fo/u fopkjr Hkh# tu] ugha vkjehks dke] foifr ns[k NksM+s rqjar e/;e eu dj ';kea iq#"k flag ladyi dj] lgrs foifr vusd] ^cuk^ u NksM+s /;s; dks] j?kqcj jk[ks VsdAA jfpr% ekuo /kez izksrk ln~q# Jh jknksm+nklth egkjkt STUDY PACKAGE Suject : Mathematics Topic : QUADRATIC EQUATIONS Inde Theory Short Revision Eercise (E to 6) Assertion & Reason 5 Que from Compt Eams 6 Yrs Que from IIT-JEE 7 0 Yrs Que from AIEEE Student s Name : Class Roll No : : ADDRESS: R-, Opp Raiway Track, New Corner Glass Building, Zone-, MP NAGAR, Bhopal : (0755) 00 000, 9890 5888, wwwtekoclassescom R QUADRATIC EQUATIONS / Page of TEKO CLASSES, HOD MATHS : SUHAG R KARIYA (S R K Sir) PH: (0755)- 00 000, 0 9890 5888, BHOPAL, (MP)
Quadratic Equation Equation v/s Identity: A quadratic equation is satisfied y eactly two values of ' ' which may e real or imaginary The equation, a + a quadratic + equation c = 0 is: if a 0 Two Roots a linear equation if a = 0, 0 One Root a contradiction if an identity if a = = 0, c 0 a = = c = 0 No Root Infinite Roots If a quadratic equation is satisfied y three distinct values of ' ', then it is an identity Solved Eample # : (i) + = 0 is a quadratic equation here a = (ii) ( + ) = + + is an identity in Solution:Here highest power of in the given relation is and this relation is satisfied y three different values = 0, = and = and hence it is an identity ecause a polynomial equation of n th degree cannot have more than n distinct roots Relation Between Roots & Co-efficients: (i) The solutions of quadratic equation, a + + c = 0, (a 0) is given y ± ac = a The epression, a c D is called discriminant of quadratic equation (ii) If α, β are the roots of quadratic equation, a + + c = 0, a 0 Then: c D (a) α + β = () α β = (c) α β = a a a (iii) A quadratic equation whose roots are α & β, is ( α) ( β) = 0 ie (sum of roots) + (product of roots) = 0 Solved Eample # : If α and β are the roots of a + + c = 0, find the equation whose roots are α+ and β+ Solution Replacing y in the given equation, the required equation is a( ) + ( ) + c = 0 ie, a (a ) + (a + c) = 0 Solved Eample # The coefficient of in the quadratic equation + p + q = 0 was taken as 7 in place of, its roots were found to e and 5 Find the roots of the original equation Solution Here q = ( ) ( 5) = 0, correct value of p = Hence original equation is + + 0 = 0 as ( + 0) ( + ) = 0 roots are 0, Self Practice Prolems : If α, β are the roots of the quadratic equation a + + c = 0 then find the quadratic equation whose roots are (i) α, β (ii) α, β (iii) α +, β + (iv) + α, α + β β (v) β α, α β (r + ) If r e the ratio of the roots of the equation a + + c = 0, show that = r ac Ans() (i) a + + c = 0 (ii) a + (ac ) + c = 0 (iii) a (a ) + a + c = 0 (iv) (a + + c) (a c) + a + c = 0 (v) ac ( ac) + ac = 0 Nature of Roots: Consider the quadratic equation, a + + c = 0 having α, β as its roots; D a c D = 0 D 0 Roots are equalα = β = /a Roots are unequal a,, c R & D > 0 a,, c R & D < 0 Roots are real Roots are imaginary α = p + i q, β = p i q a,, c Q & a,, c Q & D is a perfect square D is not a perfect square Roots are rational Roots are irrational ie α = p + q, β = p q a =,, c Ι & D is a perfect square Roots are integral Solved Eample # : For what values of m the equation ( + m) ( + m) + ( + 8m) = 0 has equal roots Solution Given equation is ( + m) ( + m) + ( + 8m) = 0 (i) Let D e the discriminant of equation (i) Roots of equation (i) will e equal if D = 0 or, ( + m) ( + m) ( + 8m) = 0 QUADRATIC EQUATIONS / Page of TEKO CLASSES, HOD MATHS : SUHAG R KARIYA (S R K Sir) PH: (0755)- 00 000, 0 9890 5888, BHOPAL, (MP)
or, ( + 9m + 6m 9m 8m ) = 0 or, m m = 0 or, m(m ) = 0 m = 0, Solved Eample # 5: Find all the integral values of a for which the quadratic equation ( a) ( 0) + = 0 has integral roots Solution: Here the equation is (a + 0) + 0a + = 0 Since integral roots will always e rational it means D should e a perfect square From (i) D = a 0a + 96 D = (a 0) = (a 0) D If D is a perfect square it means we want difference of two perfect square as which is possile only when (a 0) = and D = 0 (a 0) = ± a =, 8 Solved Eample # 6: If the roots of the equation ( a) ( ) k = 0 e c and d, then prove that the roots of the equation ( c) ( d) + k = 0, are a and Solution By given condition ( a) ( ) k ( c) ( d) or ( c) ( d) + k ( a) ( ) Aove shows that the roots of ( c) ( d) + k = 0 are a and Self Practice Prolems : Let (α ) + α = 0 (α R) e a quadratic equation Find the value of α for which (i) Both roots are real and distinct (ii) Both roots are equal (iii) Both roots are imaginary (iv) Both roots are opposite in sign (v) Both roots are equal in magnitude ut opposite in sign Find the values of a, if a + 9 = 0 has integral roots 5 If P() = a + + c, and Q() = a + d + c, ac 0 then prove that P() Q() = 0 has atleast two real roots Ans () (i) (, ) (, ) (ii) α {, } (iii) (, ) (iv) (, ) (v) φ () a =, FREE Download Study Package from wesite: wwwtekoclassescom Common Roots: Consider two quadratic equations, a + + c = 0 & a + + c = 0 (i) If two quadratic equations have oth roots common, then the equation are identical and their a co-efficient are in proportion ie = c = a c (ii) If only one root is common, then the common root ' α ' will e: α = c a c a = a a Hence the condition for one common root is: c a c a c a c a a + a a + c a a = 0 ( c a c ) = ( a a ) ( c ) a c Note : If f() = 0 & g() = 0 are two polynomial equation having some common root(s) then those common root(s) is/are also the root(s) of h() = a f() + g () = 0 Solved Eample # 7: If a + = 0 and p + q = 0 have a root in common and the second equation has equal ap roots, show that + q = Solution Given equations are : a + = 0 and p + q = 0 Let α e the common root Then roots of equation () will e α and α Let β e the other root of equation () Thus roots of equation () are α, β and those of equation () are α, α Now α + β = a (iii) αβ = (iv) α = p α = q (v) (vi) LHS = + q = αβ + α = α(α + β) (vii) ap ( α + β) α and RHS = = from (7) and (8), LHS = RHS = α (α + β) (viii) Solved Eample # 8: If a,, c R and equations a + + c = 0 and + + 9 = 0 have a common root, show that a : : c = : : 9 Solution Given equations are : + + 9 = 0 (i) and a + + c = 0 (ii) Clearly roots of equation (i) are imaginary since equation (i) and (ii) have a common root, therefore common root must e imaginary and hence oth roots will e common Therefore equations (i) and (ii) are identical a c = = a : : c = : : 9 9 Self Practice Prolems : 6 If the equation + + ac = 0 and + c + a = 0 have a common root then prove that the equation containing other roots will e given y + a + c = 0 7 If the equations a + + c = 0 and + + + = 0 have two common roots then show that a = = c 8 If a + + c = 0 and a + + c = 0 have a common root and a,, c are in GP a c,, a c c c a c c a are in AP show that QUADRATIC EQUATIONS / Page of TEKO CLASSES, HOD MATHS : SUHAG R KARIYA (S R K Sir) PH: (0755)- 00 000, 0 9890 5888, BHOPAL, (MP)
5 Factorisation of Quadratic Epressions: The condition that a quadratic epression f () = a + + c a perfect square of a linear epression, is D a The condition c = 0 that a quadratic epressionf (, y)= a² + hy + y² + g+ fy + c may e resolved into two linear factors is that; a ac + fgh af² g² ch² = 0 OR h f = 0 g f c Solved Eample # 9: Determine a such that + a and + a may have a common factor Solution Let α e a common factor of + a and + a Then = α will satisfy the equations + a = 0 and + a = 0 α α + a = 0 and α α + a = 0 Solving (i) and (ii) y cross multiplication method, we get a = Sol E 0: Show that the epression + (a + + c) + (c + ca + a) will e a perfect square if a = = c Solution Given quadratic epression will e a perfect square if the discriminant of its corresponding equation is zero ie (a + + c) (c + ca + a) = 0 or (a + + c) (c + ca + a) = 0 or ((a ) + ( c) + (c a) ) = 0 which is possile only when a = = c Self Practice Prolems : 9 For what values of k the epression ( k) + (k + ) + 8k + will e a perfect square? 0 If α e a factor common to a + + c and a + + c prove that α(a a ) = If + αy + y + a y + can e resolved into two linear factors, Prove that α is a root of the equation + a + a + 6 = 0 Ans () 0, 6 Graph of Quadratic Epression: or y = f () = a + + c D y + = a a + a the graph etween, y is always a paraola h D the co ordinate of verte are, a a If a > 0 then the shape of the paraola is concave upwards & if a < 0 then the shape of the paraola is concave downwards the paraola intersect the y ais at point (0, c) the co ordinate of point of intersection of paraola with ais are the real roots of the quadratic equation f () = 0 Hence the paraola may or may not intersect the ais at real points 7 Range of Quadratic Epression f () = a + + c (i) Asolute Range: D If a > 0 f (), a D a < 0 f (), a Hence maimum and minimum values of the epression f () is at = a (at verte) (ii) Range in restricted domain: Given [, ] (a) If [, a ] then, () If [, a ] then, f () [ min { f( ), f( ) }, ma { f( ), f( ) }] D D f () min f( ), f( ),, ma f( ), f( ), a a Solved Eample # If c < 0 and a + + c = 0 does not have any real roots then prove that (i) a + c < 0 (ii) 9a + + c < 0 Solution c < 0 and D < 0 f() = a + + c < 0 for all R f( ) = a + c < 0 and f() = 9a + + c < 0 Solved Eample # Find the maimum and minimum values of f() = 5 + 6 Solution g D a in respective cases and it occurs QUADRATIC EQUATIONS / Page of TEKO CLASSES, HOD MATHS : SUHAG R KARIYA (S R K Sir) PH: (0755)- 00 000, 0 9890 5888, BHOPAL, (MP)
minimum of f() = D a at = a 5 5 = at = = maimum of f() = Hence range is, Solved Eample # : Find the range of rational epression y = + Solution y = + + (y ) + (y + ) + y = 0 is real D 0 + if is real + + (y + ) (y ) 0 (y ) (y ) 0 y, + Solved Eample # :Find the range of y =, if is real + + 6 Solution: + y = + + 6 y + y + 6y = + is real y + (y ) + 6y = 0 D 0 (y ) 8y (6y ) 0 (y ) (y + ) 0 y, Self Practice Prolems : If c > 0 and a + + c = 0 does not have any real roots then prove that (i) a + c > 0 (ii) a + + c > 0 (a ) If f() = ( a) ( ), then show that f() For what least integral value of k the quadratic polynomial (k ) + 8 + k + > 0 R 5 Find the range in which the value of function + 7 lies R + 7 m + 6 Find the interval in which 'm' lies so that the function y = + + m R Ans () k = 5 (5) (, 5] [9, ) (6) m [, 7] 8 Sign of Quadratic Epressions: can take all real values The value of epression, f () = a + + c at = is equal to y co ordinate of a point on paraola 0 y = a + + c whose co ordinate is Hence if the point lies aove the ais for some =, then f ( ) 0 0 0 > 0 and vice versa We get si different positions of the graph with respect to ais as shown NOTE: (i) R, y > 0 only if a > 0 & D ² ac < 0 (figure ) (ii) R, y < 0 only if a < 0 & D ² ac < 0 (figure 6) 9 Solution of Quadratic Inequalities: The values of ' ' satisfying the inequality, a + + c > 0 (a 0) are: (i) If D > 0, ie the equation a + + c = 0 has two different roots α < β Then a > 0 (, α) (β, ) QUADRATIC EQUATIONS / Page 5 of TEKO CLASSES, HOD MATHS : SUHAG R KARIYA (S R K Sir) PH: (0755)- 00 000, 0 9890 5888, BHOPAL, (MP)
