Phone : 0 90 90 7779, 9890 5888 QUADRATIC EQUATIONS PART OF fo/u fopkjr Hkh# tu] ugh vkjehks dke] foifr ns[k NksM+s rqjr e/;e eu dj ';kea iq#"k flg ldyi dj] lgrs foifr vusd] ^cuk^ u NksM+s /;s; dks] j?kqcj jk[ks VsdAA jfpr% ekuo /kez iz.ksrk ln~xq# Jh j.knksm+nklth egkjkt QUADRATIC EQUATIONS Some questions (Assertion Reson type) re given elow. Ech question contins Sttement (Assertion) nd Sttement (Reson). Ech question hs choices (A), (B), (C) nd (D) out of which ONLY ONE is correct. So select the correct choice : Choices re : (A) Sttement is True, Sttement is True; Sttement is correct explntion for Sttement. (B) Sttement is True, Sttement is True; Sttement is NOT correct explntion for Sttement. (C) Sttement is True, Sttement is Flse. (D) Sttement is Flse, Sttement is True.. Sttement-: If x R, x + x + 5 is positive. Sttement-: If < 0, x + x + c, hve sme sign x R.. Sttement-: If + is root of x x = 0, then will e the other root. Sttement-: Irrtionl roots of qudrtic eqution with rtionl coefficients lwys occur in conjugte pir.. Sttement-: The roots of the eqution x + i x + = 0 re lwys conjugte pir. Sttement-: Imginry roots of qudrtic eqution with rel coefficients lwys occur in conjugte pir.. Consider the eqution ( + ) x + ( 5 + 6)x + = 0 Sttement : If =, then ove eqution is true for ll rel x. Sttement : If =, then ove eqution will hve two rel nd distinct roots. 5. Consider the eqution ( + )x + ( ) x = Sttement : Roots of ove eqution re rtionl if '' is rtionl nd not equl to. Sttement : Roots of ove eqution re rtionl for ll rtionl vlues of ''. 6. Let f(x) = x = x + ( + ) x + 5 Sttement : f(x) is positive for sme α < x < β nd for ll R Sttement : f(x) is lwys positive for ll x R nd for sme rel ''. 7. Consider f(x) = (x + x + ) (x + ) (x + x + ) = 0 Sttement : Numer of vlues of '' for which f(x) = 0 will e n identity in x is. Sttement : = the only vlue for which f(x) = 0 will represent n identity. 8. Let,, c e rel such tht x + x + c = 0 nd x + x + = 0 hve common root Sttement : = = c Sttement : Two qudrtic equtions with rel coefficients cn not hve only one imginry root common. 9. Sttement : The numer of vlues of for which ( + ) x + ( 5 + ) x + = 0 is n identity in x is. Sttement : If x + x + c = 0 is n identity in x then = = c = 0. 0. Let (, 0). Sttement : x x + < 0 for ll x R Sttement : If roots of x + x + c = 0,, c R re imginry then signs of x + x + c nd re sme for ll x R.. Let,, c R, 0. Sttement : Difference of the roots of the eqution x + x + c = 0 = Difference of the roots of the eqution x + x c = 0 Sttement : The two qudrtic equtions over rels hve the sme difference of roots if product of the coefficient of the two equtions re the sme.. Sttement : If the roots of x 5 0x + Px + Qx + Rx + S = 0 re in G.P. nd sum of their reciprocl is 0, then S =. Sttement : x. x. x.x.x 5 = S, where x, x, x, x, x 5 re the roots of given eqution.. Sttement : If 0 < α < π, then the eqution (x sin α) (x cos α) = 0 hs oth roots in (sin α, cos α) 8 of
Phone : 0 90 90 7779, 9890 5888 QUADRATIC EQUATIONS PART OF Sttement : If f nd f possess opposite signs then there exist t lest one solution of the eqution f(x) = 0 in open intervl (, ).. Sttement : If / then α < < p where α, β re roots of eqution x + x + = 0 Sttement : Roots of qudrtic eqution re rtionl if discriminnt is perfect squre. 5. Sttement- : The numer of rel roots of x + x + = 0 is zero. Sttement- : x R, x 0. 