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PREFACE It is rightly said that “Change is the only constant.” In the past few months, the world has changed drastically and adapting to these changes is vital for our progress. These are unprecedented times which have called for unprecedented measures. School is being conducted online and classrooms have shifted to computer screens. To ensure that real learning doesn’t stop even in this age of virtual classrooms; Oswaal Books has updated all its products for 2021 examinations. We have upgraded our books to supplement students’ need for extensive practice and in-depth understanding of their subjects. Our Question Banks have been redesigned keeping in mind the new & reduced syllabus and new challenges that the students are facing today. All chapters in Oswaal Question Banks are arranged 'TOPICWISE' where each topic is explained in detail and covers all typologies of Questions specified by CISCE for ICSE examinations. These books also include well-labelled and high-quality figures/diagrams for faster learning. Another unique character of this book is that it ensures in-depth study and not merely superficial learning of ICSE topics. The remarkable feature is the inclusion of Answering Tips and Examiner's Comments at the end of every question, which will empower the students to comprehend their mistakes and rectify them. We at Oswaal Books never try to teach our readers. We on the other hand, provide them the conditions in which they can learn and train their mind to think! After all Education is what remains after one has forgotten what one learned in school.
Some of the key benefits of studying from Oswaal Solved Papers are:
Strictly based on the latest CISCE Curriculum issued for Academic Year 2020-2021
Previous Years' Board Questions for in depth study
Handwritten Toppers' Answers
Answering Tips and Examiner’s Comments
Answers strictly as per the ICSE Marking Scheme
All Typology of Questions included for exam-oriented study
Revision Notes for comprehensive study
‘Mind Maps’ in each chapter for making learning simple.
We have taken due care in developing this book. There have been a lot of people who have helped us in our journey. We would like to offer heartfelt gratitude to all of them- our authors, editors, reviewers and specially students who regularly send us suggestions which helps in continuous improvement of this book
We, at Oswaal Books, wish you all Happy Learning and a successful 2020-21 !!
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Os wa al Bo ok s are rea lly mesmerizing. The features like Toppers' Answer, Mind Maps, Chapterwise and Topicw ise segregation etc. make it wo rth every penny Aryan Aggarwal The Shri Ram School, New Delhi
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The book as it C h ap te rw is states is arranged e an d To p ic There are solved paper w is e . s from 2014-2019 an teacher's co d also there are m answer shee ments, Topper's ts, tips, min d maps and commo n students. Mar errors made by ki provided w ng scheme is also hich is an amazing feature too . explained pre The answers are tt a perfect boo y well. Overall, it's k for preparat ions! Jaffery Poll ock St. Paul's S chool, Agra
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CONTENTS l
Positive Affirmations
l
Topics/Concepts Found Difficult or Confusing by Candidates
l l
Latest ICSE Reduced Syllabus issued by CISCE for Academic Year 2020 –21
ICSE Solved Paper 2020 with Topper's Answer Issued by CISCE
l
ICSE Specimen Paper 2020 Issued by CISCE
l
9 - 12
13 - 14
Suggestions for Students
l l
8 - 8
15 - 16 17 - 31
ICSE Solved Paper 2020 Issued by CISCE
32 - 43 44 - 62
ICSE Solved Paper, 2019 with Topper's Answer & Board
Marking Scheme (Qualitative Analysis) Issued by CISCE
63 - 98
l
Mind Maps
99 - 119
1. Goods and Services Tax (GST) 2. Banking
3. Linear Inequations
4. Quadratic Equations
Topic 1. Solution of Quadratic
Topic 3. Word Problems
Topic 1. Ratio
Topic 2. Proportions
7. Matrices
8. Arithmetic Progression
th
Topic 1. To find n Term of
27 - 36
37 - 41
42 - 53
54 - 70
Progression th
Topic 1. To find n term of the
Geometric Progression
Progression
Formula
Topic 1. Equation of Straight Lines
Perpendicular lines
13. Similarity
111 - 119
15. Tangents and Its Properties
129 - 136
17. Mensuration
143 - 151
