11 and 12 grade math practice the topics are divided into three parts. Part one deals with elementary **Algebra**, part two provides a basic course in **trigonometry** and part three considers elements of **two dimensional Co-ordinate Geometry** including **solid geometry and mensuration**.

Each topic that are covered in 11 and 12 grade math, concepts is
enlightened with a summarization which includes important theorems,
results and formula are discussed in each topic with numerous types of
solved examples. Sufficient number of problems have been inserted in
grade 11 and 12 practice math task worksheets beginning with easier
followed gradually by harder ones.

It is expected that students should be acquainted with the basic 11 and
12 grade math concepts relating to each topic and should be able to
apply those to simple elementary problems, preferably numerical.

**Algebra:**

In 11 and 12 grade math these are the topics which are covered in **Algebra**.

**● Variation:** Direct, inverse and joint variation, theorem of joint variation. Application to simple examples of time and work, time and distance, mensuration, physical laws, economics.

**● Arithmetical Progression: **
Definition of **A. P.**, common difference, term, summation of *n *terms. Sum of *n* natural numbers. Sum of the and cubes of first natural numbers, **A. M. **

**● Geometric Progression: **
Definition of **G. P.**, Common ratio, general term, summation of *n* terms,** G. M.**

**● Surds: **
Rational numbers. To show that √2 is not rational. Idea of irrational
numbers, surds, quadratic surds, mixed surds, conjugate surds,
properties of surds, if a + √b = 0 then a = 0, b = 0 ; if a + √b =
c + √d , then a = c, b = d. Rationalization of surds. Square root of
quadratic surds.

**● Laws of Indices:**
Proofs for fundamental laws of indices for positive integers, statement
for fractional, zero and negative indices : simple applications.

**● Logarithms:**
Definition, base, index, general properties of logarithms, common logarithm, characteristic and mantissa, antilogarithm, use of logarithmic tables.

**● Complex Numbers:**
Complex numbers, significance of the imaginary unit i, addition,
multiplication and division, properties of complex numbers ; if a + ib =
0, then a= 0, b= 0 ; if a + ib = c + id, then a = c, b = d. Argand
diagram. Modulus. Argument, complex conjugate. Square root of complex
numbers, cube roots of unity and their properties.

**● Theory of Quadratic Equations:**
Quadratic equations with real roots. Statement of fundamental theorem
of algebra. Roots (two and only two roots), relation between roots and
coefficients of a quadratic equation. Nature of roots, common roots.
Nature of the quadratic expression *ax2 + bx + c* — its sign and magnitude.

**● Permutations: **
Definition. Theorem on permutations of *n* different things taken* r* at a time, things not all different, permutation with repetitions (circular permutation excluded).

**● Combinations: **
Definition : Theorem on combination of *n *different things taken *r *at a time, things not all different. Basic identities. Division into two groups (circular combination excluded).

**● Binomial Theorem for Positive Integral Index: **
Statement of the theorem, proof by method of induction. General term,
number of terms, middle term, equidistant terms. Simple properties of
binomial coefficients.

**● Infinite Series:**
The power series Σxn. Binomial series (1 + x)n (*n* ≠ positive integer), exponential and logarithmic series with ranges of validity (statement only). Simple applications.

** Trigonometry:**

In 11 and 12 grade math these are the topics which are covered in **Trigonometry**.

**● **Revision exercises of the topics covered in the syllabus of Secondary Mathematics.

**● **The relation s = rθ.

**● The Negative and Associated angles: **
- θ, 90° ± θ, 180° ± θ, 270° ± θ, 360° ± θ.

**● Trigonometrical Ratios of Compound Angles:**
Geometrical methods (for Sine and Cosine only). Product formulae, sum & difference formulae.

**● Multiple and Sub-multiple Angles:**
Simple problems.

**● ** Identities (conditional) of Trigonometrical Ratios (Sum of angles π or π/2)

**● ** General Solutions of Trigonometrical Equations.

**● **Trigonometrical Inverses (specific mention of principal branch).

**● Graphs of Trigonometrical Functions: **
y = sin* m*x, y = cos *m*x and y = tan *m*x, where *m* is an integer with stated values.

