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Page 1 Solved Examples on Quadratic Equation JEE Mains Q1. Both the roots of given equation ( ?? - ?? ) ( ?? - ?? )+ ( ?? - ?? ) ( ?? - ?? )+ ( ?? - ?? ) ( ?? - ?? ) = ?? are always (a) Positive (b) Negative (c) Real (d) Imaginary Ans: (c) Given equation ( ?? - ?? ) ( ?? - ?? )+ ( ?? - ?? ) ( ?? - ?? )+ ( ?? - ?? ) ( ?? - ?? ) = 0 can be re- written as 3?? 2 - 2( ?? + ?? + ?? ) ?? + ( ???? + ???? + ???? ) = 0 ?? = 4[( ?? + ?? + ?? ) 2 - 3( ???? + ???? + ???? ) ] = 4[?? 2 + ?? 2 + ?? 2 - ???? - ???? - ???? ] = 2[( ?? - ?? ) 2 + ( ?? - ?? ) 2 + ( ?? - ?? ) 2 ] = 0 Hence both roots are always real. Q2. If the roots of ( ?? - ?? ) ?? ?? + ( ?? - ?? ) ?? + ( ?? - ?? ) = ?? are equal then ?? + ?? = (a) ?? ?? (b) ?? ?? (c) ?? ?? (d) ?? Ans: (a) ?? - ?? + ?? - ?? + ?? - ?? = 0 Hence one root is 1 . Also as roots are equal, other root will also be equal to 1 . Also ?? · ?? = ?? -?? ?? -?? ? 1.1 = ?? -?? ?? -?? ? ?? - ?? = ?? - ?? ? 2?? = ?? + ?? Q 3. Let ?? , ?? be the roots of ?? ?? - ?? + ?? = ?? and ?? , ?? be root of ?? ?? - ?? ?? + ?? = ?? . If ?? , ?? , ?? , ?? are in G.P., then the integral value of ?? and ?? respectively are (a) -?? , -???? (b) -?? , ?? (c) -?? , ?? (d) -?? , -???? Ans: (a) ?? + ?? = 1, ???? = ?? , ?? + ?? = 4, ???? = ?? Since ?? , ?? , ?? , ?? are in G.P. ?? = ?? /?? = ?? /?? = ?? /?? ?? + ???? = 1 ? ?? ( 1 + ?? ) = 1, ?? ( ?? 2 + ?? 3 ) = 4 ? ?? · ?? 2 ( 1 + ?? ) = 4 So ?? 2 = 4 ? ?? = ±2 If ?? = 2, ?? + 2?? = 1 ? ?? = 1/3 and ?? = -2, ?? - 2?? = 1 ? ?? = -1 But ?? = ???? ? ?? ? ?? = -2, ?? = -1 ? ?? = -2, ?? = ?? 2 ?? 5 = 1( -2) 5 = -32 Page 2 Solved Examples on Quadratic Equation JEE Mains Q1. Both the roots of given equation ( ?? - ?? ) ( ?? - ?? )+ ( ?? - ?? ) ( ?? - ?? )+ ( ?? - ?? ) ( ?? - ?? ) = ?? are always (a) Positive (b) Negative (c) Real (d) Imaginary Ans: (c) Given equation ( ?? - ?? ) ( ?? - ?? )+ ( ?? - ?? ) ( ?? - ?? )+ ( ?? - ?? ) ( ?? - ?? ) = 0 can be re- written as 3?? 2 - 2( ?? + ?? + ?? ) ?? + ( ???? + ???? + ???? ) = 0 ?? = 4[( ?? + ?? + ?? ) 2 - 3( ???? + ???? + ???? ) ] = 4[?? 2 + ?? 2 + ?? 2 - ???? - ???? - ???? ] = 2[( ?? - ?? ) 2 + ( ?? - ?? ) 2 + ( ?? - ?? ) 2 ] = 0 Hence both roots are always real. Q2. If the roots of ( ?? - ?? ) ?? ?? + ( ?? - ?? ) ?? + ( ?? - ?? ) = ?? are equal then ?? + ?? = (a) ?? ?? (b) ?? ?? (c) ?? ?? (d) ?? Ans: (a) ?? - ?? + ?? - ?? + ?? - ?? = 0 Hence one root is 1 . Also as roots are equal, other root will also be equal to 1 . Also ?? · ?? = ?? -?? ?? -?? ? 1.1 = ?? -?? ?? -?? ? ?? - ?? = ?? - ?? ? 2?? = ?? + ?? Q 3. Let ?? , ?? be the roots of ?? ?? - ?? + ?? = ?? and ?? , ?? be root of ?? ?? - ?? ?? + ?? = ?? . If ?? , ?? , ?? , ?? are in G.P., then the integral value of ?? and ?? respectively are (a) -?? , -???? (b) -?? , ?? (c) -?? , ?? (d) -?? , -???? Ans: (a) ?? + ?? = 1, ???? = ?? , ?? + ?? = 4, ???? = ?? Since ?? , ?? , ?? , ?? are in G.P. ?? = ?? /?? = ?? /?? = ?? /?? ?? + ???? = 1 ? ?? ( 1 + ?? ) = 1, ?? ( ?? 2 + ?? 3 ) = 4 ? ?? · ?? 2 ( 1 + ?? ) = 4 So ?? 2 = 4 ? ?? = ±2 If ?? = 2, ?? + 2?? = 1 ? ?? = 1/3 and ?? = -2, ?? - 2?? = 1 ? ?? = -1 But ?? = ???? ? ?? ? ?? = -2, ?? = -1 ? ?? = -2, ?? = ?? 