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# Maharahstra Board HSC class 12 Mathematics Board Exam Question Papers, School Prelim Question Papers and Chapterwise Testpapers

**Maharashtra HSC Class 12 Mathematics**

Mathematics is the language of all sciences and is perhaps the only subject which has this distinction. Mathematics is the backbone of all sciences and is used in everyday human life.

Higher Secondary is a launchpad from where students would either opt for academic education in Mathematics or professional courses like Engineering and Computer Technology, Physical and Biological Sciences. Hence to fulfil the needs of students, it is quite important to make the study of Mathematics meaningful by acquainting the student with many branches of mathematics. This will help them use Mathematical tools in the professional education. Apart from real life situations and other subject areas, major focus is on application of various concepts.

Many students find mathematics tough however Mathematics can become a high scoring subject for students if they solve their textbook at least two to three times, practice important high weightage chapters from other reference book, give a thorough revision by practicing Previous Year Maharashtra HSC Class 12 Mathematics Question Papers and Sample Papers. Students must also practice Mathematics by solving numerically rather than orally understanding question answers. Go ahead and download these question papers free of charge and practice them at your convenience:

Previous Year Maharashtra HSC Class 12 Mathematics Board Question Papers

Previous Year Maharashtra HSC Class 12 School Prelim Sample Question Papers

Previous Year Maharashtra HSC Class 12 Mathematics Chapter-wise Question Papers

Maharashtra Class 12 Mathematics syllabus* includes-:

**Mathematics HSC Class 12 Part 1**

**Mathematical Logic**

Statements - Introduction, sentences and statement, truth value of statement, open sentences, compound statement, quantifier and quantified statements, logical connectives : conjunction, disjunction, negation, implication/ conditional, biconditional, truth tables of compound statements, examples related to real life and mathematics, statement patterns and logical equivalence - tautology, contradiction, contingency, duality, negation of compound statement, contrapositive, converse, inverse, algebra of statements-idempotent law, commutative law, associative law, distributive law, involution law, complement law, identity law, DeMorgan’s laws, difference between converse, contradiction, contrapositive, application-introduction to switching circuits.

**Matrices**

Elementary transformation of a matrix-revision of cofactor and minor, elementary column transformation, elementary row transformation, inverse by elementary transformation, inverse of a matrix-existance and uniqueness of inverse of a matrix, adjoint method, application-solution of system of linear equations by – inversion method, reduction method.

**Trigonometric functions**

Trigonometric equations-general solution of trigonometric equation, Solution of a triangle: polar coordinates, sine rule, cosine rule, projection rule, area of a triangle, application, Hero’s formula, Napier Analogues, inverse trigonometric functions-definitions, domain, range, principle values, graphs of inverse trigonometric function, properties of inverse functions.

**Pair of straight lines**

Pair of lines passing through origin- combined equation, homogenous equation, theorem-the joint equation of a pair of lines passing through origin and its converse, acute angle between the lines represented by ax2+2hxy+by2=0 , condition for perpendicular lines, pair of lines not passing through origin-combined equation of any two lines, condition that the equation ax2+2hxy+by2+2gx+2fy+c=0 should represent a pair of lines (without proof), acute angle between the lines (without proof), point of intersection of two lines, condition of parallel and perpendicular lines.

**Circles**

Tangent of a circle - equation of a tangent at a point to 1) standard circle, 2) general circle, condition of tangency only for a line y=mx+c to the circle x2 + y2= a2, tangents to a circle from a point outside he circle, director circle, length of tangent segments, normal to a circle-equation of normal at a point.

**Conics**

Tangents and normals-equations of tangent and normal at a point for ellipse, parabola, hyperbola; condition of tangency for ellipse, parabola, hyperbola; tangents in terms of slope for ellipse, parabola, hyperbola, locus of points from which two tangents are mutually perpendicular, tangents from a point outside conics, properties of tangents and normals to conics.(no proof needed)

**Vectors**

Vectors Revision, Coplanarity and Collinearity of vectors : linear combination of vectors, condition of collinearity of two vectors, conditions of coplanarity of three vectors : section formula for internal and external division, centroid formula, midpoint formula, scaler triple product : definition of scaler triple product, formula of scaler triple product, properties of scaler triple product, geometrical interpretation of scalar triple product, application of vectors to geometry- medians of a triangle are concurrent, altitudes of a triangle are concurrent, angle bisectors of a triangle are concurrent, diagonals of a parallelogram bisect each other and converse, median of trapezium is parallel to the parallel sides and the length is half the sum of parallel sides, angle subtended on a semicircle is a right angle.

