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(1)Mathematics Formulae Explorer. www.MathsHomeWork123.com. MATHEMATICS FORMULAE EXPLORER. Mathematics Formulae Explorer - Page 1 of 146.

(2) Mathematics Formulae Explorer. www.MathsHomeWork123.com. This book is dedicated to my Parents – Mrs. S. Geethabai. Copies can be obtained from No. 9, New No. 29, First Street Bank Colony, MMC Chennai – 600 051, Tamilnadu. Phone : 044 – 2555 9594. Mathematics Formulae Explorer - Page 2 of 146.

(3) Mathematics Formulae Explorer. www.MathsHomeWork123.com. MATHEMATICS FORMULAE EXPLORER CONTENTS S. No.. Topics. Page No.. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34.. Algebra Analytical Geometry 3D- Analytical Geometry Boundary Value Problems Coordinate Geometry Commercial Arithmetic Complex Numbers Data Analysis Determinants Differential Calculus Differential Equations Discrete Mathematics Fourier Series Fourier Transform Graphs Integral Calculus Laplace Transform Matrices Measurement Mensuration Multiple Integrals Number Work Numbers and Operations Ordinary Differential Equations Partial Differential Equations Probability Pure Arithmetics Sets Statistics Tables Theoretical Geometry Trignometry Vector Algebra Z-Transform. 004 007 020 023 028 031 035 040 044 048 051 057 062 068 070 071 074 076 079 087 092 093 094 099 102 108 117 118 120 122 123 131 139 143. Mathematics Formulae Explorer - Page 3 of 146.

(4) Mathematics Formulae Explorer. www.MathsHomeWork123.com. 1. Algebra Expansion and Factorisation . . . . . . . . . . . .

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(6). . H.C.F x L.C.M of two expression =Product of the two expressions. Equation Two expression connected by a sign of equality is . is consistent equation, if the equality holds for some value of the variable/unknown. . an inconsistent equation, if the equality holds for no value of the variable/unknown. . an identical equation, if the equality holds for any value of the variable/unknown Mathematics Formulae Explorer - Page 4 of 146.

(7) Mathematics Formulae Explorer. www.MathsHomeWork123.com. Simultaneous Equation. Simultaneous Equation of the type

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(58) Mathematics Formulae Explorer. www.MathsHomeWork123.com. 2. Analytical Geometry. Introduction ‘Geometry’ is the study of Points, Lines, Curves, Surfaces, etc and their properties. In the 17th century AD, the methods of Algebra were applied in the study of Geometry and thereby ‘Analytical Geometry’ emerged out. The renowned French philosopher and Mathematician Rane Descartes (1596-1650) showed how the methods of Algebra could be applied to the study of Geometry. Locus The path traced by a point when it moves according to specified geometrical conditions is called the Locus of the point. Straight Lines A straight line is the simplest geometrical curve. Every straight line is associated with an equation. •. Slope-Intercept Form : y = mx + c. •. Point –Slope Form : y-y1 = m(x – x1). • Two Point Form : • • •.

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(65) ax+by+c=0 is 8 8 9 . •. The length of the perpendicular from the Origin to the line ax+by+c=0 is . 8. 9 . 8. Mathematics Formulae Explorer - Page 7 of 146.

(66) Mathematics Formulae Explorer. www.MathsHomeWork123.com. Slope of an equation ax + by + c = 0 For ax + by + c = 0, Slope m =. :(""%%(! " :(""%%(! " . Angle between two straight line. If ; is the angle between the two straight lines, then ;. !

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(70) . <. Condition for Parallel and Perpendicular •. If the two straight lines are Parallel, then their slopes are equal. i.e., m1=m2. •. If the two straight lines are Perpendicular, then the product of their slopes is -1. i.e., m1 x m2= -1.

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(82) and .. Pair of Straight Lines •. Combined. •. Pair of straight lines passing through the origin is ax +2hxy+by =0. •. The Straight line is ( i ) Real and Distinct if h2 > ab ( ii ) Coincident if h2 = ab ( iii ) Imaginary if h2 < ab. equation. of. the. ax2+2hxy+by2+2gx+2fy+c=0,. pair of straight lines is where a, b, c, f, g, h are constants. 2. 2. Mathematics Formulae Explorer - Page 8 of 146.

(83) Mathematics Formulae Explorer. www.MathsHomeWork123.com. Slopes of pair of straight line. ? . •. Sum of the slopes of pair of straight lines, m1+m2 =. •. Product of the slopes of pair of straight lines, m1m2 =. . Angle between the pair of Straight line •. • •. Angle between the pair of straight lines passing through the origin is ; !

(84) 8. 8. 9? . If the straight lines are parallel, then h2 = ab If the straight lines are perpendicular, then c(""%%(! " (""%%(! " . Condition to represent a pair of straight line •. The. general second degree equation ax +2hxy+by +2gx+2fy+c= 0 represent a pair of straight lines is 2. condition. for. a. 2. abc+2fgh-af2-bg2-ch2 = 0.. Circle Definition A circle is the locus of a point which moves in such a way that its distance from a fixed point is always constant. The fixed point is called the Centre of the Circle and the constant distance is called the Radius of the circle. •. The equation of circle when the centre is (h, k) and radius ‘r’ is (x – h)2 + (y – k)2 = r2. •. If the centre is origin, equation of circle is x2+y2 = r2. •. The equation of circle, if the end points of a diameter are given by (x – x1) (x – x2) + (y – y1) ( y – y2) = 0. •. The General equation of the circle is x2 + y2 + 2gx + 2fy + c = 0 with centre is (-g, -f) and radius is 9) " Mathematics Formulae Explorer - Page 9 of 146.

