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Natural number(N):-All the counting no. start with 1 to infinity.(1,2,3,4,5.......)., Whole number(W) :- All the counting no. start with 0 to infinity.(0,1,2,3,4,5.......)., Integers(Z):-The collection of all +ve and -ve no. of whole no.(- infinity...-3,-2,-1,0,1,2,3...infinity)., , Rational number(Q):- A number ‘r’ is called a rational number, if it can be written in the form p/q , where, p and q are integers and q ≠ 0., , Terminating number:- The decimal expansion terminates or ends after a finite number of steps., Note:-A rational no. p/q is expressible as a terminating decimal only when prime factor of q are2 or 5 only., Non-terminating:- Non-terminating decimal is a decimal no. that continues endlessy. e.g=0.33.........., Repeating or Recurring:-A decimal in which a digit or a set of digits is repeated periodically., , Ph.-9050952816, [email protected],[email protected]

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Irrational number:- A number ‘s’ is called irrational, if it cannot be written in the form p q , where p and q, are integers and q ≠ 0., , Note:- A number whose decimal expansion is non-terminating non-recurring is irrational., , Real number:-The collection of all rational and irrational numbers.(-infinity to +infinity except imaginary ), , Even number:-A number which is divisible by 2 and generates a remainder of 0 .(2,,4,6,8.....), Odd number:- A number which is not divisible by 2 .(1,3,5,7,9...), Prime number:- A prime number is a number that has only two factors, that is, 1 and the number, itself. (1,3,5,7,.....), , Ph.-9050952816, [email protected],[email protected]

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Co-prime number:- Co-prime numbers are the numbers whose common factor is only 1., e.g=(1,2),(15,8),....etc, Composite number:- Composite numbers are those numbers that have more than two factors., e.g =(2,4,6,8,9...etc), Imaginary number:-A number which is not a real or a number which is in the form of "i"=√-1., e.g =5i,√-5 etc, , Complex number:-The combination or real number and an imaginary number .e.g= a+ib, , Ph.-9050952816, [email protected],[email protected]

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Constant:- A symbol having a fixed numerical value. e.g. = 1,5,2/3,5., Variable :- A symbol which may be assigned different numerical value. e.g. = 2y+5,y is variable., Algebraic Expression:-A combination of constants and variables, connected by some or all of the, operations (+,-,⨯,÷)., , Term:- the part of an algebraic expression separated by "+" or "-" operations ., coefficient:-The numerical value with variable., , Types:- there are three type, , , , , , monomial :- An algebraic expression contain only one term with non-zero coefficient., binomial:- An algebraic expression contain only two term with non-zero coefficient., trinomial:- An algebraic expression contain only three term with non-zero coefficient., , polynomial:- An algebraic expression contain one or more than one terms with non-zero coefficient, Ph.-9050952816, [email protected],[email protected]

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Linear Expressions :-A linear expression is an algebraic expression where each term is either a, numeric constant or a variable raised only to the first power., , Linear Equations :- An equation is a mathematical statement, which has an equal sign (=), between the algebraic expression. Linear equations are the equations of degree 1., , Polynomials In One Variable,  Polynomial:- An algebraic expression in" x "of the form, , Ph.-9050952816, [email protected],[email protected]

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Type of Polynomials base on degree,  linear polynomial:- A polynomial of degree one.,  quadratic polynomial:- A polynomial of degree two.,  cubic polynomial:- A polynomial of degree three.,  biquadratic polynomial:- A polynomial of degree more than three.,  Constant polynomials:- Any constant is also a polynomial.,  Zero polynomial:- The constant polynomial 0., , Ph.-9050952816, [email protected],[email protected]

