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ICSE Class X Notes 2020 : Mathematics (Podar International School, Kalyan)

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Compound Interest A Amount P Principle n Number of years ( ) ( )( )( ) ( ( )( ) ) R Rate % per year ) ( (3 ~ 6 marks) *( ) + The Compound Interest and Simple Interest for the first year is always same C.I.1 = S.I.1 ( ) ( ) ( ) Shares and Dividend Nominal Value = Face Value = Par value Dividend = Income (3 ~ 6 marks) Investment = No. of shares x Market value per share Total Nominal value = No. of shares x Face value per share ( ) Sales Proceeds = No. of shares x selling price of each share (Market Value of each share) ( For better investment, compare the Y% both shares. Discount shares means Market Value < Face Value. Premium share means Market Value > Face Value. Share at par means Market Value. = Face Value. ) Sales Tax + VAT (2 ~ 5 marks) The tax imposed by state govt. on sale of goods within their respective states or by central government is called sales tax. If an article is to be sold for Rs.100 and 5% sales tax is applicable so the purchaser will have to pay Rs.105. i.e. sales tax (given as percentage) is to be added to the selling price of any article. Usually Selling Price Rs.100 Selling Price. + tax Rs.105 Vat to be paid = x % of ( selling price cost price ) = x % of value addition Banking (4~ 8 marks) Savings account: Calculate the principle qualifying for interest for every month. (Max. value till the tenth and then if it decreases then the smaller value) Time is always 1/12. Calculate the interest using Simple Interest formulae = PRT/100 Recurring account: Fixed amount of money is to be kept in the bank and then after a stipulated period interest and amount is calculated. ( ) ; S n no. of months Maturity value = P/month x n + S.I. OR ( ) Inequations (3 ~ 4 marks) Natural numbers (N) = {1, 2, 3, 4, 5, } Whole numbers (W) = {0, 1, 2, 3, 4, 5, } (W0N) Integers (I, Z) = {- , .-3, -2, -1, 0, 1, 2, 3, 4, 5, } Real numbers (R) = {- , .0 } (all decimals and fractions in between are included) Quadratic Equations (4 ~ 5 marks) Let the Quadratic equation be ax2 + bx + c = 0 (a 0 ) The expression ( b2 4ac ) is called discriminant = ( b2 4ac ) If = 0, roots are equal and real. If > 0 and is a perfect square, roots are real, unequal and rational. If > 0 and is not a perfect square, roots are real, unequal and irrational. If < 0, roots are imaginary. (i.e. not real) Sum of the roots of the equation = & product of the roots = A quadratic equation will have 2 solutions. Eg. x2 25 = 0 x=+5 x2 6x + 9 = 0 ( x 3) 2 = 0 x = 3, 3 Remainder & Factor Theorem (3 ~ 4 marks) Remainder Theorem : If f(x) is a polynomial, an expression with variable x and is divided by (x a); the remainder is the value of f(x) at x = a. i.e. the remainder is f(a). eg. f(x) = x 3 + 2 x 2 + 4; (x 3); f(3) = 33 + 2 x 32 + 4 = 49. Factor Theorem : When a polynomial f(x) is divided by x a, the remainder = f (a). And, if remainder f(a) = 0; x - a is a factor of polynomial f(x). OR if (x a) is a factor, the remainder is 0. Ratio & Proportion (3 ~ 5 marks) If a, b, c & d are in proportion , i.e. ad = bc (product of means = product of extremes) also called inproportion If a, b & c are in proportion , i.e. b2 = ac(where b is the mean proportion and c is the third proportion.) also continued proportion. If a, b, c, d are in continued proportion If a, b, c & d are in proportion, then using properties of ratio. By Alternendo By Invertendo By Componendo By Dividendo By Comp & Div Property of ratio , then c = dk, b = dk2, a = dk3. Matrices (3 ~ 5 marks) Size of Matrix = No. of rows x No. of columns. (Rh Rows horizontal) * * + Only same order matrix can be added or subtracted + Multiplication of matrix Matrix A m x n can only be multiplied with matrix B p x q if n = p and the resultant matrix will have m rows and q cols i.e. R m x q Matrix multiplication is not commutative. (i.e. AB BA) Matrix multiplication is associative. A (BC) = (AB) C A x I = I x A = A where I is a unit matrix of suitable order. A ( B + C ) = AB + AC (distributive property) If AB = AC then B C, A may or may not be zero. Coordinate Geometry (12 ~ 16 marks) Distance formulae : Mid Point Formulae : Section formulae : Centroid Formulae : Slope ( ) ( ) is the angle the line makes with the x - axis, +ve if it is anticlockwise ve if it is clockwise. For parallel lines m1= m2 Slopes are equal. For perpendicular lines m1 x m2 = 1 Slopes are negative reciprocal each other. Equation of a Line Slope Intercept form {m , c = y intercept} Slope point form {P(x1 , y1) & m} y = mx + c y y1 = m (x x1) Two point form {P(x1 , y1) & Q(x2 , y2 )} Equation of x - axis : y = 0 Equation of y - axis : x = 0 Slope of x-axis = tan 0 = 0 Slope of y - axis = tan 90 = undefined(1/0) Equation of line parallel to x- axis : y = a Equation of line parallel to y- axis : x = b Slope of line parallel to x- axis = 0 Slope of line || to y- axis : undefined (1/0) A point on x axis is taken as (a , 0 ) and on y axis is taken as ( 0 , b ) x coordinate is called abscissa and y is ordinate. Circumcentre is a point equidistant from the vertices of a . It is the centre of the circle drawn about a . It is the point if intersection of the bisector of sides of a . For a right angled triangle it is midpoint of the hypotenuse. Incentre is a point equidistant from the sides of a triangle. It is the centre of a circle drawn inside a triangle. It is a point of