(85) Mathematics Formulae Explorer. www.MathsHomeWork123.com. Tangent to the Circle •. Equation of the tangent to a circle at a point (x1, y1) is xx1 + yy1 + g(x + x1) + f(y + y1) + c = 0. •. Length of the tangent to the circle from a point (x1, y1) is. •. If PT2 = 0, then the point is on the Circle.. •. If PT2 > 0, then the point is outside the Circle.. •. If PT2 < 0, then the point is inside the Circle.. •. Condition for the line y = mx + c to be a tangent to the circle x2 + y2 = a2 is c2 = a2 (1 + m2). •. Point of contact of the tangent y = mx + c to be a tangent to the circle , D x2 + y2 = a2 is C 9

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(92) . •. Two tangent can be drawn from a point to a circle is. m2(x2 – a2) – 2mxy +(y2 – a2) = 0. This is a Quadratic equation in ‘m’. Thus ‘m’ has two values. But ‘m’ is the slope of the tangent. Thus, two tangents can be drawn from a point to a circle.. Mathematics Formulae Explorer - Page 10 of 146.

(93) Mathematics Formulae Explorer. www.MathsHomeWork123.com. Family of Circles Concentric Circles Two (or) more circles having the same centre are called Concentric Circle. Circles Touching Internally or Externally Two circles may touch each other either internally or externally. Let C1, C2 be the centres of the circles and r1, r2 be their radii and P, the point of contact. •. Two circle touch externally, if C1C2 = r1 + r2. •. Two circle touch internally, if C1C2 = r1 - r2. Orthogonal Circles Two circles are said to be Orthogonal if the tangent at their point of intersection are at right angles. Condition for Orthogonal •. Condition for two circles to cut orthogonal is 2g1g2 + 2f1f2 = c1+c2. Conic Definition A conic is the locus of a point which moves in a plane, so that its distance from a fixed point bears a constant ratio to its distance from a fixed straight line. The fixed point is called focus, the fixed straight line is called directrix and the constant ratio is called eccentricity, which is denoted by ‘e’. Classification with respect to the General Equation of a Conic The equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 represents either a (nondegenerate) conic or a degenerate conic. If it is a conic, then it is •. a Parabola if B2- 4AC = 0. •. an Ellipse. •. a Parabola if B2- 4AC > 0. if B2- 4AC < 0. Mathematics Formulae Explorer - Page 11 of 146.

(94) Mathematics Formulae Explorer. www.MathsHomeWork123.com. Parabola ( y2 = 4ax ) Definition The locus of a point whose distance from a fixed point is equal to the distance from a fixed line is called a Parabola. i.e., Parabola is a conic whose eccentricity is 1. Definitions •. The fixed point used to draw the parabola is called the Focus F. Here, the focus is F(a,o).. •. The fixed line used to draw a parabola is called the Directrix of the parabola. Here, the equation of the directrix is x = - a. •. The axis of the parabola is the axis of symmetry. The curve y2 = 4ax is symmetrical about x-axis and hence x-axis or y = 0 is the axis of the parabola y2 = 4ax. Note that the axis of the parabola passes through the focus and perpendicular to the directrix.. •. The point of intersection of the parabola and its axis is called its Vertex. Here, the vertex is V(0,0).. •. The Focal Distance is the distance between a point on the parabola and its focus.. •. A chord which passes through the focus of the parabola is called the Focal Chord of the parabola. •. Latus Rectum is a focal chord perpendicular to the axis of the parabola. Here, the equation of the latus rectum is x = a.. •. End points of Latus Rectum is L (a, 2a) and L/(a, -2a). •. Length of Latus Rectum = 4a. Length of Semi-Latus Rectum is 2a.. General form of the standard equation of a Parabola The General form of the standard equation of the parabola is • F ? (open rightwards) •. • •. F ? ( open leftwards). ? F (open upwards ). ? F (open downwads) Mathematics Formulae Explorer - Page 12 of 146.

(95) Mathematics Formulae Explorer. www.MathsHomeWork123.com. Ellipse . Definition. . . . . .

(96) . The locus of a point in a plane whose distance from a fixed point bears a constant ratio, less than one to its distance from a fixed line is called Ellipse. Definitions Focus : The fixed point is called focus, denoted as F(ae,0) Directrix : The fixed line is called directrix l of the ellipse and its equation is (. Major axis : The line segment AA/ is called the major axis and the length of the major axis is 2a. The equation of the major axis is y = 0. Minor axis : The line segment BB/ is called the minor axis and the length of the minor axis is 2b. The equation of the minor axis is x = 0. Centre : The point of intersection of the major axis and minor axis of the ellipse is called the Centre of the Ellipse. Vertices : The points of intersection of the ellipse and its major axis are called its vertices. Focal Distance : The focal distance with respect to any point P on the ellise is the distance of P from the referred focus. Focal Chord : A chord which passes through the focus of the ellipse is called the focal chord of the ellipse. Latus Rectum : It is the focal distance perpendicular to the major axis of the Ellipse. The equation of the latus rectum are x = + ae, x = - ae. Eccentricity : ( B

(97) . End Points of Latus Rectum are G(, E and other latus rectum are G (, E . . Length of the Latus Rectum are. . . . . Special Property : Thanks to the symmetry about the origin, it permits the second Focus F2(-ae,0) and the second directrix x = (. Mathematics Formulae Explorer - Page 13 of 146.

(98) Mathematics Formulae Explorer. www.MathsHomeWork123.com. General forms of Standard Ellipses The General forms of Standard Equation of Ellipses, if the centre C(h,k) is. ? F

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(100). a>b. Focal Property of an Ellipse The sum of the focal distances of any point on an ellipse is constant and is equal to the length of the major axis.. Hyperbola Definition. . . . .

(101) . The locus of a point in a plane whose distance from a fixed point bears a constant ratio, greater than one to its distance from a fixed line is called Hyperbola.. Definitions Focus : The fixed point is called focus, denoted as F(ae,0) Directrix : The fixed line is called directrix l of the hyperbola and its equation is (. Transverse axis : The line segment AA/ joining the vertices is called the transverse axis and the length of the transverse axis is 2a. The equation of the transverse axis is y = 0. Conjugate axis : The line segment joining the points B(0, b) and B/ (0, -b) is called the conjugate axis and the length of the conjugate axis is 2b. The equation of the conjugate axis is x = 0.. Mathematics Formulae Explorer - Page 14 of 146.