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Value of polynomial:-The value of polynomial p(x) at x= is obtained by putting x= in p(x) .it is denoted, by p( )., , Zero of a polynomial:-Let p(x) be a polynomial if p( ) =0 then we say that, , is a zero of the polynomial, , p(x) ., , Dividend = (Divisor × Quotient) + Remainder, Remainder Theorem : Let p(x) be any polynomial of degree greater than or equal to one and let a be any, real number. If p(x) is divided by the linear polynomial x – a, then the remainder is p(a)., Factor Theorem : If p(x) is a polynomial of degree n > 1 and a is any real number,, then (i) x – a is a factor of p(x), if p(a) = 0, and, , (ii) p(a) = 0, if x – a is a factor of p(x)., , Ph.-9050952816, [email protected],[email protected]

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Coordinate Geometry, Ordered Pair:-A pair of numbers a and b listed in a specific order with a at the first place and b at the, second place.(a,b), , Note:- (a,b) ≠ (b,a), Coordinate:-We represent each point in a plane by means of an ordered pair of real numbers., Coordinate Geometry:-The branch of mathematics in which geometric problems are solved through, algebra by using the coordinate system., , Coordinate axes:-The position of a point in a plane is determined with to two fixed mutually, perpendicular lines., X-axis:- The horizontal line. OR abscissa, Y-axis:-The vertical line. OR ordinate, Ph.-9050952816, [email protected],[email protected]

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Origin:-The point of intersection., Quadrants:-These axes divide the plane of the paper into four regions., Note:-Any point lying on the x-axis or y-axis does not lie in any quadrant., Note:- 1.Coordinate of a point on x-axis hence its ordinate is 0., , 2.Coordinate of a point on y-axis hence its abscissa is 0., , Ph.-9050952816, [email protected],[email protected]

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Linear Equations, 1. An equation is a mathematical statement, which has an equal sign (=) between the algebraic, expression., e.g = x + 2 = 0 , y+7x=3 , y – 3 = 6(x – 2), , linear equation in one variable :- Linear equations in one variable are those equations in which, there is only one variable present, power of the variable is 1 and there is only one solution of the, equation. e.g = x + 2 = 0 , 2x+3=8, , Solution:General form:- ax + b = 0, where, a ≠ 0 and x is the variable., Note:- A linear equation is an equation of a straight line., Ph.-9050952816, [email protected],[email protected]

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linear equation in two variables:- :- Linear equations in two variable are those equations in, which there is only two variable present and the power of the variable is 1.e.g = y+7x=0, , General form:- ax + by + c = 0, where, a ≠ 0, b ≠ 0 , x and y are the variables., , Forms of Linear Equation, , Standard Form of Linear Equation, , ax + by + c = 0, , Slope Intercept Form, , y = mx + b, , e.g 2x + 3y – 6 = 0, , Point Slope Form, , y – y1 = m(x – x1 ), slope, , slope of the line, e.g y = 3x + 7, , y-intercept, , (x1, y1)coordinates of the point, e.g y – 3 = 6(x – 2), , Ph.-9050952816, [email protected],[email protected]

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Standard Form of Linear Equation:ax + by + c = 0, where, a ≠ 0, b ≠ 0 , x and y are the variables., , e.g = x + 2y = 6, Solution:- x + 2y = 6, x = 6-2y ,put y = 0,1,2,3............infinite values then x gives infinite values., Hence , there is no end to different solutions of a linear equation in two variables. That is, a linear, equation in two variables has infinitely many solutions., , Graph of a Linear Equation in twoVariables:1. ax + by + c = 0, where, a = 0, b≠0 , x and y are the variables., e.g = 0x+y-2=0, , Ph.-9050952816, [email protected],[email protected]

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2. ax + by + c = 0, where, a≠0, b=0 , x and y are the variables., e.g = x+0y-1=0, x=1, , Note:- A line that is parallel to the y-axis is of the form 'x=k', , Ph.-9050952816, [email protected],[email protected]

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3. ax + by + c = 0, where, a≠0, b≠0 , x and y are the variables., e.g. = 2x+y = 6, y=6-2x, We can represent the solution of (1) using a table as shown below., , X, Y, , 0, 6, , 3, 0, , 1, 4, , 2, 2, , ......, ......, , Ph.-9050952816, [email protected],[email protected]