2 ?? 5 = 1( -2) 5 = -32 Q4. If the roots of the equation ?? ?? - ?? ?? + ???? = ?? are ?? , ?? and the roots of equation ?? ?? + ???? + ?? = ?? are ?? ?? + ?? ?? , ???? /?? , then (a) ?? = ?? , ?? = -???? (b) ?? = -?? , ?? = -???? (c) ?? = ?? , ?? = ???? (d) ?? = -?? , ?? = ???? Ans: (b) Since roots of the equation ?? 2 - 5?? + 16 = 0 are ?? , ?? . ? ?? + ?? = 5, ???? = 16 and ?? 2 + ?? 2 + ???? 2 = -?? ? ( ?? + ?? ) 2 - 2???? + ???? 2 = -?? ? 25 - 2( 16)+ 16 2 = -?? ? ?? = -1 and ( ?? 2 + ?? 2 )( ???? 2 ) = ?? ? [( ?? + ?? ) 2 - 2???? ] ???? 2 = ?? ? ( 25 - 32) 8 = ?? ? ?? = -56 ? ?? 2 - 19 3 ?? + 1 = 0 ? 3?? 2 - 19?? + 3 = 0 Q5. Let ?? , ?? be the roots of the equation ( ?? - ?? ) ( ?? - ?? ) = ?? , ?? ? ?? , then the roots of the equation ( ?? - ?? ) ( ?? - ?? )+ ?? = ?? are (a) a, c (b) b, c (c) ?? , ?? (d) ?? , ?? Ans: (c) Since ?? , ?? are the roots of ( ?? - ?? ) ( ?? - ?? ) = ?? i.e. of ?? 2 - ( ?? + ?? ) ?? + ???? - ?? = 0 ? ?? + ?? = ?? + ?? ? ?? + ?? = ?? + ?? and ???? = ???? - ?? ? ???? = ???? + ?? ? ?? , ?? are the roots of ?? 2 - ( ?? + ?? ) ?? + ???? + ?? = 0 ? ( ?? - ?? ) ( ?? - ?? )+ ?? = 0 Hence (c) is the correct answer Q6. If one root of the equation ?? ?? + ???? + ?? = ?? is the square of the other, then (a) ?? ?? + ?? ?? - ?? ( ?? ?? + ?? ) = ?? (b) ?? ?? + ?? ?? + ?? ( ?? + ?? ?? ) = ?? (c) ?? ?? + ?? ?? + ?? ( ?? ?? - ?? ) = ?? (d) ?? ?? + ?? ?? + ?? ( ?? - ?? ?? ) = ?? Ans: (d) Let ?? and ?? 2 be the roots then ?? + ?? 2 = -?? , ?? · ?? 2 = ?? Now ( ?? + ?? 2 ) 3 = ?? 3 + ?? 6 + 3?? 3 ( ?? + ?? 2 ) ? -?? 3 = ?? + ?? 2 - 3???? ? ?? 3 + ?? 2 + ?? ( 1 - 3?? ) = 0 Q7. If one root of a quadratic equation is ?? ?? +v ?? , then the equation is (a) ?? ?? + ?? ?? + ?? = ?? (b) ?? ?? + ?? ?? - ?? = ?? (c) ?? ?? - ?? ?? + ?? = ?? (d) None of these Ans: (b) Given root = 1 2+v 5 = 2-v 5 -1 = -2 + v 5, ? other root = -2 - v 5 Again, sum of roots = -4 and product of roots = -1. The required equation is ?? 2 + 4?? - 1 = 0 Page 3 Solved Examples on Quadratic Equation JEE Mains Q1. Both the roots of given equation ( ?? - ?? ) ( ?? - ?? )+ ( ?? - ?? ) ( ?? - ?? )+ ( ?? - ?? ) ( ?? - ?? ) = ?? are always (a) Positive (b) Negative (c) Real (d) Imaginary Ans: (c) Given equation ( ?? - ?? ) ( ?? - ?? )+ ( ?? - ?? ) ( ?? - ?? )+ ( ?? - ?? ) ( ?? - ?? ) = 0 can be re- written as 3?? 2 - 2( ?? + ?? + ?? ) ?? + ( ???? + ???? + ???? ) = 0 ?? = 4[( ?? + ?? + ?? ) 2 - 3( ???? + ???? + ???? ) ] = 4[?? 2 + ?? 2 + ?? 2 - ???? - ???? - ???? ] = 2[( ?? - ?? ) 2 + ( ?? - ?? ) 2 + ( ?? - ?? ) 2 ] = 0 Hence both roots are always real. Q2. If the roots of ( ?? - ?? ) ?? ?? + ( ?? - ?? ) ?? + ( ?? - ?? ) = ?? are equal then ?? + ?? = (a) ?? ?? (b) ?? ?? (c) ?? ?? (d) ?? Ans: (a) ?? - ?? + ?? - ?? + ?? - ?? = 0 Hence one root is 1 . Also as roots are equal, other root will also be equal to 1 . Also ?? · ?? = ?? -?? ?? -?? ? 1.1 = ?? -?? ?? -?? ? ?? - ?? = ?? - ?? ? 