(102) Mathematics Formulae Explorer. www.MathsHomeWork123.com. Centre : The point of intersection of the transverse and conjugate axes of the hyperbola is called the Centre of the Hyperbola.. Vertices : The points of intersection of the hyperbola and its transverse axis are called its vertices. Latus Rectum : It is the focal chord perpendicular to the transverse axis of the Hyperbola. The equation of the latus rectum are x = + ae, x = - ae. Eccentricity : ( B

(103) . End Points of Latus Rectum are G(, E and other latus rectum are G (, E . . Length of the Latus Rectum are. . . The other form of the Hyperbola If the transverse axis is along y-axis and the conjugate axis is along x-axis, then the equation of the hyperbola is of the form

(104) . For this type of hyperbola, we have the following points. •. Center is C(0,0). •. Vertices A(0, a) and A/(0, -a). •. Foci are F(0, ae) and F(0, -ae). •. Equation of transverse axis is x = 0. •. Equation of conjugate axis is y = 0. •. Equations of Latus rectum is E(. • • •. End points of conjugate axis is (b, 0) and (-b, 0) Equations of directrices is E. End points of Latus rectum is E. (. . , ( , E. . , (. Mathematics Formulae Explorer - Page 15 of 146.

(105) Mathematics Formulae Explorer. www.MathsHomeWork123.com. Parametric form of Conics Conic. Parametric equations. Parabola. x = at2 y = 2at. Ellipse. Hyperbola.

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(123) Mathematics Formulae Explorer. www.MathsHomeWork123.com. Equation of Tangent and Normal Conic Parabola. Ellipse. Hyperbola. Equation of Tangents at (x1, y1). Equation of Normal at (x1, y1).

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(139). Equation of Chord and Tangent at Parametric Form Conic. Parabola. Ellipse. Hyperbola. Equation of Chord at Parametric Form. Equation of Tangents at Parametric Form. Chord joining the points !

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(153) Mathematics Formulae Explorer. www.MathsHomeWork123.com. Results connected with Conics Conic. Parabola. Ellipse. Condition that y=mx+c may be a tangent to the conic . . . Point of Contact. C , D . . . . N , O . E 9 . N , O . E 9 . where . Hyperbola. Equation of any tangent is of the form. where . Asymptotes Definition An asymptote to a curve is the tangent to the curve such that the point of contact is at infinity. In particular, the asymptote touches the curve at ∞ ∞. Results regarding Asymptotes •. •. The equations of the asymptotes to the hyperbola . . . .

(154) is. The combined equation of asymptotes is i.e., . Mathematics Formulae Explorer - Page 18 of 146.

(155) Mathematics Formulae Explorer. • •. www.MathsHomeWork123.com. The asymptotes pass through the centre C(0,0) of the hyperbola.. The slopes of asymptotes are . . . . . . i.e.,. the transverse axis and. conjugate axis bisect angles between the asymptotes.. •. • •. If 7 is the angle between the asymptotes then the slope of. ! 7 . . . Angle between the asymptotes is 7 !

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(157) ( . Rectangular Hyperbola ( xy = c2 where . Definition. . ). A hyperbola is said to be a rectangular hyperbola if its asymptotes are at right angles. Results • • • • •. • • • • • •. Eccentricity of the Rectangular Hyperbola is ( √ and also b2 = a2(e2-1) The Vertices of the rectangular hyperbola are , and . √ √. . √. ,. . √. . The foci are (a, a) and (-a, -a) The equation of the transverse axis is y = x and the conjugate axis is y = - x. If the centre of the rectangular hyperbola is (h, k) then (x – h) ( y – k) = c2 The parametric equation of the rectangular hyperbola xy = c2 is x = ct, y= !. Equation of the tangent at (x1, y1) to the rectangular hyperbola xy = c2 is xy1+yx1 = 2c2 Equation of the tangent at ‘t’ is x + yt2 = 2ct Equation of normal at (x1, y1) to the rectangular hyperbola xy = c2 is xx1- yy1 = x12- y12 Equation of normal at ‘t’ is y - xt2 = ct3 !. Two tangents and four normals can be drawn from a point to a rectangular hyperbola.. Mathematics Formulae Explorer - Page 19 of 146.

(158) Mathematics Formulae Explorer. www.MathsHomeWork123.com. 3. Three Dimensional Analytical Geometry The equation of the Sphere whose centre is (a, b, c) and radius ‘r’ is P . The equation of the Sphere has the following three characteristics. • It is of second degree equation in x, y, z • The coefficients of x2, y2, z2 are equal • The terms xy, yz and zx are absent If the coefficients of x2, y2, z2 are each unity, then the coordinates of the centre of the Sphere are.

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(161) C (""%%(! " , (""%%(! " , (""%%(! " PD. and square of the radius is equal to the sum of the squares of the coordinates of the centre minus the constant term.. The equation of a Sphere whose centre is (x1, y1, z1) is P

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(167) P P Two Spheres S1 and S2 whose radii are r1 and r2 touch externally if the distance between their centres is equal to the sum of their radii. d = r1 + r2 The point of contact is the point which divides internally the line joining the centres in the ratio of the radii.. Two Spheres S1 and S2 whose radii are r1 and r2 touch internally if the distance between their centres is equal to the difference of their radii. d = r1 ~ r2. The point of contact is the point which divides externally the line joining the centres in the ratio of the radii. Mathematics Formulae Explorer - Page 20 of 146.

(168) Mathematics Formulae Explorer. www.MathsHomeWork123.com. To find the condition that the plane Ax + By + Cz + D = 0 may touch the Sphere P . ' RP is S. T' :R U S T : . ' R Condition for the two Spheres to cut Orthogonally. P . ' RP and P .

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(179) . The General Equation of a Right Circular Cylinder If the axis of the required cylinder is. 7 -. . V . . PW . and radius is ‘r’ then the. equation of a circular cylinder is - - 7 V P W 7 V P W – - The equation of the Cylinder whose generators intersect the curve ? ) " , P . and parallel to the line. -. . . . . P. . is. -P ? -P P P ) -P " P . The equation of the cylinder whose generators touch the sphere. and are parallel to the line. -. P . . . . . P. . is. - P - P . Mathematics Formulae Explorer - Page 21 of 146.