a < 0 (α, β) (ii) If D = 0, ie roots are equal, ie α = β Then a > 0 (, α) (α, ) a < 0 φ (iii) If D < 0, ie the equation a + + c = 0 has no real root Then a > 0 R a < 0 φ P() Q() R() < (iv) Inequalities of the form = A() B() C() 0 can e quickly solved using the method of > intervals, where A, B, C, P, Q, R are linear functions of ' ' + 6 7 Solved Eample # 5 Solve + Solution + 6 7 + 6 + 9 0 ( ) 0 R Solved Eample # 6: Solve + + > 0 + Solution + > 0 R { } + + > 0 D = = < 0 + + > 0 R (, ) (, ) Solved Eample # 7 + + < Solution < + + in + + D = = < 0 + + > 0 R < ( + + ) ( ) {( + + )} < 0 ( + ) ( 6 ) < 0 ( + ) ( + ) ( + ) > 0 (, ) (, ) Self Practice Prolems : 7 (i) + 5 < 0 (ii) 7 + < 8 Solve 9 + 9 Solve the inequation ( + + ) ( + ) 5 0 Find the value of parameter 'α' for which the inequality Solve 5 + + α + + + < is satisfied R Ans (7) (i) +, (ii) (, ) (8) (, ) (, ) (9) (, ] [, ] [, ) 8 5 (0) (, 5) () 0, 5, 0 Location Of Roots: Let f () = a² + + c, where a > 0 & a,, c R (i) (ii) (iii) (i) Conditions for oth the roots of f () = 0 to e greater than a specified numer 0 are (ii) ² ac 0; f ( 0 ) > 0 & ( /a) > 0 Conditions for oth the roots of f () = 0 to e smaller than a specified numer 0 are (iii) ² ac 0; f ( 0 ) > 0 & ( /a) < 0 Conditions for oth roots of f () = 0 to lie on either side of the numer 0 (in other words the numer 0 lies etween the roots of f () = 0), is f ( 0 ) < 0 (iv) (v) QUADRATIC EQUATIONS / Page 6 of TEKO CLASSES, HOD MATHS : SUHAG R KARIYA (S R K Sir) PH: (0755)- 00 000, 0 9890 5888, BHOPAL, (MP)
(iv) Conditions that oth roots of f () = 0 to e confined etween the numers and, ( < ) are ² ac 0; f ( ) > 0 ; f ( ) > 0 & < ( /a) < (v) Conditions for eactly one root of f () = 0 to lie in the interval (, ) ie < < is f ( ) f ( ) < 0 E0 (m ) + m = 0 (a) Find values of m so that oth the roots are greater than Condition - Ι D 0 (m ) m 0 m 0m + 9 0 (m ) (m 9) 0 m (, ] [9, ) (i) Condition - ΙΙ f() > 0 (m ) + m > 0 m < 0(ii), m Condition - ΙΙΙ > > m > 7(iii) a Intersection of (i), (ii) and (iii) gives m [9, 0) Ans () Find the values of m so that oth roots lie in the interval (, ) Condition - Ι D 0 m (, ] [9, ) Condition - ΙΙ f() > 0 (m ) + m > 0 > 0 m R Condition - ΙΙΙ f() > 0 m < 0 m Condition - ΙV < < < < 5 < m < 7 a intersection gives m φ Ans (c) One root is greater than and other smaller than (d) (e) Condition - Ι f() < 0 < 0 m φ Condition - ΙΙ f() < 0 m > 0 Intersection gives m φ Ans Find the value of m for which oth roots are positive Condition - Ι D 0 m (, ] [9, ) Condition - ΙΙ f(0) > 0 m > 0 m Condition - ΙΙΙ > 0 > 0 a intersection gives m [9, ) Ans m > Find the values of m for which one root is (positive) and other is (negative) Condition - Ι f(0) < 0 m < 0 Ans (f) Roots are equal in magnitude and opposite in sign sum of roots = 0 m = and f(0) < 0 m < 0 m φ Ans E0 Find all the values of 'a' for which oth the roots of the equation (a ) + a + (a + ) = 0 lies in the interval (, ) Sol Case - Ι When a > 0 a > Condition - Ι f( ) > 0 (a ) a + a + > 0 a 5 > 0 a > 5 QUADRATIC EQUATIONS / Page 7 of TEKO CLASSES, HOD MATHS : SUHAG R KARIYA (S R K Sir) PH: (0755)- 00 000, 0 9890 5888, BHOPAL, (MP)
Condition - ΙΙ f()> 0 a + > 0 a > Condition - ΙΙΙ D 0 a (a + ) (a ) 0 a 6 (a ) Condition - ΙV < > 0 a (, ) (, ) a a a a Condition - V < a (a ) > > 0 a Intersection gives a (5, 6] Ans Case-ΙΙ when a < 0 a < Condition - Ι f( ) < 0 a < 5 Condition - ΙΙ f() < 0, a < Condition - ΙΙΙ < < a (, ) (, ) a Condition - ΙV D 0 a 6 intersection gives a, complete solution is a, (5, 6] Ans Self Practice Prolems : Let (α ) + α = 0 (α R) e a quadratic equation find the value of α for which (a) Both the roots are positive () Both the roots are negative (c) (e) Both the roots are opposite in sign Both the roots are smaller than / (d) Both the roots are greater than / (f) One root is small than / and the other root is greater than / Ans (a) [, ) () φ (c) (, ) (d) φ (e) (, ] Find the values of the parameter a for which the roots of the quadratic equation (f) (, ) + (a ) + a + 5 = 0 are (i) Ans positive (i) ( 5, ] (ii) negative (iii) (ii) [, ) (iii) (, 5) opposite in sign Find the values of P for which oth the roots of the equation 0p + (5p + 5p 66) = 0 are less than Ans (, ) 5 Find the values of α for which 6 lies etween the roots of the equation + (α ) + 9 = 0 Ans, 6 Let (α ) + α = 0 (α R) e a quadratic equation find the value of α for which (i) Eactly one root lies in 0, (ii) Both roots lies in 0, (iii) At least one root lies in 0, (iv) One root is greater than / and other root is smaller than 0 Ans (i) (, ) (, ) (ii) φ (iii) (, ) (, ) (iv) φ 7 In what interval must the numer 'a' vary so that