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INTRODUCTION TO EUCLID’S GEOMETRY, Point:- A point is an exact location., Line segment:-The straight path between two points A and B., Ray:-A line segment AB when extended indefinitely in one direction., Line:- Line segment AB when extended indefinitely in both directions., Collinear points:-Three or more than three point are said to be collinear if there is a line which contains, them all., , Intersecting line:- Two line having a common point ., Concurrent line:-Three or more line intersecting at the same point., Ph.-9050952816, [email protected],[email protected]

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Plane:-A plane is a surface such that every point of the line joining any two points on it, lies on it., Parallel line:-Two line are parallel if they have no point in common., Axioms:-Self evident true statement used throughout mathematics and not specifically linked to, geometry. OR A statement that is accepted as true without proof., , Euclid’s Axioms, (1) Things which are equal to the same thing are equal to one another., (2) If equals are added to equals, the wholes are equal., (3) If equals are subtracted from equals, the remainders are equal., (4) Things which coincide with one another are equal to one another., (5) The whole is greater than the part., (6) Things which are double of the same things are equal to one another., , Ph.-9050952816, [email protected],[email protected]

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(7) Things which are halves of the same things are equal to one another., , Euclid’s five postulates,  A straight line may be drawn from any one point to any other point.,  A terminated line can be produced indefinitely.,  A circle can be drawn with any centre and any radius.,  All right angles are equal to one another.,  If a straight line falling on two straight lines makes the interior angles on the same side of it taken, together less than two right angles, then the two straight lines, if produced indefinitely, meet on that, side on which the sum of angles is less than two right angles., , Statements:- A sentence which can be decided to be true or false., Theorems:- A statement that requires a proof ., Corollary:- A statement whose truth can be easily be deduced from a theorem., , Ph.-9050952816, [email protected],[email protected]

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LINES AND ANGLES, line:- A line is a one-dimensional figure, which has length but no width. A line is made of a set of, points which is extended in opposite directions infinitely., , line-segment:- A line with two end points is called a line-segment., Ray:-A part of a line with one end point is called a ray., Collinear points:- collinear points are those points that lies on the same straight line .otherwise they are, called non-collinear points., , Angle :- An angle is formed when two rays intersect at a common end point., , Ph.-9050952816, [email protected],[email protected]

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Acute angle:- An acute angle measures between 0° and 90°., Right angle:- A right angle is exactly equal to 90°., Obtuse angle:- An angle greater than 90° but less than 180° is called an obtuse angle, , ., , Straight angle:- A straight angle is equal to 180°., Reflex angle:- An angle which is greater than 180° but less than 360° is called a reflex angle., Complementary angles:- two angles whose sum is 90° are called complementary angles., Supplementary angles:- two angles whose sum is 180° are called supplementary angles., , Ph.-9050952816, [email protected],[email protected]

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Adjacent angles:- 1. a common vertex, 2. a common arm, 3. non-common arms are on different sides of the common arm., , linear pair of angles :-linear pair is a pair of adjacent angle where non common side form a straight line, and sum of these adjacent angles is 180., , Ph.-9050952816, [email protected],[email protected]

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Vertically opposite angles:- When two lines intersect each other, then the opposite angles,, formed due to intersection are called vertically opposite angles., , Note:- vertically opposite angles are always equal. ∠2 = ∠4 , ∠1= ∠3, , Theorem: If two lines intersect each other, then the vertically opposite angles are equal., , Ph.-9050952816, [email protected],[email protected]

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Parallel Lines and a Transversal:Transversal:- A line which intersects two or more lines at distinct points is called a transversal., (a) Corresponding angles :, (i) ∠ 1 and ∠ 5 (ii) ∠ 2 and ∠ 6, (iii) ∠ 4 and ∠ 8 (iv) ∠ 3 and ∠ 7, (b) Alternate interior angles :, (i) ∠ 4 and ∠ 6 (ii) ∠ 3 and ∠ 5, (c) Alternate exterior angles:, (i) ∠ 1 and ∠ 7 (ii) ∠ 2 and ∠8, (d) Interior angles on the same side of the transversal:, (i) ∠ 4 and ∠ 5 (ii) ∠ 3 and ∠ 6, , Ph.-9050952816, [email protected],[email protected]