(180) Mathematics Formulae Explorer. www.MathsHomeWork123.com. The equation of the Cone whose vertex is at the point 7, V, W and whose generators intersect the conic is ) " , P . The equation of the Right Circular Cone whose vertex is at 7, V, W and its axis. at the line. 7 -. . V . . PW . and whose semi-vertical angle ; is. - 7 V P W - 7 V P W # ;. The equations of the enveloping cone whose vertex is at 7, V, W and whose generators touch the sphere P is. 7 V W 7 V P W 7 7 V V WP W. The equation of the tangent plane at the point (x1, y1, z1) to the cone P )P "P ? is.

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(190) Mathematics Formulae Explorer. www.MathsHomeWork123.com. 4. Boundary Value Problems VIBRATION OF STRING Equation and Conditions. Y Y . Y! Y. Initial velocity zero.. Y Y . Y! Y Initial velocity is given. Boundary Conditions , ! , Z! -, ! , Z!. Y, , [ Y! [, ". , ! , Z! -, ! , Z! , . Y, ) Y!. Correct Solution. Most General Solution. , ! :

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(194). _ _! # -. _ _! #% -. ONE DIMENSIONAL HEAT FLOW EQUATION Equation and Conditions. Boundary Conditions. Y. Y .. 7 Y! Y. ., ! , Z!. Y. Y .. 7 Y! Y. ., ! , Z!. Beginning point ‘A’ and Ending point ‘B’ are at zero temperature. Beginning point ‘A’ is at zero temperature and Ending point ‘B’ is at non-zero temperature k.. .-, ! , Z! ., ". .-, ! F, Z! ., ". Correct Solution. Most General Solution. ., ! S # & T #% & (7 & !. ., !. ., ! S # & T #% & (7 & !. ., !. ]. _ 7 \ T #% ( ^

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(197) Mathematics Formulae Explorer. www.MathsHomeWork123.com. SQUARE PLATE - Condition I Conditions of Square Plate. • ., . f(x) y=a 0o x=0. Boundary Conditions. 0o x=a. • ., . • ., . • ., ". y=0 0o. Correct Solution ., :

(198) # & : #% & : (& : (& . Most General Solution ]. ., \ : #% ^

(199). _ _ #% ? . SQUARE PLATE - Condition II Conditions of Square Plate. • ., . 0o y=a 0o x=0. Boundary Conditions. 0o x=a y=0. f(x). Correct Solution ., :

(200) # & : #% & : (& : (& . • ., . • ., . • ., " Most General Solution ]. ., \ : #% ^

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(202) Mathematics Formulae Explorer. www.MathsHomeWork123.com. SQUARE PLATE - Condition III Conditions of Square Plate. • ., . 0o y=a 0o x=0. Boundary Conditions. f(x) x=a. • ., . • ., . • ., ". y=0. 0o. Correct Solution ., :


(204). _ _ #% ? . RECTANGULAR PLATE - Condition I Conditions of Rectangular Plate. • ., . f(x) y=b 0o x=0. Boundary Conditions. 0o x=a y=0. 0o. Correct Solution ., :



(207) Mathematics Formulae Explorer. www.MathsHomeWork123.com. RECTANGULAR PLATE - Condition II Conditions of Rectangular Plate. Boundary Conditions. • ., . 0o. • ., . y=b 0o x=0. • ., . 0o x=a. • ., ". y=0. f(x). Correct Solution ., :


(209). (. _ _ _ ( ( . _ . RECTANGULAR PLATE - Condition III Conditions of Rectangular Plate. • ., . 0o y=b 0o x=0. Boundary Conditions. f(y) x=a y=0. 0o. Correct Solution ., :



(212) Mathematics Formulae Explorer. www.MathsHomeWork123.com. RECTANGULAR PLATE – Infinite Plate - Condition I Conditions of Rectangular Plate - Infinite Plate ∞. Boundary Conditions. • ., . 0o x=0. • .-, . • ., ∞ . 0o x=l. • ., ". y=0. f(x). Correct Solution. Most General Solution. ., :

(213) # & : #% & : (& : (& . ]. ., \ : #% ^

(214). _ _ ( -. RECTANGULAR PLATE – Infinite Plate - Condition II Conditions of Rectangular Plate - Infinite Plate 0o y=l f(y) y=0. ∞. 0o. Correct Solution ., :

(215) # & : #% & : (& : (& . Boundary Conditions. • ., . • ., - . • .∞, . • ., " Most General Solution ]. ., \ : #% ^

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(217) Mathematics Formulae Explorer. www.MathsHomeWork123.com. 5. Co ordinate Geometry Introduction The Modern terms Co-ordinates, abscissa and ordinate were contributed by German Mathematician Gottfried Wilhelm Von Neibliz in 1692. Rene Descartes invented co-ordinate geometry. Distance Formula between two points A(x1, y1) and B(x2, y2). U%#!(,. ST 9

(218)

(219) . Mid-Point Formula between two points A(x1, y1) and B(x2, y2).

(220)

(221) `% @%! " ST C , D. Centroid Formula between three points A(x1, y1), B(x2, y2) and C(x3, y3). :(!% C.

(222) , .

(223) D . Area of the Triangle from the given three points A(x1, y1), B(x2, y2) and C(x3, y3) S( .

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(225)

(226)

(227) . Condition for the three points A(x1, y1), B(x2, y2) and C(x3, y3) to be Collinear

(228)

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(230) Area of the Parallelogram from the given four points A(x1, y1), B(x2, y2), C(x3, y3) and D(x4, y4) S( .

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(235) . Slope (or) Gradient of the Line. If ; is the angle of inclination, then Slope, m = tan ;. Slope of the line joining two points A(x1, y1) and B(x2, y2) Slope, m =.

(236)

(237). Mathematics Formulae Explorer - Page 28 of 146.

