{"id":22069,"date":"2024-01-10T15:59:30","date_gmt":"2024-01-10T10:29:30","guid":{"rendered":"https:\/\/apboardsolutions.in\/?p=22069"},"modified":"2024-01-11T12:23:31","modified_gmt":"2024-01-11T06:53:31","slug":"inter-2nd-year-maths-2a-theory-of-equations-formulas","status":"publish","type":"post","link":"https:\/\/apboardsolutions.in\/inter-2nd-year-maths-2a-theory-of-equations-formulas\/","title":{"rendered":"Inter 2nd Year Maths 2A Theory of Equations Formulas"},"content":{"rendered":"<p>Use these <a href=\"https:\/\/apboardsolutions.in\/inter-2nd-year-maths-2a-formulas\/\">Inter 2nd Year Maths 2A Formulas<\/a> PDF Chapter 4 Theory of Equations to solve questions creatively.<\/p>\n<h2>Intermediate 2nd Year Maths 2A Theory of Equations Formulas<\/h2>\n<p>\u2192 If n is a non-negative integer and a<sub>0<\/sub>, a<sub>1<\/sub>, a<sub>2<\/sub>, &#8230;&#8230;&#8230;.. a<sub>n<\/sub> are real or complex numbers and a<sub>0<\/sub> \u2260 0, then the expression f(x) = a<sub>0<\/sub>x<sup>n<\/sup> + a<sub>1<\/sub>x<sup>n &#8211; 1<\/sup> + a<sub>2<\/sub>x<sup>n &#8211; 2<\/sup> + &#8230;&#8230;. + a<sub>n<\/sub> is called a polynomial in x of degree n.<\/p>\n<p>\u2192 f(x) = a<sub>0<\/sub>x<sup>n<\/sup> + a<sub>1<\/sub>x<sup>n &#8211; 1<\/sup> + a<sub>2<\/sub>x<sup>n &#8211; 2<\/sup> + &#8230;&#8230;. + a<sub>n<\/sub> = 0 is called a polynomial equation in x of degree n (a<sub>0<\/sub> \u2260 0). Every non-constant polynomial equation has atleast one root.<\/p>\n<p>\u2192 If f(\u03b1) = 0 then \u03b1 is called a root of the equation f(x) = 0.<\/p>\n<p>\u2192 If f(\u03b1) = 0 then (x &#8211; \u03b1) is a factor of f(x).<\/p>\n<p><img loading=\"lazy\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/07\/AP-Board-Solutions.png\" alt=\"Inter 2nd Year Maths 2A Theory of Equations Formulas\" width=\"196\" height=\"17\" \/><\/p>\n<p><strong>Relation between roots and coefficients of an equation:<\/strong><br \/>\n\u2192 If \u03b1 \u03b2 \u03b3 are the roots of x<sup>3<\/sup> + p<sub>1<\/sub>x<sup>2<\/sup> + p<sub>2<\/sub>x + p<sub>3<\/sub> = 0 then sum of the roots s<sub>1<\/sub> = \u03b1 + \u03b2 + \u03b3 = &#8211; p<sub>1<\/sub>.<br \/>\nSum of the products of two roots taken at a time s<sub>2<\/sub> = \u03b1\u03b2 + \u03b2\u03b3 + \u03b3\u03b1 = p<sub>2<\/sub>.<br \/>\nProduct of all the roots, s<sub>3<\/sub> = \u03b1\u03b2\u03b3 = &#8211; p<sub>3<\/sub>.<\/p>\n<p>\u2192 If \u03b1, \u03b2, \u03b3, \u03b4 are the roots of x<sup>4<\/sup> + p<sub>1<\/sub>x<sup>3<\/sup> + p<sub>2<\/sub>x<sup>2<\/sup> + p<sub>3<\/sub>x + p<sub>4<\/sub> = 0 then sum of the roots s<sub>1<\/sub> = \u03b1 + \u03b2 + \u03b3 + \u03b4 = &#8211; p<sub>1<\/sub>.<br \/>\nSum of the products of roots taken two at a time<br \/>\ns<sub>2<\/sub> = \u03b1\u03b2 + \u03b1\u03b3 + \u03b1\u03b4 + \u03b2\u03b3 + \u03b2\u03b4 + \u03b3\u03b4 = p<sub>2<\/sub>.