oth roots of the equation 8 a + a = 0 lies etween and Ans (, ) Find the values of k, for which the quadratic epression a + (a ) is negative for eactly two integral values of Ans [, ) Theory Of Equations: If α, α, α, α n are the roots of the equation; f() = a 0 n + a n- + a n- + + a n- + a n = 0 where a 0, a, a n are all real & a 0 0 then, a α = a, α α = + a 0 a, α α α = a 0 a,, α α α α = an n 0 ( )n a0 NOTE : (i) If α is a root of the equation f() = 0, then the polynomial f() is eactly divisile y ( α) or ( α) is a factor of f() and conversely (ii) Every equation of n th degree (n ) has eactly n roots & if the equation has more than n roots, it is an identity (iii) If the coefficients of the equation f() = 0 are all real and α + iβ is its root, then α iβ is also a root ie imaginary roots occur in conjugate pairs (iv) An equation of odd degree will have odd numer of real roots and an equation of even degree will have even numers of real roots (v) If the coef f i ci ents i n the equati on are al l rati onal & α + β i s one of i ts roots, then α β is also a root where α, β Q & β is not a perfect square (vi) If there e any two real numers 'a' & '' such that f(a) & f() are of opposite signs, then (vii) f() = 0 must have odd numer of real roots (also atleast one real root) etween ' a ' and ' ' Every equation f() = 0 of degree odd has atleast one real root of a sign opposite to that of its last term (If coefficient of highest degree term is positive) QUADRATIC EQUATIONS / Page 8 of TEKO CLASSES, HOD MATHS : SUHAG R KARIYA (S R K Sir) PH: (0755)- 00 000, 0 9890 5888, BHOPAL, (MP)
E + + 5 + 6 = 0 has roots α, β, γ then find α + β + γ, αβ + βγ + γα and αβγ 5 6 α + β + γ = = αβ + βγ + γα =, αβγ = = E Find the roots of + 0 + 6 = 0 If two roots are equal Let roots e α, α and β 0 α + α + β = α + β = 5 (i) α α + αβ + αβ = α + αβ = from equation (i) & 6 α β = α + α ( 5 α) = QUADRATIC EQUATIONS / Page 9 of FREE Download Study Package from wesite: wwwtekoclassescom α 0α α = α = /, 6 α + 0α = 0 when α = from equation (i) α β = ( 5 ) = when α = α β = 6 6 α =, β = 6 Hence roots of equation 5 6 =,, 6 Ans Self Practice Prolems : 9 Find the relation etween p, q and r if the roots of the cuic equation p + q r = 0 are such that they are in AP Ans p 9pq + 7r = 0 0 If α, β, γ are the roots of the cuic + q + r = 0 then find the equation whose roots are (a) () α + β, β + γ, γ + α αβ, βγ, γα Ans Ans + q r = 0 q r = 0 (c) α, β, γ Ans + q + q r = 0 (d) α, β, γ Ans + r + (r + q ) + r = 0 SHORT REVISION The general form of a quadratic equation in is, a + + c = 0, where a,, c R & a 0 RESULTS : ± The solution of the quadratic equation, a² + + c = 0 is given y = a The epression ac = D is called the discriminant of the quadratic equation If α & β are the roots of the quadratic equation a² + + c = 0, then; (i) α + β = /a (ii) α β = c/a (iii) α β = D / a NATURE OF ROOTS: (A) Consider the quadratic equation a² + + c = 0 where a,, c R & a 0 then ; (i) D > 0 roots are real & distinct (unequal) (ii) D = 0 roots are real & coincident (equal) (iii) (iv) D < 0 roots are imaginary If p + i q is one root of a quadratic equation, then the other must e the conjugate p i q & vice versa (p, q R & i = ) (B) Consider the quadratic equation a + + c = 0 where a,, c Q & a 0 then; (i) If D > 0 & is a perfect square, then roots are rational & unequal (ii) If α = p + q is one root in this case, (where p is rational & q is a surd) then the other root must e the conjugate of it ie β = p q & vice versa A quadratic equation whose roots are α & β is ( α)( β) = 0 ie (α + β) + α β = 0 ie (sum of roots) + product of roots = 0 5 Rememer that a quadratic equation cannot have three different roots & if it has, it ecomes an identity 6 Consider the quadratic epression, y = a² + + c, a 0 & a,, c R then ; (i) The graph etween, y is always a paraola If a > 0 then the shape of the paraola is concave upwards & if a < 0 then the shape of the paraola is concave downwards ac TEKO CLASSES, HOD MATHS : SUHAG R KARIYA (S R K Sir) PH: (0755)- 00 000, 0 9890 5888, BHOPAL, (MP)
(ii) R, y > 0 only if a > 0 & ² ac < 0 (figure ) (iii) R, y < 0 only if a < 0 & ² ac < 0 (figure 6) Carefully go through the 6 different shapes of the paraola given elow 7 SOLUTION OF QUADRATIC INEQUALITIES: a + + c > 0 (a 0) (i) If D > 0, then the equation a + + c = 0 has two different roots < Then a > 0 (, ) (, ) a < 0 (, ) (ii) If D = 0, then roots are equal, ie = In that case a > 0 (, ) (, ) a < 0 φ (iii) Inequalities of the form P ( ) 0 can e quickly solved using the method of intervals Q( ) 8 MAXIMUM & MINIMUM VALUE of y = a² + + c occurs at = (/a) according as ; ac a < 0 or a > 0 y, if a > 0 & y, ac if a < 0 a a 9 COMMON ROOTS OF QUADRATIC EQUATIONS [ONLY ONE COMMON ROOT] : Let α e the common root of a² + + c = 0 & a + + c = 0 Therefore α α a α² + α + c = 0 ; a α² + α + c = 0 By Cramer s Rule = = c c a c ac a a ca c a c c Therefore, α = = a a a c ac So the condition for a common root is (ca c a)² = (a a )(c c) 0 The condition that a quadratic function f (, y) = a² + hy + y² + g + fy + c may e resolved into two linear factors is that ; a h g ac + fgh af g ch = 0 OR h f = 0 g f c THEORY OF EQUATIONS : If α, α, α, α n are the roots of the equation; f() = a 