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ISC Class XI Board Exam 2025 : Mathematics

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Sayan Chandra
St. Stephen's School, Dum Dum, Kolkata (Calcutta)
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MATHEMATICS (860) CLASS XI There will be two papers in the subject: Paper I : Theory (3 hours) 80 marks Paper II: Project Work 20 marks PAPER I (THEORY) 80 Marks The syllabus is divided into three sections A, B and C. Section A is compulsory for all candidates. Candidates will have a choice of attempting questions from EITHER Section B OR Section C. DISTRIBUTION OF MARKS FOR THE THEORY PAPER S.No. UNIT TOTAL WEIGHTAGE SECTION A: 65 Marks 1. Sets and Functions 20 Marks 2. Algebra 24 Marks 3. Coordinate Geometry 8 Marks 4. Calculus 6 Marks 5. Statistics & Probability 7 Marks SECTION B: 15 marks 6. Conic Section 7 Marks 7. Introduction to Three-Dimensional Geometry 5 Marks 8. Mathematical Reasoning 3 Marks OR SECTION C: 15 Marks 9. Statistics 5 Marks 10. Correlation Analysis 4 Marks 11. Index Numbers & Moving Averages 6 Marks TOTAL 1 80 Marks SECTION A (iii) Trigonometry 1. Sets and Functions Positive and negative angles. Measuring angles in radians and in degrees and conversion from one measure to another. Definition of trigonometric functions with the help of unit circle. Truth of the identity sin2x+cos2x=1, for all x. Signs of trigonometric functions. Domain and range of trignometric functions and their graphs. Expressing sin (x y) and cos (x y) in terms of sinx, siny, cosx & cosy and their simple applications. Deducing the identities like the following: (i) Sets Sets and their representations. Empty set. Finite and Infinite sets. Equal sets. Subsets. Subsets of a set of real numbers especially intervals (with notations). Power set. Universal set. Venn diagrams. Union and Intersection of sets. Difference of sets. Complement of a set. Properties of Complement of Sets. (ii) Relations & Functions Ordered pairs, Cartesian product of sets. Number of elements in the cartesian product of two finite sets. Cartesian product of the set of reals with itself (upto R x R x R). Definition of relation, pictorial diagrams, domain, co-domain and range of a relation. Function as a special type of relation. Function as a type of mapping, domain, codomain and range of a function. Real valued functions, domain and range of these functions, constant, identity, polynomial, rational, modulus, signum, exponential, logarithmic and greatest integer functions. Sum, difference, product and quotient of functions. Sets: Self-explanatory. Basic concepts Functions of Relations tan (x y) = cot(x y)= sin tan x tan y , 1 tan x tan y cot x cot y 1 coty cotx sin =2sin 1 1 ( )cos ( ) 2 2 cos + cos = 2 cos 1 1 ( + ) cos ( - ) 2 2 cos - cos = - 2sin 1 ( 2 + ) sin 1 ( - ) 2 Identities related to sin 2x, cos2x, tan 2x, sin3x, cos3x and tan3x. and Angles and Arc lengths - Ordered pairs, sets of ordered pairs. - - Cartesian Product (Cross) of two sets, cardinal number of a cross product. - The relation S = r where is in radians. Relation between radians and degree. - Definition of trigonometric functions with the help of unit circle. - Truth of the identity sin2x+cos2x=1 Relations as: - an association between two sets. - a subset of a Cross Product. - Domain, Range and Co-domain of a Relation. - Functions: - As special relations, concept of writing y is a function of x as y = f(x). - Angles: Convention of sign of angles. Magnitude of an angle: Measures of Angles; Circular measure. NOTE: Questions on the area of a sector of a circle are required to be covered. Trigonometric Functions - Relationship between trigonometric functions. - Domain and range of a function. 2 Proving simple identities. - Signs of trigonometric functions. Domain and range of the trigonometric functions. Trigonometric functions of all angles. Periods of trigonometric functions. Graphs of simple trigonometric functions (only sketches). Equations reducible to quadratic form. Nature of roots b b 2 4ac x= 2a In solving quadratic equations. NOTE: Graphs of sin x, cos x, tan x, sec x, cosec x and cot x are to be included. Compound and multiple angles - Addition and subtraction formula: sin(A B); cos(A B); tan(A B); tan(A + B + C) etc., Double angle, triple angle, half angle and one third angle formula as special cases. - Sum and differences as products sin C + sin D= C+D C D 2sin cos , etc. 2 2 - Use of the formula: Product and sum of roots. Roots are rational, irrational, equal, reciprocal, one square of the other. Complex roots. Framing quadratic equations with given roots. NOTE: Questions on