QUADRATIC EQUATION. Contents


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1 QUADRATIC EQUATION Contents Topi Pge No. Theory 004 Exerise Exerise Exerise Exerise Answer Key 78 Syllus Qudrti equtions with rel oeffiients, reltions etween roots nd oeffiients, formtion of qudrti equtions with given roots, symmetri funtions of roots. Nme : Contt No. ARRIDE LEARNING ONLINE ELEARNING ACADEMY A479 indr Vihr, Kot Rjsthn Contt No
2 QUADRATIC EQUATION. Eqution v/s Identity: A qudrti eqution is stisfied y extly two vlues of ' x ' whih my e rel or imginry. The eqution, x + x + = 0 is: * qudrti eqution if ¹ 0 Two Roots * liner eqution if = 0, ¹ 0 One Root * ontrdition if = = 0, ¹ 0 No Root * n identity if = = = 0 Infinite Roots If qudrti eqution is stisfied y three distint vlues of ' x ', then it is n identity.. Reltion Between Roots & Coeffiients: (i) The solutions of qudrti eqution, x + x + = 0, ( ¹ 0) is given y x =  ±  4 The expression,  4 º D is lled disriminnt of qudrti eqution. (ii) If, re the roots of qudrti eqution, x + x + = 0, ¹ 0. Then: () + =  () = () ½  ½ = D (iii) A qudrti eqution whose roots re &, is (x  ) (x  ) = 0 i.e. x  (sum of roots) x + (produt of roots) = 0 3. Nture of Roots: Consider the qudrti eqution, x + x + = 0 hving, s its roots; D º  4 Arride lerning Online Elerning Ademy A479 Indr Vihr, Kot Rjsthn Pge No. #
3 4. Common Roots: Consider two qudrti equtions, x + x + = 0 & x + x + = 0. (i) If two qudrti equtions hve oth roots ommon, then the eqution re identil nd their oeffiient re in proportion. i.e. (ii) = =. If only one root is ommon, then the ommon root ' ' will e: =   = Hene the ondition for one ommon root is:   é  ù ê ú ë  û é  ù + ê ú ë  û + = 0 º (  ) = (  ) (  ) Note : If f(x) = 0 & g(x) = 0 re two polynomil eqution hving some ommon root(s) then those ommon root(s) is/re lso the root(s) of h(x) = f(x) + g (x) = Grph of Qudrti Expression: y = f (x) = x + x + or æ D ö æ çy + = x è 4 ø ç + è ö ø * the grph etween x, y is lwys prol. æ D * the oordinte of vertex re ö ç,  è 4 ø * If > 0 then the shpe of the prol is onve upwrds & if < 0 then the shpe of the prol is onve downwrds. * the prol interset the yxis t point (0, ). * the xoordinte of point of intersetion of prol with xxis re the rel roots of the qudrti eqution f (x) = 0. Hene the prol my or my not interset the xxis t rel points. 6. Rnge of Qudrti Expression f (x) = x + x +. (i) Asolute Rnge: é D If > 0 Þ f (x) Î ö ê, ë 4 ø æ D ù < 0 Þ f (x) Î ç,  ú è 4 û Hene mximum nd minimum vlues of the expression f (x) is  D 4 in respetive ses nd it ours t x =  (t vertex). Arride lerning Online Elerning Ademy A479 Indr Vihr, Kot Rjsthn Pge No. #
