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ICSE Class X Notes 2020 : Mathematics (HVB Academy, Mumbai) Formulae Booklet

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Instructions & Formulae, ICSE Math (X), By Rakesh Kushwaha (M.Sc.; B.Ed.): 09224190389/09930350748 Formulae Booklet With General & Special Instructions I.C.S.E. Mathematics (Grade 10) (2019 2020) By RAKESH KUSHWAHA B.Sc.(Hons); M.Sc.(Mathematics); M.A. (Lings I, English & Economics); B.Ed. (Sp. Edu.); GNIIT; S.B.T.C. Former Vice Principal (M.V.M. PUBLIC SCHOOL, ALIGARH) Coordinator, I.C.S.E. SCHOOL, MUMBAI Examiner and Team Leader, ICSE Board Examination H.O.D. MATHEMATICS, I.C.S.E. SCHOOL, MUMBAI Former Facilitator, IBDP mathematics YouTube Channel: KUSHWAH S FREE ONLINE EDUCATION Contact: 9224190389 / 9930350748 Offering 27 years of Achievement in Developing Student-Focused Environments to Maximize Learning Experiences: E-mail: rakeshpraxis@gmail.com 1 Instructions & Formulae, ICSE Math (X), By Rakesh Kushwaha (M.Sc.; B.Ed.): 09224190389/09930350748 ICSE MATHEMATICS (X) There will be one paper of 2 hours duration carrying 80 marks and Internal Assessment of 20 marks. The paper will be divided into two Sections. Section I (40 marks), Section II (40 marks). Section I: It will consist of compulsory short answer questions. Section II: Candidates will be required to answer four out of seven questions. UNITS & CHAPTERS 1. COMMERCIAL MATHEMATICS GST [Goods and Services Tax] Banking: Recurring Deposit Accounts Shares and Dividends (Brokerage and fractional shares not included) 2. ALGEBRA Linear Inequations Quadratic Equations and Solving Problems Ratio and Proportion (Applications on proportions only) Factorization of polynomials (f(x) not to exceed degree 3) Matrices Arithmetic and Geometric Progression 3. CO-ORDINATE GEOMETRY Reflection Section and Mid-Point Formula Equation of a Line 4. GEOMETRY Similarity Loci (Locus and Its Constructions) Circles Constructions (tangents to circle, circumscribing & inscribing circle on a & reg. hexagon) 5. MENSURATION Surface Area and Volume of Solids (Cylinder, Cone and Sphere) Problems on Frustum are not included. 6. TRIGONOMETRY Using Identities to solve/prove simple algebraic trigonometric expressions Heights and Distances (Cases involving more than 2 right angled excluded) 7. STATISTICS Graphical Representation (Histogram and Ogives) Measures of Central Tendency (Mean, Median, Quartiles and Mode) Probability 2 E-mail: rakeshpraxis@gmail.com Instructions & Formulae, ICSE Math (X), By Rakesh Kushwaha (M.Sc.; B.Ed.): 09224190389/09930350748 GENERAL INSTRUCTIONS: Mathematics needs Continuous Practice right from the beginning of the year. Solve each and every question by yourself with understanding. Solve similar questions from other question banks as well. Utilize all your resources. Please remember that UNDESTANDING is more IMPORTANT. If you understand one sum, you can do hundred similar sums. Be in regular touch with your Teachers. Request them for more guidance, academic help etc. Remember that without consistent practice you cannot be perfect in Mathematics. Be prepared for the worst correction. Don t give any chance to the examiner to cut your marks for silly mistakes or careless work. Don t use short cuts to get answer, make habit of writing all required steps. You never know which step has to be marked for giving marks. Marking scheme pattern tends to be changed every year. Read the full question carefully, write the data first and convert into the same unit if to be substituted for further calculation. Copy the digits carefully. Misreading is very common error and the children lose their marks for copying wrong digit. Make habit of checking the digits (data) again at the end. Write legible digits. For example, 0 and 6. It has been observed that your 6 is taken as 0 and 0 is taken as 6 by you only. Same way 11 may become 4 if written in hurry. 