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National Council of Education Research and Training Syllabus: CISCE Class XII National Council of Education Research and Training Syllabus: CISCE Class XII
National Council of Education Research and Training Syllabus: Class XII National Council of Education Research and Training Syllabus: Class XII

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12.1 Determinants and Matrices

12.1.i Determinants
- 12.1.i.1 Order.
  - Matrix vocabulary (XII-F.1)
- 12.1.i.2 Minors.
  - Determinant of a matrix (XII-F.9)
- 12.1.i.3 Cofactors.
  - Determinant of a matrix (XII-F.9)
- 12.1.i.4 Expansion.
  - Determinant of a matrix (XII-F.9)
- 12.1.i.5 Properties of determinants.
  - Determinant of a matrix (XII-F.9)
- 12.1.i.6 Simple problems using properties of determinants e.g. evaluate matrix acb, bca, cab etc.
- 12.1.i.7 Cramer's Rule
  - 12.1.i.7.1 Solving simultaneous equations in 2 or 3 variables, x = Dx/D, y = Dy/d, z = Dz/D
  - 12.1.i.7.2 Consistency, inconsistency.
  - 12.1.i.7.3 Dependent or independent.
12.1.ii Matrices
- 12.1.ii.1 Types of matrices (m x n; m, n ≤3), order; Identity matrix, Diagonal matrix.
  - Matrix vocabulary (XII-F.1)
- 12.1.ii.2 Symmetric, Skew symmetric.
- 12.1.ii.3 Operation – addition, subtraction, multiplication of a matrix with scalar, multiplication of two matrices (the compatibility).
- 12.1.ii.4 E.g. [1 1, 0 2, 1 1], [1 2, 2 2] = AB (say) bit BA is not possible.
  - Matrix operation rules (XII-F.2)
- 12.1.ii.5 Singular and non-singular matrices.
  - Is a matrix invertible? (XII-F.10)
  - Identify inverse matrices (XII-F.13)
- 12.1.ii.6 Existence of two non-zero matrices whose product is a zero matrix.
  - Multiply two matrices (XII-F.6)
- 12.1.ii.7 Inverse (2x2, 3x3) A^-1 = AdjA/|A|
  - Inverse of a 2 x 2 matrix (XII-F.11)
  - Inverse of a 3 x 3 matrix (XII-F.12)
- 12.1.ii.8 Martin's Rule (i.e. using matrices)
  - 12.1.ii.8.1 a₁x + b₁y + c₁z = d₁. a₂x + b₂y + c₂z = d₂. a₃x + b₃y + c₃z = d₃.
  - 12.1.ii.8.2 A=[a₁ b₁ c₁, a₂ b₂ c₂, a₃ b₃ c₃], B=[d₁ d₂ d₃], X=[x y z]
  - 12.1.ii.8.3 AX = B → X = A^-1 B
  - 12.1.ii.8.4 Simple problems based on above.

12.2 Boolean Algebra

12.2.1 Boolean algebra as an algebraic structure, principle of duality, Boolean function. Switching circuits, application of Boolean algebra to switching circuits.

