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ICSE Notes Mathematics: Important Tips & Formulae

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Shipra Gangwar
Gokuldham High School & Junior College (GHS), Mumbai

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Class X Subject Mathematics Formulae 1.Compound Interest ( Imp chapter) S.I = PTR 100. Amt = Pri ( 1 + Rat )Time 100 Note : If you are asked to compute the interest semi-annually ( Half yearly) the above formula is to be modified, by taking time x 2, and rate 2 If the rates are given differently for the consecutive years, then For example if the rates are 8 % , 12 % and 15 % respectively, then Amt = Pri ( 1 + 8 ) ( 1 + 12 ) ( 1 +15 ) 100 100 100. Here, you need not mention time as exponent. Depreciation Certain items value will be diminished as the time passes, then it is known as depreciation. For example the value of a car, refrigerator, machinery etc. in that case. Final Value of machine = Actual value ( 1 Rat )Time 100 In population growth problems, If present population is given and asking for i) The population n yrs ago, then take Amount as Present population , and find Principal ii) The population after n yrs, then take Principal as Present population , and find Amount 2.Sales Tax & VAT. Selling price = Marked price + x % of sales tax. Selling price = Marked price x % Discount . Tax % = Tax x 100 MP Discount % = Discount x 100 MP. While computing VAT In step1 : Take manufacturing cost and Calculate VAT on Manufacturing Cost In step2 : Take Profit 1, and Calculate VAT on Profit 1 only In step3 : Take Profit 2, and Calculate VAT on Profit 2 only In step4 : Take Profit 3, and Calculate VAT on Profit 3 only Add all to get Total VAT Selling price = Manufacturing cost +Profit 1 +Profit 2 + Profit 3 etc., + Total VAT 3.Banking Savings Bank account While taking the entries you have to bear in mind that While computing the interest always take time as 1/12, irrespective of the total number of months given. i.e in PTR / 100, take time as 1/12, instead of total no of months. If entry of a particular month is not given, then you have to take the last entry of the previous month (Here at times there is chance of making mistake, choose the value from the question.) I f you are asked to find the amount that will be obtained on closing the account Then take last entry from the Question + Interest obtained ( But DO NOT take the Total principal ) Recurring Deposits : 4.Shares & Dividends Dividend = [x % of Face Value] x No of shares No of shares = Investment . Market Val of one share Income of a person may be taken as the Dividend given by the company, therefore income % = Income x 100 Investment 5.HCF & LCM of polynomials In step1 : Factorize the given polynomials, a) Either by splitting the terms, (OR) b) Using formulae ( a + b)2 = a2 + 2ab + b2 ( a b)2 = a2 2ab + b2. a2 b2 = (a + b)(a b). a4 b4 = (a2 ) 2 (b2 ) 2. (a2 + b2 ) (a2 b2 ) (a2 + b2 ) (a b ) (a + b ) ( a + b)3 = a3 + b3 + 3ab (a +b) a3 + b3 = (a + b)( a2 + ab + b2) ( a b)3 = a3 b3 3ab (a b ) a3 b3 = (a b)( a2 + ab + b2) (or) Trial & Error method. In step2 : Take the product of Common terms as their HCF. In step3 : Take the product of All the terms , Omit, the HCF value which gives you the value of LCM. Product of LCM x HCF = Product of the two polynomials. Note: If cubical expression is given, it may be factorized by using Trial & Error method. 6.Quadratic Equations. Note: To find the value of x you may adopt either splitting the middle term or formula method , unless specified the method. Note: when x = y, is given, then use ruler to measure the vertical distance of the point from the line, and then take the same distance on the other side to obtain it s reflection. 