3 E-mail: rakeshpraxis@gmail.com Instructions & Formulae, ICSE Math (X), By Rakesh Kushwaha (M.Sc.; B.Ed.): 09224190389/09930350748 If any value is to be substituted in between the sum (e.g., value of r), don t keep in decimal form or round off form. Have it in fraction form only or else this may lead to wrong answer. If the answer is to be written up to two decimal places, find up to three decimal places and then round off at the end. Make proper equal to sign (=) for every step, sometimes it s taken as minus sign (-) which provides you wrong answer. Do Small calculation also (roughly/not mentally) parallel or adjacent to the sum. (You are suggested to do it to gain marks, e.g., 12 6, 19 + 22, 90 65 etc. have been found wrong due to carefree attitude or undue pressure). Reading time must be used to make the right choice (Best 4) in section B . Don t spend more than 5 or 6 minutes on average for a sum. This will give you spare time for a quick review at last. If you get stuck, stop! and move on. Learn to analyze and understand every mistake. Mistakes are very cruel; they always pluck marks. Never cross or cancel an answer until you revise it at last. You may get sum marks for correct steps. If you attempt a sum twice or thrice, don t cancel anything. Let the examiner decide which one is correct. This is very useful tool if you are not confident about your answer. It is always advisable to answer only that number of questions as mentioned in the question paper. It will not only save your time for revising at last but also reflect productive work. Read the questions very carefully and be very specific in answering them. Don t find unwanted stuff or extra answers in overexcitement. You will never get marks for extra information. 4 E-mail: rakeshpraxis@gmail.com Instructions & Formulae, ICSE Math (X), By Rakesh Kushwaha (M.Sc.; B.Ed.): 09224190389/09930350748 It s very important to mention clearly the formulae or the theorems used to solve a sum. All working (rough work) must be done on the same sheet with rest of the answer. You get benefit of correct rough work in case you copy wrong digits in answer. Moreover, you may lose marks if you do the sum without working (i.e., Omission of Work). When the time allotted for writing the examination gets over, arrange your answer scripts in sequence, the first page must be at the top. Check your Index Number/Unique ID for your satisfaction. Fasten all the arranged answer scripts together at the left-hand top corner and hand over them to the invigilator unfolded. 5 E-mail: rakeshpraxis@gmail.com Instructions & Formulae, ICSE Math (X), By Rakesh Kushwaha (M.Sc.; B.Ed.): 09224190389/09930350748 COMMERCIAL ARITHMETIC GST [Goods and Services Tax]: Tax is calculated on sale price. [Sale Price = Marked Price - Discount] Intra-state: within same state [GST = CGST + SGST & CGST = SGST]. inter-state: from one state to another state (GST = IGST). A consumer (end user) can t claim the GST paid by him. Only a person or an organization registered with GST, can charge and collect GST on sale / transfer of goods & services. GST is very much similar to Sales Tax Amt. paid = C.P. (for him/her) + GST Net GST payable in inter-state transaction = Output GST Input GST Banking: RD Account: Qualifying sum (P) = I= P n(n+1) r 2 12 100 ;T= n(n+1) 2 n(n+1) 2 12 , where x = monthly deposit, n = no. of months years ; P = monthly deposit, n = no. of months, r = rate% M.V. = P + I ; Maturity Value = Total deposit (monthly deposit ) + Interest Sum (money) must be written up to two decimal places if the answer comes in decimal form. Remove decimal digits and round off if asked to round off, to nearest Rupees. Make a note that the time taken (value of n) is strictly in months or else be ready to get zero. Shares and Dividend: The total money invested by the company is called its capital stock. The capital stock is