equations having common roots are to be covered. Quadratic Functions. Given , as roots then find the equation whose roots are of the form 3 , 3 , etc. Product to sum or difference i.e. 2sinAcosB = sin (A + B) + sin (A B) etc. Real roots Case I: a > 0 Complex roots Equal roots 2. Algebra Case II: a < 0 (i) Complex Numbers Real roots Complex roots, Equal roots Introduction of complex numbers and their representation, Algebraic properties of complex numbers. Argand plane and polar representation of complex numbers. Square root of a complex number. Cube root of unity. Where a is the coefficient of x2 in the equations of the form ax2 + bx + c = 0. - Conjugate, modulus and argument of complex numbers and their properties. - Sum, difference, product and quotient of two complex numbers additive and multiplicative inverse of a complex number. - Square root of a complex number. - Cube roots of unity and their properties. (ii) Quadratic Equations Statement of Fundamental Theorem of Algebra, solution of quadratic equations (with real coefficients). Sign of quadratic Sign when the roots are real and when they are complex. Inequalities - Linear Inequalities Algebraic solutions of linear inequalities in one variable and their representation on the number line. Self-explanatory. - Quadratic Inequalities Using method of intervals for solving problems of the type: 3 + -3 x2 + x 6 0 Significance of Pascal s triangle. - Binomial theorem (proof using induction) for positive integral powers, + 2 i.e. (x + y )n = A perfect square e.g. x 2 6 x + 9 0 . - Inequalities involving rational expression of type (v) Sequence and Series Sequence and Series. Arithmetic Progression (A.P.). Arithmetic Mean (A.M.) Geometric Progression (G.P.), general term of a G.P., sum of first n terms of a G.P., infinite G.P. and its sum, geometric mean (G.M.), relation between A.M. and G.M. Formulae for the following special sums n, n 2 , n 3 . (iii) Permutations and Combinations Fundamental principle of counting. Factorial n. (n!) Permutations and combinations, derivation of formulae for n Pr and n Cr and their connections, simple application. Factorial notation n! , n! =n (n-1)! Fundamental principle of counting. Permutations - nP r . - Restricted permutation. - Certain things always occur together. - Certain things never occur. - Formation of numbers with digits. - Word building - repeated letters - No letters repeated. - Permutation of alike things. - Permutation of Repeated things. - Circular permutation clockwise counterclockwise Distinguishable / not distinguishable. Arithmetic Progression (A.P.) When all things are different. - When all things are not different. Mixed problems on permutation and combinations. - T n = a + (n - 1)d - Sn = - Arithmetic mean: 2b = a + c Inserting two or more arithmetic means between any two numbers. - Three terms in A.P. : a - d, a, a + d Four terms in A.P.: a - 3d, a - d, a + d, a + 3d n {2a + (n 1)d } 2 Geometric Progression (G.P.) T n = arn-1, S n = S -= a 1 r a (r n 1) , r 1 ; r <1 Geometric Mean, b = ac Combinations - nC r , nC n =1, nC 0 = 1, nC r = nC n r , n C x = nC y , then x + y = n or x = y, n+1 C r = nC r-1 + nC r . - C0 x n + nC1 x n -1 y + ...... + nCn y n . Questions based on the above. f ( x) a . etc. to be covered. g ( x) n - Inserting two or more Geometric Means between any two numbers. - Three terms are in G.P. ar, a, ar-1 - Four terms are in GP ar3, ar, ar-1, ar-3 Special sums n, n 2 , n 3 Using these summations to sum up other related expression. (iv) Binomial Theorem 3. Coordinate Geometry History, statement and proof of the binomial theorem for positive integral indices. Pascal's triangle, General and middle term in binomial expansion, simple applications. (i) Straight Lines Brief recall of two-dimensional geometry from earlier classes. Shifting of origin. Slope 4 4. Calculus Limits and Derivatives Derivative introduced as rate of change both as that of distance function and geometrically. Intuitive idea of limit. Limits of polynomials and rational functions trigonometric, exponential and logarithmic functions. Definition of derivative relate it to scope of tangent of the curve, Derivative of sum, difference, product and quotient of functions. Derivatives of polynomial and trigonometric functions. Limits - Notion and meaning of limits. - Fundamental theorems on limits (statement only). - Limits of algebraic and trigonometric functions. NOTE: Indeterminate forms are to be introduced while calculating limits. Differentiation - Meaning and geometrical interpretation of derivative. - Derivatives of simple algebraic and trigonometric functions and their formulae. - Differentiation