0 n + a n- + a n- + + a n- + a n = 0 where a 0, a, a n are all real & a 0 0 then, α = a, α a α = + a, α 0 a α α = a,, α 0 a α α α n = ( ) a n n 0 a 0 Note : (i) If α is a root of the equation f() = 0, then the polynomial f() is eactly divisile y ( α) or ( α) is a factor of f() and conversely (ii) (iii) Every equation of nth degree (n ) has eactly n roots & if the equation has more than n roots, it is an identity If the coefficients of the equation f() = 0 are all real and α + iβ is its root, then α iβ is also a root ie imaginary roots occur in conjugate pairs (iv) If the coefficients in the equation are all rational & α + β is one of its roots, then α β is also a root where α, β Q & β is not a perfect square (v) If there e any two real numers 'a' & '' such that f(a) & f() are of opposite signs, then f() = 0 must have atleast one real root etween 'a' and '' (vi) Every equation f() = 0 of degree odd has atleast one real root of a sign opposite to that of its last term LOCATION OF ROOTS : Let f () = a + + c, where a > 0 & a,, c R (i) Conditions for oth the roots of f () = 0 to e greater than a specified numer d are ac 0; f (d) > 0 & ( /a) > d (ii) Conditions for oth roots of f () = 0 to lie on either side of the numer d (in other words the numer d lies etween the roots of f () = 0) is f (d) < 0 (iii) Conditions for eactly one root of f () = 0 to lie in the interval (d, e) ie d < < e are ac > 0 & f (d) f (e) < 0 (iv) Conditions that oth roots of f () = 0 to e confined etween the numers p & q are (p < q) ac 0; f (p) > 0; f (q) > 0 & p < ( /a) < q LOGARITHMIC INEQUALITIES (i) For a > the inequality 0 < < y & log a < log a y are equivalent (ii) For 0 < a < the inequality 0 < < y & log a > log a y are equivalent (iii) If a > then log a < p 0 < < a p (iv) If a > then log a > p > a p (v) If 0 < a < then log a < p > a p (vi) If 0 < a < then log a > p 0 < < a p EXERCISE Q If the roots of the equation [/( + p)] + [/( + q)] = /r are equal in magnitude ut opposite in sign, show that p + q = r & that the product of the roots is equal to ( /) (p + q ) Q If cos (A + B) + is a factor of the epression, + sin A sin B (cos A + cos B) + cos A cos B Then find the other factor Q α, β are the roots of the equation K ( ) + + 5 = 0 If K & K are the two values of K for which the roots α, β are connected y the relation (α/β) + (β/α) = /5 Find the value of Q (K /K ) + (K /K ) If the quadratic equations, + + c = 0 and + c + = 0 have a common root then prove that either + c + = 0 or + c + = c + + c Q5 p p If the roots of the equation q + + p( + q) + q(q ) + = 0 are equal then show that QUADRATIC EQUATIONS / Page 0 of TEKO CLASSES, HOD MATHS : SUHAG R KARIYA (S R K Sir) PH: (0755)- 00 000, 0 9890 5888, BHOPAL, (MP)
p = q Q6 If one root of the equation a + + c = 0 e the square of the other, prove that + a c + ac = ac a + ( a + ) + 9a + Q7 Find the range of values of a, such that f () = is always negative 8 + Q8 Find a quadratic equation whose sum and product of the roots are the values of the epressions (cosec 0 sec0 ) and (05 cosec0 sin70 ) respectively Also epress the roots of this quadratic in terms of tangent π of an angle lying in 0, 6 + Q9 Find the least value of for all real values of, using the theory of quadratic equations 5 8 + 7 Q0 Find the least value of (p + ) + (p ) + (p + ) for real values of p and Q If α e a root of the equation + = 0 then prove that α α is the other root Q(a) If α, β are the roots of the quadratic equation a ++c = 0 then which of the following epressions in α, β will denote the symmetric α functions of roots Give proper reasoning (i) f (α, β) = α β (ii) f (α, β) = α β + αβ (iii) f (α, β) = ln β (iv) f (α, β) = cos (α β) () If α, β are the roots of the equation p + q = 0, then find the quadratic equation the roots of which are (α β )(α β ) & α β + α β Q If α, β are the roots of a + + c = 0 & α, β are the roots of a + + c = 0, show that α, α are the roots of + a a + + c + c = 0 Q If α, β are the roots of p + = 0 & γ, δ are the roots of + q + = 0, show that (α γ) (β γ) (α + δ) (β + δ) = q p Q5 Show that if p, q, r & s are real numers & pr = (q + s), then at least one of the equations + p + q = 0, + r + s = 0 has real roots Q6 If a & are positive numers, prove that the equation + + = 0 has two real roots, one etween a/ & a/ a + and the other etween / & / Q7 If the roots of a + = 0 are real & differ y a quantity which is less than c (c > 0), prove that lies etween (/)(a c ) & (/)a Q8 At what values of 'a' do all the zeroes of the function, f () = (a ) + a + a + lie on the interval (, )? Q9 If one root of the quadratic equation a² + + c = 0 is equal to the n th power of the other, then show that (ac n ) /(n+) + (a n c) / (n+) + = 0 Q0 If p, q, r and s are distinct and different from, show that if the points with co-ordinates p p 5 q q 5,, p p r r 5,, q q s s 5, and r r, are collinear then s s pqrs = 5 (p + q + r + s) + (pqr + qrs + rsp + spq) Q The quadratic equation + p + q = 0 where p and q are integers has rational roots Prove that the roots are all integral Q If the quadratic equations + + ca = 0 & + c + a = 0 have a common root, prove that the equation containing their other root is + a + c = 0 Q If α, β are the roots of + p + q = 0 & n + p n n + q n = 0 where n is an even integer, show that α/β, β/α are the roots of n + + ( + ) n = 0 Q If α, β are the roots of the equation + = 0 otain the equation whose roots are α α + 5α, β β + β + 5 Q5 If each pair of the following three equations + p + q = 0, + p + q = 0 & + p + q = 0 has eactly one root common, prove that; (p + p + p ) = [p p + p p + p p q q q ] Q6 Show that the function z = + y + y + y + is not smaller than + Q7 Find all real numers such that, = Q8 Find the values of a for which < [( + a )/( + + )] < is valid for all real 6 6 + + 6 Q9 Find the minimum value of for > 0 + + + Q0 Find the product of the real roots of the equation, + 8 + 0 = + 8 + 5 EXERCISE Q Solve the following where R (a) ( ) + + + 5 = 0 () + = 5 (c) (e) + + = 0 (d) + + = + + For a 0, determine all real roots of the equation a a a = 0 QUADRATIC EQUATIONS / Page of TEKO CLASSES, HOD MATHS : SUHAG R KARIYA (S R K Sir) PH: (0755)- 00 000, 0 9890 5888, BHOPAL, (MP)
Q Let a,, c, d e distinct real numers and a and are the roots of quadratic equation c 5d = 0 If c and d are the roots of the quadratic equation a 5 = 0 then find the numerical value of a + + c + d p Q Let f () = a + + c = 0 has an irrational root r If u = e any rational numer, where a,, c, p and q are integer Prove that q f (u) q Q Let a,, c e real If a + + c = 0 has two real roots α & β, where α < & β > then show that + c/a + /a < 0 Q5 If α, β are the roots of the equation, a + = 0 and γ, δ are the roots of the equation, (a + ) + a (a ) = 0 such that α, β (γ, δ) then find the values of 'a' Q6 Two roots of a iquadratic 8 + k + 00 98 = 0 have their product equal to ( ) Find the value of k Q7 If y eleminating etween the equation ² + a + = 0 & y + l ( + y) + m = 0, a quadratic in y is formed whose roots are the same as those of the original quadratic in Then prove either a = l & = m or + m = al α α cosα + sin cos Q8 If e real, prove that lies etween cosβ + β and β sin cos Q9 Solve the equations, a + y + cy = + cy + ay = d g d i Q0 Find the values of K so that the quadratic equation + (K ) + K + 5 = 0 has atleast one positive root Q Find the values of '' for which the equation log + 8 = log5 has only one solution 5 Q Find all the values of the parameter 'a' for which oth roots of the quadratic equation a + = 0 elong to the interval ( 0, ) Q Find all the values of the parameters c for which the inequality has at least one solution F 7I + log + + HG K J logcc + ch Q Find the values of K for which the equation + ( K) + K = 0 ; (a) has no real solution () has one real solution Q5 Find all the values of the parameter 'a' for which the inequality a9 + (a ) + a > 0 is satisfied for all real values of Q6 Find the complete set of real values of a for which oth roots of the quadratic equation ( a 6a + 5) a + a + (6a a 8) = 0 lie on either side of the origin Q7 If g () = + p + q + r where p, q and r are integers If g (0) and g ( ) are oth odd, then prove that the equation g () = 0 cannot have three integral roots Q8 Find all numers p for each of which the least value of the quadratic trinomial p + p p + on the interval 0 is equal to Q9 Let P () = + + c, where and c are integer If P () is a factor of oth + 6 + 5 and + + 8 + 5, find the value of P() Q0 Let e a positive real Find the maimum possile value of the epression y = + + EXERCISE Solve the inequality Where ever ase is not given take it as 0 Q 5 ( log ) log 0log + 8 < 0 Q /log log < Q (log 00 ) + (log 0 ) + log Q log / ( + ) > log ( ) Q5 log log log > Q6 log /5 ( + 5 + ) < 0 Q7 log / + log > Q8 log ² ( + ) < Q9 + 5 log < 6 5 Q0 (log +6 ) log ( ) Q + log 0 + 5 Q log [(+6)/] [log {( )/( + )}] > 0 Q log ( + 7) Find out the values of 'a' for which any solution of the inequality, < is also a solution of the inequality, log( + ) + (5 a) 0a Q Solve the inequality log ( 0 + ) > 0 log Q5 Find the set of values of 'y' for which the inequality, log 05 y + log 05 y > 0 is valid for atleast one real value of '' EXERCISE QUADRATIC EQUATIONS / Page of TEKO CLASSES, HOD MATHS : SUHAG R KARIYA (S R K Sir) PH: (0755)- 00 000, 0 9890 5888, BHOPAL, (MP)
sin cos Q Prove that the values of the function do not lie from & for any real [JEE '97, 5] sin cos Q The sum of all the real roots of the equation + = 0 is [JEE '97, ] Q Let S e a square of unit area Consider any quadrilateral which has one verte on each side of S If a,, c & d denote the lengths of the sides of the quadrilateral, prove that: a + + c + d Q In a college of 00 students, every student reads 5 news papers & every news paper is read y 60 students The numer of news papers is: (A) atleast 0 (B) atmost 0 (C) eactly 5 (D) none of the aove Q5 If α, β are the roots of the equation + c = 0, then find the equation whose roots are, (α + β ) (α + β ) & α 5 β + α β 5 α β Q6(i) Let α + iβ; α, β R, e a root of the equation + q + r = 0; q, r R Find a real cuic equation, independent of α & β, whose one root is α (ii) Find the values of α & β, 0 < α, β < π/, satisfying the following equation, cos α cos β cos (α + β) = /8 [REE '99, + 6] π P Q Q7(i) In a triangle PQR, R = If tan & tan are the roots of the equation a + + c = 0 (a 0) then : (A) a + = c (B) + c = a (C) a + c = (D) = c (ii) If the roots of the equation a + a + a = 0 