8.Ratio & Proportion. Duplicate ratio of a : b is a2 : b2 ( Incase of Sub-duplicate ratio you have to take Square root ) Triplicate ratio of a : b is a3 : b3 ( Incase of Sub-triplicate ratio you have to take Cube root ) Proportion a : b = c : d, Continued Proportion a : b = b : c,(Middle value to be repeated) 1st 2nd 3rd 4th proportionals 1st 2nd 2nd 3rd proportionals Product of Means (Middle values) = Product of Extremes (Either end values) If a = b c is given, then Componendo & Dividendo is a + b = c + d d a b c d Do you have a question Where to take K method ? You may adopt it in the following situations Length on the map / model = k times the original length. Area on the map / model = k2 times the original Area. Volume of the model = k3 times the original Volume. 9.Remainder theorem. If (x 2 ) is a factor of the given expression, then 10.Matrices Some times, you may be asked to find A2 + AB + 7 is given, you have to assume it as A2 + AB + 7 I, Here, I is the Identity matrix. in which all the principal diagonal values are 1, and the rest are Zero . 11.Distance & Section Formulae Distance = \/ (x2 x1) 2 + (y2 y1) 2 . ( The same formula is to be used to find the length of line segment, sides of a triangle, square, rectangle, parallelogram etc.,) To prove co-linearity of the given three points A,B, and C, You have to find The distance of AB + The distance of BC = The distance of AC. Section formula: point (x, y) = m1 x2 + m2 x1 , m1 y 2 + m 2 y 1 m1 + m 2 Mid point = x1 + x2 2 , m1 + m 2 y1 + y 2 2 Centroid of a triangle = x1 + x2 + x3 3 , y 1 + y2 + y3 3 12.Equation of a line. If two points are given, then Slope (m) = y2 y1 x2 x1 If a point, and slope are given, then Slope (m) = y y1 x x1 If two lines are Parallel to each other then their slopes are equal i.e m1 = m2 If two lines are Perpendicular to each other then product of their slopes is 1. i.e m1 x m2 = 1 Depending upon the question You may have to use a) y = mx + c, where c is the y-intercept. 13.Similarity. If two triangles are similar then, ratio of their sides are equal. i.e if ABC ~ PQR then AB = BC = AC PQ QR PR. If ABC ~ PQR then Area of ABC = Side2 = AB2 = BC2 = AC2 Area of PQR Side2 PQ2 QR2 PR2 Size transformation: Length of the model = k times the actual length. [Here k is to be taken as 1 / 10000 ] Area of the model = k2 times the actual area. [Here k2 is to be taken as (1 / 10000) 2 ] Volume of the model = k3 times the actual volume. [Here k3 is to be taken as (1 / 10000) 3 ] 14.Symmetry. A line which divides the given figure into two identical parts is known as line of Symmetry 1. An angle has One line of symmetry. 2. A Square has 4 lines of symmetry. 3. A Rectangle has 2 lines of symmetry. 4. A Parallelogram has No lines of symmetry. 5. A Rhombus has 2 lines of symmetry. 6. An Isosceles Triangle has One line of symmetry. 7. An Equilateral Triangle has 3 lines of symmetry. 8. A Circle has Infinite lines of symmetry. 9. A Regular Polygon with n sides has n lines of symmetry. For ex: A Regular pentagon (5 sides) has 5 lines of symmetry A Regular hexagon (6 sides) has 6 lines of symmetry. Note: Angle of a regular polygon = ( 2n 4 )x 90 n. Here n refers number of sides of a polygon. ( This formula may be used to construct a regular Polygon.) 15.Loci. The Locus of a line segment is it s Perpendicular bisector. The Locus of an angle is it s Angle bisector. For solving most of the Locus problems, the above two points are good enough. In addition to these points, You should have the basic knowledge of geometrical constructions. Also look at the given figure in terms of either line segments or angles 16.Circles, & Tangents. Equal chords of a circle are equi distant from the center. The sum of opposite angles of a cyclic quadrilateral is always 1800. 17.Circumference & Area of a Circle. r2. Area of a Circle = Perimeter of a Circle = 2 r Area of sector = Length of an arc = Area of ring = Distance moved by a wheel in one revolution = Circumference of the wheel. Number of revolutions = . x r2. 360 x 2 r. 360 . ( R2 r2 ) Total distance moved Circumference of the wheel. . Area of an equilateral triangle = \/3 Side2. 4 . Note: While solving Mensuration problems, take care of the following. 1. If diameter of a circle is given, then find the radius first (Have you made mistake earlier by taking d as radius and solved the problem ?) 2. Check the units of the entire data. If the units are different, then convert them to the same units. For Example: Diameter = 14 cm, and Height = 3 m Therefore Diameter = 14 cm, and Height = 300 cm (Have you ever committed such mistake ?) 