using first principles. - Derivatives of sum/difference. - Derivatives of product of functions. Derivatives of quotients of functions. of a line and angle between two lines. Various forms of equations of a line: parallel to axis, point-slope form, slopeintercept form, two-point form, intercept form and normal form. General equation of a line. Equation of family of lines passing through the point of intersection of two lines. Distance of a point from a line. Basic concepts of Points and their coordinates. The straight line - Slope or gradient of a line. - Angle between two lines. Condition of perpendicularity and parallelism. - Various forms of equation of lines. Slope intercept form. - Two-point slope form. - Intercept form. - Perpendicular /normal form. General equation of a line. - Distance of a point from a line. - Distance between parallel lines. Equation of lines bisecting the angle between two lines. - Equation of family of lines Definition of a locus. - Equation of a locus. 5. Statistics and Probability (i) Statistics Measures of dispersion: range, mean deviation, variance and standard deviation of ungrouped/grouped data. Mean deviation about mean. Standard deviation - by direct method, short cut method and step deviation method. NOTE: Mean, Median and Mode of grouped and ungrouped data are required to be covered. (ii) Probability Random experiments; outcomes, sample spaces (set representation). Events; occurrence of events, 'not', 'and' and 'or' events, exhaustive events, mutually exclusive (ii) Circles Equations of a circle in: - Standard form. - Diameter form. - General form. - Parametric form. Given the equation of a circle, to find the centre and the radius. Finding the equation of a circle. - Given three non collinear points. Given other sufficient data for example centre is (h, k) and it lies on a line and two points on the circle are given, etc. 5 events, Axiomatic (set theoretic) probability, connections with other theories studied in earlier classes. Probability of an event, probability of 'not', 'and' and 'or' events. Random experiments and their outcomes. Events: sure events, impossible events, mutually exclusive and exhaustive events. - Definition of probability of an event - Laws of probability addition theorem. centre; and equations of directrices and the axes. - - 6. Conic Section Cases when coefficient y2 is negative and coefficient of x2 is negative. - Rough sketch of the above. - Focal property i.e. SP - S P = 2a. - Transverse and Conjugate axes; Latus rectum; coordinates of vertices, foci and centre; and equations of the directrices and the axes. 7. Introduction to three-dimensional Geometry Coordinate axes and coordinate planes in three dimensions. Coordinates of a point. Distance between two points and section formula. - As an extension of 2-D - Distance formula. - Section and midpoint form 8. Mathematical Reasoning Conics as a section of a cone. - Definition of Foci, Directrix, Latus Rectum. - PS = ePL where P is a point on the conics, S is the focus, PL is the perpendicular distance of the point from the directrix. (i) Parabola e =1, y2 = 4ax, x2 = 4ay, y2 = -4ax, x2 = -4ay. - Rough sketch of the above. - The latus rectum; quadrants they lie in; coordinates of focus and vertex; and equations of directrix and the axis. - Mathematically acceptable statements. Connecting words/ phrases - consolidating the understanding of "if and only if (necessary and sufficient) condition", "implies", "and/or", "implied by", "and", "or", "there exists" and their use through variety of examples related to the Mathematics and real life. Validating the statements involving the connecting words, Difference between contradiction, converse and contrapositive. Finding equation of Parabola when Foci and directrix are given, etc. Application questions based on the above. Self-explanatory. (ii) Ellipse - x2 y2 + 2 = 1 , e <1, b 2 = a 2 (1 e 2 ) 2 a b - Cases when a > b and a < b. - Rough sketch of the above. Major axis, minor axis; latus rectum; coordinates of vertices, focus and x2 y2 = 1 , e > 1, b2 = a 2 ( e 2 1) a 2 b2 - Sections of a cone, ellipse, parabola, hyperbola, a point, a straight line and a pair of intersecting lines as a degenerated case of a conic section. Standard equations and simple properties of parabola, ellipse and hyperbola. - Focal property i.e. SP + SP = 2a. (iii) Hyperbola SECTION B Finding equation of ellipse when focus and directrix are given. Simple and direct questions based on the above. 6 SECTION C - Calculation of moving averages with the given periodicity and plotting them on a graph. - If the period is even, then the centered moving average is to be found out and plotted. 