**Three dimensional geometry**

Direction cosines and direction ratios: direction angles, direction cosines, direction ratios, relation between direction ratio and direction cosines, angle between two lines, condition of perpendicular lines.

**Line**

Equation of line passing through given point and parallel to given vector, equation of line passing through two given points, distance between two skew lines, distance of a point from a line, distance between two parallel lines (with a vector approach).

**Plane**

Equation of plane in normal form, equation of plane passing through the given point and perpendicular to given vector, equation of plane passing through the given point and parallel to two given vectors, equation of plane passing through three non- collinear points, equation of plane passing through the intersection of two given planes, angle between two planes, angle between line and plane, distance of a point from a plane, condition for the coplanarity of two lines (vector approach).

**Linear programming problems**

Introduction of L.P.P. definition of constraints, definition of objective function, optimization, constraint equations, non- negativity restrictions, infeasible and feasible region, feasible solutions. Mathematical formulation-mathematical formulation of L.P.P. different types of L.P.P. problems, optimum feasible solution, graphical solutions for problem in two variables.

**Mathematics HSC Class 12 Part II**

**Continuity**

Continuity of a function at a point : left hand limit, right hand limit, definition of continuity of a function at a point, discontinuity of a function, types of discontinuity, continuity in interval-definition, algebra of continuous functions, continuity of some standard functions- rational, trigonometric, polynomial, logarithmic and exponential function.

**Differentiation**

Revision of derivative, relationship between continuity and differentiability- right hand derivative and left hand derivative (concept and need), every differentiable function is continuous but converse is not true, Derivative of composite function-chain rule, derivative of inverse function, derivative of inverse trigonometric function : Derivative of implicit function definition and examples, derivative of parametric function – definition of parametric function , exponential and logarithmic function- derivative of functions which are expressed in one of the following form a) product of functions, b) quotient of functions, c) higher order derivative, second order derivative d) [f(x)][g(x)]

**Applications of derivative**

Geometrical application-tangent and normal at a point, Rolle's theorem, Mean value theorem and their geometrical interpretation (proof not required), derivative as a rate measure-introduction, increasing and decreasing function, approximation (without proof), Maxima and minima- introduction of extrema and extreme values, maxima and minima in a closed interval, first derivative test, second derivative test.

**Integration**

Indefinite integrals-methods of integration, substitution method, integrals of the various types, integration by parts (reduction formulae are not expected), integration by partial fraction-factors involving repeated and non-repeated linear factors, non-repeated quadratic factors, definite integral-definite integral as a limit of sum, fundamental theorem of integral calculus (without proof), evaluation of definite integral 1) by substitution, 2) integration by parts, properties of definite integrals

**Applications of definite integral**

Area under the curve : area bounded by curve and axis (simple problems), area bounded by two curves, volume of solid of revolution-volume of solid obtained by revolving the area under the curve about the axis (simple problems).

**Differential equation**

Definition-differential equation, order, degree, general solution, particular solution of differential equation, formation of differential equation-formation of differential equation by eliminating arbitrary constants (at most two constants), solution of first order and first degree differential equation-variable separable method, homogeneous differential equation (equation reducible to homogeneous form are not expected), Linear differential equation, applications : population growth, bacterial colony growth, surface area, Newton’s laws of cooling, radioactive decay.

**Statistics**

Bivariate frequency distribution -bivariate data and its tabulation, covariance of ungrouped data, scatter diagram, covariance of ungrouped data, covariance for bivariate frequency distribution, Karl Pearson’s coefficient of correlation.

**Probability distribution**

Probability distribution of a random variable-definition of a random variable, discrete and continuous random variable, probability mass function (p.m.f.), probability distribution of a discrete random variable, cumulative probability distribution of a discrete random variable, expected value, variance and standard deviation of a discrete random variable, probability density function (p.d.f.), distribution function of a continuous random variable.

**Bernoulli trials and Binomial distribution**

Definition of Bernoulli trial, conditions for Binomial distribution, binomial distribution (p.m.f.), mean, variance and standard deviation, calculation of probabilities (no proof required), Normal distribution : p.d.f., mean, variance and standard deviation, standard normal variable, simple problems (without proof).

*For exact syllabus please refer to Maharashtra Board Website

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