# MATHS X STD. Try, try and try again you will succeed atlast. P.THIRU KUMARESA KANI M.A., M.Sc.,B.Ed., (Maths)

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1 MATHS X STD Try, try and try again you will succeed atlast P.THIRU KUMARESA KANI M.A., M.Sc.,B.Ed., (Maths) Govt.Girls High School,Konganapuram Salem (Dt.) Cell No and 1

2 1. SETS AND FUNCTIONS 1. Commutative property AUB = BUA A B = B A 2. Associative property AU( BUC) = (AUB)UC A ( B C) = (A B) C 3. Distributive property AU( B C) = (AUB) (AUC) A ( BUC) = (A B)U(A C) 4. De Morgan s laws i) (AUB) = A B ii) (A B) = A U B iii) A (BUC) = (A B) (A C) iv) A (B C) = (A B)U (A C) 5. Cardinality of sets i) n(aub) = n(a) +n(b) n(a Β) ii) n(aubuc) = n(a) + n(b) + n(c) n(a B) n(b C) n(a C) + n(a B C) 6. Representation of functions a set of ordered pairs, a table, an arrow diagram, a graph 7. Types of functions 1. One-One function Every element in A has an image in B. 2 Onto function Every element in B has a pre-image in A. 3. One-One and onto function Both a one-one and an onto function. 4. Constant function Every element of A has the same image in B. 5. Identity function An identity function maps each element of A into itself. 2. SEQUENCES AND SERIES OF REAL NUMBERS Arithmetic sequence or Arithmetic Progression (A.P.) 1. General form a, a+d, a+2d, a+3d, Three consecutive terms a d, a, a + d 3. The number of terms n = General term t n = a + (n 1 )d 5. The sum of the first n terms (if the common difference d is given.) S n = [ 2a + (n 1)d ] 6. The sum of the first n terms (if the last term l is given.) S n = [ a + l] 2

3 Geometric Sequence or Geometric Progression (G.P.) 7. General form a, ar, ar 2,ar 3,...,ar n 1, ar n, General term t n = ar n 1 9. Three consecutive terms a / r, a, ar 10. The sum of the first n terms 1 1 Special series 11. The sum of the first n natural numbers, n = 12. The sum of the first n odd natural numbers, ( 2k 1 ) = n The sum of first n odd natural numbers (when the last term l is given) l = 14. The sum of squares of first n natural numbers, k 2 = 15. The sum of cubes of the first n natural numbers, k 3 = 1 (a + b) 2 = a 2 + 2ab + b 2 2 (a - b) 2 = a 2-2ab + b 2 3 a 2 - b 2 = (a + b) (a-b) 4 a 2 + b 2 = (a + b) 2-2ab 5 a 2 + b 2 = (a - b) 2 + 2ab 8 a 3 + b 3 = (a + b) (a 2 ab + b 2 ) 9 a 3 - b 3 = (a - b) (a 2 + ab + b 2 ) 10 a 3 + b 3 = (a + b) 3 3ab (a + b) 11 a 3 - b 3 = (a - b) 3 + 3ab (a - b) 12 a 4 +b 4 = (a 2 +b 2 ) 2-2 a 2 b 2 13 a 4 - b 4 =(a +b)(a - b)(a 2 + b 2 ) 3. ALGEBRA 14 (a + b + c) 2 = a 2 + b 2 +c 2 + 2(ab + bc +ca) 15 (x +a) (x+b) = x 2 + (a+b) x + ab 16 (x +a)(x+b)(x+c) = x 3 + (a+b+c) x 2 + (ab+bc+ca) x + abc 3

4 17 Quadratic polynomials ax 2 + bx + c = 0 18 sum of zeros ( α + β ) = - coefficient of x / coefficient of x 2 = ( 19 product of zeros ( α β ) = constant term / coefficient of x 2 = ( ) ) 20 Quadratic polynomials with zeros α and β. : x 2 - ( α + β ) x + ( α β ) 20 Relation between LCM and GCD : L CM x GCD = f(x) x g(x) 21 Solution of quadratic equation by formula method x = 22 Nature of roots Δ = b 2-4ac Δ > 0 Real and unequal Δ = 0 Real and equal. Δ < 0 No real roots. (It has imaginary roots) 23 Formation of quadratic equation when roots are given X 2 ( sum of roots) x + ( product of roots ) = 0 4. MATRICES 1 Row matrix : A matrices has only one row. 2. Column matrix : A matrices has only one column. 3 Square matrix : A matrix in which the number of rows and the number of columns are equal 4 Diagonal matrix : A square matrix in which all the elements above and below the leading diagonal are equal to zero 5 Scalar matrix : A diagonal matrix in which all the elements along the leading diagonal are equal to a non-zero constant 6. Unit matrix : A diagonal matrix in which all the leading diagonal entries are 1 7 Null matrix or Zero-matrix : A matrices has each of its elements is zero. 8 Transpose of a matrix : A matrices has interchanging rows and columns of the matrix 9 Negative of a matrix : The negative of a matrix A is - A 10 Equality of matrices : Two matrices are same order and each element of A is equal to the corresponding element of B 4