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Note:- If a transversal intersects two parallel lines, then each pair of, (a) Corresponding angles are equal, (i) ∠ 1 = ∠ 5 (ii) ∠ 2 = ∠ 6, (iii) ∠ 4 = ∠ 8 (iv) ∠ 3 = ∠ 7, (b) Alternate interior angles are equal, (i) ∠ 4 = ∠ 6 (ii) ∠ 3 = ∠ 5, (c) Alternate exterior angles are equal, (i) ∠ 1 = ∠ 7 (ii) ∠ 2 = ∠ 8, (d) Interior angles on the same side of the transversal are equal, (i) ∠ 4 = ∠ 5 (ii) ∠ 3 = ∠ 6, ., , Ph.-9050952816, [email protected],[email protected]

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Theorem : If a transversal intersects two parallel lines, then each pair of interior angles on the same side, of the transversal is supplementary., ∠ 3 + ∠ 6=180 and ∠ 4 + ∠ 5 =180, , Theorem : The sum of the angles of a triangle is 180º., , Ph.-9050952816, [email protected],[email protected]

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Types of a triangles, By side, scalene triangle, , isosceles triangles, , By angle, acute angle triangle, , obtuse angle triangle, , Right triangle, equilateral triangle, , Ph.-9050952816, [email protected],[email protected]

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Median of a triangle:-, , A median of a triangle is a line segment joining a, , vertex to the midpoint of the opposite side, thus bisecting that side., , Centroid of a triangle :- The centroid of a triangle is the intersection of the three, medians of the triangle., , Altitude of a triangle:- An altitude of a triangle is a line segment through, a vertex and perpendicular to the opposite side., , Orthocenter of a triangle :- The point of intersection of all the three altitude of a triangle., , Ph.-9050952816, [email protected],[email protected]

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Incenter of a triangle:-The point of intersection of the internal bisector, of the angle of a triangle., , Circumcenter of a triangle :-The point of intersection of the, perpendicular bisectors of the sides of a triangle., , Ph.-9050952816, [email protected],[email protected]

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Criteria for Congruence of Triangles:1) (SSS congruence rule):- Side-side-side If all the three sides of one triangle are equivalent to the, corresponding three sides of the second triangle, then the two triangles are said to be congruent by SSS, rule., 2) (SAS congruence rule):-side-angle-side If any two sides and the angle included between the sides of, one triangle are equivalent to the corresponding two sides and the angle between the sides of the second, triangle, then the two triangles are said to be congruent by SAS rule., 3) (ASA congruence rule):-angle-side-angle If any two angles and the side included between the angles, of one triangle are equivalent to the corresponding two angles and side included between the angles of, the second triangle, then the two triangles are said to be congruent by ASA rule., 4) (AAS Congruence Rule):-angle angle side When two angles and a non-included side of a triangle are, equal to the corresponding angles and sides of another triangle, then the triangles are said to be, congruent., 5) (RHS congruence rule):-right angle-Hypotenuse side If the hypotenuse and a side of a right- angled, triangle is equivalent to the hypotenuse and a side of the second right- angled triangle, then the two right, triangles are said to be congruent by RHS rule., , Ph.-9050952816, [email protected],[email protected]

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Some Properties of a Triangle, Theorem : Angles opposite to equal sides of an isosceles triangle are equal., AB=AC Then, ∠B=∠ c, Theorem : In any triangle, the side opposite to the larger (greater) angle is longer., AC ˃ BC So ∠B ˃ ∠c, , Theorem : The sum of any two sides of a triangle is greater than the third side., xy + xz ˃ yz, , Ph.-9050952816, [email protected],[email protected]