are real & less than then (A) a < (B) a (C) < a (D) a > [JEE '99, + ] Q8 If α, β are the roots of the equation, ( a)( ) + c = 0, find the roots of the equation, ( α) ( β) = c [REE 000 (Mains), ] Q9(a) For the equation, + p + = 0, p > 0 if one of the roots is square of the other, then p is equal to: (A) / (B) (C) (D) / () If α & β (α < β), are the roots of the equation, + + c = 0, where c < 0 <, then (A) 0 < α < β (B) α < 0 < β < α (C) α < β < 0 (D) α < 0 < α < β (c) If > a, then the equation, ( a) ( ) = 0, has : (A) oth roots in [a, ] (B) oth roots in (, a) (C) oth roots in [, ) (D) one root in (, a) & other in (, + ) [JEE 000 Screening, + + out of 5] (d) If α, β are the roots of a + + c = 0, (a 0) and α + δ, β + δ, are the roots of, A + B + C = 0, (A 0) for some constant δ, then prove that, ac B AC = [JEE 000, Mains, out of 00] a A Q0 The numer of integer values of m, for which the co-ordinate of the point of intersection of the lines + y = 9 and y = m + is also an integer, is [JEE 00, Screening, out of 5] (A) (B) 0 (C) (D) Q Let a,, c e real numers with a 0 and let α, β e the roots of the equation a + + c = 0 Epress the roots of a + ac + c = 0 in terms of α, β [JEE 00, Mains, 5 out of 00] Q The set of all real numers for which + + > 0, is (A) (, ) U (, ) (B) (, ) U (, ) (C) (, ) U (, ) (D) (, ) [JEE 00 (screening), ] Q If + (a ) + ( a ) = 0 where a, R then find the values of a for which equation has unequal real roots for all values of [JEE 00, Mains- out of 60] [ Based on M R test] Q(a) If one root of the equation + p + q = 0 is the square of the other, then (A) p + q q(p + ) = 0 (B) p + q + q( + p) = 0 (C) p + q + q(p ) = 0 (D) p + q + q( p) = 0 () If + a + 0 a > 0 for all R, then (A) 5 < a < (B) a < 5 (C) a > 5 (D) < a < 5 [JEE 00 (Screening)] + 5 π π Q5 Find the range of values of t for which sin t =, t, [JEE 005(Mains), ] Q 6(a) Let a,, c e the sides of a triangle No two of them are equal and λ R If the roots of the equation + (a + + c) + λ(a + c + ca) = 0 are real, then 5 5 (A) λ < (B) λ > 5 (C) λ, (D) λ, [JEE 006, ] () If roots of the equation 0c d = 0 are a, and those of 0a = 0 are c, d, then find the value of a + + c + d (a,, c and d are distinct numers) [JEE 006, 6] EXERCISE 5 QUADRATIC EQUATIONS / Page of TEKO CLASSES, HOD MATHS : SUHAG R KARIYA (S R K Sir) PH: (0755)- 00 000, 0 9890 5888, BHOPAL, (MP)
Part : (A) Only one correct option The roots of the quadratic equation (a + c) (a c) + (a + c) = 0 are (A) a + + c and a + c (C) a + c and a + c The roots of the equation + = 9 are given y (B) and a + c (D) none of these (A) log, (B) log (/), (C), (D), Two real numers α & β are such that α + β = & α β =, then α & β are the roots of the quadratic equation: (A) 7 = 0 (B) + 7 = 0 (C) + 5 = 0 (D) none of these Let a, and c e real num ers such that a + + c = 0 and a > 0 Then the equat i on a + + c = 0 has (A) real roots (B) imaginary roots (C) eactly one root (D) none of these 5 If e cos e cos =, then the value of cos is (A) log ( 5 ) + (B) log ( + 5 ) (C) log ( + 5 ) log log (D) none of these 6 The numer of the integer solutions of + 9 < ( + ) < 8 + 5 is : (A) (B) (C) (D) none 7 If ( + ) is greater than 5 & less than 7 then the integral value of is equal to (A) (B) (C) (D) 8 The set of real ' ' satisfying, is: (A) [0, ] (B) [, ] (C) [, ] (D) [, ] 9 Let f() = + + Then (A) f() > 0 for all (B) f() > when 0 (C) f() when (D) f() = f( ) for all + 0 If is real and k = + + (A) then: k (B) k 5 (C) k 0 (D) none + c If is real, then can take all real values if : + + c (A) c [0, 6] (B) c [ 6, 0] (C) c (, 6) (0, ) (D) c ( 6, 0) The solution set of the inequality + 0 0 is: (A) (, 5) (, ) (6, ) {0} (B) (, 5) [, ] (6, ) {0} (C) (, 5] [, ] [6, ) {0} (D) none of these If y and y are two factors of the epression y + λy + µy, then (A) λ =, µ = (B) λ =, µ = (C) λ =, µ = (D) none of these If α, β are the roots of the equation, m + m = 0 then the range of values of m for which α, β (, ) is: (A) (, ) (B) (, ) (C) (, ) ((, ) (D) none 5 If the inequality (m ) + 8 + m + > 0 is satisfied for all R then the least integral m is: (A) (B) 5 (C) 6 (D) none 6 For all R, if m 9m + 5m + > 0, then m lies in the interval (A) (/6, 0) (B) [0, /6) (C) (/6, 6/) (D) ( 6/, 0] 7 Let a > 0, > 0 & c > 0 Then oth the roots of the equation a + + c = 0 (A) are real & negative (B) have negative real parts (C) are rational numers (D) none 8 The value of 'a' for which the sum of the squares of the roots of the equation, (a ) a = 0 assume the QUADRATIC EQUATIONS / Page of TEKO CLASSES, HOD MATHS : SUHAG R KARIYA (S R K Sir) PH: (0755)- 00 000, 0 9890 5888, BHOPAL, (MP)