divided into a number of equal units. Each unit is a called a share. Nominal Value is also called Register Value, Printed Value, and Face Value. The FV of a share always remains the same, while its MV goes on changing. The part of the profit of a company which is distributed amongst the shareholders is known as dividend. If the MV = NV, the share is said to be at par. 6 E-mail: rakeshpraxis@gmail.com Instructions & Formulae, ICSE Math (X), By Rakesh Kushwaha (M.Sc.; B.Ed.): 09224190389/09930350748 If the MV >NV, the share is said to be at premium. If the MV <NV, the share is said to be at discount. Investment TotalDividend No. of shares = MV of each share = Dividend per share Annual Dividend/income = Rate of Dividend FV No. of shares ; AI = Return % = Annual Income Investment 100 100 % Rate of dividend% NV = Return % MV ; DN = PM or DF = PM New Dividend % increase in return on original investment = Original Investment 100 % % increase in return = New Dividend Old Dividend Old dividend 100 % Sum (money) must be written up to two decimal places. Don t leave the answer up to 1dp. Annual dividend or Annual income is always calculated on Face Value. Return% is also called profit%. ALGEBRA Linear Inequations: The signs >, < , and are called signs of inequality. On transferring + ve term becomes ve and vice versa. If each term is multiplied or divided by + ve number, the sign of inequality remains the same. The sign of inequality reverses: If each term is multiplied or divided by same negative number. If the sign of each term on both the sides of an inequation is changed. On taking reciprocals of both sides, in case both the sides are positive or negative. Always, write the solution set for the inequation, e.g.,{x : x 3, x N}, solution set = {1, 2, 3} To represent the solution on a number line: Put arrow sign on both the ends of the line and keep extra integers beyond the range. Use dark dots on the line for each element of N, W and Z. (WIN) For Q, R: mark range with solid circle (for or ), hollow circle (for < and >.) and means Intersection (only common elements of the sets). Common range for R or means Union (all elements of the sets without repetition). 2 Don t use decimal values e.g., 1.66 or 1.33 on number line, use fraction form 13 etc only. 7 E-mail: rakeshpraxis@gmail.com Instructions & Formulae, ICSE Math (X), By Rakesh Kushwaha (M.Sc.; B.Ed.): 09224190389/09930350748 1 Show the number on the number line which you are representing, e.g., 12. Solution must be written inside curly brackets { } only. 1 Show the fraction (1 2) on number line or use an arrow to indicate the written value. Remember that solution set for R and Q is always written in set-builder form only. Quadratic Equations: 1. Quadratic equation is an equation with one variable, the highest power of the variable is 2. 2. Some useful results: a) (a + b)2 = a2 + b2 + 2ab b) (a - b)2 = a2 + b2 - 2ab c) a 2 b2= (a + b) (a b) d) (a + b)2 - (a - b)2 = 4ab e) a3 + b3 =(a + b)(a2- ab+ b2) f) a3 - b3 =(a - b)(a2+ ab+ b2) g) (a + b)3 = a3 + b3 + 3ab(a + b) h) (a - b)3 = a3 - b3 - 3ab(a - b) i) (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca j) a3 + b3 + c3 3abc = (a + b + c) (a2 + b2 + c2 ab bc ca) 3. Steps for solving quadratic equation by factorization: (Product Sum Integers system) a. Clear all fractions and brackets if necessary. b. Bring it to the form ax2 + bx + c = 0 by transposing terms. c. Factorize the expression by splitting the middle term as a sum of product of a and c. 4. Discriminant (D) = 2 4 a. if D > 0, then the roots are real and unequal b. if D = 0, then the roots are real and equal c. if D < 0, then the roots are not real (imaginary). 5. The roots of the quadratic equation ax2 + bx + c = 0 ; a 0 can be obtained by using the formula: x= 2 4 2 6. Use powers and factorial page to find the square root of the number and round off only in the end as asked in the question. 