**● Properties of Triangles:**
Basic relations between sides, angles, circus-radius and in-radius.
Area of triangles in different forms. Simple and direct applications.

**Plane Analytical Geometry, Mensuration & Solid Geometry:**

In 11 and 12 grade math these are the topics which are covered in **Plane Analytical Geometry, Mensuration & Solid Geometry**.

**● Rectangular Cartesian Co-ordinates:**
Directed line and directed line segment, co-ordinate system on a
directed line and rectangular Cartesian co-ordinate system in a plane.

**● Polar Co-ordinates: **Notion of directed angles and polar co-ordinate system. (Radius vector o be taken as positive.)

**● ****Transformation** from Cartesian to Polar Co-ordinates and vice-versa.

**● Distance between Two Points:** Division of a line segment in a given ratio. Area of a triangle (all in terms of rectangular Cartesian co-ordinates). Application to geometrical properties. Verification of Apollonius’ Theorem.

**● Locus:** Concept of locus by simple illustration. Equation of locus in term of rectangular Cartesian co-ordinates.

**● Equations of Straight Lines (in rectangular Cartesian co-ordinates only):**
Notion of inclination and slope of a line. Slope in terms of
co-ordinates of two points on it. Equations of co-ordinate axes,
equations of lines parallel to co-ordinate axes, slope-intercept form,
point-slope form, equation of the line through two given points,
intercept form, symmetric form, normal form. Every first degree
equation represents a straight line.

**● Angle between Two Lines: ** Conditions of perpendicularity and
parallelism of two lines. Equation of a line parallel to a given line.
Equation of a line perpendicular to a given line, conditions that two
lines may be identical.

**● Distance of a Point from a Given Line:**
Notion of a signed distance of a point from a line, position of a point
with respect to a line, sides of a line. Equations of bisectors of
angles between two lines, equation of bisector of an angle that contains
the origin.

**● Conic Section:** Idea of conic sections as sections of cone.
Focus— Directrix definitions of a conic section, eccentricity, classification according to the value of eccentricity.

**● Ellipse and Hyperbola: **Standard equations only. Conjugate hyperbola. Elementary properties. Parametric equation.

**● **To investigate whether a point is inside, on or outside a
conic. Intersection of a straight line with a conic, equation of chord
of a conic with respect to the middle point.

**● Diameters of Conic:** Definition, equation of a diameter.
Equation of a conjugate diameter: elementary properties of conjugate
diameter (statement only).

**● Solid Geometry:**
Incidence relations between points and planes, lines and planes,
coplanarity, skew lines, parallel planes. Intersecting planes—Two
intersecting planes cut one another in a straight line and in no point
outside it, perpendicular to a plane, projection of a line segment on a
line and on a plane. Dihedral angle.

Corollary: Three straight lines intersecting pair wise or two parallel lines and its transversal lie in the same plane.

**● Theorems:** Theorem 1:
If a straight line is perpendicular to each of two intersecting
straight lines at their point of intersection, it is also perpendicular
to the plane in which they lie. (Apollonius’ Theorem may be used.)

Theorem 2: All straight lines drawn perpendicular to a given straight line at a given point are co-planar.

Theorem 3:
If two straight lines are parallel and if one of them is perpendicular
to a plane, then the other is also perpendicular to the same plane and
its converse.

Theorem 3: Theorem of Three Perpendiculars

Surface areas and volumes of prism and pyramid

**●** **Formula**

**Basic Math Formulas**

**Math Formula Sheet on Co-Ordinate Geometry**

**All Math Formula on Mensuration**

**Simple Math Formula on Trigonometry**

**●** **Mathematical Induction**

**Mathematical Induction**

**Problems on Principle of Mathematical Induction**

**Proof by Mathematical Induction**

**Induction Proof**

**●** **Variation**

**What is Variation?**

**Direct Variation**

**Inverse Variation**

**Joint Variation**

**Theorem of Joint Variation**

**Worked out Examples on Variation**

**Problems on Variation**

**●** **Surds**

**Definitions of Surds****Order of a Surd****Equiradical Surds****Pure and Mixed Surds****Simple and Compound Surds****Similar and Dissimilar Surds****Comparison of Surds****Addition and Subtraction of Surds****Multiplication of Surds****Division of Surds****Rationalization of Surds****Conjugate Surds****Product of two unlike Quadratic Surds****Express of a Simple Quadratic Surd****Properties of Surds****Rules of Surds****Problems on Surds**