6. Sttement-: If ll rel vlues of x otined from the eqution x ( ) x + ( ) = 0 re non-positive, then (, 5] Sttement-: If x + x + c is non-positive for ll rel vlues of x, then c must e ve or zero nd must e ve. 7. Sttement-: If,, c, d R such tht < < c < d, then the eqution (x ) (x c) + (x ) (x d) = 0 re rel nd distinct. Sttement-: If f(x) = 0 is polynomil eqution nd, re two rel numers such tht f f < 0 hs t lest one rel root. x + x + 8. Sttement-: f(x) = > 0 x R x + x + 5 Sttement-: x + x + c > 0 x R if > 0 nd c < 0. 9. Sttement-: If + + c = 0 then x + x + c = 0 must hve s root of the eqution Sttement-: If + + c = 0 then x + x + c = 0 hs roots of opposite sign. 0. Sttement-: x + x + C = 0 is qudrtic eqution with rel coefficients, if + is one root then other root cn e ny other rel numer. Sttement-: If P + q is rel root of qudrtic eqution, then P - q is other root only when the coefficients of eqution re rtionl. Sttement-: If px + qx + r = 0 is qudrtic eqution (p, q, r R) such tht its roots re α, β & p + q + r < 0, p q + r < 0 & r > 0, then [α] + [β] =, where [ ] denotes G.I.F. Sttement-: If for ny two rel numers &, function f(x) is such tht f.f < 0 f(x) hs t lest one rel root lying etween (, ). Sttement-: If x = + is root of qudrtic eqution then nother root of this eqution must e x = + Sttement-: If x + x + c = 0,,, c Q, hving irrtionl roots then they re in conjugte pirs.. Sttement-: If roots of the qudrtic eqution x + x + c = 0 re distinct nturl numer then oth roots of the eqution cx + x + = 0 cnnot e nturl numers. Sttement-: If α, β e the roots of x + x + c = 0 then, α β re the roots of cx + x + = 0.. Sttement-: The (x p) (x r) + λ (x q) (x s) = 0 where p < q < r < s hs non rel roots if λ > 0. Sttement-: The eqution (p, q, r R) βx + qx + r = 0 hs non-rel roots if q pr < 0. 5. Sttement-: One is lwys one root of the eqution (l m)x + (m n) x + (n l ) = 0, where l, m, n R. Sttement-: If + + c = 0 in the eqution x + x + c = 0, then is the one root. 6. Sttement-: If ( ) x + ( + ) x + ( 7 + 0) = 0 is n identity, then the vlue of is. Sttement-: If = = 0 then x + x + c = 0 is n identity. 7. Sttement-: x + x + > 0 x R Sttement-: x + x + c > 0 x R if c < 0 nd > 0. 8. Sttement-: Mximum vlue of is x x / + Sttement-: Minimum vlue of x + x + c ( > 0) occurs t x =. 9. Sttement-: If qudrtic eqution x + x = 0 hve non-rel roots then < 0 Sttement-: For the qudrtic expression f(x) = x + x + c if c < 0 then f(x) = 0 hve non rel roots. 0. Sttement-: Roots of eqution x 5 0x + Px + Qx + Rx + S = 0 re in G.P. nd sum of their reciprocl is equl to 0 then s =. Sttement-: If x, x, x, x re roots of eqution x + x + cx + dx + e = 0 ( 0) x + x + x + x = / c xx = d e xxx = xx xx = 9 of
Phone : 0 90 90 7779, 9890 5888 QUADRATIC EQUATIONS PART OF. Sttement-: The rel vlues of form which the qudrtic eqution x ( + 8 ) + = 0. Possesses roots of opposite signs re given y 0 < <. Sttement-: Disc 0 nd product of root is < ANSWER KEY. A. A. D. C 5. C 6. C 7. D 8. A 9. A 0. D. C. C. D. B 5. A 6. B 7. A 8. A 9. C 0. A. A. A. A. D 5. A 6. C 7. A 8. A 9. A 0. A. A Solution 5. Oviously x = is one of the root Other root = = rtionl for ll rtionl. + (C) is correct option. 6. Here f(x) is downwrd prol D = ( + ) + 0 > 0 From the grph clerly st () is true ut st () is flse - α β 7. f(x) = 0 represents n identity if 6 = 0 =, 6 = 0 =, = 0 =, =0 =, = is the only vlues. Ans.: D 8. (A) x + x + = 0 D = < 0 x + x + = 0 nd x + x + c = 0 hve oth the roots common = = c. 9. (A) ( + ) x + ( 5 + 6) x + = 0 Clerly only for =, it is n identify. 