8 E-mail: rakeshpraxis@gmail.com Instructions & Formulae, ICSE Math (X), By Rakesh Kushwaha (M.Sc.; B.Ed.): 09224190389/09930350748 7. For word problems, keep first the fraction with small denominator. e.g., smaller denominator) and in 2500 5 2500 2500 2500 +12 = 20 , here (x is ( x 5 is smaller denominator) = 12 8. In speed sums, convert minutes to hours because the speeds of train, plane or cars are in km/hr. There is mark for correct formula with correct substitution (with proper sign). Wrong formula (or wrong substitution) leads to zero. Use log table to find square root. Don t find square root by division method, it s risky and leads to wrong answer. There is difference between answers 2 decimal places and 2 significant figures. (2.21 is up to 2 d.p. and 2.2 is up to 2 s.f.) Ratio and Proportion: A ratio is a comparison of the sizes of two or more quantities of the same kind by division. Since ratio is a number, so it has no units. To find the ratio between two quantities, change them to the same units. To compare two ratios, convert them into like fractions. In the ratio, a : b, a is called antecedent and b is called consequent. Proportion- An equality of two ratios is called a proportion. Written as: a : b :: c : d or = Product of extreme terms = product of middle terms, if a, b, c, d are in proportion then ad = bc Continued Proportion- a : b :: b : c or a : b = b : c ; mean proportion (b) = 2 = Invertendo: If a: b = c: d, then b: a = d: c Alternendo : If a : b = c : d, then a : c = b : d Componendo : If a : b = c : d, then a + b : b = c + d : d Dividendo : If a : b = c : d, then a - b : b = c - d : d Componendo and Dividendo : = = k (say) , then a = bk and c = dk used for proving sums = = = ( ), then d = d, c = dk, b = dk2 , a = dk3 used for proving sums If a : b = c : d, then a + b : a b = c + d : c d 25 Write final answer in ratio form 25 : 16 . (16 ) Ratio must be written in simplified form.(15 25 = 3 5) In using C & D whichever term (variable with coefficient) is cancelled in numerator, doesn t get cancelled in denominator (+ or -) 9 E-mail: rakeshpraxis@gmail.com Instructions & Formulae, ICSE Math (X), By Rakesh Kushwaha (M.Sc.; B.Ed.): 09224190389/09930350748 In left or right hand side, if denominator is missing, use denominator as 1 before using C & D or else be prepared to get straight zero mark. Remainder and Factor Theorem: 1. If f(x) is a polynomial, which is divisible by (x a), a R, then the remainder is f (a). 2. If the remainder on dividing a polynomial f(x) by (x a), f (a) = 0, then (x - a) is a factor of f(x). 3. When f(x) is divided by (ax + b), then remainder is f( ), a 0 4. When f(x)is divided by (ax - b), then remainder is f( ), a 0 3 If (2x + 3) is a factor of f (x), then never skip to write f ( 2 ) = 0 If remainder is given, don t dare to replace it by zero. Write all factors in product form, e.g., (x + 1) (2x+3) (x 3). [(x + 1), (2x+3), (x 3) is wrong] Matrices: A rectangular arrangement of numbers, in the form of horizontal (rows) and vertical lines (columns) is called a matrix. Each number of a matrix is called its element. The elements of a matrix are enclosed in brackets [ ]. The order of a matrix = No. of rows No. of columns Row matrix: Only 1 row. E.g., [ Column matrix: Only 1 column. E.g., [ ] Square matrix: No. of rows = No. of columns. E.g., [ Rectangular matrix: No. of rows No. of columns. E.g., [ 0 Zero matrix: All elements are zero. E.g., [ 0 Diagonal matrix: A square matrix with all the elements zero except the elements on the leading diagonal. E.g., [0 0 0 0 ] ] ] 0 0 ] 0 0 0 0] Unit matrix (I): A diagonal matrix with all the elements on the leading diagonal = 1; I = [ 1 0 ] 0 1 10 E-mail: rakeshpraxis@gmail.com Instructions & Formulae, ICSE Math (X), By Rakesh Kushwaha (M.Sc.; B.Ed.): 09224190389/09930350748 1 1 2 5 ] then At = [2 2 3 6 5 2 3] 6 Transpose of a matrix: If A = [ Addition