4 (ii) Rnge in restrited domin: Given x Î [x, x ] () If  Ï [x, x ] then, () If  Î [x, x ] then, f (x) Î [ min { f x ), f( x )}, mx { f( x ), f( )}] ( x é ì D ü ì f (x) Î ê min í f( x), f( x),  ý, mx í f( x), f( x),  ë î 4þ î 7. Sign of Qudrti Expressions: The vlue of expression, f (x) = x + x + t x = x is equl to yoordinte of point on prol 0 y = x + x + whose xoordinte is x. Hene if the point lies ove the xxis for some x = x, 0 0 then f (x 0 ) > 0 nd vievers. We get six different positions of the grph with respet to xxis s shown. D 4 ü ù ý ú þ û NOTE: (i) " x Î R, y > 0 only if > 0 & D º ²  4 < 0 (figure 3). (ii) " x Î R, y < 0 only if < 0 & D º ²  4 < 0 (figure 6). 8. Solution of Qudrti Inequlities: The vlues of ' x ' stisfying the inequlity, x + x + > 0 ( ¹ 0) re: (i) If D > 0, i.e. the eqution x + x + = 0 hs two different roots <. Then > 0 Þ x Î (, ) È (, ) < 0 Þ x Î (, ) (ii) If D = 0, i.e. roots re equl, i.e. =. Then > 0 Þ x Î (, ) È (, ) < 0 Þ x Î f (iii) If D < 0, i.e. the eqution x + x + = 0 hs no rel root. Then > 0 Þ x Î R < 0 Þ x Î f (iv) P(x) Q(x) R(x)... Inequlities of the form A(x) B(x) C(x)... < = 0 n e quikly solved using the > method of intervls, where A, B, C..., P, Q, R... re liner funtions of ' x '. Arride lerning Online Elerning Ademy A479 Indr Vihr, Kot Rjsthn Pge No. # 3
5 9. Lotion Of Roots: Let f (x) = x² + x +, where > 0 &,, Î R. (i) (ii) (iii) (i) Conditions for oth the roots of f (x) = 0 to e greter thn speified numer x 0 re ²  4 ³ 0; f (x 0 ) > 0 & ( /) > x 0. (ii) Conditions for oth the roots of f (x) = 0 to e smller thn speified numer x 0 re ²  4 ³ 0; f (x 0 ) > 0 & ( /) < x 0. (iii) Conditions for oth roots of f (x) = 0 to lie on either side of the numer x 0 (in other words the numer x 0 lies etween the roots of f (x) = 0), is f (x 0 ) < 0. (iv) (v) (iv) (v) Conditions tht oth roots of f (x) = 0 to e onfined etween the numers x nd x, (x < x ) re ²  4 ³ 0; f (x ) > 0 ; f (x ) > 0 & x < ( /) < x. Conditions for extly one root of f (x) = 0 to lie in the intervl (x, x ) i.e. x < x < x is f (x ). f (x ) < Theory Of Equtions: If,, 3,... n re the roots of the eqution; f(x) = 0 x n + x n + x n n x + n = 0 where 0,,... n re ll rel & 0 ¹ 0 then, å = , å = +, å 0 = ,...,... = 3. n ()n NOTE : (i) If is root of the eqution f(x) = 0, then the polynomil f(x) is extly divisile y (x  ) or (x  ) is ftor of f(x) nd onversely. (ii) Every eqution of n th degree (n ³ ) hs extly n roots & if the eqution hs more thn n roots, it is n identity. (iii) If the oeffiients of the eqution f(x) = 0 re ll rel nd + i is its root, then  i is lso root. i.e. imginry roots our in onjugte pirs. (iv) An eqution of odd degree will hve odd numer of rel roots nd n eqution of even degree will hve even numers of rel roots. (v) If the oeffiients in the eqution re ll rtionl & + n 0 is one of its roots, then  is lso root where, Î Q & is not perfet squre. (vi) If there e ny two rel numers '' & '' suh tht f() & f() re of opposite signs, then f(x) = 0 must hve odd numer of rel roots (lso tlest one rel root) etween ' ' nd ' '. (vii) Every eqution f(x) = 0 of degree odd hs tlest one rel root of sign opposite to tht of its lst term. Arride lerning Online Elerning Ademy A479 Indr Vihr, Kot Rjsthn Pge No. # 4