(238) Mathematics Formulae Explorer. www.MathsHomeWork123.com. Slope of the line ax+by+c = 0. Slope =. :(""%%(! " . :(""%%(! " . Equation of Straight Line with Slope m and y-intercept c Equation of straight line is y = m x + c. Equation of Straight Line with Slope m and point A(x1, y1) Equation of straight line is y – y1 = m (x – x1) Equation of Straight Line with Slope m and joining two points A(x1, y1) and B(x2, y2) Equation of straight line is.

(239).

(240). .

(241).

(242). Equation of Straight Line with x intercept a and y intercept b Equation of straight line is. . .

(243) . . Condition for two lines to be Parallel. Two lines are Parallel, then their slopes are equal. i.e., m1 = m2 Condition for two lines to be Perpendicular Two lines are Perpendicular, then their product of their slopes gives -1 i.e., m1 x m2 = -1 Equation of Straight Lines with different cases •. Any line parallel to ax + by + c = 0 is ax + by + k = 0 (differ only by constant). •. Any line parallel to x-axis is y=k ( k is constant). •. Any line parallel to y-axis is x = c ( c is constant). •. The line which is perpendicular to the line ax + by + c = 0 is of the form bx – ay + k = 0. Mathematics Formulae Explorer - Page 29 of 146.

(244) Mathematics Formulae Explorer. www.MathsHomeWork123.com. Circumcentre, Centroid and Orthocentre Circumcentre : The perpendicular bisector of the sides of a triangle are concurrent. The point of concurrence is called circumcentre. Centroid of a triangle : The medians of a triangle meet at a point. This point is known as centroid. Orthocentre of a triangle : The altitudes of a triangle meet at a point. This point is called Orthocentre. Slope of both axes •. The Slope of x-axis = 0. •. The Slope of y-axis = not defined. Concurrency of Three Lines. Condition that the lines

(245)

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(247) , and may be concurrent if,

(248)

(249)

(250)

(251)

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(253) . Intersection of Two Straight Lines. The two lines if not parallel in a plane intersect in a unique point.. Mathematics Formulae Explorer - Page 30 of 146.

(254) Mathematics Formulae Explorer. www.MathsHomeWork123.com. 6. Commercial Arithmetic Basic Definitions. Percentage . % . .

(255) .

(256) % . Mb # &((!)( " . .

(257) . Profit and Loss . Profit = Selling Price (S.P) – Cost Price (C.P). . Loss = Cost Price (C.P) – Selling Price (S.P). . Selling Price = Cost Price + Profit. . Selling Price = Cost Price - Loss. . Cost Price = Selling Price – Profit. . Cost Price = Selling Price + Loss. . Profit (in percent) =. . Loss (in percent) =. . Selling Price = Cost Price + x% of Cost Price,. ,. if Profit is x%.. .

(258) Cost Price = Selling Price x d e, if Profit is x%.

(259) . . Selling Price = Cost Price - x% of Cost Price,. ,. if Loss is x%.. . A!- @"%! :#! @%(. A!- G##. :#! @%(. x 100 =. x 100 =. c.@:.@. :#! @%(. :.@c.@. :#! @%(. x100. x100.

(260) Cost Price = Selling Price x d e, if Loss is x%.


(262) Mathematics Formulae Explorer. www.MathsHomeWork123.com. Discount and Market Price . Discount = Marked Price – Actual Selling Price. . Discount in Percent = Marked Price –. . Actual Selling Price = Marked Price – Discount = Marked Price -. . Marked Price = d.

(263) U%#.! %. • • • • • •. `F( @%(. %U%#.!

(264) . x 100. x Marked Price. e c(--%) @%(. Successive (2nd) discount is calculated on the balance after deduction of the first discount from the marked price and so on.. Simple Interest •.

(265) . S!.- c(--%) @%(. Simple Interest (S.I) =. @fg

(266) . = PNi, where P is the Principal, N is the Period. in years and R% is the rate of interest for 1 year. % unit principal for one year. @. h

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(268) . f. g.

(269) . = interest for. fg. h

(270) @g. @f. Amount (A) = Principal + Interest. @. S

(271) .

(272) fg. Interest = Amount - Principal. Mathematics Formulae Explorer - Page 32 of 146.

(273) Mathematics Formulae Explorer. www.MathsHomeWork123.com. Compound Interest (C.I) f. •. Compound Interest (C.I) = @ d

(274)

(275) e @, where P is the Principal, N is the Period in years and R% is the rate percent annually.. •. Amount, A = @ d

(276) . g. g.

(277) . e. f. •. Principal = Amount – Compound Interest. •. Difference between C.I and S.I for 2 years = @ d. •. e

(278) g. Difference between C.I and S.I for 3 years =

(279) d

(280) e @g. g. Recurring Deposit (R.D). Recurring Deposit is a special type of deposit in which a person deposits a fixed sum every month over a period of years and receives a large sum at the end of the specified number of years. Since the deposit is made month after month, it is called Recurring Deposit. Recurring Deposits are also known as Cumulative Term Deposits. The amount deposited every month is called the Monthly Deposit.. Total Interest =. @fg

(281) . , where N =.

(282)

(283) . ,. P be the Monthly Instalments, R % be the rate of Interest and ‘n’ be the number of monthly instalments.. Amount Due = Amount Deposited + Total Interest. Mathematics Formulae Explorer - Page 33 of 146.

(284) Mathematics Formulae Explorer. www.MathsHomeWork123.com. Fixed Deposit Fixed Deposit are deposits for a fixed period of time and the depositor can withdraw his money only after the expiry of the fixed period. It is also known as Term Deposits. However, in the case of necessity, the depositor can get his fixed deposit terminated earlier to get a loan from the bank under terms laid down by the bank. There are two types of fixed deposits, namely . Short Term Deposits Long Term Deposits. Short Term Fixed Deposits are accepted by the banks for a short period ranging from 46 days to one year. The interest paid on this deposit is Simple Interest. Long Term Fixed Deposits are accepted by the banks for a period of one year or more. The interest paid on this type of deposit is Compound Interest.. Quarterly Interest =. @g. . Half Yearly Interest =. @g. . Mathematics Formulae Explorer - Page 34 of 146.