<br \/>\nSum of the products of roots taken three at a time .<br \/>\ns<sub>3<\/sub> = \u03b1\u03b2\u03b3 + \u03b2\u03b3\u03b4 + \u03b3\u03b4\u03b1 + \u03b4\u03b1\u03b2 = &#8211; p<sub>3<\/sub>.<br \/>\nProduct of the roots, s<sub>4<\/sub> = \u03b1\u03b2\u03b3\u03b4 = p<sub>4<\/sub>.<\/p>\n<p>\u2192 For a cubic equation, when the roots are<\/p>\n<ul>\n<li>In A.P., then they are taken as a &#8211; d, a, a + d.<\/li>\n<li>In G.P., then they are taken as \\(\\frac{a}{r}\\), a, ar.<\/li>\n<li>In H.P., then they are taken as \\(\\frac{1}{a-d}, \\frac{1}{a^{\\prime}}, \\frac{1}{a+d}\\).<\/li>\n<\/ul>\n<p>\u2192 For a bi quadratic equation, if the roots are<\/p>\n<ul>\n<li>In A.P., then they are taken as a &#8211; 3d, a &#8211; d, a + d, a + 3d.<\/li>\n<li>In C.P., then they are taken as \\(\\frac{1}{a-3 d}, \\frac{1}{a-d}, \\frac{1}{a+d}, \\frac{1}{a+3 d}\\).<\/li>\n<\/ul>\n<p>\u2192 In an equation with real coefficients, imaginary roots occur in conjugate pairs.<\/p>\n<p>\u2192 In an equation with rational coefficients, irrational roots occur in pairs of conjugate surds.<\/p>\n<p>\u2192 The equation whose roots are those of the equation f(x) = 0 with contrary signs is f(- x) = 0.<\/p>\n<p>\u2192 The equation whose roots are multiplied by kfa 0) of those of the proposed equation f(x) = 0 is f \\(\\left(\\frac{x}{k}\\right)\\) = 0.<\/p>\n<p>\u2192 The equation whose roots are reciprocals of the roots of f(x) = 0 is f \\(\\left(\\frac{1}{x}\\right)\\) = 0.<\/p>\n<p>\u2192 The equation whose roots are exceed by h than those of f(x) = 0 is f(x &#8211; h) = 0.<\/p>\n<p>\u2192 The equation whose roots are diminished by h than those of f(x) 0 is f(x + h) = 0.<\/p>\n<p>\u2192 The equation whose roots are the square of the roots of f(x) = 0 is obtained by eliminating square root from f(\u221ax) = 0.<\/p>\n<p><img loading=\"lazy\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/07\/AP-Board-Solutions.png\" alt=\"Inter 2nd Year Maths 2A Theory of Equations Formulas\" width=\"196\" height=\"17\" \/><\/p>\n<p>\u2192 If f(x) = p<sub>0<\/sub>x<sup>n<\/sup> + p<sub>1<\/sub>x<sup>n &#8211; 1<\/sup> + p<sub>2<\/sub>x<sup>n &#8211; 2<\/sup> + &#8230;&#8230;. + p<sub>n<\/sub> = 0 then to eliminate the second term,<br \/>\nf(x) = 0 can be transformed to f(x + h) = 0 where h = \\(\\frac{-p_{1}}{n \\cdot p_{0}}\\).<\/p>\n<p>\u2192 If an equation is unaltered by changing x into \\(\\frac{1}{x}\\) then it is a reciprocal equation.<\/p>\n<p>\u2192 A reciprocal equation f(x) = p<sub>0<\/sub>x<sup>n<\/sup> + p<sub>1<\/sub>x<sup>n &#8211; 1<\/sup> + &#8230;&#8230; + p<sup>n<\/sup> = 0 is said to be a reciprocal equation of first class p<sub>i<\/sub> = p<sub>n &#8211; i<\/sub> for all i.