12.3 Conics

12.3.1 As a section of a cone.
12.3.2 Definition of Foci, Directrix, Latus Rectum.
12.3.3 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
12.3.i Parabola
- 12.3.i.1 e = 1, y² = 4ax, x² = 4ay, y² = -4ax, x² = -4ay, (y -β)² = 4a (x - α), (x - α)² = 4a (y - β).
- 12.3.i.2 Rough sketch of the above.
  - Graph parabolas (XII-J.3)
- 12.3.i.3 The latus rectum; quadrants they lie in; coordinates of focus and vertex; and equations of directrix and the axis.
  - Find properties of parabolas (XII-J.1)
- 12.3.i.4 Finding equation of Parabola when Foci and directrix are given.
  - Write equations of parabolas in vertex form (XII-J.2)
- 12.3.i.5 Simple and direct questions based on the above.
12.3.ii Ellipse
- 12.3.ii.1 x²/a² + y²/b²=1, e<1, b²=a²(1-e²)
  - Find the eccentricity of an ellipse (XII-J.8)
- 12.3.ii.2 Cases when a > b and a < b.
- 12.3.ii.3 Rough sketch of the above.
- 12.3.ii.4 Major axis, minor axis; latus rectum; coordinates of vertices, focus and centre; and equations of directrices and the axes.
  - Find properties of ellipses (XII-J.7)
- 12.3.ii.5 Finding equation of ellipse when focus and directrix are given.
  - Write equations of ellipses in standard form (XII-J.9)
- 12.3.ii.6 Simple and direct questions based on the above.
- 12.3.ii.7 Focal property i.e. SP + SP' = 2a.
12.3.iii Hyperbola
- 12.3.iii.1 x²/a² - y²/b² =1, e> 1, b² = a² (e² - 1)
  - Find the eccentricity of a hyperbola (XII-J.11)
- 12.3.iii.2 Cases when coefficient y² is negative and coefficient of x² is negative.
- 12.3.iii.3 Rough sketch of the above.
- 12.3.iii.4 Focal property i.e. SP - S'P = 2a.
- 12.3.iii.5 Transverse and Conjugate axes; Latus rectum; coordinates of vertices, foci and centre; and equations of the directrices and the axes.
  - Find properties of hyperbolas (XII-J.10)
  - Write equations of hyperbolas in standard form (XII-J.12)
- 12.3.iii.6 General second degree equation ax²+ 2hxy + by² + 2gx +2fy + c = 0 represents a parabola if h² = ab, ellipse if h² < ab, and hyperbola if h² > ab.
- 12.3.iii.7 Condition that y = mx + c is a tangent to the conics.

12.4 Inverse Trigonometric Function

12.4.1 Principal values.
- Inverses of trigonometric functions (XII-G.8)
12.4.2 sin^-1x, cos^-1x, tan^-1x etc. and their graphs.
12.4.3 sin^-1x, cos^-1√(1-x²) = tan^-1x/√(1-x²)
12.4.4 sin^-1x= cosec^-11/x; sin^-1x + cos^-1x = π/2 and similar relations for cot^-1x, tan^-1x, ect.
12.4.5 Addition formulae.
- 12.4.5.1 sin^-1x ± sin^-1(x√(1 - y²) ± y √(1 - x²))
- 12.4.5.2 cos^-1x ± cos^-1y=cos¹(xy √(1 - y²) √(1 - x²))
- 12.4.5.3 similarly tan^-1x ± tan¹y= tan¹ (x ± y)/(1 ± xy), < 1
- 12.4.5.4 Similarly, establish formulae for 2sin^-1x, 2cos^-1x, 2tan^-1x, 3tan^-1x etc. using the above formula.
12.4.6 Application of these formulae.

12.5 Calculus

12.5.i Differential Calculus
- 12.5.i.1 Revision of topics done in Class XI - mainly the differentiation of product of two functions, quotient rule, etc.
- 12.5.i.2 Derivatives of trigonometric functions.
  - Find derivatives of trigonometric functions I (XII-T.3)
  - Find derivatives of trigonometric functions II (XII-T.4)
- 12.5.i.3 Derivatives of exponential functions.
  - Find derivatives of exponential functions (XII-T.5)
- 12.5.i.4 Derivatives of logarithmic functions.
  - Find derivatives of logarithmic functions (XII-T.6)
- 12.5.i.5 Derivatives of inverse trigonometric functions - differentiation by means of substitution.
  - Find derivatives of inverse trigonometric functions (XII-T.7)
- 12.5.i.6 Derivatives of implicit functions and chain rule for composite functions.
- 12.5.i.7 Derivatives of Parametric functions.
- 12.5.i.8 Differentiation of a function with respect to another function e.g. differentiation of sinx³ with respect to x³.
- 12.5.i.9 Logarithmic Differentiation - Finding dy/dx when y = x to the x to the x power
  - Find derivatives using logarithmic differentiation (XII-U.3)
- 12.5.i.10 Successive differentiation up to 2nd order.
  - Find second derivatives of trigonometric, exponential and logarithmic functions (XII-V.3)
- 12.5.i.11 L'Hospital's theorem.
- 12.5.i.12 0/0 form, ∞/∞ form, 0°form, ∞ to the infinity power form etc.
- 12.5.i.13 Rolle's Mean Value Theorem - its geometrical interpretation.
- 12.5.i.14 Lagrange's Mean Value Theorem - its geometrical interpretation.
- 12.5.i.15 Maxima and minima.
12.5.ii Integral Calculus
- 12.5.ii.1 Revision of formulae from Class XI.
- 12.5.ii.2 Integration of 1/x, e to the x power.
- 12.5.ii.3 Integration by simple substitution.
- 12.5.ii.4 Integrals of the type f' (x)[f (x)]n, f'(x)/f(x)
- 12.5.ii.5 Integration of tanx, cotx, secx, cosecx.
- 12.5.ii.6 Integration by parts.
- 12.5.ii.7 Integration by means of substitution.
- 12.5.ii.8 Integration using partial fractions, Expressions of the form f(x)/g(x) when degree of f(x)< degree of g(x) - E.g. (x+2)/((x-3)(x+1)) = A/(x-3)+ B/(x+1), (x+2)/((x-2)(x-1)²)=A/(x-1) + B/(x-1)² + c/(x-2), (x+1)/((x²+3)(x-1)) = (Ax+B)/(X²+3) + C/(x-1)
- 12.5.ii.9 When degree of f (x) ≥ degree of g(x), e.g. (x² + 1)/(x² + 3x + 2) = 1 - [(3x + 1)/(x² + 3x + 2)].
- 12.5.ii.10 Integrals of the type: ∫dx/(x² ± a²),∫dx/√(x² ± a²), ∫(px + q)/(ax² + bx + c)dx, ∫(px + q)/√(ax² + bx + c) and expressions reducible to this form.
- 12.5.ii.11 Integrals of the form: ∫dx/(acosx + bsinx), ∫dx/(a + bcosx), ∫dx/(a + bsinx), ∫(1 ± x²)/(1 + x?)dx, ∫dx/(1 + x?), ∫√(tanx)dx, ∫√(cotx)dx.
- 12.5.ii.12 Properties of definite integrals. Problems based on the following properties of definite integrals are to be covered.
- 12.5.ii.13 Application of definite integrals - area bounded by curves, lines and coordinate axes is required to be covered.

12.6 Correlation and Regression

12.6.1 Definition and meaning of correlation and regression coefficient.
- Match correlation coefficients to scatter plots (XII-Y.1)
- Interpret regression lines (XII-Y.4)
12.6.2 Coefficient of Correlation by Karl Pearson.
- 12.6.2.1 If x -x, y-y are small non-fractional numbers we use r = (∑(x-x)(y-y))/√(∑(x-x)²) √((y-y)²), If x and y are small numbers, we use r = (∑xy - 1/N∑x∑y)/√(∑x² - 1/N(∑x)²)√(y² - 1/N(∑y)²), Otherwise, we use assumed means A and B, where u = x-A, v = y-B r = (∑uv - 1/N (∑u)(∑v))/√(∑u² - 1/N(∑u)²)√(∑v² - 1/N(∑v)²)
  - Calculate correlation coefficients (XII-Y.2)
- 12.6.2.2 Rank correlation by Spearman's (Correction included).
- 12.6.2.3 Lines of regression of x on y and y on x.
12.6.i The method of least squares.
- Find the equation of a regression line (XII-Y.3)
12.6.ii Lines of best fit.
- Find the equation of a regression line (XII-Y.3)
12.6.iii Regression coefficient of x on y and y on x.
12.6.iv bxy x byx = r², 0 ≤ b×y x byx ≤ 1
12.6.v Identification of regression equations