(285) Mathematics Formulae Explorer. www.MathsHomeWork123.com. 7. Complex Numbers The Complex Number System •. A Complex number is of the form a+ib, where ‘a’ and ‘b’ are real numbers and ‘I’ is called the imaginary unit, having the property i2 = -1.. •. If z = a+ib then ‘a’ is called the real part of z, denoted by Re(z) and ‘b’ is called the imaginary part of z and is denoted by Im(z).. •. If z = a+ib is a complex number then the negative of z is denoted by –z and it is defined as –z = -a + i (-b).. •. Basic Algebraic Operations with Complex Numbers (a + ib) + (c + id) = (a + c) + i (b + d) (a + ib) - (c + id) = (a - c) + i (b - d) (a + ib) (c + id) = (ac - bd) + i (ad + bc). •. If z = a + i b, then the conjugate of z is denoted by Pi and. •. Properties of Complex Numbers. is defined by Pi %.. PPi % % :j.)!( " Pi P. %. (. , Pi P. Z is real / the imaginary part is zero. g(P . hP . PPi. PPi %. Mathematics Formulae Explorer - Page 35 of 146.

(286) Mathematics Formulae Explorer. www.MathsHomeWork123.com. Conjugate of the sum is the sum of their conjugates iiiiiiiiii i

(287) Pi P

(288) P P. Conjugate of the product of two complex numbers is the product of their conjugates. iiiiii i

(289) Pi P

(290) P P. The conjugate of the quotient of two complex numbers is the quotient of their conjugates.. P iii i P. •. iiiii P P iii

(291)

(292) P. P iii. The Modulus (or) Absolute value of z = a+ib is denoted by |P| is defined by √ . • The Amplitude (or) Argument of z = a+ib is denoted by arg z or arg z is defined by ; !

(293) . •. . It is obvious that |Pi| |P|. Also, |P| √PPi. •. g(P [ |P| and hP [ |P|. •. The Modulus of a product of two complex numbers is equal to the. •. The above result can be extended to any finite number of complex. •. The Modulus of a quotient of two complex numbers is equal to the. product of their moduli. |P

(294) P | |P

(295) ||P |. numbers. i.e., |P

(296) P … . P | |P

(297) ||P ||P | … |P | quotient of their moduli. < < P

(298) P. |P

(299) |. |P |. Mathematics Formulae Explorer - Page 36 of 146.

(300) Mathematics Formulae Explorer. •. www.MathsHomeWork123.com. Triangle Inequality The Modules of sum of two complex numbers is always less than or equal to the sum of their moduli.. |P

(301) P | [ |P

(302) | |P |. |P

(303) P | [ |P

(304) | |P |. |P

(305) P l … P | [ |P

(306) | |P | l |P | •. The Modulus of the difference of two complex numbers is always greater. •. Polar form of a Complex Number P % # ; % #% ;. •. than or equal to the difference of their moduli. |P

(307) P | m |P

(308) | |P |. For any two complex numbers P

(309) P. |P

(310) P | |P

(311) ||P |. )P

(312) P ) P

(313) ) P )

(314) ) P

(315) ) P P P. •. The above result can be extended to any finite number of complex numbers .. |P

(316) P … … . P | |P

(317) ||P | … … . . |P |. )P

(318) P … … … . P ) P

(319) ) P l … … . . ) P. •. The Exponential form of a Complex Number (%; is known as Euler’s. Formula and is defined by (%; # ; % #% ;. Mathematics Formulae Explorer - Page 37 of 146.

(320) Mathematics Formulae Explorer. General Rule for determining the argument ;. Let z = a + ib where a, b n R. In First Quadrant,. ; 7. In Third Quadrant,. ; _ 7. www.MathsHomeWork123.com. Take 7 !

(321) || ||. In Second Quadrant, ; _ 7. In Fourth Quadrant, ; 7. Both cos ; and sin ; are positive.. Z lies in the first quadrant.. Sin ; is positive and cos ; is negative. Z lies in the second quadrant.. Both cos ; and sin ; are negative.. Z lies in the third quadrant.. Sin ; is negative and cos ; is positive. Z lies in the fourth quadrant.. ; 7. ; _ 7. ; _ 7. ; 7. Theorem For any polynomial equation P(x) = 0 with real coefficients, imaginary (complex) roots occur in conjugate pairs. De Moivre’s Theorem For any rational number n, opq rs t qtr rs is the value or one of the values of # ; % #% ; Mathematics Formulae Explorer - Page 38 of 146.

(322) Mathematics Formulae Explorer. www.MathsHomeWork123.com. Roots of a Complex Number Working Rule to find the nth roots of a Complex Number •. Write the given number in polar form. •. Add 2k_ to the argument. •. Apply De Moivre’s theorem. •. Put k=0, 1, 2,3, …upto (n-1). nth roots of unity • •. R

(323). Sum of the roots is zero.. •. The roots are in Geometric Progression with common ratio w.. •. The arguments are in Arithmetic Progression with common difference. •. Product of the roots =

(324)

(325). _ . Cube Roots of Unity •. If R .

(326) %√. , then R .

(327) %√. •. The sum of the cube roots of unity is zero.. •. w3 = 1. 1+w+w2 = 0.. Fourth Roots of Unity •. 1+w+w2 +w 3= 0.. •. w4 = 1. •. The values of w used in cube roots of unity and in fourth roots of unity are different. Mathematics Formulae Explorer - Page 39 of 146.