6 PART  I : OBJECTIVE QUESTIONS * Mrked Questions re hving more thn one orret option. Setion (A) : Identity & Reltion etween the roots nd oeffiients A. Numer of vlues of ' p ' for whih the eqution (p  3p + ) x  (p  5p + 4) x + p  p = 0 possess more thn two roots, is: (A) 0 (B) (C) (D) none A. If, re the roots of qudrti eqution x + p x + q = 0 nd g, d re the roots of x + p x r = 0, then (  g). (  d) is equl to : (A) q + r (B) q r (C) (q + r) (D) (p + q + r) A3. A4. A5. A6. Two rel numers & re suh tht + = 3 & ½  ½ = 4, then & re the roots of the qudrti eqution: (A) 4x  x  7 = 0 (B) 4x  x + 7 = 0 (C) 4x  x + 5 = 0 (D) none of these If, re the roots of the eqution (x ) + x = 0, then the eqution whose roots re  nd  is (A) x + 6x + 9 = 0 (B) x + 6x 9 = 0 (C) x + 6x 9 = 0 (D) x + x = If x = then the vlue of x 4 x 3 x + 3x + is equl to (A) (B) 3 (C) 5 (D) 0 If 4 x 4 x = 4 then (x) 5/ hs the vlue equl to (A) 5 5 (B) 5 (C) 5 5 (D) 5 A7. The vlue of is (A) 0 (B) 6 (C) 8 (D) none of these Setion (B) : Nture of Roots nd Common Roots B. B. B3. If,, re integers nd = 4( + 5d ), d Î N, then roots of the eqution x + x + = 0 re (A) Irrtionl (B) Rtionl & different (C) Complex onjugte (D) Rtionl & equl Consider the eqution x + x n = 0, where n Î N nd n Î [5, 00]. Totl numer of different vlues of 'n' so tht the given eqution hs integrl roots, is (A) 4 (B) 6 (C) 8 (D) 3 If P(x) = x + x + & Q (x) = x + dx +, where ¹ 0, then P(x). Q(x) = 0 hs (A) extly one rel root (B) tlest two rel roots (C) extly three rel roots (D) ll four re rel roots Arride lerning Online Elerning Ademy A479 Indr Vihr, Kot Rjsthn Pge No. # 5
7 B4. If the equtions x + px + q = 0 nd x + qx + p = 0 hve extly one root in ommon then the eqution ontining their other root is (A) x x + pq = 0 (B) x + x + pq = 0 (C) x x pq = 0 (D) x + x pq = 0 B5. x + x + = 0 nd x + x + = 0 hs ommon root, (,, Î R) then (A) = K, = K, = K, K¹ 0 (B) = K, = K, = 3K, K¹ 0 (C) = K, = K, = K, K¹ 0 (D) = K, = K, = K, K¹ 0 Setion (C) : Grph nd Rnge C. The entire grph of the expression y = x + kx x + 9 is stritly ove the xxis if nd only if (A) k < 7 (B) 5 < k < 7 (C) k > 5 (D) none C. Whih of the following grph represents the expression f(x) = x + x + ( ¹ 0) when > 0, < 0 & < 0? (A) (B) (C) (D) C3. If y = x 6x + 9, then (A) mximum vlue of y is nd it ours t x = (B) minimum vlue of y is nd it ours t x = (C) mximum vlue of y is 3.5 nd it ours t x =.5 (D) minimum vlue of y is 3.5 nd it ours t x =.5 x  x + C4. If 'x' is rel nd k =, then : x + x + (A) 3 k 3 (B) k ³ 5 (C) k 0 (D) none C5. Let, nd e rel numers suh tht = 0 nd > 0. Then the eqution x + x + = 0 hs (A) rel roots (B) imginry roots (C) extly one root (D) none of these Setion (D) : Lotion of Roots D. If the inequlity ( m ) x + 8x + m + 4 > 0 is stisfied for ll x Î R then the lest integrl m is (A) 4 (B) 5 (C) 6 (D) none D. For ll 'x' x + x > 0, then the intervl in whih '' lies is (A) < 5 (B) 5 < < (C) > 5 (D) < < 5 Arride lerning Online Elerning Ademy A479 Indr Vihr, Kot Rjsthn Pge No. # 6