or subtraction of matrices is possible if and only if they are of the same order. Addition of two matrices: [ + + ]=[ ] + + Multiplication of matrix by a real number: i [ ]=[ ] ]+[ Multiplication of 2 matrices: [ ]x y [ ]b a , y = b ,order of the product matrix = ( x a) , Multiplication process: [ ] [ ( ) + ( r) ]=[ ( ) + ( r) ( ) + ( s) ] , run & fall ( q) + (d s) Don t use curly brackets for writing matrices. Matrix M2 is not the square of each element in the matrix M. Two matrices can t de divided by one another. [ ] [ ] is wrong. Working for multiplication is must. Remember RUN and FALL DOWN (Row with Column) Be very careful in doing multiplication, even one wrong element will lead to zero [M0, A0]. Be very careful in addition and subtraction. Easy addition or subtraction goes wrong if it is done carelessly. (It s surprising to find silly mistake like 3 + 3 = 9!) A.P. & G.P.: A.P. In an A.P., the difference between 2 consecutive terms is called its common difference (d). d = t 2 t1 or d = tn+1 tn tn = a + (n - 1) d tn = nth term, a = first term (t1), n = number of terms, d = common difference Sum of n terms of an A.P.: Sn = [2a + (n - 1) d] or 2 Sn = (a + l) 2 Sn = Sum of n terms, l = last term G.P. In a G.P., tn+1 = r tn or r= +1 t or r= 2 1 11 E-mail: rakeshpraxis@gmail.com Instructions & Formulae, ICSE Math (X), By Rakesh Kushwaha (M.Sc.; B.Ed.): 09224190389/09930350748 General term: tn = a r n-1 Sum of n terms of an A.P.: where a = first term, r is common ratio Sn =na Sn = when r = 1 (1 ) Sn = 1 ( 1) 1 when |r| < 1 when |r| > 1 A.P. has common difference and G.P. has common ratio between any two consecutive terms. Apply the correct formulae with correct values for nth term and sum of n terms. Use simultaneous equations to find a and d when two random terms of A.P. are given. Use method of proportion to find a and r when two random terms of G.P. are given COORDINATE GEOMETRY Reflection: Mx (x, -y) My (-x, y) Mo (-x, -y) X- axis means y = 0 Y- axis means x = 0 Any point that remains unaltered under a given transformation is called an invariant point. Mx=a(x, y) (2a - x, y ) Mx=b(x, y) (x, 2b - y) = = Write co-ordinates of the reflected point on the graph paper and name it specifically. Don t write co-ordinates in square brackets. 12 E-mail: rakeshpraxis@gmail.com Instructions & Formulae, ICSE Math (X), By Rakesh Kushwaha (M.Sc.; B.Ed.): 09224190389/09930350748 Learn to recognize the geometrical shapes: kite, arrow, arrow head, hexagon, octagon, trapezium, rhombus, etc. Also learn to find the areas of triangle and quadrilaterals. There is difference between arrow and arrow head. Section and Mid-Point Formulae: 1 2 + 2 1 1 2 + 2 1 Section formula: Coordinates of a point P(x, y) = ( Midpoint formula: Coordinates of the midpoint M(x, y) of a line segment = ( The co-ordinates of the centroid of a triangle G(x, y) = ( 1 + 2 , 1 + 2 + 3 3 1 + 2 , ) ; ratio = m1 : m2 1 + 2 1 + 2 + 3 3 , 2 1 + 2 2 ) ) Substitute the correct values with sign in correct formula. Check the sign of coordinates carefully while writing the final answer. Sign Mistake is very common error. Equation of a Line: Every straight line can be represented by a linear equation. Any point, which satisfies the equation of a line, lies on that line. Inclination of a line is the angle which the part of the line makes with x-axis. Inclination is positive in anti-clockwise direction and negative in clockwise direction. Slope or gradient of any inclined plane is ratio of vertical rise and horizontal distance. Slope of a line (m) = = tan Inclination of x-axis and every line parallel to it is 0 . Inclination of y-axis and every line parallel to it is 90 . The slope of a vertical line segment is not defined. Slope of a line which passes through any two points P(x1, y1) and Q(x2, y2) = ( 2 1 ). Slopes of two parallel lines are equal or m1 = m2. Product of the slopes of two perpendicular