(328) Mathematics Formulae Explorer. www.MathsHomeWork123.com. 8. Data Analysis Statistics is the study of the methods of collecting, organizing and analyzing quantitative data, and drawing conclusions. The data are collected on samples from various populations of people, animals and things by different methods such as observations, interviews, etc. Statistics is used in almost every field such as business, education, science, psychology, research, etc. The word ‘data’ is the plural form of datum, which means facts and figures. Data Data represent factual information (in the form of measurements or statistics) which is used as a basis for reasoning, discussion or calculation. Data are classified as either Primary or Secondary. Primary Data Primary data are the data which are collected directly for a specific purpose for the first time and they are original in character. Examples : Questionnaires, Interviews, etc., Secondary Data Secondary data are data already collected, analyzed and presented in written form ready for people to use. Examples : Government reports, books, articles, maps, etc., Types of Data Data can be qualitative or quantitative. Names of persons, marital status, etc., are examples of qualitative data. Quantitative Data Quantitative data are measurements expressed in terms of numbers. Income of individuals, production of a car company, exports in units of a garment company, marks of students, etc., are all quantitative data. Quantitative data can further be classified as continuous data and discrete data. Continuous Data : Takes numerical values within a certain range. Example : Height of a person. Discontinuous (or) Discrete Data : Takes only whole-number values. Example : The number of boys in each class can be expressed only in whole numbers.. Mathematics Formulae Explorer - Page 40 of 146.

(329) Mathematics Formulae Explorer. www.MathsHomeWork123.com. Displaying Data Tables, Charts and Graphs are examples of visual representation of data. Graphs or Charts show the relationship between changing things and are used to make facts clearer and more understandable. Line Graph A Line Graph is used to show continuous data. The dependent data is plotted along the y-axis and the independent data along the x-axis. Multiple-Line Graph A multiple-line graph can effectively compare similar data over the same period of time. Pie Chart A pie chart is a circular chart divided into segments. Each segment illustrates relative magnitudes or frequencies. It shows the component parts of a whole. A pie chart uses percentages to compare information since they are the easiest way to represent a whole (100%). In a Pie chart, the arc length, central angle and area of each segment is proportional to the quantity it represents. Exploded Pie Chart A chart with one or more segments separated from the rest of the disc is called an exploded pie chart. Formation of Frequency Tables Classification and Tabulation Collection of data in the form of numbers alone will not help us to make decisions or form conclusions. Since just a huge collection of numbers does not have any meaning, it is necessary to classify the numbers as values and pictures before presentation. Classification is the process of grouping data according to their common characteristics. Tabulation is the process of arranging the classified data in tabular form. Notes •. The number of times a particular observation or a variable ‘x’ occurs in a data set is called its frequency which is denoted by ‘f’. Mathematics Formulae Explorer - Page 41 of 146.

(330) Mathematics Formulae Explorer. www.MathsHomeWork123.com. •. Frequency distributions show the actual number of observations falling in each range of observations.. •. In a continuous distribution the data are obtained by measurement.. •. The vertical bar ‘|’ which represents each occurrence of a variable ‘x’ or observation is called a tally mark.. •. The mid-value of a class interval is called its class mark.. •. Class boundaries are actual or true limits of a class interval in a grouped distribution table and are continuous.. Measures of Central Tendency The classification and tabulation of statistical data is a process of condensing the entire data. The graphs / charts give a visual presentation and make the comparisons easier. But for analysis of given numerical data, some description of the given data is needed. The statistical average is a numerical value around which the greatest proportion of the data concentrates. For example, if we say in a class of 40 students, the mathematics marks vary from 40 to 95, but most of them secured 70 marks then 70 is the statistical average marks of the class. Such values are called measures of central tendency. The three important measures of central tendency are • • •. Arithmetic mean (or) Average Median Mode. Arithmetic Mean (A.M) The Arithmetic Mean of a collection of data is a measure of central tendency and it helps in interpreting the data. The arithmetic mean (or) AM is commonly known as the mean or the average of a given set of data. Arithmetic Mean (A.M) of Ungrouped Data. The formula used is S. `,. Median of Ungrouped Data. u . c. " v#('!%#. f.( " v#('!%#. . ∑ . Median is the middle value or the mean of the middle two values, when a set of observed data is arranged in numerical order. Median divides the distribution into two equal halves such that there are as many observations less than it as there are greater than it. Mathematics Formulae Explorer - Page 42 of 146.

(331) Mathematics Formulae Explorer. www.MathsHomeWork123.com. In a set of N observations, when N is odd, the arranged data in the numerical order is the median.. !?. f

(332) !?. . In a set of N observations, when N is even, the average of.

(333) f. median.. and. . observation of. f !?. . observation. observation of the arranged data in numerical order is the. Mode of Ungrouped Data Mode is the data which occurs most frequently in the given set of observations (data). It is possible to have more than one mode. Range of Ungrouped Data The difference between the highest and lowest values of the observed data is called the Range.. Mathematics Formulae Explorer - Page 43 of 146.

(334) Mathematics Formulae Explorer. www.MathsHomeWork123.com. 9. Determinants Singular / Non Singular. A Square Matrix A is said to be Singular if |S| . Otherwise it is said to be Non-Singular. Adjoint of A Let A = [ aij ] be a square matrix of order n. Let Aij be the cofactor of aij. The adjoint of A is nothing but the transpose of the cofactor matrix [Aij ] of A. Theorem If A is a Square matrix of order n, then. A (Adjoint A) = |S| In = (adjoint A) A. where In is the identity matrix of order n.. Theorem If a matrix A possesses an inverse then it must be unique. Theorem If A is a non singular matrix, there exists an inverse which is given by. S

(335) .

(336). |S|. j%! S. Reversal Law for Inverses If A, B are any two non-singular matrices of the same order, then AB is also non-singular and ST

(337) T

(338) S

(339). Reversal Law for Transposes. If A and B are matrices conformable to multiplication, then STA TA SA Inverses and Transposes. For any non-singular matrix A, SA

(340) S

(341) A Matrix Inversion Method. For a system of n linear non-homogeneous equations in ‘n’ unknowns is represented by AX = B, then its unique solution is given by X = A-1B. Mathematics Formulae Explorer - Page 44 of 146.

