JEE Exam > JEE Notes > Mathematics (Maths) for JEE Main & Advanced > Quadratic Equation Solved Examples
Quadratic Equation Solved Examples | Mathematics (Maths) for JEE Main & Advanced PDF Download
| Download, print and study this document offline |
Please wait while the PDF view is loading
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
|
209 videos|443 docs|143 tests
|
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.
Related Exams
About this Document
|
4.62/5 Rating |
|
Dec 26, 2024 Last updated |
Document Description: Quadratic Equation Solved Examples for JEE 2024 is part of Mathematics (Maths) for JEE Main & Advanced preparation.
The notes and questions for Quadratic Equation Solved Examples have been prepared according to the JEE exam syllabus. Information about Quadratic Equation Solved Examples covers topics
like and Quadratic Equation Solved Examples Example, for JEE 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Quadratic Equation Solved Examples.
Introduction of Quadratic Equation Solved Examples in English is available as part of our Mathematics (Maths) for JEE Main & Advanced
for JEE & Quadratic Equation Solved Examples in Hindi for Mathematics (Maths) for JEE Main & Advanced course.
Download more important topics related with notes, lectures and mock test series for JEE
Exam by signing up for free. JEE: Quadratic Equation Solved Examples | Mathematics (Maths) for JEE Main & Advanced
Description
Full syllabus notes, lecture & questions for Quadratic Equation Solved Examples | Mathematics (Maths) for JEE Main & Advanced - JEE | Plus excerises question with solution to help you revise complete syllabus for Mathematics (Maths) for JEE Main & Advanced | Best notes, free PDF download
Information about Quadratic Equation Solved Examples
In this doc you can find the meaning of Quadratic Equation Solved Examples defined & explained in the simplest way possible. Besides explaining types of
Quadratic Equation Solved Examples theory, EduRev gives you an ample number of questions to practice Quadratic Equation Solved Examples tests, examples and also practice JEE
tests
|
209 videos|443 docs|143 tests
|
Download as PDF
|
Explore Courses for JEE exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
Related Searches
shortcuts and tricks
,Sample Paper
,Objective type Questions
,Free
,Important questions
,Exam
,MCQs
,Semester Notes
,past year papers
,Previous Year Questions with Solutions
,mock tests for examination
,Viva Questions
,study material
,Quadratic Equation Solved Examples | Mathematics (Maths) for JEE Main & Advanced
,practice quizzes
,ppt
,Summary
,video lectures
,Extra Questions
,Quadratic Equation Solved Examples | Mathematics (Maths) for JEE Main & Advanced
,Quadratic Equation Solved Examples | Mathematics (Maths) for JEE Main & Advanced
;
Additional Information about Quadratic Equation Solved Examples for JEE Preparation
Quadratic Equation Solved Examples Free PDF Download
The Quadratic Equation Solved Examples is an invaluable resource that delves deep into the core of the JEE exam.
These study notes are curated by experts and cover all the essential topics and concepts, making your preparation more efficient and effective.
With the help of these notes, you can grasp complex subjects quickly, revise important points easily,
and reinforce your understanding of key concepts. The study notes are presented in a concise and easy-to-understand manner,
allowing you to optimize your learning process. Whether you're looking for best-recommended books, sample papers, study material,
or toppers' notes, this PDF has got you covered. Download the Quadratic Equation Solved Examples now and kickstart your journey towards success in the JEE exam.
Importance of Quadratic Equation Solved Examples
The importance of Quadratic Equation Solved Examples cannot be overstated, especially for JEE aspirants.
This document holds the key to success in the JEE exam.
It offers a detailed understanding of the concept, providing invaluable insights into the topic.
By knowing the concepts well in advance, students can plan their preparation effectively.
Utilize this indispensable guide for a well-rounded preparation and achieve your desired results.
Quadratic Equation Solved Examples Notes
Quadratic Equation Solved Examples Notes offer in-depth insights into the specific topic to help you master it with ease.
This comprehensive document covers all aspects related to Quadratic Equation Solved Examples.
It includes detailed information about the exam syllabus, recommended books, and study materials for a well-rounded preparation.
Practice papers and question papers enable you to assess your progress effectively.
Additionally, the paper analysis provides valuable tips for tackling the exam strategically.
Access to Toppers' notes gives you an edge in understanding complex concepts.
Whether you're a beginner or aiming for advanced proficiency, Quadratic Equation Solved Examples Notes on EduRev are your ultimate resource for success.
Quadratic Equation Solved Examples JEE Questions
The "Quadratic Equation Solved Examples JEE Questions" guide is a valuable resource for all aspiring students preparing for the
JEE exam. It focuses on providing a wide range of practice questions to help students gauge
their understanding of the exam topics. These questions cover the entire syllabus, ensuring comprehensive preparation.
The guide includes previous years' question papers for students to familiarize themselves with the exam's format and difficulty level.
Additionally, it offers subject-specific question banks, allowing students to focus on weak areas and improve their performance.
Study Quadratic Equation Solved Examples on the App
Students of JEE can study Quadratic Equation Solved Examples alongwith tests & analysis from the EduRev app,
which will help them while preparing for their exam. Apart from the Quadratic Equation Solved Examples,
students can also utilize the EduRev App for other study materials such as previous year question papers, syllabus, important questions, etc.
The EduRev App will make your learning easier as you can access it from anywhere you want.
The content of Quadratic Equation Solved Examples is prepared as per the latest JEE syllabus.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup