PSEB Class 12 Maths study plan

Test continuity of a piecewise function at its joining point by computing the left-hand limit, right-hand limit and f(a) separately, then state the equality that must hold.

Differentiate composite functions with the chain rule without first expanding them, covering nested trigonometric, exponential, logarithmic and inverse-trigonometric forms.

Carry out implicit differentiation on relations such as x^2 + xy + y^2 = 7 and express dy/dx in terms of both x and y.

Apply logarithmic differentiation to products of several factors and to functions of the form [f(x)]^g(x), and differentiate parametric functions using dy/dx = (dy/dt)/(dx/dt), including the second derivative.

Distinguish differentiability from continuity by constructing a function that is continuous but not differentiable at a point, and explain the failure in one line.

Interpret dy/dt as a rate of change and solve related-rates problems on an expanding sphere, a sliding ladder and a draining cone, stating which variable is being differentiated with respect to time.

Find the equation of the tangent and the normal at a given point on an explicit curve and on an implicitly defined curve.

Determine intervals of strict increase and decrease from the sign of f'(x), then classify each critical point as local maximum, local minimum or neither using the first or second derivative test.

Locate the absolute maximum and minimum of a continuous function on a closed interval and solve word problems that maximise volume or area and minimise cost or surface area.

Verify Rolle's theorem and Lagrange's mean value theorem for a stated function on a stated interval and find the value of c guaranteed by each.

Evaluate indefinite integrals using standard forms and substitution, including completing the square to reach forms involving a^2 + x^2, a^2 - x^2 and x^2 - a^2.

Integrate by parts using a deliberate choice of first and second function, and handle repeated integration by parts and the looping case such as the integral of e^x sin x.

Decompose rational functions into partial fractions for distinct linear, repeated linear and irreducible quadratic factors, then integrate each piece.

Evaluate definite integrals with the fundamental theorem and exploit the property integral from 0 to a of f(x) dx equals the integral from 0 to a of f(a - x) dx to simplify symmetric integrands.

Evaluate a definite integral as the limit of a Riemann sum for a linear or quadratic integrand, and integrate products of sines and cosines after applying power-reduction identities.

Sketch the region bounded by given curves and decide whether to integrate with respect to x or with respect to y before writing the definite integral for the area.

Compute the area bounded by a parabola and a line, by a circle and by an ellipse using definite integration, and give the answer in square units.

Find the area between two intersecting curves by solving for the intersection points and integrating upper curve minus lower curve over the correct interval.

Split an integral at the point where a curve crosses the axis so that area is never counted as negative, and explain in writing why the split is required.

Use the accumulated integral techniques to evaluate the antiderivatives that appear in these area problems without a separate revision round.

Classify any given differential equation by order, degree and linearity, and justify each classification in one sentence.

Solve variable-separable equations and model one growth-or-decay situation and one Newton's-law-of-cooling situation with them.

Solve homogeneous first-order equations by substituting y = vx and reducing to a separable form, then restoring y.

Solve a first-order linear equation by computing the integrating factor as e to the power of the integral of P dx, and apply an initial condition to fix the constant.

Verify that a proposed function satisfies a given differential equation by direct substitution, and form the differential equation from a family of curves with one or two parameters.

Compute dot and cross products and use them to find the angle between two vectors, the projection of one vector on another, and the area of a triangle or parallelogram.

Apply the scalar triple product to test whether three vectors are coplanar and to find the volume of a parallelepiped.

Write the equation of a line in both vector and Cartesian form for a line through a given point parallel to a given vector and for a line through two points, and convert between the two forms.

Write the equation of a plane in normal form and in general form, and find the distance of a point from a plane and the shortest distance between two skew lines.

Find the angle between two lines, between two planes, and between a line and a plane, and state the perpendicularity and parallelism conditions for each pair.

Test a relation on a small finite set for reflexivity, symmetry and transitivity, and list the equivalence classes when it is an equivalence relation.

Decide whether a function is one-one and onto, and compute the inverse of a bijection and verify that composition in both orders gives the identity.

Evaluate inverse trigonometric expressions using principal value ranges, and simplify composite forms such as sin inverse of 2x over 1 plus x squared and tan inverse combinations using the addition identity.

Compute determinants up to order 3 by cofactor expansion and by using row or column operations to create zeros, and find the inverse of a 3x3 matrix through its adjoint.

Solve a system of linear equations by the matrix-inverse method and by Cramer's rule, and interpret the determinant-equals-zero case as no unique solution.

Formulate a linear programming problem from a word description by defining the decision variables, writing the objective function and listing every constraint with its non-negativity condition.

Graph the feasible region, evaluate the objective function at each corner point, and identify when the problem has no feasible region or an unbounded solution.

Compute conditional probability P(A given B), apply the multiplication rule for independent and dependent events, and test two events for independence.

Apply Bayes' theorem to a two-stage problem such as defective items from two machines and interpret the posterior probability in words.

Solve binomial distribution problems by computing P(X = r), the mean np and the variance npq, and recognise when a situation is not binomial.