(342) Mathematics Formulae Explorer. www.MathsHomeWork123.com. Properties of Determinants •. The Value of a determinant is unaltered by interchanging its rows and columns. •. If any two rows (columns) of a determinant are interchanged the determinant changes its sign but its numerical value is unaltered.. •. If two rows (columns) of a determinant are identical then the value of the determinant is zero.. •. If every element in a row (or column) of a determinant is multiplied by a constant “K” then the value of the determinant is multiplied by K.. •. If every element in any row (column) can be expressed as the sum of two quantities then given determinant can be expressed as the sum of two determinants of the same order with the elements of the remaining rows (columns) of both being the same.. •. A determinant is unaltered when to each element of any row (column) is added to those of several other rows (columns) multiplied respectively by constant factors.. Mathematics Formulae Explorer - Page 45 of 146.

(343) Mathematics Formulae Explorer. www.MathsHomeWork123.com. Rank of a Matrix The matrix A is said to be of rank r, if •. A has atleast one minor of order r which does not vanish. •. Every minor of A of order (r+1) and higher order vanishes. In other words, the rank of a matrix is the order of any highest order non vanishing minor of the matrix. The rank of A is denoted by xS.. The rank of an m x n matrix A cannot exceed the minimum of m and n. i.e., xS [ % , . Elementary Transformation on a Matrix Let A be an mxn matrix. An elementary row (column) operation on A is of any one of the following three types. • • •. The interchange of any two Ith and jth rows (columns). i.e., g% y gj. Multiplication of a Ith row (column) by a non zero constant C. i.e., g% z : g% Addition of any multiple of one row (column) with any other row (column). i.e., g% y g% Fgj. Echelon Form A matrix A (of order m x n) is said to be in Echelon form (Triangular form) if • Every row of A which has all its entries 0 occurs below every row which has a non-zero entry. •. The first non zero entry in each non zero row is 1.. •. The Number of zeros before the first non zero element in a row is less than the number of such zeros in the next row.. Note : • Any matrix can be brought to Echelon matrix form. • The Rank of a matrix in Echelon form is equal to number of non zero rows of the matrix. Mathematics Formulae Explorer - Page 46 of 146.

(344) Mathematics Formulae Explorer. www.MathsHomeWork123.com. Cramer’s Rule Method ( Determinant Rule) For a system of non-homogeneous equation with 3 unknowns, the system is Consistent and has Unique Solution, if %" ∆ . Solution is . . ∆ ∆. , . ∆ ∆. P . ∆P ∆. .. Consistency for a given System of Equations by using Rank Method • • •. xS xS, T , !?( (|.!%# ( %#%#!(! ?# #-.!%.. xS xS, T f.( " .FR# , !?( (|.!%# ( #%#!(! ?'( %"%%!( .( " #-.!%#. xS xS, T f.( " .FR# , !?( (|.!%# ( #%#!(! ?'( .%|.( #-.!%.. Consistency for a System of Homogeneous Equation A System of Homogeneous equations is always consistent. • •. xS xS, T f.( " .FR# , !?( !%'%- #-.!% %# !?( .%|.( #-.!% xS xS, T f.( " .FR# , !?( ##!( ?# !%'%- #-.!%. Mathematics Formulae Explorer - Page 47 of 146.

(345) Mathematics Formulae Explorer. www.MathsHomeWork123.com. 10. Differential Calculus Derivatives of Standard Functions :. #!! ( ( . .

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(399) Mathematics Formulae Explorer. www.MathsHomeWork123.com. Curvature of Curve The rate of bending of a curve in any interval is called the Curvature of the curve in that interval. Cartesian Curve y = f(x) Polar Curve r = f(ϴ ϴ) 1 . 1 . Sin Ψ =. cos Ψ =. 1 . sin =. . . tan =. . . cos =. . . tan Ψ =. . Radius of Curvature. . . . p= r sin . Parametric Form. Implicit Form. The reciprocal of the Let x=f(t) and y=g(t) be the Let f(x,y)=o be the implicit Curvature of a curve at parametric equations of form of the given curve. any point is called the the given curve. Radius of Curvature at the point and is denoted by x.

(400) . . . /. . Polar Form. Let r = f(ϴ ϴ) be the given curve in polar coordinates.

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(408) , . . u x u . Local Maxima and Minima for functions of one variable. Given y=f(x), (i) if f/(c)=0 and f//(c)>0, then f has a local minimum at c. (ii) if f/(c)=0 and f//(c)<0, then f has a local maximum at c. Mathematics Formulae Explorer - Page 49 of 146.

(409) Mathematics Formulae Explorer. Maxima and Minima for functions of two variables Necessary Condition Let fx(a,b)=0 and. fy(a,b)=0. www.MathsHomeWork123.com. Method of Lagrangian Multiplier. To find the maximum and minimum values of f(x,y,z) where x,y,z are subject to a constraint equation g(x,y,z)=o, we define a function. Sufficient Condition F(x,y,z) = f(x,y,z) + λ g(x,y,z), If fx(a,b)=0, fy(a,b)=0 and fxx(a,b)=A, fxy(a,b)=B, fyy(a,b)=C then i) f(a,b) is maximum value if AC-B2 > 0 and A<0 (or B<0) ii) f(a,b) is minimum value if AC-B2 > 0 and A>0 (or B>0) iii) f(a,b) is not an extremum if AC-B2 < 0 and iv) If AC-B2 > 0, the test is inconclusive.. where λ is called Lagrange Multiplier which is independent of x,y,z, The necessary condition for a maximum or minimum are Y" , Y. Y" , Y. Y" YP. Solving the above equations for four unknowns λ, x, y, z, we obtain the point (x,y,z). The point may be a maxima, A function f(x,y) at (a,b) or f(a,b) is said minima or neither which is decided by the to be a Stationary Value of f(x,y) if physical consideration. fx(a,b)=0 and fy(a,b)=0.. Stationary Value. Jacobians. Properties of Jacobian. If u1, u2, u3, …….un are functions of n 1. If u and v are the functions of x and y, Y.,' Y, variables x1, x2, x3, …xn, then the then 1. Y, Y.,' Jacobian of the transformation from x1, x2, x3, …xn to u1, u2, u3, …….un is 2. If u,v are the functions of x,y and x,y defined by are themselves functions of r,s then Y.

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