**●** **Arithmetic Progression**

**Definition of Arithmetic Progression****General Form of an Arithmetic Progress****Arithmetic Mean****Sum of the First n Terms of an Arithmetic Progression****Sum of the Cubes of First n Natural Numbers****Sum of First n Natural Numbers****Sum of the Squares of First n Natural Numbers****Properties of Arithmetic Progression****Selection of Terms in an Arithmetic Progression****Arithmetic Progression Formulae****Problems on Arithmetic Progression****Problems on Sum of 'n' Terms of Arithmetic Progression**

**●** **Geometric Progression**

**Definition of****Geometric Progression****General Form and General Term of a Geometric Progression****Sum of n terms of a Geometric Progression****Definition of Geometric Mean****Position of a term in a Geometric Progression****Selection of Terms in Geometric Progression****Sum of an infinite Geometric Progression****Geometric Progression Formulae****Properties of Geometric Progression****Relation between Arithmetic Means and Geometric Means**

**●** **Logarithm**

**Mathematics Logarithms**

**Convert Exponentials and Logarithms**

**Logarithm Rules or Log Rules**

**Solved Problems on Logarithm**

**Common Logarithm and Natural Logarithm**

**Antilogarithm**

Trigonometry

**●** **Measurement of Angles**

**Sign of Angles****Trigonometric Angles****Measure of Angles in Trigonometry****Systems of Measuring Angles****Important Properties on Circle****S is Equal to R Theta****Sexagesimal, Centesimal and Circular Systems****Convert the Systems of Measuring Angles****Convert Circular Measure****Convert into Radian****Problems Based on Systems of Measuring Angles****Length of an Arc****Problems based on S R Theta Formula**

**●** **Trigonometric Functions**

**Basic Trigonometric Ratios and Their Names****Restrictions of Trigonometrical Ratios****Reciprocal Relations of Trigonometric Ratios****Quotient Relations of Trigonometric Ratios****Limit of Trigonometric Ratios****Trigonometrical Identity****Problems on Trigonometric Identities****Elimination of Trigonometric Ratios****Eliminate Theta between the equations****Problems on Eliminate Theta****Trig Ratio Problems****Proving Trigonometric Ratios****Trig Ratios Proving Problems****Verify Trigonometric Identities****Trigonometrical Ratios of 0°****Trigonometrical Ratios of 30°****Trigonometrical Ratios of 45°****Trigonometrical Ratios of 60°****Trigonometrical Ratios of 90°****Trigonometrical Ratios Table****Problems on Trigonometric Ratio of Standard Angle****Trigonometrical Ratios of Complementary Angles****Rules of Trigonometric Signs****Signs of Trigonometrical Ratios****All Sin Tan Cos Rule****Trigonometrical Ratios of (- θ)****Trigonometrical Ratios of (90° + θ)****Trigonometrical Ratios of (90° - θ)****Trigonometrical Ratios of (180° + θ)****Trigonometrical Ratios of (180° - θ)****Trigonometrical Ratios of (270° + θ)****Trigonometrical Ratios of (270° - θ)****Trigonometrical Ratios of (360° + θ)****Trigonometrical Ratios of (360° - θ)****Trigonometrical Ratios of any Angle****Trigonometrical Ratios of some Particular Angles****Trigonometric Ratios of an Angle****Trigonometric Functions of any Angles****Problems on Trigonometric Ratios of an Angle****Problems on Signs of Trigonometrical Ratios**