14. Circles
16. Constructions
Topic 1. Surface areas and Volumes
One Type of Metallic Solid
into Another
Topic 1. Mean, Median and Mode
137 - 142
152 - 157
158 - 166
167 - 194
Topic 2. Calculation of Median and Mode through Graphs
21. Probability
195 - 200 qq
Scan the QR Code to download the fully Board Solved Paper 2020 English -1
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120 - 128
Topic 2. Problem Involving Converting
20. Statistics
83 - 90
97 - 110
Topic 2. Parallel and
19. Heights and Distances 71 - 82
91 - 96
12. Equation of Straight Lines
18. Trigonometric Identities
Topic 2. Sum of a Geometric
10. Reflection
11. Section And Mid-Point
Topic 2. Sum of an Arithmetic
16 - 26
the Arithmetic Progression
9. Geometric Progression
10 - 15
Topic 2. Nature of Roots
6. Factorisation of Polynomials
5 - 9
Equations
5. Ratio and Proportion
1 - 4
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Reduced Syllabus for the Academic year (2020-21)
Mathematics CLASS 10
There will be one paper of two and a half 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: Will consist of compulsory short answer questions. Section II: Candidates will be required to answer four out of seven questions. 1. Commercial Mathematics (i) Goods and Services Tax (GST) Computation of tax including problems involving discounts, list-price, profit, loss, basic/cost price including inverse cases. Candidates are also expected to find price paid by the consumer after paying State Goods and Service Tax (SGST) and Central Goods and Service Tax (CGST) - the different rates as in vogue on different types of items will be provided. Problems based on corresponding inverse cases are also included. (ii) Banking Recurring Deposit Accounts : computation of interest and maturity value using the formula: n( n + 1) r × I= P 2 × 12 100 MV = P × n + I 2. Algebra (i) Linear Inequations Linear Inequations in one unknown for x Î N, W, Z, R. Solving l Algebraically and writing the solution in set notation form. l Representation of solution on the number line. (ii) Quadratic Equations in one variable (a) Nature of roots l Two distinct real roots if b2 – 4ac > 0 l Two equal real roots if b2 – 4ac = 0 l No real roots if b2 – 4ac < 0 (b) Solving Quadratic equations by: l Factorisation l Using Formula. (c) Solving simple quadratic equation problems. (iii) Ratio and Proportion (a) Proportion, Continued proportion, mean proportion. (b) Componendo, dividendo, alternendo, invertendo properties and their combinations. (iv) Factorisation of polynomials: (a) Factor Theorem.
(b) Remainder Theorem. (c) Factorising a polynomial completely after obtaining one factor by factor theorem. Note : f (x) not to exceed degree 3. (v) Matrices (a) Order of a matrix. Row and column matrices. (b) Compatibility for addition and multiplication. (c) Null and Identity matrices. (d) Addition and subtraction of 2×2 matrices. (e) Multiplication of a 2×2 matrix by l a non-zero rational number l a matrix. (vi) Arithmetic Progression l Finding their General term. l Finding Sum of their first ‘n’ terms. (vii) Co-ordinate Geometry (a) Reflection (i) Reflection of a point in a line : x=0, y =0, x= a, y=a, the origin. (ii) Reflection of a point in the origin. (iii) Invariant points. (b) Co-ordinates expressed as (x,y), Section formula, Midpoint formula, Concept of slope, equation of a line, Various forms of straight lines. (i) Section and Mid-point formula (Internal section only, co-ordinates of the centroid of a triangle included). (ii) Equation of a line: l Slope –intercept form y = mx + c l Two- point form (y–y1) = m(x–x1) Geometric understanding of ‘m’ as slope/ gradient tanq where q is the angle the line makes with the positive direction of the x axis. Geometric understanding of ‘c’ as the y-intercept/the ordinate of the point where the line intercepts the y axis/ the point on the line where x=0. l Conditions for two lines to be parallel or perpendicular. Simple applications of all the above. 3. Geometry (a) Similarity Similarity, conditions of similar triangles. (i) Comparison with congruency, keyword being proportionality. (ii) Three conditions: SSS, SAS, AA. Simple applications (proof not included). (iii) Applications of Basic Proportionality Theorem. (iv) Areas of similar triangles are proportional to the squares of corresponding sides.