lines= - 1 or m1 m2 = -1. The equation of x-axis is y = 0 and the equation of y-axis is x = 0, 2 1 Equation of a line: The equation of a line can be written in various ways according to the given conditions: 13 E-mail: rakeshpraxis@gmail.com Instructions & Formulae, ICSE Math (X), By Rakesh Kushwaha (M.Sc.; B.Ed.): 09224190389/09930350748 o y = mx + c : o (y y1) = m (x x1) o = (Slope-intercept form : m = slope, c = y-intercept) :(Slope-point form: (x1, y1) = co-ordinates of the point) :[Two points form (x1, y1) ,(x2, y2)] simplified form GEOMETRY Similarity: Criteria for similarity 1. AA or AAA A drawn from vertex of a rt- d divides the into 2 similar , also to original triangle. BPT A line drawn || to any side of a divides other two sides proportionally. The areas of 2 similar are proportional to the square of their corresponding sides. Median divides a triangle into 2 of equal area. If have common vertex & are between same ||, ratio of their areas = ratio of bases. Scale factor = k, k = 2. SAS ; k2 = 3. SSS ; k3 = . Loci: The locus is the set of all points which satisfy the given geometrical condition. Locus of a point equidistant from 2 fixed points A and B is bisector of line segment joining them. Locus of a point equidistant from 2 intersecting lines AB and BC is angle bisector between the lines. Locus of a point A at a constant distance from a fixed point is circle. Locus of a point equidistant from a given line AB is a pair of lines parallel to the given line and at the given distance from it. For equilateral triangle, centroid = incentre = circumcentre = orthocentre 14 E-mail: rakeshpraxis@gmail.com Instructions & Formulae, ICSE Math (X), By Rakesh Kushwaha (M.Sc.; B.Ed.): 09224190389/09930350748 Circle: Angle at the centre = 2 angle on the circumference. Angles in the same segment are equal. Angle in a semicircle is a right angle. The opposite angles of a cyclic quadrilateral are supplementary. If the opposite angles of a quadrilateral are supplementary, then the quadrilateral is cyclic. Angle in the major segment is acute and in the minor segment is obtuse. Exterior angle of a cyclic quadrilateral = Interior opposite angle. In equal or same circle. If two arcs subtend equal angle at the centre, then they are equal. In equal circle, if two arcs are equal, then they subtend equal angle at the centre. In equal circle, if two chords are equal, they cut off equal arcs. In equal circle, if two arcs are equal, the chords of the arcs are also equal. The tangent at any point of a circle & the radius through this point are to each other. If two tangents are drawn to a circle from an exterior point, o The tangents are equal, o They subtend equal angle at the centre of the circle, o They are equally inclined to the line joining the point and the centre of the circle. If two chords of a circle intersect internally/externally, the product of their segments is equal. Angles in the alternate segment are equal. Tangent2 = product of the lengths of the segments of the chord. Incentre : Point of intersection of the angle bisectors. Cicumcentre : Point of intersection of the bisectors of the sides. Make habit of copying diagram to find the values of different angles. Mark 90 for angle in semi-circle; equal angles in same segment; central angle is twice the angle on circumference, opposite angles of cyclic; exterior angle of cyclic, etc. in the diagram. Reasons must be written wherever possible. (No reason leads to no mark.) Constructions: Standard Angles should be made by using scale and compass only not by using protractor. Two angle bisectors and a Perpendicular from centre to any side of regular polygon are required to draw incircle. Any two side perpendicular bisectors are required to draw circumcircle. 