2?? = ?? + ?? Q 3. Let ?? , ?? be the roots of ?? ?? - ?? + ?? = ?? and ?? , ?? be root of ?? ?? - ?? ?? + ?? = ?? . If ?? , ?? , ?? , ?? are in G.P., then the integral value of ?? and ?? respectively are (a) -?? , -???? (b) -?? , ?? (c) -?? , ?? (d) -?? , -???? Ans: (a) ?? + ?? = 1, ???? = ?? , ?? + ?? = 4, ???? = ?? Since ?? , ?? , ?? , ?? are in G.P. ?? = ?? /?? = ?? /?? = ?? /?? ?? + ???? = 1 ? ?? ( 1 + ?? ) = 1, ?? ( ?? 2 + ?? 3 ) = 4 ? ?? · ?? 2 ( 1 + ?? ) = 4 So ?? 2 = 4 ? ?? = ±2 If ?? = 2, ?? + 2?? = 1 ? ?? = 1/3 and ?? = -2, ?? - 2?? = 1 ? ?? = -1 But ?? = ???? ? ?? ? ?? = -2, ?? = -1 ? ?? = -2, ?? = ?? 2 ?? 5 = 1( -2) 5 = -32 Q4. If the roots of the equation ?? ?? - ?? ?? + ???? = ?? are ?? , ?? and the roots of equation ?? ?? + ???? + ?? = ?? are ?? ?? + ?? ?? , ???? /?? , then (a) ?? = ?? , ?? = -???? (b) ?? = -?? , ?? = -???? (c) ?? = ?? , ?? = ???? (d) ?? = -?? , ?? = ???? Ans: (b) Since roots of the equation ?? 2 - 5?? + 16 = 0 are ?? , ?? . ? ?? + ?? = 5, ???? = 16 and ?? 2 + ?? 2 + ???? 2 = -?? ? ( ?? + ?? ) 2 - 2???? + ???? 2 = -?? ? 25 - 2( 16)+ 16 2 = -?? ? ?? = -1 and ( ?? 2 + ?? 2 )( ???? 2 ) = ?? ? [( ?? + ?? ) 2 - 2???? ] ???? 2 = ?? ? ( 25 - 32) 8 = ?? ? ?? = -56 ? ?? 2 - 19 3 ?? + 1 = 0 ? 3?? 2 - 19?? + 3 = 0 Q5. Let ?? , ?? be the roots of the equation ( ?? - ?? ) ( ?? - ?? ) = ?? , ?? ? ?? , then the roots of the equation ( ?? - ?? ) ( ?? - ?? )+ ?? = ?? are (a) a, c (b) b, c (c) ?? , ?? (d) ?? , ?? Ans: (c) Since ?? , ?? are the roots of ( ?? - ?? ) ( ?? - ?? ) = ?? i.e. of ?? 2 - ( ?? + ?? ) ?? + ???? - ?? = 0 ? ?? + ?? = ?? + ?? ? ?? + ?? = ?? + ?? and ???? = ???? - ?? ? ???? = ???? + ?? ? ?? , ?? are the roots of ?? 2 - ( ?? + ?? ) ?? + ???? + ?? = 0 ? ( ?? - ?? ) ( ?? - ?? )+ ?? = 0 Hence (c) is the correct answer Q6. If one root of the equation ?? ?? + ???? + ?? = ?? is the square of the other, then (a) ?? ?? + ?? ?? - ?? ( ?? ?? + ?? ) = ?? (b) ?? ?? + ?? ?? + ?? ( ?? + ?? ?? ) = ?? (c) ?? ?? + ?? ?? + ?? ( ?? ?? - ?? ) = ?? (d) ?? ?? + ?? ?? + ?? ( ?? - ?? ?? ) = ?? Ans: (d) Let ?? and ?? 2 be the roots then ?? + ?? 2 = -?? , ?? · ?? 2 = ?? Now ( ?? + ?? 2 ) 3 = ?? 3 + ?? 6 + 3?? 3 ( ?? + ?? 2 ) ? -?? 3 = ?? + ?? 2 - 3???? ? ?? 3 + ?? 2 + ?? ( 1 - 3?? ) = 0 Q7. If one root of a quadratic equation is ?? ?? +v ?? , then the equation is (a) ?? ?? + ?? ?? + ?? = ?? (b) ?? ?? + ?? ?? - ?? = ?? (c) ?? ?? - ?? ?? + ?? = ?? (d) None of these Ans: (b) Given root = 1 2+v 5 = 2-v 5 -1 = -2 + v 5, ? other root = -2 - v 5 Again, sum of roots = -4 and product of roots = -1. The required equation is ?? 2 + 4?? - 1 = 0 Q8. If ?? , ?? , ?? are in G.P. then the equations ?? ?? ?? + ?? ???? + ?? = ?? and ?? ?? ?? + ?? ???? + ?? = ?? have a common root if ?? ?? , ?? ?? , ?? ?? are in (a) A.P. (b) G.P. (c) H.P. (d) None of these Ans: (a) As given, ?? 2 = ???? ? ?? ?? 2 + 2???? + ?? = 0 can be written as ?? ?? 2 + 2v ???? ?? + ?? = 0 ? ( v ?? ?? + v ?? ) 2 = 0 ? ?? = -v ?? ?? This must be common root by hypothesis So it must satisfy the equation, ?? ?? 