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...contd. (b) Circles (i) Angle Properties l The angle that an arc of a circle subtends at the centre is double that which it subtends at any point on the remaining part of the circle. l Angles in the same segment of a circle are equal (without proof). l Angle in a semi-circle is a right angle. (ii) Cyclic Properties: l Opposite angles of a cyclic quadrilateral are supplementary. l The exterior angle of a cyclic quadrilateral is equal to the opposite interior angle (without proof). (iii) Tangent and Secant Properties: l The tangent at any point of a circle and the radius through the point are perpendicular to each other. l From any point outside a circle, two tangents can be drawn and they are equal in length. l If a line touches a circle and from the point of contact, a chord is drawn, the angles between the tangent and the chord are respectively equal to the angles in the corresponding alternate segments. Note : Proofs of the theorems given above are to be taught unless specified otherwise. (iv) Constructions (a) Construction of tangents to a circle from an external point. (b) Circumscribing and inscribing a circle on a triangle and a regular hexagon. 4. Mensuration Area and volume of solids – Cylinder, Cone and Sphere. Three-dimensional solids - right circular cylinder, right circular cone and sphere: Area (total surface and curved surface) and Volume. Direct application problems including cost, Inner and Outer volume and melting and recasting method to find the volume or surface area of a new solid. Combination of solids included. Note : Problems on Frustum are not included. 5. Trigonometry (a) Using Identities to solve/prove simple algebraic trigonometric expressions sin2 A + cos2 A = 1 2 2 1 + tan A = sec A 2 2 1+cot A = cosec A; 0 £ A £ 90°
(b) Heights and distances: Solving 2-D problems involving angles of elevation and depression using trigonometric tables. Note: Cases involving more than two right angled triangles excluded. 6. Statistics Statistics – basic concepts, Mean, Median, Mode. Histograms and Ogive. (a) Computation of : l Measures of Central Tendency : Mean, median, mode for raw and arrayed data. Mean*, median class and modal class for grouped data. (both continuous and discontinuous). * Mean by all 3 methods included: Direct :
åfx åf
åfd Short-cut : A + åf where d = x – A
Step-deviation: A +
åft × i where t = x − A åf i
(b) Graphical Representation. Histograms and Less than ogive. l Finding the mode from the histogram, the upper quartile, lower Quartile and median etc. from the ogive. l Calculation of inter Quartile range. 7. Probability l Random experiments l Sample space l Events l Definition of probability l Simple problems on single events Note: SI units, signs, symbols and abbreviations (1) Agreed conventions (a) Units may be written in full or using the agreed symbols, but no other abbreviation may be used. (b) The letter 's' is never added to symbols to indicate the plural form. (c) A full stop is not written after symbols for units unless it occurs at the end of a sentence. (d) When unit symbols are combined as a quotient, e.g. metre per second, it is recommended that they be written as m/s, -1 or as m s . (e) Three decimal signs are in common international use: the full point, the midpoint and the comma. Since the full point is sometimes used for multiplication and the comma for spacing digits in large numbers, it is recommended that the mid-point be used for decimals.
( 10 )
...contd. (2) Names and symbols In general Implies that Þ is logically equivalent to Û Identically equal to º is approximately equal to >> In set language Belongs to Î Is equivalent to « union È universal set x natural (counting) N numbers integers Z In measures Kilometre Centimetre Kilogram Litre Square kilometre Square centimetre Cubic metre Kilometres per hour
does not belong to Ï is not equivalent to « / intersection Ç is contained in Ì the empty set Æ whole numbers W real numbers R
km Metre cm Millimetre kg Gram 1 Centilitre km2 Square meter cm2 Hectare m3 Cubic centimetre km/h Metres per second
m mm g cL m2 ha cm3 m/s
INTERNAL ASSESSMENT The minimum number of assignments : Two assignments as prescribed by the teacher. Suggested Assignments l Comparative newspaper coverage of different items. l Survey of various types of Bank accounts, rates of interest offered. l Planning a home budget. l Conduct a survey in your locality to study the mode of conveyance / Price of various essential commodities/ favourite sports. Represent the data using a bar graph / histogram and estimate the mode. l To use a newspaper to study and report on shares and dividends. l Set up a dropper with ink in it vertical at a height say 20 cm above a horizontally placed sheet of plain paper. Release one ink drop; observe the pattern, if any, on the paper. Vary the vertical distance and repeat. Discover any pattern of relationship between the vertical height and the ink drop observed. l You are provided (or you construct a model as shown) - three vertical sticks (size of a pencil) stuck to a horizontal board. You should also have discs of varying sizes with holes (like a doughnut). Start with one disc; place it on (in) stick A. Transfer it to another stick (B or C); this is one move (m). Now try with two discs placed in A such that the large disc is below and the smaller disc is above (number of discs = n=2 now). Now transfer them one at a time in B or C to obtain similar situation (larger disc below). How many moves? Try with more discs (n = 1, 2, 3, etc.) and generalise.
A
B
C
l The board has some holes to hold marbles, red on one side and blue on the other. Start with one pair. Interchange the positions by making one move at a time. A marble can jump over another to fill the hole behind. The move (m) equal 3. Try with 2 (n=2) and more. Find the relationship between n and m.