intersection of the bisectors of the angles of a triangle. Centroid is the point of intersection of the medians of a triangle. Centroid divides the median in the ratio 2 : 1 Orthocentre is the point of intersection of the altitudes of a triangle. For a parallelogram x1 + x3 = x2 + x4 and y1 + y3 = y2 + y4 where x and y are coordinates of the parallelogram. For any points P( a ,b ) Mx P (a , b) Reflected in x axis Invariant point is a point which P( a ,b ) My P ( a , b) Reflected in y axis does not change under reflection. P( a ,b ) Mo P ( a , b) Reflected in origin eg. P( 2 , 0 ) Mx P (2 , 0) Symmetry (3 ~ 5 marks) If a figure is divided into two congruent parts with respect to a line, then the figure is said to be symmetric about the line and the line is called the axis of symmetry. Any figure may have line and point symmetry. Similarity and Size Transformation (6 ~ 8 marks) ABC ~ PQR [ ] A1 = A2 By A. A. test or S.A.S test or by S.S.S. test C. P. S. T. Areas of ~ s are proportional to squares of their corresponding sides. A median divides a into 2 s with equal area. Areas of s meeting at common vertex and base through the same st line is proportional to their bases. Loci (5 ~ 8 marks) It is a path traced by a moving particle which fulfils a given condition. Theorem 1 : Locus of points equidistant from two fixed points is the perpendicular bisector of the line joining the two fixed points. Theorem 2 : Locus of points equidistant from two fixed lines is the angle bisector of the angle formed where the two lines intersect. Theorem 3 : Locus of point B such that AB2 + BC2 = AC2 is the circle with AC as diameter. Circles (4~ 8 marks) Chord Properties: (all theorems are applied to same circle or equal circles i.e. circles with equal radius) A straight line drawn from centre of a circle bisecting the chord is to it. A straight line drawn from centre of a circle bisects the chord. Only one circle can be drawn passing through three non collinear points. Equal chords are equidistant from the centre of the circle. Equidistant chords are equal. Angle Properties: The angle which an arc of a circle subtends at the centre is double that which it subtend at any point on the remaining part of the circumference. Angles in the same segment are equal. The angle subtended by a semi circle is a right angle. Opposite angles of a cyclic quadrilateral are supplementary. Exterior angle of a cyclic quadrilateral is equal to its interior opposite angle. Arc Properties: If two arcs subtend equal angles at the center they are equal. If two arcs are equal they subtend equal angles at the center. If two chords are equal they cut off equal arcs. If two arcs are equal they cut off equal chords. Tangent Properties: The tangent to any circle and radius through the point of contact are to each other. Two tangents drawn to a circle from an external point are i) equal in length, ii) equally inclined to each other through the central line, iii) equally inclined at the centre. If two circles touch each other the point of contact lies on the straight line drawn through the centres. If two chords intersect each other then the product of their lengths of their segments is equal. If a tangent and a chord intersect then the product of the lengths of the segments of the chord is equal to square of the length of the tangent from the point of contact to point of intersection. The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment. (Alternate segment theorem T. C. A. S. A.) Mensuration (12 ~ 14 marks) Area Circle Circumference r 2 = D2/4 2 r= D Semi Circle Ring Perimeter 2 r= D r+2r ( R2 r 2 ) = ( R r) (R + r) 2 ( R + r ) 2 (R+r) Sector Distance travelled by a wheel = No. of rev. x circumference of wheel Cylinder Volume r2h Curved Surface Area Total Surface Area 2 rh 2 r ( h + r ) Hollow Cylinder / Pipe (R2 r2) h 2 h (R + r) 2 (Rh + rh + R2 r2) Cone rL r (L + r) Sphere 4 r2 4 r2 Hemisphere 2 r2 3 r2 4 ( R2 + r 2 ) 4 ( R2 + r 2 ) 2 ( R2 + r 2 ) ( 3 R2 + r2 ) Spherical Shell Hemispherical Shell ( ) ( ) Volume of Big Sphere = No. of lead shots x Volume of each lead shot. Volume = Area of cross section x length ( height ) 1m3 = 1000 ltrs = 106 cm3 Trigonometry (10 ~ 16 marks) oSh aCh oTa Trigometric Identities Complementary Angle Sin = Cos ( 90 ) Cos = Sin ( 90 ) Tan A = Cot ( 90 ) Cosec = Sec (90 ) Sec = Cosec (90 ) Cot = Tan (90 ) Standard Angles 0o 30o 45o 60o 90o Sin 0 Cos Tan 1 1 0 Statistics (8 ~ 16 marks) Mean ( ) : The average value of the given data is mean Ungrouped Grouped a = assumed mean Short cut method d = x a and Step Deviation method and Mean = ( i = class width) Median (Q2) The exact middle value of the given data. Raw data Odd , - Ungrouped frequency distribution Median (Q2) { } Lower Quratile (Q1) Upper Quartile (Q3) , { - Even ( , , - ( ) } { } Grouped frequency distribution (all values from OGIVE) Median (Q2) , { } Lower Quartile (Q1) Upper Quartile (Q3) { { , - } ( Inter Quartile Range = Q3 Q1 Semi Inter Quartile Range = ) } { } ) Mode : (z) The most occurring value in a ungrouped data and the class mark with highest frequency in grouped data. Raw Data = the most occurring number Ungrouped Mean = the number with the highest frequency. Grouped draw a histogram and the highest bar gives the mode. Probability (3 ~ 5 marks) Probability of an impossible event = 0 All possible outcomes added = 1 Sample space denotes all possible outcomes. Complementary event P (A) + P (A) = 1 or P(A) = 1 P(A)

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