0. Sttement II is true s if x + x + c = 0 hs imginry roots, then for no rel x, x + x + c is zero, mening therey x + x + c is lwys of one sign. Further lim ( + + ) x x c = signum. x sttement I is flse, ecuse roots of x x + = 0 re rel for ny (-, 0) nd hence x x + tkes zero, positive nd negtive vlues. Hence (d) is the correct nswer.. Sttement I is true, s Difference of the roots of qudrtic eqution is lwys D, D eing the discriminnt of the qudrtic eqution nd the two given equtions hve the sme discriminnt. Sttement II is flse s if two qudrtic equtions over rels hve the sme product of the coefficients, their discriminents need not e sme. Hence is the correct nswer.. Roots of the eqution x 5 0x + px + qx + rx + s = 0 re in G.P., let roots e, r, r, r, r + r + r + r + r = 0... (i) nd + + + + r r r r = 0... (ii) from (i) nd (ii); r = ±... (iii) Now, - S = product of roots = 5 r 0 = (r ) 5 = ±. s =. Hence is the correct nswer.. Let, f(x) = (x sin α) (x cos α) then, f(sin α) = - < 0; f(cos α) = - < 0 Also s 0 < α < π ; sin α < cos α There-fore eqution f(x) = 0 hs one root in (-, sin α) nd other in (cos α, ) Hence is the correct nswer. 0 of
Phone : 0 90 90 7779, 9890 5888 QUADRATIC EQUATIONS PART OF sin α cos α Hence (d) is the correct nswer.. (B) x x = 0 g() < 0 > / 5. eqution cn e written s ( x ) ( ) x ( ) = 0 x = & x = 6. (A) Let f(x) = (x ) (x c) + (x ) (x d) Then f = ( ) ( d) > 0 Since x 0 nd x = [ x is non positive] f = ( ) ( c) < 0 0 < < 5 f(d) = (d ) (d ) > 0 i.e., (, 5] Hence root of f(x) = 0 lies etween & nd nother Hence ns. (B). root lies etween ( & d). 7. x + x + > 0 x R = > 0 c = = - < 0 x + x + 5 > 0 x R = > 0 c = 0 = -6 < 0 x + x + So > 0 x R is correct x + x + 5 9. (A) If the coefficients of qudrtic eqution re not rtionl then root my e + nd +. Hence the roots of the given eqution re rel nd distinct. 8. x + x + c = 0 Put x = + + c = 0 which is given So clerly is the root of the eqution Nothing cn e sid out the sign of the roots. c is correct. 0. (D) R is oviously true. So test the sttement let f(x) = (x p) (x r) + λ (x q) (x s) = 0 Then f(p) = λ (p q) (p s) f(r) = λ (r q) (r s) If λ > 0 then f(p) > 0, f(r) < 0 There is root etween p & r Thus sttement- is flse.. (A) Both Sttement- nd Sttement- re true nd Sttement- is the correct explntion of Sttement-.. (C) Clerly Sttement- is true ut Sttement- is flse. x + x + c = 0 is n identity when = = c = 0.. (A) for x + x + > 0 nd D < 0. (A) x x + = x + 5. The roots of the given eqution will e of opposite signs. If they re rel nd their product is negtive D 0 nd product of root is < 0 ( 8 ) 8( ) 0 nd < 0 < 0 0 < <. Ans.. If x = + + +... to infinity, then x = + 5 ± 5 Que. from Compt. Exms 5 (d) None of these. For the eqution x + x 6 = 0, the roots re [EAMCET 988, 9] One nd only one rel numer Rel with sum one Rel with sum zero (d) Rel with product zero. If x + x + c = 0, then x = [MP PET 995] of
Phone : 0 90 90 7779, 9890 5888 QUADRATIC EQUATIONS PART OF ± ± c c c ± c (d) None of these. If the equtions x + x + 5λ = 0 nd x + x + λ = 0 hve common root, then = 0 0, (d), λ [RPET 989] 5. If the eqution x + λx + µ = 0 hs equl roots nd one root of the eqution x + λx = 0 is, then ( λ, µ ) = (, ) (,) (, ) (d) (, ) x x + 6. If x is rel nd k =, then [MNR 99; RPET 997] x + x + k k 5 k 0 (d) None of these 7. If < < c < d, then the roots of the eqution ( x )( x c) + ( x )( x d) = 0 re [IIT 98] Rel nd distinct Rel nd equl Imginry (d) None of these 8. If the roots of the eqution qx + px + q = 0 where p, q re rel, e complex, then the roots of the eqution x qx + p = 0 re Rel nd unequl Rel nd equl Imginry (d) None of these 9. The vlues of '' for which ( ) x + ( ) x + is positive for ny x re [UPSEAT 00] > (d) < or > x x m 0. If the roots of eqution = re equl ut opposite in sign, then the vlue of m will e x c m + [RPET 