2 + 2???? + ?? = 0 ? ?? ( ?? ?? ) - 2?? v ?? ?? + ?? = 0 ?? ?? + ?? ?? = 2?? ?? v ?? ?? = 2?? v ?? · v ?? ? ?? ?? + ?? ?? = 2?? ?? Hence ?? ?? , ?? ?? , ?? ?? are in A.P. Hence ?? ?? , ?? ?? , ?? ?? are in A.P. Q9. If ?? , ?? ( ?? < ?? ) are roots of the equation ?? ?? + ???? + ?? = ?? where ( ?? < ?? < ?? ) then (a) ?? < ?? < ?? (b) ?? < ?? < ?? < |?? | (c) ?? < ?? < ?? (d) ?? < ?? < ?? |< ?? Ans: (b) Since ?? ( 0) = 0 + 0 + ?? = ?? < 0 ? Roots will be of opposite sign, ?? + ?? = -?? = -???? ( ?? > 0) It is given that ?? < ?? So, ?? + ?? = -???? is possible only when |?? | > ?? ? ?? < 0, ?? > 0, |?? | > ?? ? ?? < 0 < ?? < |?? | Q10. If the roots of the equation ?? ?? - ?? ???? + ?? ?? + ?? - ?? = ?? are real and less than 3 , then (a) ?? < ?? (b) ?? = ?? = ?? (c) ?? < ?? = ?? (d) ?? > ?? Ans: (a) Given equation is ?? 2 - 2???? + ?? 2 + ?? - 3 = 0 If roots are real, then ?? = 0 ? 4?? 2 - 4( ?? 2 + ?? - 3) = 0 ? -?? + 3 = 0 ? ?? - 3 = 0 ? ?? = 3 As roots are less than 3 , hence ?? ( 3) > 0 9 - 6?? + ?? 2 + ?? - 3 > 0 ? ?? 2 - 5?? + 6 > 0 ? ( ?? - 2) ( ?? - 3) > 0 ? ?? < 2, ?? > 3. Hence ?? < 2 satisfy all the conditions. Page 4 Solved Examples on Quadratic Equation JEE Mains Q1. Both the roots of given equation ( ?? - ?? ) ( ?? - ?? )+ ( ?? - ?? ) ( ?? - ?? )+ ( ?? - ?? ) ( ?? - ?? ) = ?? are always (a) Positive (b) Negative (c) Real (d) Imaginary Ans: (c) Given equation ( ?? - ?? ) ( ?? - ?? )+ ( ?? - ?? ) ( ?? - ?? )+ ( ?? - ?? ) ( ?? - ?? ) = 0 can be re- written as 3?? 2 - 2( ?? + ?? + ?? ) ?? + ( ???? + ???? + ???? ) = 0 ?? = 4[( ?? + ?? + ?? ) 2 - 3( ???? + ???? + ???? ) ] = 4[?? 2 + ?? 2 + ?? 2 - ???? - ???? - ???? ] = 2[( ?? - ?? ) 2 + ( ?? - ?? ) 2 + ( ?? - ?? ) 2 ] = 0 Hence both roots are always real. Q2. If the roots of ( ?? - ?? ) ?? ?? + ( ?? - ?? ) ?? + ( ?? - ?? ) = ?? are equal then ?? + ?? = (a) ?? ?? (b) ?? ?? (c) ?? ?? (d) ?? Ans: (a) ?? - ?? + ?? - ?? + ?? - ?? = 0 Hence one root is 1 . Also as roots are equal, other root will also be equal to 1 . Also ?? · ?? = ?? -?? ?? -?? ? 1.1 = ?? -?? ?? -?? ? ?? - ?? = ?? - ?? ? 2?? = ?? + ?? Q 3. Let ?? , ?? be the roots of ?? ?? - ?? + ?? = ?? and ?? , ?? be root of ?? ?? - ?? ?? + ?? = ?? . If ?? , ?? , ?? , ?? are in G.P., then the integral value of ?? and ?? respectively are (a) -?? , -???? (b) -?? , ?? (c) -?? , ?? (d) -?? , -???? Ans: (a) ?? + ?? = 1, ???? = ?? , ?? + ?? = 4, ???? = ?? Since ?? , ?? , ?? , ?? are in G.P. ?? = ?? /?? = ?? /?? = ?? /?? ?? + ???? = 1 ? ?? ( 1 + ?? ) = 1, ?? ( ?? 2 + ?? 3 ) = 4 ? ?? · ?? 2 ( 1 + ?? ) = 4 So ?? 2 = 4 ? ?? = ±2 If ?? = 2, ?? + 2?? = 1 ? ?? = 1/3 and ?? = -2, ?? - 2?? = 1 ? ?? = -1 But ?? = ???? ? ?? ? ?? = -2, ?? = -1 ? ?? = -2, ?? = ?? 2 ?? 5 = 1( -2) 5 = -32 Q4. If the roots of the equation ?? ?? - ?? ?? + ???? = ?? are ?? , ?? and the roots of equation ?? ?? + ???? + ?? = ?? are ?? ?? + ?? ?? , ???? /?? , then (a) ?? = ?? , ?? = -???? (b) ?? = -?? , ?? = -???? (c) ?? = ?? , ?? = ???? (d) ?? = -?? , ?? = ???? Ans: (b) Since roots of the equation ?? 2 - 5?? + 16 = 0 are ?? , ?? . ? ?? + ?? = 5, ???? = 16 and ?? 2 + ?? 2 + ???? 2 = -?? ? ( ?? + ?? ) 2 - 2???? + ???? 2 = -?? ? 25 - 2( 16)+ 16 2 = -?? ? ?? = -1 and ( ?? 2 + ?? 2 )( ???? 2 ) = ?? ? [( ?? + ?? ) 2 - 2???? ] ???? 2 = ?? ? ( 25 - 32) 8 = ?? ? ?? = -56 ? ?? 2 - 19 3 ?? + 1 = 0 ? 3?? 