15 E-mail: rakeshpraxis@gmail.com Instructions & Formulae, ICSE Math (X), By Rakesh Kushwaha (M.Sc.; B.Ed.): 09224190389/09930350748 MENSURATION Surface Area and Volume: Solid Cuboid Shape Curved Surface Area Total Surface Area Volume Area of 4 walls 2(lb + bh + hl) l b h 6 side 2 side3 2(l + b) h Cube Area of 4 walls 4 side 2 Cylinder 2 rh 2 r(h + r) r2h Cylinder 2 h(R+r) 2 rh + 2 Rh + (R2 - r2)h (solid External + Internal 2 (R2 - r2) enclosed between) Cone rl r(l + r) 1 Sphere -- 4 r2 4 Hemi- 2 r2 3 r2 2 -- 4 ( 2 + r2) 4 3 3 3 r2h r3 r3 sphere Spherical Shell Hemi- External + Internal 2 (R2 + r2) 2 (R2 + r2) + Spherical (R2 r2) Shell = (3R2 + r2) 3 (R3 - r3) (solid enclosed between) 2 3 (R3 - r3) 16 E-mail: rakeshpraxis@gmail.com Instructions & Formulae, ICSE Math (X), By Rakesh Kushwaha (M.Sc.; B.Ed.): 09224190389/09930350748 Diagonal of a cuboid = + + Diagonal of a cube = a Slant height of a right circular cone, l = + Marks are given for correct answer only, so don t leave the answer in decimal form if asked to round off to integer. 3 1 Make habit of writing all possible ways, e.g., 2 = 1 2 = 1.5, if money then 1.50, let the examiner choose the correct one. There is no harm in writing decimal as well as fraction form. There are always exceptions, somewhere fraction is demanded, elsewhere decimal. Trigonometry: sin | cos | OR ; | sec | SOH CAH TOA or OSH ACH OTA Opp side Trigonometric ratios of standard angles Hypotenuse Adj side sin 0 30 0 =0 4 cos tan 1 45 1 1 4 2 = 3 2 1 0 60 2 1 4 2 = 3 = 4 1 90 3 2 4 =1 4 0 2 1 2 1 3 n.d. 3 = = = = , , = , = , = = 17 E-mail: rakeshpraxis@gmail.com Instructions & Formulae, ICSE Math (X), By Rakesh Kushwaha (M.Sc.; B.Ed.): 09224190389/09930350748 2 + 2 = 1 (mutual understanding) 2 - 2 = 1 or 1 + 2 = 2 (cosec is big brother) 2 - 2 = 1 or 1 + 2 = 2 (sec is big brother) 1 1 3 1 sin 3A 3 sinA sin3A , if A = 30 , then sin 3A = sin 90 = 1, 3 sinA= 3 2 , sin3A = (2) = 8. In proving sums, don t assume LHS = RHS in the beginning, Solve Separately. Transfer of digits from one side to other is strictly prohibited. Don t do direct conversion of trigonometric ratios of complementary angles. Show the steps. Write reasons for the conversion without fail. Don t forget to write the identity whichever is used. Height and Distance: angle > angle Angle of depression Angle of elevation Bigger angle is always inside and smaller is outside. Height is vertical length and distance is horizontal length. tan = Remember the values of tan 30 , 45 and 60 . It s very easy to remember. Just remember (1 45 , 1 3 = = 1 3 . 30 , 3 60 ) Diagram is must in Heights and Distances. Mark angles correctly. Outside angle is always smaller. Angle of elevation decreases when distance increases. Make habit of writing the standard values of as per given in the question. E.g., you get 1 mark just for writing tan 45 = 1. 18 E-mail: rakeshpraxis@gmail.com Instructions & Formulae, ICSE Math (X), By Rakesh Kushwaha (M.Sc.; B.Ed.): 09224190389/09930350748 You may easily get 3 marks in heights and distances sums: 1 mark for correct diagram, 1mark for the value of tan 30 , 45 or 60 and 1 mark for writing correct formula with correct substitution. There is only one mark for big calculation. So don t omit this important topic. Use log table to write the values of angles (22 , 32 , 48 etc) other than standard angles in finding heights and distances. Keep unknown side as numerator while using , to avoid division by big decimal number. Multiplication is always easier. Use complementary angle if required. STATISTICS Statistics: Arithmetic mean of non-tabulated data: = Arithmetic mean of tabulated data(Direct Method): = Arithmetic mean by Short-cut Method: = Arithmetic mean by Step-deviation Method: = ( ) + A ; i= class width , t = For raw data, if n is odd, Median = ( ; x = mid value (C.I.) + A ; A = assumed mean , d = x A +1 2 ) term 2 2 (( ) term + ( + 1) term) For raw data, if n is even, Median = 2 From ogive, Median = ( 2) term. From ogive Lower quartile, Q1 = ( 4) term From ogive Upper quartile, Q3 = ( 4 ) term Inter Quartile Range, IQR = Q3 Q1 Semi Inter Quartile Range = Mode is the variate which has the maximum frequency. The class with maximum frequency is called the modal class. (e.g., 20 30) To estimate mode from histogram: draw two straight lines from the corners of the rectangles 3 3 1 2 on either sides of the highest rectangle to the opposite corners of the highest rectangle. 