<\/p>\n<p>\u2192 A reciprocal equation f(x) = p<sub>0<\/sub>x<sup>n<\/sup> + p<sub>1<\/sub>x<sup>n &#8211; 1<\/sup> + &#8230;&#8230; + p<sup>n<\/sup> = 0 is said to be a reciprocal equation of second class if p<sub>i<\/sub> = &#8211; p<sub>n &#8211; i<\/sub> for all i.<\/p>\n<p>\u2192 For an odd degree reciprocal equation of class one, &#8211; 1 is a root and for an odd degree reciprocal equation of class two, 1 is a root.<\/p>\n<p>\u2192 For an even degree reciprocal equation of class two, 1 and &#8211; 1 are roots.<\/p>\n<p>\u2192 If f(x) = 0 is an equation of degree &#8216;n&#8217; then to eliminate r<sup>th<\/sup> term, f(x) = 0 can be transformed to f(x + h) = 0 where h is a constant such that f<sup>(n &#8211; r + 1)<\/sup>(h) = 0 i.e.,(n &#8211; r + 1)<sup>th<\/sup> derivative of f(h) is zero.<\/p>\n<p>\u2192 Every n<sup>th<\/sup> degree equation has exactly n roots real or imaginary.<\/p>\n<p>\u2192 Relation between, roots and coefficients of an equation.<\/p>\n<p>(i) If \u03b1, \u03b2, \u03b3 are the roots of x<sup>3<\/sup> + p<sub>1<\/sub>x<sup>2<\/sup> + p<sub>2<\/sub>x + p<sub>3<\/sub> = 0 the sum of the roots s<sub>1<\/sub> = \u03b1 + \u03b2 + \u03b3 = -p<sub>1<\/sub>.<br \/>\nSum of the products of two roots taken at a time s<sub>2<\/sub> = \u03b1\u03b2 + \u03b2\u03b3 + \u03b3\u03b1 = -p<sub>2<\/sub><br \/>\nProduct of all the roots, s<sub>3<\/sub> = \u03b1\u03b2\u03b3= &#8211; p<sub>3<\/sub>.<\/p>\n<p>(ii) If \u03b1, \u03b2, \u03b3, \u03b4 are the roots of x<sup>4<\/sup> + p<sub>1<\/sub>x<sup>3<\/sup> + p<sub>2<\/sub>x<sup>2<\/sup> + p<sub>3<\/sub>x + p<sub>4<\/sub> = 0 then<\/p>\n<ul>\n<li>Sum of the roots s<sub>1<\/sub> = a+P+y+S = -p<sub>1<\/sub>.<br \/>\ns<sub>2<\/sub> = \u03b1\u03b2 + \u03b1\u03b3 + \u03b1\u03b4 + \u03b2\u03b1 + \u03b2\u03b4 + \u03b3\u03b4 = p<sub>2<\/sub>.<\/li>\n<li>Sum of the products of roots taken three at a time<br \/>\ns<sub>3<\/sub> = \u03b1\u03b2\u03b3 + \u03b2\u03b3\u03b4 + \u03b3\u03b4\u03b1 + \u03b4\u03b1\u03b2 = &#8211; p<sub>3<\/sub>.<\/li>\n<li>Product of the roots, s<sub>4<\/sub> = \u03b1\u03b2\u03b3\u03b4 = p<sub>4<\/sub><\/li>\n<\/ul>\n<p>\u2192 For the equation x<sup>n<\/sup> + p<sub>1<\/sub>x<sup>n-1<\/sup> + p<sub>2<\/sub>x<sup>n-2<\/sup> + &#8230;&#8230;&#8230; + p<sub>n<\/sub> = 0<\/p>\n<ul>\n<li>\u03a3 \u03b1<sub>2<\/sub> = p<sub>1<\/sub><sup>2<\/sup> &#8211; 2p<sub>2<\/sub><\/li>\n<li>\u03a3 \u03b1<sub>3<\/sub> = -p<sub>1<\/sub><sup>3<\/sup> + 3p<sub>1<\/sub>p<sub>2<\/sub> &#8211; 3p<sub>3<\/sub><\/li>\n<li>\u03a3 \u03b1<sub>4<\/sub> =p<sub>1<\/sub><sup>4<\/sup> &#8211; 4p<sub>1<\/sub><sup>2<\/sup>p<sup>2<\/sup> + 2p<sub>2<\/sub><sup>2<\/sup> + 4p<sub>1<\/sub>p<sub>3<\/sub> &#8211; 4p<sub>4<\/sub><\/li>\n<li>\u03a3 \u03b1<sub>2<\/sub>\u03b2 = 3p<sub>3<\/sub> &#8211; p<sub>1<\/sub>p<sub>2<\/sub><\/li>\n<li>\u03a3 \u03b1<sub>2<\/sub>\u03b2\u03b3 = p<sub>1<\/sub>p<sub>3<\/sub> &#8211; 4p<sub>4<\/sub><\/li>\n<\/ul>\n<p>Note: For the equation x<sup>3<\/sup> + p<sub>1<\/sub>x<sup>2<\/sup> + p<sub>2<\/sub>x + p<sub>3<\/sub> = 0 \u03a3\u03b1<sub>2<\/sub>\u03b2<sub>2<\/sub> \u2014 p<sub>2<\/sub> -2p<sub>1<\/sub>p<sub>3<\/sub><\/p>\n<p>\u2192 To remove the second term from a nth degree equation, the roots must be diminished by \\(\\frac{-\\mathrm{a}_{1}}{\\mathrm{na}_{0}}\\) and the resultant equation will not contain the term with x<sup>n-1<\/sup>.<\/p>\n<p>\u2192 If \u03b1<sub>1<\/sub> , \u03b1<sub>2<\/sub> &#8230;&#8230;&#8230;&#8230;&#8230;&#8230;. , \u03b1<sub>n<\/sub> are the roots of f(x) = 0, the equation<\/p>\n<ul>\n<li>Whose roots are \\(\\) is f\\(\\left(\\frac{1}{x}\\right)\\) = 0<\/li>\n<li>Whose roots are k\u03b1<sub>1<\/sub>, k\u03b1<sub>2<\/sub> &#8230;,k\u03b1<sub>n<\/sub> is f\\(\\left(\\frac{x}{h}\\right)\\) = 0<\/li>\n<li>Whose roots are \u03b1<sub>1<\/sub> &#8211; h, \u03b1<sub>2<\/sub> &#8211; h, &#8230;. \u03b1<sub>n<\/sub> &#8211; h is f(x + h) = 0.<\/li>\n<li>Whose roots are \u03b1<sub>1<\/sub> + h, \u03b1<sub>2<\/sub> + h, &#8230;&#8230;&#8230;&#8230;.. \u03b1<sub>n<\/sub> + h is f(x &#8211; h) = 0<\/li>\n<li>Whose roots are \u03b1<sub>1<\/sub><sup>2<\/sup>, \u03b1<sub>2<\/sub><sup>2<\/sup>&#8230;. \u03b1<sub>1<\/sub><sup>2<\/sup> is f (f\u221ay) = 0<\/li>\n<\/ul>\n<p>\u2192 In any equation with rational coefficients, irrational roots occur in conjugate pairs.<\/p>\n<p>\u2192 In any equation with real coefficients, complex roots occur in conjugate pairs.<\/p>\n<p><img loading=\"lazy\" src=\"https:\/\/apboardsolutions.com\/wp-content\/uploads\/2022\/07\/AP-Board-Solutions.png\" alt=\"Inter 2nd Year Maths 2A Theory of Equations Formulas\" width=\"196\" height=\"17\" \/><\/p>\n<p>\u2192 If \u03b1 is r &#8211; multiple root of f(x) = 0, then a is a (r &#8211; 1) &#8211; multiple root of f<sup>1<\/sup>(x) = 0 and (r-2) &#8211; Multiple root of f <sup>11<\/sup>(x) = 0 and non multiple root of f<sup>r-1<\/sup>(x) =0.<\/p>\n<p>\u2192 If f(x) = x<sup>n<\/sup> + p<sub>1<\/sub>x<sup>n-1<\/sup> + &#8230;&#8230;&#8230;&#8230;. + p<sub>n-1<\/sub>x + p<sub>n<\/sub> and f(a) and f(b) are of opposite sign, then at least<br \/>\none real root of f(x) =0 lies between a and b.<\/p>\n<p>(a) For a cubic equation, when the roots are<\/p>\n<ul>\n<li>In A.P., then they are taken as a &#8211; d, a, a + d<\/li>\n<li>In G.P., then are taken as \\(\\frac{a}{r}\\), a, ar<\/li>\n<li>In H.P., then they are taken as \\(\\frac{1}{a-d}, \\frac{1}{a}, \\frac{1}{a+d}\\)<\/li>\n<\/ul>\n<p>(b) For a bi quadratic equation, if the roots are<\/p>\n<ul>\n<li>In A.P., then they are taken as a &#8211; 3d, a + d, a + 3d<\/li>\n<li>In G.P., then they are \\(\\frac{a}{d^{3}}, \\frac{a}{d}\\), ad, ad<sup>3<\/sup><\/li>\n<li>In H.P., then they are taken as \\(\\frac{1}{a-3 d}, \\frac{1}{a-d}, \\frac{1}{a+d}, \\frac{1}{a+3 d}\\)<\/li>\n<\/ul>\n<p>\u2192 It an equation is unaltered by changing x into \\(\\frac{1}{x}\\), then it is a reciprocal equation.