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Parallelogram:- A parallelogram is a special type of quadrilateral whose opposite sides equal, and parallel., , Properties: The opposite sides are parallel and equal.,  Opposite angle are equal.,  Sum of any two adjacent angle is 180.,  Diagonals bisect each other.,  Diagonals need not be equal in length.,  Diagonals need not bisect at right angle.,  Each diagonal divides a parallelogram into two congruent triangles.,  Line joining the mid-points of the adjacent sides of a quadrilateral form a parallelogram.,  The parallelogram that is inscribed in a circle is a rectangle.,  The parallelogram that is circumscribed about a circle is a rhombus., Ph.-9050952816, [email protected],[email protected]

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 Parallelogram that lie on the same base and between the same parallel lines are equal in, area.,  A parallelogram is a rectangle if its diagonals are equal., , Rectangle:-A parallelogram, , in which all four angles at vertices are 90., , Properties: Opposite sides are parallel and equal.,  Opposite angle are equal and of 90.,  Diagonals are equal and bisect each other, but not necessarily at 90.,  Every rectangle is a parallelogram .,  When a rectangle is inscribed in a circle, the diameter of the circle is equal to the, diagonal of the rectangle., , , Ph.-9050952816, [email protected],[email protected]

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Rhombus:- A parallelogram, , in which all sides are equal., , Properties:, , , , , , , , Opposite sides are parallel and equal., Opposite angle are equal., Diagonals bisect each other at 90, but they are not necessarily equal., Diagonals bisect the vertex angles., Sum of any two adjacent angle is 180., Figure formed by joining the mid-points of the adjacent sides of a rhombus is a rectangle., A parallelogram is a rhombus if its diagonals are perpendicular to each other., , Ph.-9050952816, [email protected],[email protected]

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Square :-A rectangle whose all sides are equal., Properties:, , , , , , All sides are parallel and equal., All angle are 90., Diagonals are equal and bisect each other at 90., Figure formed by joining the mid-points of the adjacent sides of a square is a square., Each square is a parallelogram, a rectangle and a rhombus., , Ph.-9050952816, [email protected],[email protected]

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Trapezium:- A Quadrilateral whose only one pair of sides is parallel and other two sides, are non parallel., , Properties:, , , , , , , One pair of opposite sides are parallel, The diagonals intersect each other proportionally, in the ratio of lengths of parallel sides., The line that joins the mid-points of the non-parallel sides is always parallel to the, bases or parallel sides which is equal to half of the sum of parallel sides, If the non parallel sides are equal, then the diagonals will also be equal to each other., By joining the mid-point of adjacent sides of a trapezium ,four similar triangles are, obtained., If a trapezium is inscribed in a circle, then it is an isosceles trapezium with equal sides., , Ph.-9050952816, [email protected],[email protected]

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kite:-pair of adjacent sides are equal., Properties:, , , , JK = JM And KL = LM, Diagonals intersect AT 90., Shorter diagonal is bisected by the longer diagonal., , The Mid-point Theorem:- The line segment joining the mid-points of two sides of a triangle is, parallel to the third side and is also half of the length of the third side., , DE|| BC & BE = (1/2)BC, , Ph.-9050952816, [email protected],[email protected]

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Parallelogram:Area of parallelogram = Base × Height, Area of any parallelogram without height = ab× sin(α), Area of any parallelogram using its diagonal = ½ × d1 × d2 sin (y), Note:- y = any angle between at the intersection point of the, diagonals, Area of parallelogram in vector form = | a × b|, Area of parallelogram when diagonals are given in the vector form = 1/2 (d1 × d2), Perimeter = 2 (a + b), , Rectangle:- Area of rectangle = l × b, Perimeter = 2 (a + b), Area of rectangle using its diagonal, , = L × (D2 - L2 )1/2 OR B× (D2 - B2 )1/2, , Ph.-9050952816, [email protected],[email protected]

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Trapezium:- ½ x (a + b) x h, Area of trapezium:- ½ x Sum of parallel sides x Distance between the parallel sides, Perimeter of trapezium = a + b + c + d, , kite:Area of a kite:- ½ × d1 × d2, Perimeter of the kite = 2(a + b), , Ph.-9050952816, [email protected],[email protected]