8 D3. The set of vlues of 'm' for whih the eqution x (m + )x + m + m 8 = 0 hs root in the intervl (, ) nd the other in the intervl (, ) is (A) (, ) (B) (, ) (C) (, ) (D) (, ) D4. If oth roots of the eqution x (m + ) x + m + 4 = 0 re rel nd negtive, then set of vlues of 'm' is (A) 3 < m (B) 4 < m 3 (C) 3 m 5 (D) 3 ³ m or m ³ 5 D5. If oth roots of the qudrti eqution ( x) (x +) = p re distint & positive then p must lie in the intervl: (A) p > 9 (B) < p < (C) p <  (D) < P < 4 D6. The vlue of p for whih oth the roots of the qudrti eqution, 4x 0px + (5p + 5p 66) = 0 re less thn lies in : (A) (4/5, ) (B) (, ) (C) (, 4/5) (D) (, ) D7. D8. The rel vlues of '' for whih the qudrti eqution x  ( ) x = 0 possesses roots of opposite sign is given y: (A) > 5 (B) 0 < < 4 (C) > 0 (D) > 7 If, re the roots of the qudrti eqution x  p (x  4)  5 = 0, then the set of vlues of p for whih one root is less thn & the other root is greter thn is: (A) (7/3, ) (B) (, 7/3) (C) x Î R (D) none Setion (E) : Theory of Eqution E. The ondition tht x 3 px + qx r = 0 my hve two of its roots equl to eh other ut of opposite signs is (A) r = pq (B) r = p 3 + pq (C) r = p q (D) none of these E. If, & g re the roots of the eqution x 3  x  = 0 then, g +  g hs the vlue equl to: (A) zero (B)  (C)  7 (D) E3. Let,, g e the roots of (x ) (x ) (x ) = d, d ¹ 0 then the roots of the eqution (x ) (x ) (x g) + d = 0 re : (A) +, +, + (B),, (C),, (D),, E4. If,, g, d re the roots of the eqution, x 4 Kx 3 + Kx + Lx + M = 0 where K, L & M re rel numers then the minimum vlue of + + g + d is : (A) 0 (B)  (C) (D) Arride lerning Online Elerning Ademy A479 Indr Vihr, Kot Rjsthn Pge No. # 7
9 PART  II : MISCELLANEOUS OBJECTIVE QUESTIONS Comprehensions # : In the given figure DOBC is n isoseles right tringle in whih AC is medin, then nswer the following questions : Y C y = x + x + O A B X. Roots of y = 0 re (A) {, } (B) {4, } (C) {, /} (D) {8, 4}. The eqution whose roots re ( + ) & ( ), where, ( > ) re roots otined in previous question, is (A) x 4x + 3 = 0 (B) x 8x + = 0 (C) 4x 8x + 3 = 0 (D) x 6x + 48 = 0 3. Minimum vlue of the qudrti expression orrespoinding to the qudrti eqution otined in Q. No. ours t x = (A) 8 (B) (C) 4 (D) Comprehensions # : Consider the eqution x 4 lx + 9 = If the eqution hs four rel nd distint roots, then l lies in the intervl (A) (, 6) È (6, ) (B) (0, ) (C) (6, ) (D) (, 6) 5. If the eqution hs no rel root, then l lies in the intervl (A) (, 0) (B) (, 6) (C) (6, ) (D) (0, ) 6. If the eqution hs only two rel roots, then set of vlues of l is (A) (, 6) (B) ( 6, 6) (C) {6} (D) f Mth The Column : 7. For the qudrti eqution x (k 3)x + k = 0, then mth the following olumns ColumnI ColumnII (A) Both roots re positive (P) (, ) (B) Both roots re negtive (Q) (9, ) (C) Both roots re rel (R) (0, ) (D) One root <, the other > Arride lerning Online Elerning Ademy A479 Indr Vihr, Kot Rjsthn Pge No. # 8