Red Blue l Take a square sheet of paper of side 10 cm. Four small squares are to be cut from the corners of the square sheet and then the paper folded at the cuts to form an open box. What should be the size of the squares cut so that the volume of the open box is maximum? l Take an open box, four sets of marbles (ensuring that marbles in each set are of the same size) and some water. By placing the marbles and water in the box, attempt to answer the question: do larger marbles or smaller marbles occupy more volume in a given space? l An eccentric artist says that the best paintings have the same area as their perimeter (numerically). Let us not argue whether such sizes increases the viewer’s appreciation, but only try and find what sides (in integers only) a rectangle must have if its area and perimeter are to be equal (note: there are only two such rectangles). l Find by construction the centre of a circle, using only a 60-30 setsquare and a pencil. l Various types of “cryptarithm”.
EVALUATION The assignments/project work are to be evaluated by the subject teacher and by an External Examiner. (The External Examiner may be a teacher nominated by the Head of the school, who could be from the faculty, but not teaching the subject in the section/class. For example, a teacher of Mathematics of Class VIII may be deputed to be an External Examiner for Class X, Mathematics projects.) The Internal Examiner and the External Examiner will assess the assignments independently. Award of marks (20 Marks) Subject Teacher (Internal Examiner) : 10 marks External Examiner : 10 marks The total marks obtained out of 20 are to be sent to the Council by the Head of the school. The Head of the school will be responsible for the online entry of marks on the Council's CAREERS portal by the due date.
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...contd. INTERNAL ASSESSMENT IN MATHEMATICS- GUIDELINES FOR MARKING WITH GRADES Criteria
Preparation
Concepts
Computation
Presentation
Understanding
Grade I
Exhibits and selects a well-defined problem. Appropriate of techniques.
Admirable use of mathematical concepts and methods and exhibits competency in using extensive range of mathematical techniques.
Careful and accurate work with appropriate computation, construction and measurement with correct units.
Presents well stated conclusions; mathematical language, symbols, conventions, tables, diagrams, graphs, etc.
Shows strong personal contribution; demonstrate knowledge and understanding of assignment and can apply the same in different situations.
4 marks for each criterion
Grade II
Exhibits and selects routine approach. Fairly good techniques.
Appropriate use of mathematical concepts and methods and shows adequate competency in using limited range of techniques.
Commits negligible errors in computation, construction and measurement.
Some statements of conclusion; uses appropriate math language, symbols, conventions, tables, diagrams, graphs, etc.
Neat with average amount of help; assignment shows learning of mathematics with a limited ability to use it.
3 marks for each criterion
Grade III Exhibits and selects trivial problems. Satisfactory techniques.
Uses appropriate mathematical concepts and shows competency in using limited range of techniques.
Commits a few errors in computation, construction and measurement.
Assignment is presentable thought it is disorganized in some places.
Lack of ability to conclude without help; shows some learning of mathematics with a limited ability to use it.
2 marks for each criterion
Grade IV Exhibits and selects an insignificant problem. Uses some unsuitable techniques.
Uses inappropriate mathematical concepts for the assignment.
Commits many mistakes in computation and measurement.
Presentation made is somewhat disorganized and untidy.
Lack of ability to conclude even with considerable help; assignment contributes to mathematical learning to a certain extent.
1 mark for each criterion
Grade V
Not able to use mathematical concepts.
Inaccurate computation, construction and measurement.
Presentation made is completely disorganized untidy and poor.
Assignment does not contribute to mathematical learning and lacks practical applicability.
0 marks
Exhibits and selects a completely irrelevant problem. Uses unsuitable techniques.
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Marks
Topics Found Confusing / Difficult By Candidates Examination Paper 2019 l Linear inequation and representation. l Shares and Dividend. l Rationalization. l Algebraic division of polynomials by binomial and simplification. l Rounding off answers to the desired form. l Significant figures. l Arithmetic and Geometrical Progression. l Section formula and points on x and y axes. l Properties of Ratio and Proportion. l Geometry constructions. l Trigonometry. l Matrix- scalar multiplication and multiplication of 2 Matrices. l Graphs – Axes and choice of scale. l Size Transformation. l Mensuration formula. l Circle theorems. l Statistics- Calculation of Central tendency of non-grouped frequency l Calculation of Mean by Short-cut Method. l Remainder-Factor theorem. l
distribution.
Solving Geometry problems on circles and similar triangles using the appropriate property
Examination Paper 2018 l Arithmetic Progression and Geometric Progression. l Shares and Dividend. l Geometry based problems using properties of circle and similar triangles. l Geometrical Constructions. l Section formula. l Trigonometrical Identities and Heights and Distances. l Properties of Ratio and Proportion. l Approximation: to given significant figures or to nearest whole number. l Factorization in various problems. l Maps and models. l Maturity value and interest in Banking sums. l Use of Remainder and Factor Theorem. l Forming correct equation in Quadratic word problem. l
Inequation solving and representing solution on number line.
Examination Paper 2017 l l l l l
Value Added Tax (VAT). Shares and Dividend. Use of Remainder and Factor Theorem. Inequation solving and representing solution. Geometry : Constructions and solving problems using properties of circle and similar triangles. l Application of Circle Theorems. l Finding out minimum balance in a month of a Savings Bank account. l Coordinate geometry: Section formula. l Trigonometry, complementary angles and Heights and Distances. ( 13 )
...contd Topics Found Confusing / Difficult By Candidates l
Properties of Ratio and Proportion. Approximation to given significant figures or to nearest whole number. l Statistics: finding mean by using step-deviation Method. l Representing solutions on number line. l Probability. l Problems based on graph. Choosing correct axis and scale. l
Examination Paper 2016 l VAT. l l l l l l l
Shares and Dividend. Geometry solving problems using properties of circle and similar triangles. Geometry Constructions. Coordinate geometry: Section formula. Trigonometry, complementary angles and Heights and Distances. Properties of Ratio and Proportion. Approximation: to given significant figures or to nearest whole number.
Examination Paper 2015 l
Value Added Tax (VAT). l Compound Interest inverse problems. l Trigonometry. l Similarity. l Rounding off final result e.g. significant figures. l Theorems on properties of circle. l Properties of proportion. l Constructing a circle about a constructed Hexagon. l Calculation of mean. l Coordinate geometry, Section formula and identifying points on x or y axis. l Quadratic equation problem. l Inequation: writing solution and representation on number line.
Examination Paper 2014 l
Value Added Tax (VAT). Compound Interest inverse problems. l Trigonometry. l Similarity. l Rounding off final result e.g. significant figures. l Theorems on properties of circle. l Properties of proportion. l Construction of incircle. l Short cut method of calculation of mean. l Coordinate geometry, Section formula and identifying points on x or y axis. l Conditions of collinear. l Quadratic equation problem. l
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Suggestions for Students ð Prove trigonometrical identities by starting from left hand side and then proving it equal to right hand side or vice a versa. ð Write reasons while solving Geometry problems. ð Use mathematical tables to find square root, cube root of a number etc. to save time. ð Recheck the calculation to avoid calculation error. ð Be careful in conversion of units. ð Utilise reading time must be used judiciously to make the right choice of questions. ð Practise rounding off answers correct to required number of decimal places. ð Practise significant figures and express the answer up to asked digits in the question. ð Show clearly all steps of working including rough work on the same answer sheet. ð Show clearly all traces of construction while working with Geometrical Constructions. ð Practise thoroughly all formulae related to Coordinate geometry, mensuration etc. ð Utilize the reading time to make the right choice of question and to be thorough with the given data. ð Practise approximation like significant figures, rounding off to a certain place of decimal or nearest whole number extensively. ð Practise plotting graphs using the correct scale, reading values/interpretation from graph. ð Clearly show the traces of geometrical constructions. ð Give reasons while working out Geometry based problems. ð Practise trigonometric basic identities comprehensively. ð Practise the concepts of Coordinate Geometry- section formula, slope of a parallel/ perpendicular line, equation of a line. ð Concepts of Arithmetic Progression(AP) and Geometric Progression(GP) must be done meticulously. ð Use Mathematical tables to find square roots and ratios other than that of standard angles. ð All steps of working including rough work must be clearly shown on same answer page. ð Study the entire syllabus completely and revise from time to time. ð Revise the concepts learnt in Class IX and integrated with the Class X syllabus. ð Develop logical and reasoning skills to have a clear understanding of the concepts. ð Reading time must be utilised to make the right choice of question and to be thorough with the given data. ð Round off the answers where asked. ð Choose the correct scale while drawing graphs and special care must be taken while marking the axes and plotting points. ð While solving geometry problems reasons must be given. ð Show clearly all traces of constructions. ð To avoid calculation errors, all calculations must be rechecked. ð Make wise choices from the options available in the question paper and manage time wisely. ð Study the entire syllabus thoroughly and revise from time to time. Concepts of Class IX must be revised and integrated with the Class X syllabus. ð Develop logical and reasoning skills to have a clear understanding. ð Revise all topics and formulae involved and make a chapter wise or topic-wise list of these. ( 15 )