12.7 Probability

12.7.1 Random experiments and their outcomes.
12.7.2 Events: sure events, impossible events, mutually exclusive events, independent events and dependent events.
- Introduction to probability (XII-W.1)
- Identify independent events (XII-W.6)
12.7.3 Definition of probability of an event.
- Introduction to probability (XII-W.1)
12.7.4 Laws of probability: addition and multiplication laws, conditional probability (excluding Bayes' theorem).
- Independence and conditional probability (XII-W.8)
- Find probabilities using the addition rule (XII-W.10)

12.8 Complex Numbers

12.8.1 Argument and conjugate of complex numbers.
12.8.2 Sum, difference, product and quotient of two complex numbers additive and multiplicative inverse of a complex number.
12.8.3 Simple locus question on complex number; proving and using
- 12.8.3.1 z.z=|z|²; z₁ ± z₂= z₁ ± z₂ and (z₁/z₂) = z₁/z₂
12.8.4 Triangle inequality.
12.8.5 Square root of a complex number.
12.8.6 Demoivre's theorem and its simple applications.
12.8.7 Cube roots of unity: 1,ω,ω²; application problems.

12.9 Differential Equations

12.9.1 Differential equations, order and degree.
12.9.2 Solution of differential equations.
12.9.3 Variable separable.
12.9.4 Homogeneous equations and equations reducible to homogeneous form.
12.9.5 Linear form dy/dx + Py = Q where P and Q are functions of x only. Similarly for dx/dy.

12.10 Vectors

12.10.1 Scalar (dot) product of vectors.
12.10.2 Cross product - its properties - area of a triangle, collinear vectors.
12.10.3 Scalar triple product - volume of a parallelopiped, co-planarity.
12.10.4 Proof of Formulae (Using Vectors)
- 12.10.4.1 Sine rule.
- 12.10.4.2 Cosine rule
- 12.10.4.3 Projection formula
- 12.10.4.4 Area of a δ = ½absinC

12.11 Co-ordinate geometry in 3-Dimensions

12.11.i Lines
- 12.11.i.1 Cartesian and vector equations of a line through one and two points.
- 12.11.i.2 Coplanar and skew lines.
- 12.11.i.3 Conditions for intersection of two lines.
- 12.11.i.4 Shortest distance between two lines.
12.11.ii Planes
- 12.11.ii.1 Cartesian and vector equation of a plane.
- 12.11.ii.2 Direction ratios of the normal to the plane.
- 12.11.ii.3 One point form.
- 12.11.ii.4 Normal form.
- 12.11.ii.5 Intercept form.
- 12.11.ii.6 Distance of a point from a plane.
- 12.11.ii.7 Angle between two planes, a line and a plane.
- 12.11.ii.8 Equation of a plane through the intersection of two planes i.e. - P₁ + kP₂ = 0.
- 12.11.ii.9 Simple questions based on the above.

12.12 Probability

12.12.1 Bayes' theorem; theoretical probability distribution, probability distribution function; binomial distribution – its mean and variance.

12.13 Discount

12.13.1 True discount; banker's discount; discounted value; present value; cash discount, bill of exchange.

12.14 Annuities

12.14.1 Meaning, formulae for present value and amount; deferred annuity, applied problems on loans, sinking funds, scholarships.

12.15 Linear Programming

12.15.1 Introduction, definition of related terminology such as constraints, objective function, optimization, isoprofit, isocost lines; advantages of linear programming; limitations of linear programming; application areas of linear programming; different types of linear programming (L.P.), problems, mathematical formulation of L.P problems, graphical method of solution for problems in two variables, feasible and infeasible regions, feasible and infeasible solutions, optimum feasible solution.
- Linear programming (XII-D.4)

12.16 Application of derivatives in Commerce and Economics in the following:

12.16.1 Cost function, average cost, marginal cost, revenue function and break even point.

12.17 Index numbers and moving averages

12.17.1 Price index or price relative.
12.17.2 Simple aggregate method.
12.17.3 Weighted aggregate method.
12.17.4 Simple average of price relatives.
12.17.5 Weighted average of price relatives (cost of living index, consumer price index).
12.17.6 Meaning and purpose of the moving averages.
12.17.7 Calculation of moving averages with the given periodicity and plotting them on a graph.
12.17.8 If the period is even, then the centered moving average is to be found out and plotted.

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12.1 Determinants and Matrices

12.1.i Determinants

12.1.i.1 Order.

12.1.i.2 Minors.

12.1.i.3 Cofactors.

12.1.i.4 Expansion.

12.1.i.5 Properties of determinants.

12.1.i.6 Simple problems using properties of determinants e.g. evaluate matrix acb, bca, cab etc.

12.1.i.7 Cramer's Rule

12.1.i.7.1 Solving simultaneous equations in 2 or 3 variables, x = Dx/D, y = Dy/d, z = Dz/D

12.1.i.7.2 Consistency, inconsistency.

12.1.i.7.3 Dependent or independent.

12.1.ii Matrices

12.1.ii.1 Types of matrices (m x n; m, n ≤3), order; Identity matrix, Diagonal matrix.

12.1.ii.2 Symmetric, Skew symmetric.

12.1.ii.3 Operation – addition, subtraction, multiplication of a matrix with scalar, multiplication of two matrices (the compatibility).

12.1.ii.4 E.g. [1 1, 0 2, 1 1], [1 2, 2 2] = AB (say) bit BA is not possible.

12.1.ii.5 Singular and non-singular matrices.

12.1.ii.6 Existence of two non-zero matrices whose product is a zero matrix.

12.1.ii.7 Inverse (2x2, 3x3) A-1 = AdjA/|A|

12.1.ii.8 Martin's Rule (i.e. using matrices)

12.1.ii.8.1 a1x + b1y + c1z = d1. a2x + b2y + c2z = d2. a3x + b3y + c3z = d3.

12.1.ii.8.2 A=[a1 b1 c1, a2 b2 c2, a3 b3 c3], B=[d1 d2 d3], X=[x y z]

12.1.ii.8.3 AX = B → X = A-1 B

12.1.ii.8.4 Simple problems based on above.

12.2 Boolean Algebra

12.2.1 Boolean algebra as an algebraic structure, principle of duality, Boolean function. Switching circuits, application of Boolean algebra to switching circuits.

12.3 Conics

12.3.1 As a section of a cone.

12.3.2 Definition of Foci, Directrix, Latus Rectum.

12.3.3 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

12.3.i Parabola

12.3.i.1 e = 1, y² = 4ax, x² = 4ay, y² = -4ax, x² = -4ay, (y -β)² = 4a (x - α), (x - α)² = 4a (y - β).

12.3.i.2 Rough sketch of the above.

12.3.i.3 The latus rectum; quadrants they lie in; coordinates of focus and vertex; and equations of directrix and the axis.

12.3.i.4 Finding equation of Parabola when Foci and directrix are given.

12.3.i.5 Simple and direct questions based on the above.

12.3.ii Ellipse

12.3.ii.1 x²/a² + y²/b²=1, e<1, b²=a²(1-e²)

12.3.ii.2 Cases when a > b and a < b.

12.3.ii.3 Rough sketch of the above.

12.3.ii.4 Major axis, minor axis; latus rectum; coordinates of vertices, focus and centre; and equations of directrices and the axes.

12.3.ii.5 Finding equation of ellipse when focus and directrix are given.

12.3.ii.6 Simple and direct questions based on the above.

12.3.ii.7 Focal property i.e. SP + SP' = 2a.

12.3.iii Hyperbola

12.3.iii.1 x²/a² - y²/b² =1, e> 1, b² = a² (e² - 1)

12.3.iii.2 Cases when coefficient y² is negative and coefficient of x² is negative.