2 - 19?? + 3 = 0 Q5. Let ?? , ?? be the roots of the equation ( ?? - ?? ) ( ?? - ?? ) = ?? , ?? ? ?? , then the roots of the equation ( ?? - ?? ) ( ?? - ?? )+ ?? = ?? are (a) a, c (b) b, c (c) ?? , ?? (d) ?? , ?? Ans: (c) Since ?? , ?? are the roots of ( ?? - ?? ) ( ?? - ?? ) = ?? i.e. of ?? 2 - ( ?? + ?? ) ?? + ???? - ?? = 0 ? ?? + ?? = ?? + ?? ? ?? + ?? = ?? + ?? and ???? = ???? - ?? ? ???? = ???? + ?? ? ?? , ?? are the roots of ?? 2 - ( ?? + ?? ) ?? + ???? + ?? = 0 ? ( ?? - ?? ) ( ?? - ?? )+ ?? = 0 Hence (c) is the correct answer Q6. If one root of the equation ?? ?? + ???? + ?? = ?? is the square of the other, then (a) ?? ?? + ?? ?? - ?? ( ?? ?? + ?? ) = ?? (b) ?? ?? + ?? ?? + ?? ( ?? + ?? ?? ) = ?? (c) ?? ?? + ?? ?? + ?? ( ?? ?? - ?? ) = ?? (d) ?? ?? + ?? ?? + ?? ( ?? - ?? ?? ) = ?? Ans: (d) Let ?? and ?? 2 be the roots then ?? + ?? 2 = -?? , ?? · ?? 2 = ?? Now ( ?? + ?? 2 ) 3 = ?? 3 + ?? 6 + 3?? 3 ( ?? + ?? 2 ) ? -?? 3 = ?? + ?? 2 - 3???? ? ?? 3 + ?? 2 + ?? ( 1 - 3?? ) = 0 Q7. If one root of a quadratic equation is ?? ?? +v ?? , then the equation is (a) ?? ?? + ?? ?? + ?? = ?? (b) ?? ?? + ?? ?? - ?? = ?? (c) ?? ?? - ?? ?? + ?? = ?? (d) None of these Ans: (b) Given root = 1 2+v 5 = 2-v 5 -1 = -2 + v 5, ? other root = -2 - v 5 Again, sum of roots = -4 and product of roots = -1. The required equation is ?? 2 + 4?? - 1 = 0 Q8. If ?? , ?? , ?? are in G.P. then the equations ?? ?? ?? + ?? ???? + ?? = ?? and ?? ?? ?? + ?? ???? + ?? = ?? have a common root if ?? ?? , ?? ?? , ?? ?? are in (a) A.P. (b) G.P. (c) H.P. (d) None of these Ans: (a) As given, ?? 2 = ???? ? ?? ?? 2 + 2???? + ?? = 0 can be written as ?? ?? 2 + 2v ???? ?? + ?? = 0 ? ( v ?? ?? + v ?? ) 2 = 0 ? ?? = -v ?? ?? This must be common root by hypothesis So it must satisfy the equation, ?? ?? 2 + 2???? + ?? = 0 ? ?? ( ?? ?? ) - 2?? v ?? ?? + ?? = 0 ?? ?? + ?? ?? = 2?? ?? v ?? ?? = 2?? v ?? · v ?? ? ?? ?? + ?? ?? = 2?? ?? Hence ?? ?? , ?? ?? , ?? ?? are in A.P. Hence ?? ?? , ?? ?? , ?? ?? are in A.P. Q9. If ?? , ?? ( ?? < ?? ) are roots of the equation ?? ?? + ???? + ?? = ?? where ( ?? < ?? < ?? ) then (a) ?? < ?? < ?? (b) ?? < ?? < ?? < |?? | (c) ?? < ?? < ?? (d) ?? < ?? < ?? |< ?? Ans: (b) Since ?? ( 0) = 0 + 0 + ?? = ?? < 0 ? Roots will be of opposite sign, ?? + ?? = -?? = -???? ( ?? > 0) It is given that ?? < ?? So, ?? + ?? = -???? is possible only when |?? | > ?? ? ?? < 0, ?? > 0, |?? | > ?? ? ?? < 0 < ?? < |?? | Q10. If the roots of the equation ?? ?? - ?? ???? + ?? ?? + ?? - ?? = ?? are real and less than 3 , then (a) ?? < ?? (b) ?? = ?? = ?? (c) ?? < ?? = ?? (d) ?? > ?? Ans: (a) Given equation is ?? 2 - 2???? + ?? 2 + ?? - 3 = 0 If roots are real, then ?? = 0 ? 4?? 2 - 4( ?? 2 + ?? - 3) = 0 ? -?? + 3 = 0 ? ?? - 3 = 0 ? ?? = 3 As roots are less than 3 , hence ?? ( 3) > 0 9 - 6?? + ?? 2 + ?? - 3 > 0 ? ?? 2 - 5?? + 6 > 0 ? ( ?? - 2) ( ?? - 3) > 0 ? ?? < 2, ?? > 3. Hence ?? < 2 satisfy all the conditions. ?????? . If ?? ?? ?? ?? ?? +?? ?? +?? > ?? ?? +?? , then (a) -?? > ?? > -?? (b) -?? = ?? = -?? (c) -?? < ?? < -?? (d) -?? < ?? = -?? Solution: (c) Given 2?? 2?? 2 +5?? +2 - 1 ?? +1 > 0 ? 2?? 2 +2?? -2?? 