18.Solids. 1. Cylinder: Volume of a cylinder = r2h Curved surface area = 2 rh Total surface area = 2 r h + 2 r2 Volume of hollow cylinder = TSA of hollow cylinder 2 r (h +r ) R2h r2h ( R2 r2) h = Outer CSA + Inner CSA + 2 x Area of ring. 2 Rh + 2 rh + 2x [ R2 r2] ( Of course, If you want, you may take common ) 2. Cone: Volume of a Cone = r2h. CSA of a Cone = r l ( Here l refers to Slant height which may be obtained by using Pythagoras theorem ) TSA of a Cone = r l + r2. r ( l + r ) 3. Sphere: Surface area of a Sphere = 4 r2. ( In case of Sphere, there is no CSA, TSA separately) Volume of hemi sphere = r3 [Take half the volume of a sphere] CSA of hemisphere = 2 r2 TSA of hemisphere = 2 r2 + r2 [Take half the SA of a sphere] 3 r2 While solving the combination of solids it would be better if you take common If a solid is melted and, recast into number of other small solids, then Volume of the larger solid = No x Volume of the smaller solid For Ex: A cylinder is melted and cast into smaller spheres. Find the number of spheres Volume of Cylinder = No x Volume of sphere. If an Ice cream cone with hemispeherical top is given then you have to take a) Total Volume = Volume of Cone + Volume of Hemisphere b) Surface area = CSA of Cone + CSA of hemisphere (usually Surface area will not be asked) 19.Trigonometric Identities. Wherever Square appears think of using the identities i) Sin2 + Cos2 = 1 ii) Sec2 Tan2 = 1 iii) Coseec2 Cot2 = 1 Try to convert all the values of the given problem in terms of Sin and Cos Cosec may be written as 1/Sin Sec may be written as 1/Cos Cot may be written as 1/Tan Tan may be written as Sin / Cos Wherever fractional parts appears then think taking their LCM Think of using ( a + b )2, ( a b )2, ( a + b )3, ( a b )3formulae etc., Rationalize the denominator [ If a + b, (or) a b format is given in the denominator] You may separate the denominator For Ex : Sin + Cos Sin as Sin + Cos Sin Sin 1 + Cot If you are not able to solve the LHS part completely, Do the problem to such an extent you can solve, then start working with RHS, and finally you will end up the problem at a step where LHS = RHS Sin ( 90 ) = Cos Sec( 90 ) = Cosec : Cosec ( 90 ) = Sec Tan ( 90 ) = Cot : Cos ( 90 ) : Cot ( 90 ) = Sin . = Tan 20.Heights & Distances. 21.Graphical Representation. Don t forget to write the scale on x-axis, and on y-axis. To find the Lower quartile take N/4 [Here N is f] then take the corresponding point on X-axis To find the Upper quartile take 3N/4, then take the corresponding point on Xaxis To find the Median take N/2, then take the corresponding point on X-axis 22.Measures of Central Tendency. For un-grouped data Arithmetic Mean = Sum of observations No of observations Mode = The most frequently occurred value of the raw data. To find the Median first of all arrange the data in Ascending or Descending order, then Median = (N+1)/2 term value of the given data, in case of the data is having odd no of observations. Median = [(N/2) + (N+1)/2)] / 2 term value of the given data, in case of the data is having even number of observations. For grouped data Arithmetic Mean = fx (Direct method) f Arithmetic Mean = a + fd (short cut method) f Arithmetic Mean = a + fu x c (step-deviation method) f 23.Probability. Probability of an event : P(event) = Number of favorable outcomes Total number of outcomes In a deck of playing cards, there are four symbols (Spades in Black colour) having A, 2,3,4,5,6,7,8,9,10,J,K, and Q (Clubs in Black colour) (Hearts in Red colour) total 13 cards having A, 2,3,4,5,6,7,8,9,10,J,K, and Q total 13 cards having A, 2,3,4,5,6,7,8,9,10,J,K, and Q total 13 cards (Diamond in Red colour) having A, 2,3,4,5,6,7,8,9,10,J,K, and Q total 13 cards 52 cards Jack, King and Queen are known as Face Cards , As these cards are having some pictures on. Note: Drawing rough sketches is always advisable, though, you may not get marks for them, but, they will give a clear cut idea to solve the problem. All The Best

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