<\/p>\n<ul>\n<li>A reciprocal equation f (x) = p<sub>0<\/sub>x<sup>n<\/sup> + p<sub>1<\/sub>x<sup>n-1<\/sup> + &#8230;&#8230;&#8230;&#8230;&#8230;&#8230;.. + p<sub>n<\/sub> = 0 is said to be a reciprocal equation of first class p<sub>i<\/sub> = p<sub>n-i<\/sub> for all i.<\/li>\n<li>A reciprocal equation f (x) = p<sub>0<\/sub>x<sup>n<\/sup> + p<sub>1<\/sub>x<sup>n-1<\/sup> + &#8230;&#8230;&#8230;&#8230;&#8230;&#8230;.. + p<sub>n<\/sub> = 0 = 0 is said to be a reciprocal equation of second class p<sub>i<\/sub> = p<sub>n-i<\/sub> for all i.<\/li>\n<li>For an odd degree reciprocal equation of class one, -1 is a root and for an odd degree reciprocal equation of class two, 1 is a root.<\/li>\n<li>For an even degree reciprocal equation of class two, 1 and -1 are roots.<\/li>\n<\/ul>\n<p>\u2192 If f(x) = 0 is an equation of degree \u2018n\u2019 then to eliminate rth term, .f(x) = 0 can be transformed to f(x+h) = 0 where h is a constant such that f<sup>(n-r+1)<\/sup>(h) =0 i.e., (n &#8211; r + 1)<sup>th<\/sup> derivative of f(h) is zero.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Use these Inter 2nd Year Maths 2A Formulas PDF Chapter 4 Theory of Equations to solve questions creatively. Intermediate 2nd Year Maths 2A Theory of Equations Formulas \u2192 If n is a non-negative integer and a0, a1, a2, &#8230;&#8230;&#8230;.. an are real or complex numbers and a0 \u2260 0, then the expression f(x) = a0xn &#8230; <a title=\"Inter 2nd Year Maths 2A Theory of Equations Formulas\" class=\"read-more\" href=\"https:\/\/apboardsolutions.in\/inter-2nd-year-maths-2a-theory-of-equations-formulas\/\" aria-label=\"More on Inter 2nd Year Maths 2A Theory of Equations Formulas\">Read more<\/a><\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[15],"tags":[],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/posts\/22069"}],"collection":[{"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/comments?post=22069"}],"version-history":[{"count":1,"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/posts\/22069\/revisions"}],"predecessor-version":[{"id":22070,"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/posts\/22069\/revisions\/22070"}],"wp:attachment":[{"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/media?parent=22069"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/categories?post=22069"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/apboardsolutions.in\/wp-json\/wp\/v2\/tags?post=22069"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}