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Skills available for India class XI maths curriculum
Objectives are in black and IXL maths skills are in dark green. Hold your mouse over the name of a skill to view a sample question. Click on the name of a skill to practise that skill.
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- National Council of Education Research and Training Syllabus: CISCE Class XI National Council of Education Research and Training Syllabus: CISCE Class XI
- National Council of Education Research and Training Syllabus: Class XI National Council of Education Research and Training Syllabus: Class XI
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11.1 Mathematical Reasoning
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11.1.1 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 real life and Mathematics. Validating the statements involving the connecting words, differences between contradiction, converse and contrapositive.
11.2 Algebra
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11.2.i Complex Numbers
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11.2.i.1 Complex numbers as an ordered pair of real numbers in the form a + ib, (a, b)
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11.2.i.2 Geometrical representation in complex plane - Argand diagram for z (a complex number), 1/z, z and z; equality of two complex numbers; absolute value (modulus).
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11.2.ii Quadratic Equations
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11.2.ii.1 Use of the formula:
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11.2.ii.1.1 x = (-b ± √(b² - 4ac))/2a in solving quadratic equations.
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11.2.ii.2 Equations reducible to quadratic form.
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11.2.ii.3 Nature of roots
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11.2.ii.3.1 Product and sum of roots.
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11.2.ii.3.2 Roots are rational, irrational, equal, reciprocal, one square of the other.
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11.2.ii.3.3 Imaginary numbers.
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11.2.ii.3.4 Complex roots.
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11.2.ii.3.5 Framing quadratic equations with given root.
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11.2.ii.4 Quadratic Functions.
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11.2.ii.4.i Given α, β as roots then find the equation whose roots are of the form a³, β³, etc.
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11.2.ii.4.ii Case I: a > 0; Real roots, Complex roots, Equal roots
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11.2.ii.4.iii Case II: a < 0; Real roots, Complex roots, Equal roots
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11.2.ii.4.1 where 'a' is the coefficient of x2 in the equations of the form ax2 + bx + c = 0.
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11.2.ii.5 Understanding the fact that a quadratic expression (when plotted on a graph) is a parabola.
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11.2.ii.6 Sign of quadratic
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11.2.ii.6.1 Sign when the roots are real and when they are complex.
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11.2.ii.7 Quadratic inequalities.
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11.2.ii.7.1 Using method of intervals for solving problems of the type:
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11.2.ii.7.1.a x² + x - 6 ≤ 0
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11.2.ii.7.1.b A perfect square e.g. x² -6x + 9 ≥ 0
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11.2.ii.7.1.c (3x² - 2x - 5)/(x² - 2x + 5) < 2
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11.2.ii.7.1.d (x² - 2x + 5)/(3x² - 2x - 5) > 1/2
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11.2.iii Finite and Infinite Sequences
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11.2.iii.a Arithmetic Progression (A.P.)
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11.2.iii.a.1 Tn = a + (n - 1)d
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11.2.iii.a.2 Sn = n/2 {2a = (n-1)d}
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11.2.iii.a.3 Arithmetic mean: 2b = a + c
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11.2.iii.a.3.1 Inserting 2 or 3 arithmetic mean between any two numbers.
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11.2.iii.a.3.2 Three numbers in A.P..: a - d, a, a + d
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11.2.iii.a.3.3 Four in A.P.: a - 3d, a - d, a + d, a + 3d
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11.2.iii.b Geometric Progression (G.P.)
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11.2.iii.b.1 Tn = arn-¹, Sn = a(rn - 1)/(r-1), S∞ = a/(1-r); |r| < 1
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11.2.iii.b.1.1 Geometric Mean, b = √ac
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11.2.iii.b.1.2 Inserting 2 or 3 Geometric Mean between any two numbers.
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11.2.iii.b.2 Three numbers are in G.P. ar, a, ar-¹
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11.2.iii.b.3 Four ar³, ar, ar-¹, ar-³
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11.2.iii.c Harmonic Progression
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11.2.iii.c.1 a, b, c are in H.P then 1/a, 1/b, 1/c are in A.P.
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11.2.iii.c.2 Harmonic mean b = 2ac/(a + c)
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11.2.iii.d Arithmetico Geometric Series
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11.2.iii.d.1 Identifying series as A.P. x G.P. (when we substitute d = 0 in the series, we get a G.P. and when we substitute r =1 the A.P.)