least value is: (A) 0 (B) (C) (D) 9 Consider y =, then the range of epression, y + + y is: (A) [, ] (B) [0, ] (C) [ 9/, 0 ] (D) [ 9/, ] 0 If oth roots of the quadratic equation + + p = 0 eceed p where p R then p must lie in the interval: (A) (, ) (B) (, ) (C) (, ) (0, /) (D) (, ) If a,, p, q are non zero real numers, the two equations, a a + = 0 and p + pq + q = 0 have: (A) no common root (B) one common root if a + = p + q (C) two common roots if pq = a (D) two common roots if q = ap + α + β + γ If α, β & γ are the roots of the equation, = 0 then, + + has the value equal to: α β γ (A) zero (B) (C) 7 (D) The equations + 5 + p + q = 0 and + 7 + p + r = 0 have two roots in common If the third root of each equation is represented y and respectively, then the ordered pair (, ) is: (A) ( 5, 7) (B) (, ) (C) (, ) (D) (5, 7) If α, β are roots of the equation a + + c = 0 then the equation whose roots are α + β and α + β is (A) a (a + ) c + (a + ) = 0 (B) ac (a + c) + (a + c) = 0 (C) ac + (a + c) (a + c) = 0 (D) none of these 5 If coefficients of the equation a + + c = 0, a 0 are real and roots of the equation are non-real comple and a + c <, then (A) a + c > (B) a + c < (C) a + c = (D) none of these 6 The set of possile values of λ for which (λ 5λ + 5) + (λ λ ) = 0 has roots, whose sum and product are oth less than, is 5 5 5 (A), (B) (, ) (C), (D), 7 Let conditions C and C e defined as follows : C : ac 0, C : a,, c are of same sign The roots of a + + c = 0 are real and positive, if (A) oth C and C are satisfied (B) only C is satisfied (C) only C is satisfied (D) none of these Part : (B) May have more than one options correct 8 If a, are non-zero real numers, and α, β the roots of + a + = 0, then (A) (C) α, β are the roots of ( a ) + a = 0 (B), are the roots of α β + a + = 0 α β, are the roots of β α + ( a ) + = 0 (D) α, β are the roots of + a = 0 9 + + is a factor of a + + c + d = 0, then the real root of aove equation is (a,, c, d R) (A) d/a (B) d/a (C) ( a)/a (D) (a )/a 0 If ( + + ) + ( + + ) + ( + + 5) + + ( + 0 + 9) = 500, then is equal to: (A) 0 (B) 0 (C) 05 (D) 05 cos α is a root of the equation 5 + 5 = 0, < < 0, then the value of sin α is: (A) /5 (B) /5 (C) /5 (D) 0/5 If the quadratic equations, + a + c = 0 and + ac + = 0 have a common root then the equation containing their other roots is/are: (A) + a ( + c) a c = 0 (B) a ( + c) + a c = 0 (C) a ( + c) ( + c) + ac = 0 (D) a ( + c) + ( + c) ac = 0 EXERCISE 6 Solve the equation, ( + ) ( + ) ( + ) = 0 Solve the following where R (a) ( ) ² + + ² + 5 = 0 () ( + ) + + + + = 0 (c) ( + ) ( + ) + + 5 = 0 (d) + + = + + If ' ' is real, show that, ( ) ( + ) ( + ) ( + 6) + 5 0 7 + 8 + QUADRATIC EQUATIONS / Page 5 of TEKO CLASSES, HOD MATHS : SUHAG R KARIYA (S R K Sir) PH: (0755)- 00 000, 0 9890 5888, BHOPAL, (MP)
Find the value of which satisfy inequality > + 5 Find the range of the epression f() = sin sin + R 6 Find the range of the quadratic epression f() = + [0, ] 7 Prove that the function y = (² + + )/(² + ) cannot have values greater than / and values smaller than / for R 8 If e real, show that + 9 lies in + + 9, 9 For what values of k the epression + y + y + + y + k can e resolved into two linear factors 0 Show that one of the roots of the equation, a + + c = 0 may e reciprocal of one of the roots of a + + c = 0 if (a a c c ) = ( c a ) ( c a ) Let α + iβ ; α, β R, e a root of the equation + q + r = 0; q, r R Find a real cuic equation, independent of α and β, whose one root is α If a, are the roots of + p + = 0 and c, d are the roots of + q + = 0 Show that q p = (a c) ( c) (a + d) ( + d) If α, β are the roots of the equation ² - p + q = 0, then find the quadratic equation the roots of which are (α β ) (α β ) & α β + α β If ' ' is real, find values of ' k ' for which, + k + < is valid + + 5 Solve the inequality, + < 0 6 The equations a + = 0 & p + q = 0, where 0, q 0 have one common root & the second equation has two equal roots Prove that (q + ) = ap 7 Find the real values of m for which the equation, + (m ) + + m = 0 has atleast one real root? 8 Let a and e two roots of the equation +p + q + r = 0 satisfying the relation a + = 0 Prove that r + pr + q + = 0 ANSWER KEY EXERCISE F I HG K J Q + cos (A B) Q 5 Q7 a, π 5π Q8 + = 0 ; α = tan ; β = tan Q9 Q0 minimum value when = and p = 0 Q (a) (ii) and (iv) ; () p(p 5p q + 5q ) + p q (p q) (p q) = 0 Q8, {} (5, 6] Q + = 0 Q7 = Q9 y min = 6 Q0 0 EXERCISE 5 + Q (a) = ; () = or 5; (c) = or ; (d) or = ; (e) = ( ) a or ( 6 ) a Q 0 Q5 a ( ), Q6 k = 86 Q9 = y = d/(a++c) ; /(c a) = y/(a ) = K where K²a (a² + ² + c² a c ca) = d Q 0 K Q (, ) { }, Q (0, 8] Q (a) K < or K > 5/ () K = Q 5 [, ) Q 6 (, ] [ 0, ) (, ) (5, ) Q8 a = or 5 + 0 L NM I KJ Q Q8 < a < a < QUADRATIC EQUATIONS / Page 6 of TEKO CLASSES, HOD MATHS : SUHAG R KARIYA (S R K Sir) PH: (0755)- 00 000, 0 9890 5888, BHOPAL, (MP)
Q9 P () = Q 0 ( ) Q, ( 8,6) where = EXERCISE Q (0,) (, 0 / 0 ) Q 6 8 Q < < 5 or + 5 0 0 9 < < Q 5 < < - ; < < Q 6 (, 5) (0, ) Q 7 0 < < / log (where ase of log is ) Q 8 <<, <<0, 0<<, > Q 9 < < Q 0 < 7, 5 <, Q ; Q ( 6, 5) (, ) Q a 5, Q (,5 ) ( 7, ) Q 5 ( ), 0 0, EXERCISE (, ) Q Q C Q5 ( + ) + = 0 where = ( c) ( c) ; = c ( c) Q6 (i) + q r = 0, (ii) α = β = π/, Q7 (i) A, (ii) A, Q8 (a, ) Q9 (a) C, () B, (c) D Q0 A Q γ = α β and δ = αβ or γ = αβ and δ = α β Q B Q a > π π π π Q (a) D ; () A Q5,, 0 0 EXERCISE 5 Q6 (a) A, () 0 D D A A 5 D 6 D 7 C 8 B 9 C 0 A B B C A 5 B 6 B 7 B 8 B 9 C 0 B A C A D 5 B 6 D 7 A 8 BC 9 AD 0 AD AC BD EXERCISE 6 {, 5} (a) = () = ( 7 7 )/ (c) =,, (+ ) (d), = (, ), (, ) 5, 6 [, ] 9 k = 8 p(p 5p q + 5q ) + p q (p q) (p q) = 0 k (0, ) 7 5 5 (, ) (, ) (, ) (, 6) (7, ) 7, 6 + q r = 0 for Yrs Que of IIT-JEE & 0 Yrs Que of AIEEE we have distriuted already a ook QUADRATIC EQUATIONS / Page 7 of TEKO CLASSES, HOD MATHS : SUHAG R KARIYA (S R K Sir) PH: (0755)- 00 000, 0 9890 5888, BHOPAL, (MP)