5 11 Two matrices of the same order, then the addition of A and B is a matrix C 12 If A is a matrix of order m x n and B is a matrix of order n x p, then the product matrix AB is m x p. 13 Properties of matrix addition Commutative A +B = B + A Associative A + (B + C) = (A + B) +C Existence of additive identity A + O = O + A =A Existence of additive inverse A + ( A) = ( A) + A = O 14 Properties of matrix multiplication Not commutative in general A B = BA Associative A(BC) = (AB)C distributive over addition A(B + C) = AB + AC (A + B)C = AC + BC Existence of multiplicative identity A I = I A = A Existence of multiplicative inverse AB = BA = I 15 (A T ) T = A ; (A +B) T = A T + B T ; (AB) T = B T A T 5. COORDINATE GEOMETRY 1 Distance between Two points AB= 2 The line segment joining the two points A(x 1,y 1 ), and B(x 2,y 2 ) internally in the ratio l : m is P (, ) 3 The line segment joining the two points A(x 1,y 1 ), and B(x 2,y 2 ) externally in the ratio l : m is P (, 4 The midpoint of the line segment M = ( 5 The centroid of the triangle G = ( ), ), 6 Area of a triangle A = sq. unit or A = sq. unit 7 Area of the Quadrilateral A = sq. unit 7 Collinear of three points (or) Slope of AB = Slope of BC or slope of AC. 8 If a line makes an angle θ with the positive direction of x- axis, then the slope m = tan θ 9 Slope of the non-vertical line passing through the points m = 10 Slope of the line ax + by + c =0 is m = 11 The straight line ax + by + c =0, y-intercept c y = - 12 Two lines are parallel if and only if their slopes are equal. : m 1 = m 2 ) 5

6 13 Two lines are perpendicular if and only if the product of their slopes is -1 : m 1 m 2 = - 1 Equation of straight lines 14 x-axis y = 0 15 y-axis x = 0 16 Parallel to x-axis y = k 17 Parallel to y-axis x = k 18 Parallel to ax+by+c =0 ax+by+k=0 19 Perpendicular to ax+by+c =0 bx ay+k=0 20 Passing through the origin y =mx 21 Slope m, y-intercept c y = mx+c 22 Slope m, a point (x1, y1) y y 1 = m(x x 1 ) 23 Passing through two points 24 x-intercept a, y-intercept b 1 6 GEOMETRY 1 Basic Proportionality theorem or Thales Theorem If a straight line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides the two sides in the same ratio. 2 Converse of Basic Proportionality Theorem ( Converse of Thales Theorem) If a straight line divides any two sides of a triangle in the same ratio, then the line must be parallel to the third side. 3 Angle Bisector Theorem The internal (external) bisector of an angle of a triangle divides the opposite side internally (externally) in the ratio of the corresponding sides containing the angle. 4 Converse of Angle Bisector Theorem If a straight line through one vertex of a triangle divides the opposite side internally (externally) in the ratio of the other two sides, then the line bisects the angle internally (externally) at the vertex. 5 Similar triangles corresponding angles are equal (or) corresponding sides have lengths in the same ratio 1. AA( Angle-Angle ) similarity criterion If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar. 2. SSS (Side-Side-Side) similarity criterion for Two Triangles In two triangles, if the sides of one triangle are proportional (in the same ratio) to the sides of the other triangle, then their corresponding angles are equal 3. SAS (Side-Angle-Side) similarity criterion for Two Triangles 6