10 8. Mth the following Column ColumnI ColumnII (A) (x ) ( x 3) + k(x ) (x 4) = 0 (P) ( 5, ) (k Î R), hs rel roots for k Î x  (B) Rnge of the funtion does not (Q) f x  k + ontin ny vlue in the intervl [, ] for k Î (C) The eqution, sex + osex = k (R) (, ) æ 5 ö hs rel roots for xî ç0,,if k Î è ø (D) The eqution x + (k )x + k + 5 = 0 hs (S) [, ) positive nd distint roots, if k Î Assertion / Reson : Diretion : Eh question hs 5 hoies (A), (B), (C), (D) nd (E) out of whih ONLY ONE is orret. (A) Sttement is True, Sttement is True; Sttement is orret explntion for Sttement. (B) Sttement is True, Sttement is True; Sttement is NOT orret explntion for Sttement. (C) Sttement is True, Sttement is Flse. (D) Sttement is Flse, Sttement is True. (E) Sttement nd Sttement oth re Flse. 9. STATEMENT  : Mximum vlue of log /3 (x 4x + 5) is '0'. STATEMENT  : log x 0 for x ³ nd 0 < <. 0. Let, e the roots of f(x) = 3x 4x + 5 = 0. STATEMENT : The eqution whose roots re, is given y 3x + 8x 0 = 0. STATEMENT : To otin, from the eqution f(x) = 0, hving roots nd, the eqution hving roots, one needs to hnge x to x in f(x) = 0. PART  I : MIXED OBJECTIVE * Mrked Questions re hving more thn one orret option.. If the roots of the equtions x + x += 0 re rel nd of the form  nd +, then the vlue of ( + + ) is (A) 4 (B) (C) (D) None of these. The eqution, p x =  x + 6x  9 hs: (A) no solution (B) one solution (C) two solutions (D) infinite solutions 3. If, Î R, ¹ 0 nd the qudrti eqution x  x + = 0 hs imginry roots then + + is: (A) positive (B) negtive (C) zero (D) depends on the sign of Arride lerning Online Elerning Ademy A479 Indr Vihr, Kot Rjsthn Pge No. # 9
11 4. If, e the roots of 4x 6x + l = 0, where l Î R, suh tht < < nd < < 3, then the numer of integrl solutions of l is (A) 5 (B) 6 (C) (D) 3 5. If oth roots of the qudrti eqution (  x) (x + ) = p re distint & positive, then p must lie in the intervl: (A) (, ) (B) (, 9/4) (C) (, ) (D) (, ) 6. The vlue of '' for whih the sum of the squres of the roots of the eqution x  (  ) x   = 0 ssume the lest vlue is: (A) 0 (B) (C) (D) 3 7. The vlues of k, for whih the eqution x + (k  ) x + k + 5 = 0 possess tlest one positive root, re: (A) [4, ) (B) (,  ] È [4, ) (C) [, 4] (D) (,  ] 8. If >, then the eqution (x  ) (x  ) + = 0, hs: (A) oth roots in (, ) (B) oth roots in (, ) (C) oth roots in (, ) (D) one root in (, ) & other in (, ) 9. If (l + l )x + (l + ) x < for ll x Î R, then l elongs to the intervl ö (A) (, ) (B) ê é æ ö , ë 5 (C) ç, ø è 5 ø (D) none of these 0. If the roots of the eqution x + x + = 0 re rel nd distint nd they differ y t most m, then lies in the intervl (A) ( m, ) (B) [ m, ) (C) (, + m ) (D) none of these. If < < 3 < 4 < 5 < 6, then the eqution (x )(x 3 )(x 5 )+3(x )(x 4 )(x 6 )=0 hs (A) three rel roots (B) root is (, ) (C) no rel root in (, ) (D) no rel root in ( 5, 6 ). For every x Î R, the polynomil x 8 x 5 + x x + is : (A) Positive (B) never positive (C) positive s well s negtive (D) negtive 3. The gretest vlue of lest vlue of the qudrti trinomil, x + x + ( + ), is (A) 9 4 (B) 7 4 (C) 0 (D) 3 (A),,  (B),, (C),, (D),, 5. Roots of the eqution x 3 + x + x + = 0 re 3 onseutive positive integer, the vlue of 4. If (x ) is ftor of x 3 + x +, then roots of the eqution x 3 + x + = 0 re is (A) 5 (B) 7 (C) 9 (D) If,, re rel numers stisfying the ondition + + = 0 then the roots of the qudrti equtions, 3x + 5x + 7 = 0 re : (A) positive (B) negtive (C) rel & distint (D) imginry Arride lerning Online Elerning Ademy A479 Indr Vihr, Kot Rjsthn Pge No. # 0