**Proof of Compound Angle Formula sin (α + β)****Proof of Compound Angle Formula sin (α - β)****Proof of Compound Angle Formula cos (α + β)****Proof of Compound Angle Formula cos (α - β)****Proof of Compound Angle Formula sin \(^{2}\) α - sin \(^{2}\) β****Proof of Compound Angle Formula cos \(^{2}\) α - sin \(^{2}\) β****Proof of Tangent Formula tan (α + β)****Proof of Tangent Formula tan (α - β)****Proof of Cotangent Formula cot (α + β)****Proof of Cotangent Formula cot (α - β)****Expansion of sin (A + B + C)****Expansion of sin (A - B + C)****Expansion of cos (A + B + C)****Expansion of tan (A + B + C)****Compound Angle Formulae****Problems using Compound Angle Formulae****Problems on Compound Angles**

**●**** Converting Product into Sum/Difference and Vice Versa**

**Converting Product into Sum or Difference****Formulae for Converting Product into Sum or Difference****Converting Sum or Difference into Product****Formulae for Converting Sum or Difference into Product****Express the Sum or Difference as a Product****Express the Product as a Sum or Difference**

**sin 2A in Terms of A****cos 2A in Terms of A****tan 2A in Terms of A****sin 2A in Terms of tan A****cos 2A in Terms of tan A****Trigonometric Functions of A in Terms of cos 2A****sin 3A in Terms of A****cos 3A in Terms of A****tan 3A in Terms of A****Multiple Angle Formulae**

**Trigonometric Ratios of Angle \(\frac{A}{2}\)****Trigonometric Ratios of Angle****\(\frac{A}{3}\)****Trigonometric Ratios of Angle \(\frac{A}{2}\) in Terms of cos A****tan \(\frac{A}{2}\) in Terms of tan A**-
**Exact value of sin 7½°** -
**Exact value of cos 7½°** -
**Exact value of tan 7½°** **Exact Value of cot 7½°****Exact Value of tan 11¼°****Exact Value of sin 15°****Exact Value of cos 15°****Exact Value of tan 15°****Exact Value of sin 18°****Exact Value of cos 18°****Exact Value of sin 22½°****Exact Value of cos 22½°****Exact Value of tan 22½°****Exact Value of sin 27°****Exact Value of cos 27°****Exact Value of tan 27°****Exact Value of sin 36°****Exact Value of cos 36°****Exact Value of sin 54°****Exact Value of cos 54°****Exact Value of tan 54°****Exact Value of sin 72°****Exact Value of cos 72°****Exact Value of tan 72°****Exact Value of tan 142½°****Submultiple Angle Formulae****Problems on Submultiple Angles**

**●**** Conditional Trigonometric Identities**

**Identities Involving Sines and Cosines****Sines and Cosines of Multiples or Submultiples****Identities Involving Squares of Sines and Cosines****Square of Identities Involving Squares of Sines and Cosines****Identities Involving Tangents and Cotangents****Tangents and Cotangents of Multiples or Submultiples**

**● ****Graphs of Trigonometrical Functions**

**Graph of y = sin x****Graph of y = cos x****Graph of y = tan x****Graph of y = csc x****Graph of y = sec x****Graph of y = cot x**

**General solution of the equation sin x = ½****General solution of the equation cos x = 1/√2****G****eneral solution of the equation tan x = √3****General Solution of the Equation sin θ = 0****General Solution of the Equation cos θ = 0****General Solution of the Equation tan θ = 0****General Solution of the Equation sin θ = sin ∝****General Solution of the Equation sin θ = 1****General Solution of the Equation sin θ = -1****General Solution of the Equation cos θ = cos ∝****General Solution of the Equation cos θ = 1****General Solution of the Equation cos θ = -1****General Solution of the Equation tan θ = tan ∝****General Solution of a cos θ + b sin θ = c****Trigonometric Equation Formula****Trigonometric Equation using Formula****General solution of Trigonometric Equation****Problems on Trigonometric Equation**