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Theorem : Parallelograms on the same base and between the same parallels, are equal in area., , Note:- two figures are said to be on the same base and between the same parallels,, if they have a common base (side) and the vertices (or the vertex) opposite to the, common base of each figure lie on a line parallel to the base., , Theorem : Triangle on the same base and between the same parallels, are equal in area., ar (ADC) = ar (BCD), , Ph.-9050952816, [email protected],[email protected]

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SURFACE AREAS AND VOLUMES, Cuboid:2, , Surface Area of a Cuboid(s) = 2(lb + bh + hl) = ( l+b+h) -d, , 2, , lateral surface area of the cuboid(ls):- 2lh + 2bh or 2(l + b)h, Principle Diagonal of a cuboid(d):-(l2 + b2 + h2 )1/2, Diagonal of face of a cuboid (d)=(l2 + b2 )1/2, Perimeter of a cuboid(P) = 4( l+b+h), Volume of a cuboid(v) = l × b × h, Note:-The rise or fall of liquid level in a container =, , total volume of objects submerged or taken out, cross-section area of container, , Ph.-9050952816, [email protected],[email protected]

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Cube :Surface Area of a cube(s) = 6a2 OR 2d2, lateral surface area of the cube(ls)= 4a2, Principle Diagonal of a cube(d) = √3a, Diagonal of face of a cube (d)= √2a, Perimeter of a cube(P) = 12 a, Volume of a cube(v) = a3 OR (d/√3 )2 OR (√s/6)3, , Note:- For two cube, , , , , , Ratio of volumes = (Ratio of sides)3, Ratio of surface areas = (Ratio of side)2, (Ratio of surface areas) = (Ratio of volumes)2, , Ph.-9050952816, [email protected],[email protected]

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Cylinder:Total Surface Area of a Cylinder = 2πr(r + h) or 2πr2 +2πrh, Curved Surface Area of a Cylinder = 2πrh, Volume of a Cylinder = πr2h, Note:- 1. for two cylinders,  when radii are equal, Ratio of volumes = Ratio of heights, Ratio of volumes = Ratio of curved surface areas, Ratio of curved surface areas = Ratio of heights,  when height are equal, Ratio of volumes = (Ratio of radii)2, Ratio of volumes = (Ratio of curved surface areas)2, Radii of curved surface areas = Ratio of radii, , Ph.-9050952816, [email protected],[email protected]

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 when volume are equal, Ratio of radii = √inverse Ratio of heights, Ratio of curved surface areas = inverse Ratio of radii, Ratio of curved surface areas = √ Ratio of heights,  when curved surface areas are equal, Ratio of radii = inverse Ratio of heights, Ratio of volumes = inverse Ratio of heights, Ratio of volumes = Ratio of radii, 2. for a cylinder, Ratio of radii = (Ratio of curved surfaces) × (inverse Ratio of heights), Ratio of heights = (Ratio of curved surfaces) × (inverse Ratio of radii), Ratio of curved surfaces = (Ratio of radii) × (Ratio of heights), , Ph.-9050952816, [email protected],[email protected]

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if the ratio of heights and ratio of radii of two right circular cylinders are given,then, Ratio of curved surfaces = (Ratio of radii) × (Ratio of heights), if the ratio of heights and ratio of curved surfaces of two right circular cylinders are given,then, Ratio of radii = (Ratio of curved surfaces) × (inverse Ratio of heights), if the ratio of radii and ratio of curved surfaces of two right circular cylinders are given,then, Ratio of heights = (Ratio of curved surfaces) × (inverse Ratio of radii), , Cone:-, , Slant height(l) = (h2 + r2)1/2, 2, , Surface Area of cone = πr + πrl, lateral surface area of the cube(ls)= πrl, Volume of a cone = 1/3 πr2 h, Note:- 1. for two cone,  when radii are equal = Ratio of volumes = Ratio of heights, , Ph.-9050952816, [email protected],[email protected]