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11.2.iii.e Special sums ∑n, ∑n², ∑n³
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11.2.iii.e.1 Using these summations to sum up other related expression.
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11.2.iv Permutations Combinations
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11.2.iv.1 Factorial notation n!, n! =n(n-1)!
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11.2.iv.2 Fundamental principle of counting.
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11.2.iv.2.a Permutations
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11.2.iv.2.a.1 nPr
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11.2.iv.2.a.2 Restricted permutation.
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11.2.iv.2.a.3 Certain things always occur together.
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11.2.iv.2.a.4 Certain things never occur.
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11.2.iv.2.a.5 Formation of numbers with digits.
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11.2.iv.2.a.6 Word building – repeated letters – No letters repeated.
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11.2.iv.2.a.7 Permutation of alike things.
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11.2.iv.2.a.8 Permutation of Repeated things.
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11.2.iv.2.a.9 Circular permutation – clockwise counterclockwise – Distinguishable / not distinguishable.
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11.2.iv.2.b Combinations
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11.2.iv.2.b.1 nCr, nCn = 1, nC0 = 1, nCr = nCn-r, nCx = nCy, then x + y = n or x = y, n+1Cr = nCr-1 + nCr.
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11.2.iv.2.b.2 When all things are different.
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11.2.iv.2.b.3 When all things are not different.
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11.2.iv.2.b.4 Division into groups - e.g. distinct groups, identical groups.
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11.2.iv.2.b.5 Mixed problems on permutation and combinations.
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11.2.v Mathematical induction
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11.2.v.1 Using induction to prove various summations and divisibility.
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11.2.vi Binomial Theorem
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11.2.vi.a Significance of Pascal's triangle.
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11.2.vi.b Binomial theorem (proof using induction) for positive integral powers, i.e. (x + y)n = nC0Xn + nC1Xn-1y....nCnYn. Simple direct questions based on the above.
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11.2.vi.c Binomial theorem for negative or fractional indices (1 + x)n = 1 + nx + n(n-1)/2! x² + (n (n-1)(n-2))/3! x³.....
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11.2.vi.c.1 When ||x| <1
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11.2.vi.c.2 Simple questions on the application of the above.
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11.2.vi.c.3 Finding the rth term for the above (Tr).
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11.2.vi.c.4 Applying the theorem on approximations e.g. (0.99)8 = (1 - 0.01)8.
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11.2.vii Properties of Binomial Coefficients.
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11.2.vii.1 C0 + C1 +...... Cn = 2n; C0 + C2 + C4..... = C1 + C3 + C5.... = 2n-1 Simple problems involving application of the above.
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11.3 Trigonometry
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11.3.i Angles and Arc lengths
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11.3.i.1 Angles: Convention of sign of angles.
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11.3.i.2 Magnitude of an angle: Measures of Angles; Circular measure.
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11.3.i.3 The relation S = rθ where θ is in radians.
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11.3.i.4 Relation between radians and degree.
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11.3.i.5 Definition of trigonometric functions with the help of unit circle.
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11.3.i.6 Truth of the identity sin² x + cos² x = 1.
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11.3.ii Trigonometric Functions
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11.3.ii.1 Relationship between trigonometric functions.
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11.3.ii.2 Proving simple identities.
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11.3.ii.3 Signs of trigonometric functions.
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11.3.ii.4 Domain and range of the trigonometric functions.
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11.3.ii.5 Trigonometric functions of all angles.
- Trigonometric ratios: sin, cos and tan (XI-R.3)
- Trigonometric ratios: csc, sec and cot (XI-R.4)
- Trigonometric ratios in similar right triangles (XI-R.5)
- Find trigonometric ratios using the unit circle (XI-R.6)
- Sin, cos and tan of special angles (XI-R.7)
- Csc, sec and cot of special angles (XI-R.8)
- Find trigonometric functions using a calculator (XI-R.9)
- Law of Sines (XI-R.17)
- Law of Cosines (XI-R.18)
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11.3.ii.6 Periods of trigonometric functions.
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11.3.ii.7 Graphs of simple trigonometric functions (only sketches).
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11.3.iii Compound and multiple angles
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11.3.iii.1 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.
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11.3.iii.2 Sum and differences as products sinC + sinD = 2sin(C + D/2) cos(C - D/2), etc.
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11.3.iii.3 Product to sum or difference i.e. 2sinAcosB = sin(A + B) + sin(A – B) etc.