7 If one angle of a triangle is equal to one angle of the other triangle and if the corresponding sides including these angles are proportional, then the two triangles are similar. 6 Pythagoras theorem (Bandhayan theorem) In a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. 7 Converse of Pythagorous theorem In a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite to the first side is a right angle. 8 Tangent-Chord theorem If from the point of contact of tangent (of a circle), a chord is drawn, then the angles which the chord makes with the tangent line are equal respectively to the angles formed by the chord in the corresponding alternate segments. 9 Converse of Theorem If in a circle, through one end of a chord, a straight line is drawn making an angle equal to the angle in the alternate segment, then the straight line is a tangent to the circle. 10 If two chords of a circle intersect either inside or out side the circle, the area of the rectangle contained by the segments of the chord is equal to the area of the rectangle contained by the segments of the other P A X PB = PC X PD Circles and Tangents 11 A tangent at any point on a circle is perpendicular to the radius through the point of contact. 12 Only one tangent can be drawn at any point on a circle. However, from an exterior point of a circle two tangents can be drawn to the circle. 13 The lengths of the two tangents drawn from an exterior point to a circle are ual. 14 If two circles touch each other, then the point of contact of the circles lies on the line joining the centres. 15 If two circles touch externally, the distance between their centres is equal to the sum of their radii. 16 If two circles touch internally, the distance between their centres is equal to the difference of their radii. 7 Trigonometry 01 sin θ cosec θ = 1 ; sin θ = 1/ cosec θ ; cosec θ = 1/ sin θ 02 cos θ sec θ = 1 ; cos θ = 1/ sec θ ; sec θ = 1/ cos θ 03 tan θ cot θ = 1 ; tan θ = 1/ cot θ ; cot θ =1/ tan θ 04 tan θ = sin θ / cos θ cot θ = cos θ / sin θ 7

8 05 sin 2 θ + cos 2 θ = 1 ; sin 2 θ = 1- cos 2 θ ; cos 2 θ = 1 -sin 2 θ 06 sec 2 θ tan 2 θ = 1 ; sec 2 θ = 1+ tan 2 θ ; tan 2 θ = sec 2 θ cosec 2 θ cot 2 θ = 1 ; cosec 2 θ =1+ cot 2 θ ; cot 2 θ = cosec 2 θ 1 08 sin (90 θ)= cos θ cosec (90 θ)= sec θ 09 cos (90 θ)= sin θ sec (90 θ) = cosec θ 10 tan (90 θ)= cot θ cot (90 θ) = tan θ 11 Componendo and dividendo rule then angle Sin 0 Cos 1 Tan MENSURATION Sl. No Name Surface Area (sq.units) Total Surface Area (sq.units) Volume (cu.units) 1 Solid right circular cylinder 2πrh 2πr(h+r) πr 2 h 2 Right circular hollow cylinder 2π(R+r) h 2π(R+r)(R-r+h) π (R 2 - r 2 ) h 3 Solid right circular cone πrl πr(l + r) πr2 h 4 Frustum - - (R2 + r 2 + Rr) h 5 Sphere 4πr 2 - πr3 6 Hollow sphere - - π (R3 - r 3 ) 7 Solid Hemisphere 2πr 2 3πr 2 πr3 8 Hollow Hemisphere 2π(R 2 + r 2 ) π(3r 2 + r 2 ) π (R3 - r 3 ) 8

9 9 Cone l = ; h = ; r = 10 CSA of a cone = Area of the sector πrl = r2 11 Length of the sector = Base circumference of the cone L = 2πr 12 Volume of water flows out through a pipe = {Cross section area x Speed x Time } Volume of the solid which is melted 13 No. of new solids obtained by recasting = volume of one solid which is made 14 1 m 3 = 1000 litres 1000 litres = 1 k.l 1 d.m 3 = 1 litres 1000 cm 3 = 1 litres 11 STATISTICS 1 Range R = 2 coefficient of range Q = 3 Standard deviation (Ungrouped) 1. Direct method 2. Actual mean method 3. Assumed mean method 4. Step deviation method 4 Standard deviation (Grouped ) 1. Actual mean Method Here d = x Here d = x A x C Here d = Here d = x 2. Assumed mean method 3. Step deviation method Here d = x A x C Here d = 5 Standard deviation of the first n natural numbers, 6 Variance is the square of standard deviation. 7 Standard deviation of a collection of data remains unchanged when each value is added or subtracted by a constant. 9

10 8 Standard deviation of a collection of data gets multiplied or divided by the quantity k, if each item is multiplied or divided by k. 9 Coefficient of variation, C.V = 100 It is used for comparing the consistency of two or more collections of data. 12 PROBABILITY 1 Tossing an unbiased coin once S = { H, T } 2 Tossing an unbiased coin twice S = { HH, HT, TH, TT } 3 Rolling an unbiased die once S = { 1, 2, 3, 4, 5, 6 } 4 The probability of an event A lies between 0 and 1,both inclusive The probability of the sure event is 1. P(S)= 1 7 The probability of an impossible event is 0. P() = 0 8 The probability that the event A will not occur 1 9 P(A) += Addition theorem on probability P(AUB) = P(A) +P(B) P(A B) 12 If A and B are mutually exclusive events, Then P(A B) = Thus P(AUB) = P(A) +P(B) Try, try and try again you will succeed atlast Wish you all the Best P.THIRU KUMARESA KANI M.A., M.Sc.,B.Ed., (Maths) Govt.Girls High School, Konganapuram Edappady (Tk.) Salem (Dt.) Cell No and 10

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