⟹ 𝑎2 = 2𝑏2, ⟹ 2 divides 𝑎2, ⟹ 2 divides 𝑎, Let 𝑎 = 2𝑘, ⟹ (2𝑘)2 = 2𝑏2, ⟹ 4𝑘 2 = 2𝑏2, ⟹ 2𝑘 2 = 𝑏2, ⟹ 𝑏2 = 2𝑘 2, ⟹ 2 divides 𝑏2, , Verify by the method of, contradiction that √2 is irrational, , ⟹ 2 divides 𝑏, , Ans :, To prove √2 is irrational, Assume that √2 is rational, , Thus we get 2 divides 𝑎 and 2 divides 𝑏, i.e., 2 is a common factor of 𝑎 and 𝑏, Which is a contradiction to our assumption., ∴ √2 is irrational, , 𝑎, , ∴ √2 = , where 𝑎 and 𝑏 are integers, 𝑏, , PRACTICE PROBLEMS, (a) If the integer n is even, then n2 is even, (b) If 𝑥 is prime number , then 𝑥 is odd, , 1. Write the negation of the following, statements, (a) Both diagonals of a rectangle have, same length, (b) Chennai is the capital of Tamil, Nadu, 2. Write the converse and contrapositive, of the following statements, , 3., , Verify by the method of contradiction, that √5 is irrational, , , , VIJAYABHERI DISTRICT PANCHAYATH & MAM MALAPPURAM, , 60

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𝑓𝑖, 7, 10, 15, 10, 6, 48, , 𝑥𝑖, 3, 8, 13, 18, 23, , 𝑥𝑖2, , 𝑓𝑖 𝑥𝑖, 21, 80, 195, 180, 138, 614, , 𝑓𝑖 𝑥𝑖2, 63, 640, 2535, 3240, 3174, 9652, , 9, 64, 169, 324, 529, 1095, , Σ𝑓𝑖 𝑥𝑖, , (a) Mean =, , 4., 𝑁, , (b) Variance = 𝜎, , =, =, , =, , 2, , Σ𝑓𝑖 𝑥𝑖2, 𝑁, 9652, 48, , 614, 48, , 2655, 45, , = 12.79, , = 59, , Σ𝑓𝑖 𝑥𝑖2, , =, , 𝑁 = 48, Σ𝑓𝑖 𝑥𝑖, , =, , (b) Variance = 𝜎 2 =, , Ans :, , (a) Mean 𝑥̅ =, , 𝑁, , − (𝑥̅ )2, , 𝑁, 162525, 45, , − 592, , = 130.66, SD = √variance, = √130.66, = 11.43, Find the mean deviation about mean for, the following data :, x, 2, 5, 6, 8, 10, 12, , − (𝑥̅ )2, , − 12.792, , = 37.45, SD = √variance, = √37.45, = 6.12, , f, 2, 8, 10, 7, 8, 5, , Ans :, 3., , Consider the Following distribution., Classes, Frequencies, , 3040, 3, , 4050, 7, , 5060, 12, , 6070, 15, , 7080, 8, , 8090, 3, , 90100, 2, , (a) Find the mean., (b) Find the variance and standard, deviation, Ans :, 𝑓𝑖, 3, 7, 12, 15, 8, 45, , 𝑥𝑖, 35, 45, 55, 65, 75, , 𝑓𝑖 𝑥𝑖, 105, 315, 660, 975, 600, 2655, , 𝑥𝑖2, 1225, 2025, 3025, 4225, 5625, , 𝑓𝑖 𝑥𝑖2, 3675, 14175, 36300, 63375, 45000, 162525, , 𝑥, 2, 5, 6, 8, 10, 12, , 𝑓, 2, 8, 10, 7, 8, 5, N=40, 𝑥̅ =, , 𝑓𝑥, 4, 40, 60, 56, 80, 60, 300, , |𝑥 − 𝑥̅ |, 5.5, 2.5, 1.5, 0.5, 2.5, 4.5, , 𝑓|𝑥 − 𝑥̅ |, 11, 20, 15, 3.5, 20, 22.5, 92, , Σ𝑓𝑖 𝑥𝑖, , =, 𝑀𝐷 =, , 𝑁, 300, 40, , = 7.5, , Σ𝑓𝑖 |𝑥𝑖 −𝑥̅ |, 𝑁, , VIJAYABHERI DISTRICT PANCHAYATH & MAM MALAPPURAM, , 92, , = 40 = 2.3, , 62

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PRACTICE PROBLEMS, 1. Find the mean , mean deviation about mean, variance and standard deviation for the following, data, (a) 6,7,10,12,13,4,8,12, (b) 6, 8,10,12,14,16,18,20,22 ,24, 2. Find the mean , variance and standard deviation for the following data, (i), , 𝑥𝑖, 𝑓𝑖, , 5, 7, , (ii), , 10 15 20 25, 4 6 3 5, , 𝑥𝑖, 𝑓𝑖, , 4, 3, , 8, 5, , 11 17 20 24 32, 9 5 4 3 1, , 3. Find the mean ,mean deviation about mean, variance and standard deviation for the following, data, (i), Classes, 0-10 10-20 20-30 30-40 40-50, (ii), , Frequencies 5, , 8, , 15, , 16, , 6, , Classes, , 2030, 15, , 3040, 13, , 4050, 7, , 5060, 9, , 1020, Frequencies 6, , For commerce only:, 1. Find the mean deviation about mean for the data, 6,7,10,12,13,4,8,12, 2. Find the mean deviation about mean for the data, 𝑥𝑖 5 10 15 20 25, 𝑓𝑖 7 4 6 3 5, 3. Find the mean deviation about mean for the data, Classes, 0-10, Frequencies 5, , 10-20 20-30 30-40 40-50, 8, 15, 16, 6, , , , VIJAYABHERI DISTRICT PANCHAYATH & MAM MALAPPURAM, , 63