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11.3.iv Trigonometric Equations
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11.3.iv.1 Solution of trigonometric equations (General solution and solution in the specified range).
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11.3.iv.1.a Equations in which only one function of a single angle is involved e.g. sin 5θ =0
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11.3.iv.1.b Equations expressible in terms of one trigonometric ratio of the unknown angle.
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11.3.iv.1.c Equations involving multiple and sub-multiple angles.
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11.3.iv.1.d Linear equations of the form acosθ + bsinθ = c, where |c| ≤ √(a² + b²) and a, b 0≠
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11.4 Calculus
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11.4.i Basic Concepts of Relations and Functions
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11.4.i.a Ordered pairs, sets of ordered pairs.
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11.4.i.b Cartesian Product (Cross) of two sets, cardinal number of a cross product. Relations as:
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11.4.i.b.1 an association between two sets.
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11.4.i.b.2 a subset of a Cross Product.
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11.4.i.c Types of Relations: reflexive, symmetric, transitive and equivalence relation.
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11.4.i.d Binary Operation.
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11.4.i.e Domain, Range and Co-domain of a Relation.
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11.4.i.f Functions:
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11.4.i.f.1 As special relations, concept of writing "y is a function of x" as y = f(x).
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11.4.i.f.2 Types: one to one / many to one, into/onto.
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11.4.i.f.3 Domain and range of a function.
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11.4.i.f.4 Composite function.
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11.4.i.f.5 Inverse of a function.
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11.4.i.f.6 Classification of functions.
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11.4.i.f.7 Sketches of graphs of exponential function, logarithmic function, mod function, step function.
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11.4.ii Differential calculus
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11.4.ii.a Limits
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11.4.ii.a.1 Notion and meaning of limits.
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11.4.ii.a.2 Fundamental theorems on limits (statement only).
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11.4.ii.a.3 Limits of algebraic and trigonometric functions.
- Find limits using graphs (XI-X.1)
- Find one-sided limits using graphs (XI-X.2)
- Determine if a limit exists (XI-X.3)
- Find limits using addition, subtraction and multiplication laws (XI-Y.1)
- Find limits using the division law (XI-Y.2)
- Find limits using power and root laws (XI-Y.3)
- Find limits using limit laws (XI-Y.4)
- Find limits of polynomials and rational functions (XI-Y.5)
- Find limits involving factorisation and rationalisation (XI-Y.6)
- Find limits at vertical asymptotes using graphs (XI-Z.1)
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11.4.ii.b Continuity
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11.4.ii.b.1 Continuity of a function at a point x = a.
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11.4.ii.b.2 Continuity of a function in an interval.
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11.4.ii.b.3 Removable discontinuity.
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11.4.ii.c Differentiation
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11.4.ii.c.1 Meaning and geometrical interpretation of derivative.
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11.4.ii.c.2 Concept of continuity and differentiability of x, [x], etc.
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11.4.ii.c.3 Derivatives of simple algebraic and trigonometric functions and their formulae.
- Average rate of change I (XI-CC.1)
- Average rate of change II (XI-CC.2)
- Find instantaneous rates of change (XI-CC.3)
- Velocity as a rate of change (XI-CC.4)
- Find values of derivatives using limits (XI-CC.5)
- Find derivatives of polynomials (XI-DD.6)
- Find derivatives of rational functions (XI-DD.7)
- Find derivatives of trigonometric functions I (XI-DD.8)
- Find derivatives of trigonometric functions II (XI-DD.9)
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11.4.ii.c.4 Differentiation using first principles.
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11.4.ii.c.5 Derivatives of sum/difference.
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11.4.ii.c.6 Derivatives of product of functions.
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11.4.ii.c.7 Derivatives of quotients of functions.
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11.4.ii.c.8 Derivatives of composite functions.
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11.4.ii.d Application of derivatives
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11.4.ii.d.1 Equation of Tangent and Normal approximation.
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11.4.ii.d.2 Rate measure.
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11.4.ii.d.3 Sign of derivative.
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11.4.ii.d.4 Monotonocity of a function.
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11.4.iii Integral Calculus
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11.4.iii.1 Indefinite integral
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11.4.iii.1.1 Integration as the inverse of differentiation.
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11.4.iii.1.2 Anti-derivatives of polynomials and functions (ax +b)n, sinx, cosx, sec2x, cosec2x.
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11.4.iii.1.3 Integrals of the type sin2x, sin3x, sin4x, cos2x, cos3x, cos4x.