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 when height are equal = Ratio of volumes = (Ratio of radii)2,  when volume are equal = Ratio of radii = √inverse Ratio of heights,  when curved surface areas are equal = Ratio of radii = inverse Ratio of heights, if the ratio of volumes and the ratio of heights of two right circular cone are given ,then, Ratio of radii =√(Ratio of volumes)( inverse Ratio of heights), if the ratio of heights and the ratio of diameters(radii) of two right circular cone are given ,then, Ratio of volumes =(Ratio of radii)2 × (Ratio of heights), if the ratio of radii (diameter) and the ratio of volumes of two right circular cone are given ,then, Ratio of heights =(inverse Ratio of radii)2 × (Ratio of volumes), , Ph.-9050952816, [email protected],[email protected]

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, , (Ratio of surface areas)3 = (Ratio of volumes)2, , Hemisphere :2, , Surface area of a sphere = 3π r, Lateral surface area of the cube(ls)= 2π r2, Volume of a Sphere = 2/3(π r2 ), , Important concept, , , Carpeting the floor of a room, if the length and breadth of a room are "l" and "b" resp. and a carpet of width "w "is used to cover the, floor, the required length of the carpet = (l× b)/w, , Ph.-9050952816, [email protected],[email protected]

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, , Number of square tiles requied for flooring, , If the length and breadth of a room are "l" and "b" resp. , then the least number of a square tiles, required to cover the floor = (l× b)/H.C.F(l,b), ,  Path around a rectangular space, 1. A rectangular garden l m long and b m broad is surrounded by a path w m wide. the area of, the path is given by = 2w(l + b+2w) sq.m., 2. A rectangular garden l m long and b m broad is surrounded by a path w m wide constructed, inside it along its boundary. the area of the path is given by = 2w(l + b-2w) sq.m., 3. A rectangular Park is l m long and b m broad. two path w m wide are perpendicular to each, other inside the park. the area of the paths = w(l+b-w) sq.m., also area of the park minus the paths = (l-w)(b-w) sq. m., 4. A circular ground of radius r has a pathway of width w around it on its outside. The area of, circular pathway is given by = πw(2r+w), Ph.-9050952816, [email protected],[email protected]

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5. A circular ground of radius r has a pathway of width w around it on its inside. The area of circular, , pathway is given by = πw(2r-w), 6.If the area of a square is a sq. cm then the area of the circle formed by the same perimeter is, (4a/ π)sq.cm, 7. The area of the largest circle that can be inscribed in a square a is = (πa2)/4, 8. Area of a square inscribed in a circle of radius r is 2r2 and the side of the square is √2r., , Soilds inscribed circumscribing other solids, 1., 2., 3., , If a largest possible sphere is circumscribed by a cube of edge a cm then the radius of the sphere, = a/2, If a largest possible cube is inscribed in a sphere of radius a cm. then the edge of the cube = 2a/√3, If a largest possible sphere is inscribed in a cylinder of radius a cm and height h cm,then the radius, , of the sphere = { a for h ˃ a, , Ph.-9050952816, [email protected],[email protected]

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{h/2 for a ˃ h, 4., , If a largest possible sphere is inscribed in a cone of radius a cm and slant height equal to the, diameter of the base, then radius of the sphere = a/√3, , 5., , If a largest possible cube is inscribed in a hemisphere of radius a cm. then the edge of the cube =, a√2/3, , 6., if a sphere of radius R is melted to form smaller spheres each of radius r, then the, number, of smaller spheres = (R/r)3, 7., If by melting n sphere ,each of radius r, a big sphere is made ,then radius of the big sphere = r3√n, 8., , if a cylinder is melted to form smaller spheres each of radius r , then the number of small sphere, volume of cylinder, volume of 1 sphere, , 9., , If a sphere of radius r is melted and cone of height h is made, then radius of the cone = 2(√r3/h), , Ph.-9050952816, [email protected],[email protected]