2 -5?? -2 ( 2?? +1) ( ?? +2) ( ?? +1) > 0 ? -3?? -2 ( 2?? +1) ( ?? +2) ( ?? +1) > 0 ? -3( ?? + 2/3) ( ?? + 1) ( ?? + 2) ( 2?? + 1) > 0 ? ( ?? + 2/3) ( ?? + 1) ( ?? + 2) ( 2?? + 1) < 0 Equating each factor equal to 0 , We get ?? = -2, -1, -2/3, -1/2 ? ?? ?] - 2, -1[?] - 2/3, -1/2[? -2/3 < ?? < -1/2 or -2 < ?? < -1 Q12. The number of integral solution of ?? +?? ?? ?? +?? > ?? ?? is (a) 1 (b) 2 (c) 5 (d) None of these Ans: (c) ?? + 1 ?? 2 + 2 - 1 4 > 0 ? ?? 2 - 4?? - 2 ?? 2 + 2 < 0 ? ( ?? - ( 2 + v 6) ) ( ?? - ( 2 - v 6) ) < 0 ? 2 - v 6 < ?? < 2 + v 6 Approximately, - 0.4 < ?? < 4.4 Hence, integral values of ?? are 0,1,2,3,4 Hence, number of integral solution = 5 Approximately, -0.4 < ?? < 4.4 Hence, integral values of ?? are 0,1,2,3,4 Hence, number of integral solution = 5 Q13. The roots of |?? - ?? | ?? + |?? - ?? | - ?? = ?? are (a) 0,4 (b) -?? , ?? (c) 4,2 (d) 5, 1 Ans: (a) We have |?? - 2| 2 + |?? - 2| - 6 = 0 Let |?? - 2| = ?? Page 5 Solved Examples on Quadratic Equation JEE Mains Q1. Both the roots of given equation ( ?? - ?? ) ( ?? - ?? )+ ( ?? - ?? ) ( ?? - ?? )+ ( ?? - ?? ) ( ?? - ?? ) = ?? are always (a) Positive (b) Negative (c) Real (d) Imaginary Ans: (c) Given equation ( ?? - ?? ) ( ?? - ?? )+ ( ?? - ?? ) ( ?? - ?? )+ ( ?? - ?? ) ( ?? - ?? ) = 0 can be re- written as 3?? 2 - 2( ?? + ?? + ?? ) ?? + ( ???? + ???? + ???? ) = 0 ?? = 4[( ?? + ?? + ?? ) 2 - 3( ???? + ???? + ???? ) ] = 4[?? 2 + ?? 2 + ?? 2 - ???? - ???? - ???? ] = 2[( ?? - ?? ) 2 + ( ?? - ?? ) 2 + ( ?? - ?? ) 2 ] = 0 Hence both roots are always real. Q2. If the roots of ( ?? - ?? ) ?? ?? + ( ?? - ?? ) ?? + ( ?? - ?? ) = ?? are equal then ?? + ?? = (a) ?? ?? (b) ?? ?? (c) ?? ?? (d) ?? Ans: (a) ?? - ?? + ?? - ?? + ?? - ?? = 0 Hence one root is 1 . Also as roots are equal, other root will also be equal to 1 . Also ?? · ?? = ?? -?? ?? -?? ? 1.1 = ?? -?? ?? -?? ? ?? - ?? = ?? - ?? ? 2?? = ?? + ?? Q 3. Let ?? , ?? be the roots of ?? ?? - ?? + ?? = ?? and ?? , ?? be root of ?? ?? - ?? ?? + ?? = ?? . If ?? , ?? , ?? , ?? are in G.P., then the integral value of ?? and ?? respectively are (a) -?? , -???? (b) -?? , ?? (c) -?? , ?? (d) -?? , -???? Ans: (a) ?? + ?? = 1, ???? = ?? , ?? + ?? = 4, ???? = ?? Since ?? , ?? , ?? , ?? are in G.P. ?? = ?? /?? = ?? /?? = ?? /?? ?? + ???? = 1 ? ?? ( 1 + ?? ) = 1, ?? ( ?? 2 + ?? 3 ) = 4 ? ?? · ?? 2 ( 1 + ?? ) = 4 So ?? 2 = 4 ? ?? = ±2 If ?? = 2, ?? + 2?? = 1 ? ?? = 1/3 and ?? = -2, ?? - 2?? = 1 ? ?? = -1 But ?? = ???? ? ?? ? ?? = -2, ?? = -1 ? ?? = -2, ?? = ?? 2 ?? 5 = 1( -2) 5 = -32 Q4. If the roots of the equation ?? ?? - ?? ?? + ???? = ?? are ?? , ?? and the roots of equation ?? ?? + ???? + ?? = ?? are ?? ?? + ?? ?? , ???? /?? , then (a) ?? = ?? , ?? = -???? (b) ?? = -?? , ?? = -???? (c) ?? = ?? , ?? = ???? (d) ?? = -?? , ?? = ???? Ans: (b) Since roots of the equation ?? 2 - 5?? + 16 = 0 are ?? , ?? . ? ?? + ?? = 5, ???? = 16 and ?? 2 + ?? 2 + ???? 2 = -?? ? ( ?? + ?? ) 2 - 2???? + ???? 2 = -?? ? 25 - 2( 16)+ 16 2 = -?? ? ?? = -1 and ( ?? 2 + ?? 2 )( ???? 2 ) = ?? ? [( ?? + ?? ) 2 - 2???? ] ???? 2 = ?? ? ( 25 - 32) 8 = ?? ? ?? = -56 ? ?? 2 - 19 3 ?? + 1 = 0 ? 3?? 2 - 19?? + 3 = 0 Q5. Let ?? , ?? be the roots of the equation ( ?? - ?? ) ( ?? - ?? ) = ?? , ?? ? ?? , then the roots of the equation ( ?? - ?? ) ( ?? - ?? )+ ?? = ?? are (a) a, c (b) b, c (c) ?? , ?? (d) ?? , ?? Ans: (c) Since ?? , ?? are the roots of ( ?? - ?? ) ( ?? - ?? ) = ?? i.e. of ?? 