**●** Inverse Trigonometric Functions

**General and Principal Values of sin\(^{-1}\) x****General and Principal Values of cos\(^{-1}\) x****General and Principal Values of tan\(^{-1}\) x****General and Principal Values of csc\(^{-1}\) x****General and Principal Values of sec\(^{-1}\) x****General and Principal Values of cot\(^{-1}\) x****Principal Values of Inverse Trigonometric Functions****General Values of Inverse Trigonometric Functions****arcsin(x) + arccos(x) = \(\frac{π}{2}\)****arctan(x) + arccot(x) = \(\frac{π}{2}\)****arctan(x) + arctan(y) = arctan(\(\frac{x + y}{1 - xy}\))****arctan(x) - arctan(y) = arctan(\(\frac{x - y}{1 + xy}\))****arctan(x) + arctan(y) + arctan(z)= arctan\(\frac{x + y + z – xyz}{1 – xy – yz – zx}\)****arccot(x) + arccot(y) = arccot(\(\frac{xy - 1}{y + x}\))****arccot(x) - arccot(y) = arccot(\(\frac{xy + 1}{y - x}\))****arcsin(x) + arcsin(y) = arcsin(x \(\sqrt{1 - y^{2}}\) + y\(\sqrt{1 - x^{2}}\))****arcsin (x) - arcsin(y) = arcsin (x \(\sqrt{1 - y^{2}}\) - y\(\sqrt{1 - x^{2}}\))****arccos (x) + arccos(y) = arccos(xy - \(\sqrt{1 - x^{2}}\)\(\sqrt{1 - y^{2}}\))****arccos(x) - arccos(y) = arccos(xy + \(\sqrt{1 - x^{2}}\)\(\sqrt{1 - y^{2}}\))****2 arcsin(x) = arcsin(2x\(\sqrt{1 - x^{2}}\))****2 arccos(x) = arccos(2x\(^{2}\) - 1)****2 arctan(x) = arctan(\(\frac{2x}{1 - x^{2}}\)) = arcsin(\(\frac{2x}{1 + x^{2}}\)) = arccos(\(\frac{1 - x^{2}}{1 + x^{2}}\))****3 arcsin(x) = arcsin(3x - 4x\(^{3}\))****3 arccos(x) = arccos(4x\(^{3}\) - 3x)****3 arctan(x) = arctan(\(\frac{3x - x^{3}}{1 - 3 x^{2}}\))****Inverse Trigonometric Function Formula****Principal Values of Inverse Trigonometric Functions****Problems on Inverse Trigonometric Function**

**The Law of Sines or The Sine Rule****Theorem on Properties of Triangle****Projection Formulae****Proof of Projection Formulae****The Law of Cosines or The Cosine Rule****Area of a Triangle****Law of Tangents****Properties of Triangle Formulae****Problems on Properties of Triangle**