9. Statistics Combined mean and standard deviation. The Median, Quartiles and Mode of grouped and ungrouped data. 10. Correlation Analysis Definition and meaning of covariance. Coefficient of Correlation by Karl Pearson. If x - x, y - y are small non - fractional PAPER II PROJECT WORK 20 Marks Candidates will be expected to have completed two projects, one from Section A and one from either Section B or Section C. Mark allocation for each Project [10 marks]: numbers, we use ( x - x )( y - y ) r= (x - x ) 2 (y - y) 2 If x and y are small numbers, we use xy r= 2 x 1 x y N 1 ( x )2 y 2 1 ( y )2 N N (i) Index Numbers Simple aggregate method. - Weighted aggregate method. - Simple average of price relatives. - Weighted average of price relatives (cost of living index, consumer price index). 2 marks Viva-voce based on the Project 3 marks Total 10 marks 3. Using Venn diagram, verify the distributive law for three given non-empty sets A, B and C. 4. Identify distinction between a relation and a function with suitable examples and illustrate graphically. 5. Establish the relationship between the measure of an angle in degrees and in radians with suitable examples by drawing a rough sketch. 6. Illustrate with the help of a model, the values of sine and cosine functions for different angles which are multiples of /2 and . (ii) Moving Averages - Findings 2. Verify that for two sets A and B, n(A B) = pq, where n(A) = p and n(B)= q, the total number of relations from A to B is 2pq. 11. Index Numbers and Moving Averages - 4 marks 1. Using a Venn diagram, find the number of subsets of a given set and verify that if a set has n number of elements, the total number of subsets is 2n . 1 uv ( u )( v ) N 2 2 2 1 2 1 u ( u) v ( v) N N Price index or price relative. Content Section A A and B, where u = x-A, v = y-B - 1 mark List of suggested assignments for Project Work: Otherwise, we use assumed means r= Overall format Meaning and purpose of the moving averages. 7. Draw the graphs of sin x, sin 2x, 2 sin x, and sin x/2 on the same graph using same coordinate axes and interpret the same. 7 17. Use focal property of ellipse to construct ellipse. 8. Draw the graph of cos x, cos 2x, 2 cos x, and cos x/2 on the same graph using same coordinate axes and interpret the same. 18. Use focal property of hyperbola to construct hyperbola. 9. Using argand plane, interpret geometrically, and its integral the meaning of powers. of quadratic function . From the graph find maximum/minimum value of the function. Also determine the sign of the expression. 19. Write geometrical significance of X coordinate, Y coordinate, and Z coordinate in space. Using the above, find the distance of the point in space from x-axis/y-axis/z-axis. Explain the above using a three-dimensional model/ power point presentation. 11. Construct a Pascal s triangle to write a binomial expansion for a given positive integral exponent. 20. Obtain truth values of compound statements of the type by using switch connection in series. 12. Obtain a formula for the sum of the squares/sum of cubes of n natural numbers. 21. Obtain truth values of compound statements of the type by using switch connection in parallel. 10. Draw the graph 13. Obtain the equation of the straight line in the normal form, for (the angle between the perpendicular to the line from the origin and the x-axis) for each of the following, on the same graph: (i) < 90 (ii) 90 < < 180 Section C 22. Explain the statistical significance of percentile and draw inferences of percentile for a given data. 23. Find median from the point of intersection of cumulative frequency curves (less than and more than cumulative frequency curves). (iii) 180 < < 270 24. Describe the limitations of Spearman s rank correlation coefficient and illustrate with suitable examples. (iv) 270 < < 360 14. Identify the variability and consistency of two sets of statistical data using the concept of coefficient of variation. 25. Identify the purchasing power using the concept of cost of living index number. 15. Construct the tree structure of the outcomes of a random experiment, when elementary events are not equally likely. Also construct a sample space by taking a suitable example. 26. Identify the purchasing power using the concept of weighted aggregate price index number. 27. Calculate moving averages with the given even Periodicity. Plot them and as well as the original data on the same graph. Section B 16. Construct different types of conics by PowerPoint Presentation, or by making a model, using the concept of double cone and a plane. 8

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