2 - ( ?? + ?? ) ?? + ???? - ?? = 0 ? ?? + ?? = ?? + ?? ? ?? + ?? = ?? + ?? and ???? = ???? - ?? ? ???? = ???? + ?? ? ?? , ?? are the roots of ?? 2 - ( ?? + ?? ) ?? + ???? + ?? = 0 ? ( ?? - ?? ) ( ?? - ?? )+ ?? = 0 Hence (c) is the correct answer Q6. If one root of the equation ?? ?? + ???? + ?? = ?? is the square of the other, then (a) ?? ?? + ?? ?? - ?? ( ?? ?? + ?? ) = ?? (b) ?? ?? + ?? ?? + ?? ( ?? + ?? ?? ) = ?? (c) ?? ?? + ?? ?? + ?? ( ?? ?? - ?? ) = ?? (d) ?? ?? + ?? ?? + ?? ( ?? - ?? ?? ) = ?? Ans: (d) Let ?? and ?? 2 be the roots then ?? + ?? 2 = -?? , ?? · ?? 2 = ?? Now ( ?? + ?? 2 ) 3 = ?? 3 + ?? 6 + 3?? 3 ( ?? + ?? 2 ) ? -?? 3 = ?? + ?? 2 - 3???? ? ?? 3 + ?? 2 + ?? ( 1 - 3?? ) = 0 Q7. If one root of a quadratic equation is ?? ?? +v ?? , then the equation is (a) ?? ?? + ?? ?? + ?? = ?? (b) ?? ?? + ?? ?? - ?? = ?? (c) ?? ?? - ?? ?? + ?? = ?? (d) None of these Ans: (b) Given root = 1 2+v 5 = 2-v 5 -1 = -2 + v 5, ? other root = -2 - v 5 Again, sum of roots = -4 and product of roots = -1. The required equation is ?? 2 + 4?? - 1 = 0 Q8. If ?? , ?? , ?? are in G.P. then the equations ?? ?? ?? + ?? ???? + ?? = ?? and ?? ?? ?? + ?? ???? + ?? = ?? have a common root if ?? ?? , ?? ?? , ?? ?? are in (a) A.P. (b) G.P. (c) H.P. (d) None of these Ans: (a) As given, ?? 2 = ???? ? ?? ?? 2 + 2???? + ?? = 0 can be written as ?? ?? 2 + 2v ???? ?? + ?? = 0 ? ( v ?? ?? + v ?? ) 2 = 0 ? ?? = -v ?? ?? This must be common root by hypothesis So it must satisfy the equation, ?? ?? 2 + 2???? + ?? = 0 ? ?? ( ?? ?? ) - 2?? v ?? ?? + ?? = 0 ?? ?? + ?? ?? = 2?? ?? v ?? ?? = 2?? v ?? · v ?? ? ?? ?? + ?? ?? = 2?? ?? Hence ?? ?? , ?? ?? , ?? ?? are in A.P. Hence ?? ?? , ?? ?? , ?? ?? are in A.P. Q9. If ?? , ?? ( ?? < ?? ) are roots of the equation ?? ?? + ???? + ?? = ?? where ( ?? < ?? < ?? ) then (a) ?? < ?? < ?? (b) ?? < ?? < ?? < |?? | (c) ?? < ?? < ?? (d) ?? < ?? < ?? |< ?? Ans: (b) Since ?? ( 0) = 0 + 0 + ?? = ?? < 0 ? Roots will be of opposite sign, ?? + ?? = -?? = -???? ( ?? > 0) It is given that ?? < ?? So, ?? + ?? = -???? is possible only when |?? | > ?? ? ?? < 0, ?? > 0, |?? | > ?? ? ?? < 0 < ?? < |?? | Q10. If the roots of the equation ?? ?? - ?? ???? + ?? ?? + ?? - ?? = ?? are real and less than 3 , then (a) ?? < ?? (b) ?? = ?? = ?? (c) ?? < ?? = ?? (d) ?? > ?? Ans: (a) Given equation is ?? 2 - 2???? + ?? 2 + ?? - 3 = 0 If roots are real, then ?? = 0 ? 4?? 2 - 4( ?? 2 + ?? - 3) = 0 ? -?? + 3 = 0 ? ?? - 3 = 0 ? ?? = 3 As roots are less than 3 , hence ?? ( 3) > 0 9 - 6?? + ?? 2 + ?? - 3 > 0 ? ?? 2 - 5?? + 6 > 0 ? ( ?? - 2) ( ?? - 3) > 0 ? ?? < 2, ?? > 3. Hence ?? < 2 satisfy all the conditions. ?????? . If ?? ?? ?? ?? ?? +?? ?? +?? > ?? ?? +?? , then (a) -?? > ?? > -?? (b) -?? = ?? = -?? (c) -?? < ?? < -?? (d) -?? < ?? = -?? Solution: (c) Given 2?? 2?? 2 +5?? +2 - 1 ?? +1 > 0 ? 2?? 2 +2?? -2?? 