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11.5 Coordinate Geometry
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11.5.i Basic concepts of Points and their coordinates.
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11.5.ii The straight line
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11.5.ii.1 Slope and gradient of a line.
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11.5.ii.2 Angle between two lines.
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11.5.ii.3 Condition of perpendicularity and parallelism.
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11.5.ii.4 Various forms of equation of lines.
- Slope-intercept form: find the slope and y-intercept (XI-D.4)
- Slope-intercept form: graph an equation (XI-D.5)
- Slope-intercept form: write an equation from a graph (XI-D.6)
- Slope-intercept form: write an equation (XI-D.7)
- Write equations in standard form (XI-D.12)
- Standard form: find x- and y-intercepts (XI-D.13)
- Standard form: graph an equation (XI-D.14)
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11.5.ii.5 Slope intercept form.
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11.5.ii.6 Two point slope form.
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11.5.ii.7 Intercept form.
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11.5.ii.8 Perpendicular /normal form.
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11.5.ii.9 General equation of a line.
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11.5.ii.10 Distance of a point from a line.
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11.5.ii.11 Distance between parallel lines.
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11.5.ii.12 Equation of lines bisecting the angle between two lines.
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11.5.ii.13 Definition of a locus.
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11.5.ii.14 Methods to find the equation of a locus.
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11.5.iii Circles
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11.5.iii.1 Equations of a circle in:
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11.5.iii.1.1 Standard form.
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11.5.iii.1.2 Diameter form.
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11.5.iii.1.3 General form.
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11.5.iii.1.4 Parametric form.
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11.5.iii.1.5 Given the equation of a circle, to find the centre and the radius.
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11.5.iii.2 Finding the equation of a circle.
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11.5.iii.2.1 Given three non collinear points.
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11.5.iii.2.2 Given other sufficient data that the centre is (h, k) and it lies on a line and two points on the circle are given.
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11.5.iii.3 Tangents:
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11.5.iii.3.1 Tangent to a circle when the slope of the tangent is given: y = mx ± a √(1 + m²)
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11.5.iii.4 Intersection:
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11.5.iii.4.1 Circle with a line hence to find the length of the chord.
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11.5.iii.5 Finding the equation of a circle through the intersection of two circles i.e. S1 + kS2 = 0.
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11.6 Statistics
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11.6.1 Measures of central tendency.
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11.6.2 Standard deviation - by direct method, short cut method and step deviation method.
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11.6.3 Combined mean and standard deviation.
11.7 Vectors
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11.7.1 As directed line segments.
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11.7.2 Magnitude and direction of a vector.
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11.7.3 Types: equal vectors, unit vectors, zero vector.
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11.7.4 Position vector.
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11.7.5 Components of a vector.
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11.7.6 Vectors in two and three dimensions.
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11.7.7 i, j, k as unit vectors along the x, y and the z axes; expressing a vector in terms of the unit vectors.
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11.7.8 Operations: Sum and Difference of vectors; scalar multiplication of a vector.
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11.7.9 Section formula.
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11.7.10 Simple questions based on the above e.g. A line joining the mid point of any two sides of a triangle is parallel to the third side and half of it, concurrency of medians.
11.8 Co-ordinate geometry in 3-Dimensions
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11.8.1 As an extension of 2-D.
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11.8.2 Distance formula.
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11.8.3 Section and midpoint formula.
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11.8.4 Equation of x-axis, y-axis, z axis and lines parallel to them.
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11.8.5 Equation of xy – plane, yz – plane, zx – plane.
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11.8.6 Direction cosines, direction ratios.
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11.8.7 Angle between two lines in terms of direction cosines /direction ratios.
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11.8.8 Condition for lines to be perpendicular/ parallel.
11.9 Statistics
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11.9.1 Median - direct and by using the formula.
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11.9.2 Quartiles - direct and by using the formula.
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11.9.3 Deciles - direct and by using the formula.
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11.9.4 Percentiles - direct and by using the formula.
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11.9.5 Mode - graphically, direct method and by using the formula.
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11.9.6 Estimation of median/quartiles from Ogives.
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11.9.7 The Median, Quartiles, Deciles and Percentiles of grouped and ungrouped data;
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11.9.8 Mode – grouped and ungrouped data; estimation of mode by using graphical method. (Bimodal data not included).
11.10 Average Due Date
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11.10.1 Zero date.
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11.10.2 Equated periods.