988, 00; MP PET 996, 00; P. CET 000] + + (d) + +. The coefficient of x in the eqution x + px + q = 0 ws tken s 7 in plce of, its roots were found to e nd 5, The roots of the originl eqution re [IIT 977, 79], 0, 0 5, 8 (d) None of these. If one root of the eqution x + x + c = 0 e n times the other root, then n = c( n + ) n = c( n + ) nc = ( n + ) (d) None of these. If one root of the qudrtic eqution x + x + c = 0 is equl to the n th power of the other root, then the vlue of n ( n n c ) n+ + ( c) + = [IIT 98] n+ (d) n+. If sin α, cos α re the roots of the eqution x + x + c = 0, then [MP PET 99] + c = 0 ( c) = + c + c = 0 (d) + + c = 0 5. If oth the roots of the qudrtic eqution x kx + k + k 5 = 0 re less thn 5, then k lies in the intervl [AIEEE 005] (, ) [, 5] (5, 6] (d) (6, ) 6. If the roots of the equtions x x + c = 0 nd x cx + = 0 differ y the sme quntity, then + c is equl to [BIT Rnchi 969; MP PET 99] 0 (d) 7. If the product of roots of the eqution log k x kx + e = 0 is 7, then its roots will rel when [IIT 98] k = k = k = (d) None of these 8. If root of the given eqution ( c) x + ( c ) x + c( ) = 0 is, then the other will e [RPET 986] ( c) ( c ) c( ) (d) None of these ( c ) ( c) ( c) 9. In tringle ABC the vlue of A is given y 5 cos A + = 0, then the eqution whose roots re sin A nd tn A will e [Roorkee 97] 5 x 8 x + 6 = 0 5 x + 8 x 6 = 0 5 x 8 x + 6 = 0 (d) 5 x 8 x 6 = 0 0. If one root of the eqution x + x + c = 0 the squre of the other, then ( c ) = cx, where X is + ( ) (d) None of these of
Phone : 0 90 90 7779, 9890 5888 QUADRATIC EQUATIONS PART OF. If 8, re the roots of x + x + β = 0 nd, re the roots of x + α x + = 0, then the roots of x + x + = 0 re 8, 9, 8, (d) 9, [EAMCET 987] x +. The set of vlues of x which stisfy 5 x + < x + 8 nd <, x (, ) (,) (, ) (,) (d) (, ) n n is [EAMCET 989]. If α, β re the roots of x x + = 0 nd if α + β = Vn, then [RPET 995; Krntk CET 000; P. CET 00] V n+ = Vn + Vn V n+ = Vn + Vn V n+ = Vn Vn (d) Vn+ = Vn Vn 7. The vlue of c for which α β =, where α nd β re the roots of x + 7 x + c = 0, is 0 6 (d) 5. For wht vlue of λ the sum of the squres of the roots of x + ( + λ) x ( + λ) = 0 is minimum [AMU 999] / / (d) / 6. The product of ll rel roots of the eqution x x 6 = 0 is [Roorkee 000] 9 6 9 (d) 6 7. For the eqution x + px + = 0, p > 0 if one of the root is squre of the other, then p is equl to [IIT Screening 000] (d) 8. If α, β e the roots of x + px + q = 0 nd α + h, β + h re the roots of x + rx + s = 0, then [AMU 00] p q p r = h = + p q = r s (d) pr = qs r s q s 9. If x + px + q = 0 is the qudrtic eqution whose roots re nd where nd re the roots of x x + = 0, then [Kerl (Engg.) 00] =, q = p =, q = 5 p 5 p =, q = (d) None of these 0. The vlue of for which one root of the qudrtic eqution ( 5 + ) x + ( ) x + = 0 is twice s lrge s the other, is [AIEEE 00]. If,, c re in G.P., then the equtions x + x + c = 0 nd dx + ex + f = 0 hve common root if [IIT 985; P. CET 000; DCE 000] (d) d e f,, re in c A.P. G.P. H.P. (d) None of these. The vlue of for which the equtions x x + = 0 nd x + x = 0 hve common root is [P. CET 999] (d). If ( x + ) is fctor of x ( p ) x ( p 5) x + ( p 7) x + 6, then p = [IIT 975] (d) None of these. The roots of the eqution x x + 57 x + 8 x 5 = 0, If one of them is + i 6, re i 6, ± i 6, ± i 6, ± (d) None of these 5. The vlues of for which x ( + ) x + ( + ) = 0 my hve one root less thn nd other root greter thn re given y > 0 [UPSEAT 00] < < > 0 0 (d) > 0 or < ANSWER KEY(Que. from Compt. Exms) c c c 5 6 7 8 9 d 0 5 6 d 7 8 c 9 0 d c c 5 c 6 7 c 8 c 9 d 0 d c 5 d 6 d 7 8 c 9 0 of