2 -5?? -2 ( 2?? +1) ( ?? +2) ( ?? +1) > 0 ? -3?? -2 ( 2?? +1) ( ?? +2) ( ?? +1) > 0 ? -3( ?? + 2/3) ( ?? + 1) ( ?? + 2) ( 2?? + 1) > 0 ? ( ?? + 2/3) ( ?? + 1) ( ?? + 2) ( 2?? + 1) < 0 Equating each factor equal to 0 , We get ?? = -2, -1, -2/3, -1/2 ? ?? ?] - 2, -1[?] - 2/3, -1/2[? -2/3 < ?? < -1/2 or -2 < ?? < -1 Q12. The number of integral solution of ?? +?? ?? ?? +?? > ?? ?? is (a) 1 (b) 2 (c) 5 (d) None of these Ans: (c) ?? + 1 ?? 2 + 2 - 1 4 > 0 ? ?? 2 - 4?? - 2 ?? 2 + 2 < 0 ? ( ?? - ( 2 + v 6) ) ( ?? - ( 2 - v 6) ) < 0 ? 2 - v 6 < ?? < 2 + v 6 Approximately, - 0.4 < ?? < 4.4 Hence, integral values of ?? are 0,1,2,3,4 Hence, number of integral solution = 5 Approximately, -0.4 < ?? < 4.4 Hence, integral values of ?? are 0,1,2,3,4 Hence, number of integral solution = 5 Q13. The roots of |?? - ?? | ?? + |?? - ?? | - ?? = ?? are (a) 0,4 (b) -?? , ?? (c) 4,2 (d) 5, 1 Ans: (a) We have |?? - 2| 2 + |?? - 2| - 6 = 0 Let |?? - 2| = ?? ?? 2 + ?? - 6 = 0 ? ?? = -1 ± v 1 + 24 2 = 2, -3 ? ?? = 2 and ?? = -3 ? |?? - 2| = 2 and |?? - 2| = -3, which is not possible. ? ?? - 2 = 2 or ?? - 2 = -2 ? ?? = 4 or ?? = 0 Q14. The greatest value of ?? ( ?? ) = ?? ?? - ?? ?? if ?? ?? + ???? = ???? ?? ?? , is (A) 18 (B) 2 (C) -2 (D) -18 Ans: (A) ?? ( ?? ) = ?? 3 - 3?? ? ?? ' ( ?? ) = 3?? 2 - 3 ? ?? 4 - 13?? 2 + 36 = 0 ? ?? ? [-3, -2],? [2,3] ? ?? ( ?? ) increases in the interval ( -8, -1)? ( 1, 8) Finding the values of ?? ( ?? ) at ?? = -3, -2,2,3. We get ?? ( 3) = 18. which is greatest. Q15. If ?? ?? + ???? - ?????? ?? = ?? has integral roots, where ?? is a prime number, then the value (s) of ?? is (are) (A) 2 (B) 3 (C) 2, 3 and 37 (D) 37 Ans: (D) The equation has integral roots hence ?? = -?? ±v?? 2 +4×444?? 2 since ?? = 2 does not give the integral roots. ? ?? must be perfect square of on odd integer i.e. D 2 = ?? 2 + 1776 ?? = ?? ( ?? + 1776 ) since D is a perfect square ? ?? + 1776 must be a multiple of ?? ? 1776 must be a multiple of ?? Now 1776 = 2 4 . 3.37 where ?? = 2 or 3 or 37 (i) ?? = 2 then ?? ( ?? + 1776 )= 2( 2 + 1776 )= 3556 = 4 × 7 × 127 Which is not a perfect square (ii) ?? = 3 then ?? ( ?? + 1776 ) = 3( 3 + 1776 )= 5337 which is not a perfect square as its last digit is 7 . (iii) ?? = 37 then ?? ( ?? + 1776 ) = 37( 37 + 1776 )= 37 2 . 7 2 which is odd. Hence ?? = 37.Read More
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FAQs on Quadratic Equation Solved Examples - Mathematics (Maths) for JEE Main & Advanced
| 1. What is a quadratic equation? |  |
Ans. A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0.
| 2. How do you solve a quadratic equation? | |
Ans. Quadratic equations can be solved using methods such as factoring, completing the square, or using the quadratic formula.
| 3. What is the quadratic formula? | |
Ans. The quadratic formula is x = (-b ± √(b^2 - 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0.
| 4. Can a quadratic equation have more than two solutions? | |
Ans. No, a quadratic equation can have at most two real solutions, which can be equal or distinct.
| 5. How are quadratic equations used in real life? | |
Ans. Quadratic equations are used in various real-life applications such as physics, engineering, economics, and computer science to model and solve problems involving quadratic relationships.
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