**●**** Trigonometrical Table**

**●**** Co-ordinate Geometry**

**What is Co-ordinate Geometry?**

**Rectangular Cartesian Co-ordinates**

**Polar Co-ordinates**

**Relation between Cartesian and Polar Co-Ordinates**

**Distance between Two given Points**

**Distance between Two Points in Polar Co-ordinates**

**Division of Line Segment****: Internal & External**

**Area of the Triangle Formed by Three co-ordinate Points**

**Condition of Collinearity of Three Points**

**Medians of a Triangle are Concurrent**

**Apollonius' Theorem**

**Quadrilateral form a Parallelogram**

**Problems on Distance Between Two Points**

**Area of a Triangle Given 3 Points**

**Worksheet on Quadrants**

**Worksheet on Rectangular – Polar Conversion**

**Worksheet on Line-Segment Joining the Points**

**Worksheet on Distance Between Two Points**

**Worksheet on Distance Between the Polar Co-ordinates**

**Worksheet on Finding Mid-Point**

**Worksheet on Division of Line-Segment**

**Worksheet on Centroid of a Triangle**

**Worksheet on Area of Co-ordinate Triangle**

**Worksheet on Collinear Triangle**

**Worksheet on Area of Polygon**

**Worksheet on Cartesian Triangle**

**●**** Locus**

**Concept of Locus**

**Concept of Locus of a Moving Point**

**Locus of a Moving Point**

**Worked-out Problems on Locus of a Moving Point**

**Worksheet on Locus of a Moving Point**

**Worksheet on Locus**

**●**** The Straight Line**

**Straight Line****Slope of a Straight Line****Slope of a Line through Two Given Points****Collinearity of Three Points****Equation of a Line Parallel to x-axis****Equation of a Line Parallel to y-axis****Slope-intercept Form****Point-slope Form****Straight line in Two-point Form****Straight Line in Intercept Form****Straight Line in Normal Form****General Form into Slope-intercept Form****General Form into Intercept Form****General Form into Normal Form****Point of Intersection of Two Lines****Concurrency of Three Lines****Angle between Two Straight Lines****Condition of Parallelism of Lines****Equation of a Line Parallel to a Line****Condition of Perpendicularity of Two Lines****Equation of a Line Perpendicular to a Line****Identical Straight Lines****Position of a Point Relative to a Line****Distance of a Point from a Straight Line****Equations of the Bisectors of the Angles between Two Straight Lines****Bisector of the Angle which Contains the Origin****Straight Line Formulae****Problems on Straight Lines****Word Problems on Straight Lines****Problems on Slope and Intercept**

**●** **The Circle**

**Definition of Circle****Equation of a Circle****General Form of the Equation of a Circle****General Equation of Second Degree Represents a Circle****Centre of the Circle Coincides with the Origin****Circle Passes through the Origin****Circle Touches x-axis****Circle Touches y-axis****Circle Touches both x-axis and y-axis****Centre of the Circle on x-axis****Centre of the Circle on y-axis****Circle Passes through the Origin and Centre Lies on x-axis****Circle Passes through the Origin and Centre Lies on y-axis****Equation of a Circle when Line Segment Joining Two Given Points is a Diameter****Equations of Concentric Circles****Circle Passing Through Three Given Points****Circle Through the Intersection of Two Circles****Equation of the Common Chord of Two Circles****Position of a Point with Respect to a Circle****Intercepts on the Axes made by a Circle****Circle Formulae****Problems on Circle**

**● ****The Parabola**

**Concept of Parabola****Standard Equation of a Parabola****Standard form of Parabola y\(^{2}\) = - 4ax****Standard form of Parabola x\(^{2}\) = 4ay****Standard form of Parabola x\(^{2}\) = -4ay****Parabola whose Vertex at a given Point and Axis is Parallel to x-axis****Parabola whose Vertex at a given Point and Axis is Parallel to y-axis****Position of a Point with respect to a Parabola****Parametric Equations of a Parabola****Parabola Formulae****Problems on Parabola**

**● ****The Ellipse**

**Definition of Ellipse****Standard Equation of an Ellipse****Two Foci and Two Directrices of the Ellipse****Vertex of the Ellipse****Centre of the Ellipse****Major and Minor Axes of the Ellipse****Latus Rectum of the Ellipse****Position of a Point with respect to the Ellipse****Ellipse Formulae****Focal Distance of a Point on the Ellipse****Problems on Ellipse**

**● ****The ****Hyperbola**

**Definition of Hyperbola****Standard Equation of an Hyperbola****Vertex of the Hyperbola****Centre of the Hyperbola****Transverse and Conjugate Axis of the Hyperbola****Two Foci and Two Directrices of the Hyperbola****Latus Rectum of the Hyperbola****Position of a Point with Respect to the Hyperbola****Conjugate Hyperbola****Rectangular Hyperbola****Parametric Equation of the Hyperbola****Hyperbola Formulae****Problems on Hyperbola**

**●** **Solid Geometry**

**Solid Geometry**

**Worksheet on Solid Geometry**

**Theorems on Solid Geometry**

**Theorems on Straight Lines and Plane**

**Theorem on Co-planar**

**Theorem on Parallel Lines and Plane**

**Theorem of Three Perpendiculars**

**Worksheet on Theorems of Solid Geometry**

**●**** Mensuration**

**Formulas for 3D Shapes**

**Volume and Surface Area of the Prism**

**Worksheet on Volume and Surface Area of Prism**

**Volume and Whole Surface Area of Right Pyramid**

**Volume and Whole Surface Area of Tetrahedron**

**Volume of a Pyramid**

**Volume and Surface Area of a Pyramid**

**Problems on Pyramid**

**Worksheet on Volume and Surface Area of a Pyramid**

**Worksheet on Volume of a Pyramid**

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