19 E-mail: rakeshpraxis@gmail.com Instructions & Formulae, ICSE Math (X), By Rakesh Kushwaha (M.Sc.; B.Ed.): 09224190389/09930350748 Through the point of intersection of the two straight lines, draw a vertical line to meet the xaxis at the point M (say). The variate at the point M is the required mode. To find median of grouped data, draw ogive (cumulative frequency curve). For finding mean of raw data, be very careful in counting number of data (n). If assumed mean is given in the question itself, don t be silly to change it. Use kink (if required), show scale and label the axes to get marks. Don t dare to change the scale if already given in the question. For finding median, Q1, Q3 or number of desired range (below or above) using ogive, perpendiculars are must. No mark is allotted without perpendiculars (lines on graph). Write the answer in decimal form not in fraction. In Ogive, it should be the smooth curve without halt. Probability: Probability is a measure of uncertainty. An Experiment is an action which results in some (well-defined) outcomes. Sample space is the collection of all possible outcomes of an experiment. n(S) An Event is a subset of the sample space associated with a random experiment. n(E) An Event occurs when the outcome of an experiment satisfies the condition mentioned in the event. The outcomes which ensure the occurrence of an event are called favorable outcomes to that event. The probability of an event E, written as P(E), is defined as P (E) = P(E) = The value of probability is always between 0 and 1. The probability of sure (certain) event is 1. The probability of an impossible event is 0. An elementary event is an event which has one (favorable) outcome from the sample space. A Compound event is an event which has more than one outcome from the sample space. ( ) ( ) 20 E-mail: rakeshpraxis@gmail.com Instructions & Formulae, ICSE Math (X), By Rakesh Kushwaha (M.Sc.; B.Ed.): 09224190389/09930350748 . If E is an event, then the event not E is complementary event of E and denoted by E 0 ( ) 1 ) = 1 P(E) + P(E In a pack (deck) of playing cards, there are 52 cards which are divided into 4 suits of 13 cards each spades ( ), hearts ( ), diamonds ( ) and clubs ( ). Spades and clubs are black in colour, while hearts and diamonds are of red colour. The cards in each suit are ace, king, queen, jack, 10, 9, 8, 7, 6, 5, 4, 3, 2. Kings, queens and jacks are called face (picture/court) cards. The cards bearing number 10, 9, 8, 7, 6, 5, 4, 3, 2 are called numbered cards. Thus a pack of playing cards has 4 aces, 12 face cards and 36 numbered cards. The aces together with face cards (= 16) are called cards of honour. When a coin is tossed, it may show head (H) up or tail (T) up. Thus the outcomes are: {H, T}. When two coins are tossed simultaneously, then the outcomes are: {HH, HT, TH, TT}. [n(S) = 2n] When a die is thrown once the outcomes are: {1, 2, 3, 4, 5, 6}. When two dice are thrown simultaneously, then the outcomes are: {(1, 1), (1, 2) .(6, 6)}. 9 Write the answer in simplified form. (30 = 3 10 [n(S) = 6n] ) Don t forget to write sample space. At times, there is special 1 mark for it, which is ignored by most of the students. Please share the document with your contacts. Your one click may change someone s life. Feel free to share your feedback and comments through my e-mail. Watch my videos for better understanding! Type: Rakesh Kushwaha Math Thanks!! 21 E-mail: rakeshpraxis@gmail.com

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