12.3.iii.3 Rough sketch of the above.

12.3.iii.4 Focal property i.e. SP - S'P = 2a.

12.3.iii.5 Transverse and Conjugate axes; Latus rectum; coordinates of vertices, foci and centre; and equations of the directrices and the axes.

12.3.iii.6 General second degree equation ax²+ 2hxy + by² + 2gx +2fy + c = 0 represents a parabola if h² = ab, ellipse if h² < ab, and hyperbola if h² > ab.

12.3.iii.7 Condition that y = mx + c is a tangent to the conics.

12.4 Inverse Trigonometric Function

12.4.1 Principal values.

12.4.2 sin-1x, cos-1x, tan-1x etc. and their graphs.

12.4.3 sin-1x, cos-1√(1-x²) = tan-1x/√(1-x²)

12.4.4 sin-1x= cosec-11/x; sin-1x + cos-1x = π/2 and similar relations for cot-1x, tan-1x, ect.

12.4.5 Addition formulae.

12.4.5.1 sin-1x ± sin-1(x√(1 - y²) ± y √(1 - x²))

12.4.5.2 cos-1x ± cos-1y=cos¹(xy √(1 - y²) √(1 - x²))

12.4.5.3 similarly tan-1x ± tan¹y= tan¹ (x ± y)/(1 ± xy), < 1

12.4.5.4 Similarly, establish formulae for 2sin-1x, 2cos-1x, 2tan-1x, 3tan-1x etc. using the above formula.

12.4.6 Application of these formulae.

12.5 Calculus

12.5.i Differential Calculus

12.5.i.1 Revision of topics done in Class XI - mainly the differentiation of product of two functions, quotient rule, etc.

12.5.i.2 Derivatives of trigonometric functions.

12.5.i.3 Derivatives of exponential functions.

12.5.i.4 Derivatives of logarithmic functions.

12.5.i.5 Derivatives of inverse trigonometric functions - differentiation by means of substitution.

12.5.i.6 Derivatives of implicit functions and chain rule for composite functions.

12.5.i.7 Derivatives of Parametric functions.

12.5.i.8 Differentiation of a function with respect to another function e.g. differentiation of sinx³ with respect to x³.

12.5.i.9 Logarithmic Differentiation - Finding dy/dx when y = x to the x to the x power

12.5.i.10 Successive differentiation up to 2nd order.

12.5.i.11 L'Hospital's theorem.

12.1.ii.7 Inverse (2x2, 3x3) A^-1 = AdjA/|A|

12.1.ii.8.1 a₁x + b₁y + c₁z = d₁. a₂x + b₂y + c₂z = d₂. a₃x + b₃y + c₃z = d₃.

12.1.ii.8.2 A=[a₁ b₁ c₁, a₂ b₂ c₂, a₃ b₃ c₃], B=[d₁ d₂ d₃], X=[x y z]

12.1.ii.8.3 AX = B → X = A^-1 B

12.4.2 sin^-1x, cos^-1x, tan^-1x etc. and their graphs.

12.4.3 sin^-1x, cos^-1√(1-x²) = tan^-1x/√(1-x²)

12.4.4 sin^-1x= cosec^-11/x; sin^-1x + cos^-1x = π/2 and similar relations for cot^-1x, tan^-1x, ect.

12.4.5.1 sin^-1x ± sin^-1(x√(1 - y²) ± y √(1 - x²))

12.4.5.2 cos^-1x ± cos^-1y=cos¹(xy √(1 - y²) √(1 - x²))

12.4.5.3 similarly tan^-1x ± tan¹y= tan¹ (x ± y)/(1 ± xy), < 1

12.4.5.4 Similarly, establish formulae for 2sin^-1x, 2cos^-1x, 2tan^-1x, 3tan^-1x etc. using the above formula.

12.8.3.1 z.z=|z|²; z₁ ± z₂= z₁ ± z₂ and (z₁/z₂) = z₁/z₂