12 7. x + x + = 0 hs rel nd distint roots nd ( > ). Further > 0, > 0 nd < 0, then (A) 0 < < (B) 0 < < (C) + > 0 (D) + = 8. If l, m, n re rel, l ¹ m, then the roots of the eqution : (l m) x 5(l + m)x (l m) = 0 re (A) rel nd equl (B) Complex (C) rel nd unequl (D) none of these 9. If the roots of eqution x x + = 0 re two onseutive integers, then 4 equls (A) (B) 3 (C) (D) 0. Whih of the following sttements is true out qudrti eqution x + x + = 0, where,, Î R, ¹ 0 (A) If < 0 then roots re imginry (B) If + + = 0 then roots re rel (C) If,, re equl, roots re equl (D) If < 0 roots re essentilly rel..* If the roots of the eqution x+ p x+ q re equl is mgnitude nd opposite in sign, then r (A) p + q = r (B) p + q = r (C) produt of roots =  ( p + q ) (D) sum of roots =.* The djoining figure shows the grph of y = x + x +. Then y Vertex x x x (A) > 0 (B) > 0 (C) > 0 (D) < 4 3.* Let Q (x) = x + x + = 0, Q (x) = x + x + = 0 e two qudrti equtions, then (A) they hve ommon root if = (B) they hve ommon root if = (C) they hve t lest one ommon root for = nd = (D) they hve omplex ommon root if = 4.* If the differene of the roots of the eqution x + hx + 7 = 0 is 6, then possile vlue(s) of h re (A) 4 (B) 4 (C) 8 (D) 8 5.* For the eqution x + x 6 = 0, the orret sttement (s) is (re) : (A) sum of roots is 0 (B) produt of roots is 4 (C) there re 4 roots (D) there re only roots 6.* If, re the roots of x + x + = 0, nd + h, + h re the roots of px + qx + r = 0, (where h ¹ 0 ), then (A) p = q = r (B) h = æ qö ç  è p ø (C) h = æ q ö ç + è p ø (D)  4 q  4pr = p Arride lerning Online Elerning Ademy A479 Indr Vihr, Kot Rjsthn Pge No. #
13 7.* If, re nonzero rel numers nd, the roots of x + x + = 0, then (A), re the roots of x ( ) x + = 0 (B), re the roots of x + x + = 0 (C), re the roots of x + ( ) x + = 0 (D) ( ), ( ) re the roots of the eqution x + x ( + ) = 0 8. If the roots of the eqution x 3 + Px + Qx  9 = 0 re eh one more thn the roots of the equton x 3  Ax + Bx  C = 0, where A, B, C, P & Q re onstnts, then the vlue of A + B + C is equl to : (A) 8 (B) 9 (C) 0 (D) none PART  II : SUBJECTIVE QUESTIONS. If nd re the roots of the eqution x + x + = 0, then find the eqution whose roots re given y : (i) +, + (ii) +, +. If ¹ ut = 5 3, = 5 3, then find the eqution whose roots re nd. 3. In opying qudrti eqution of the form x + px + q = 0, the oeffiient of x ws wrongly written s 0 in ple of nd the roots were found to e 4 nd 6. Find the roots of the orret eqution. 4. If one root of the eqution x + x + = 0 is equl to n th power of the other root, show tht ( n ) /(n + ) + ( n ) /(n + ) + = For wht vlues of k the expression kx + (k + )x + will e perfet squre of liner polynomil. 6. If,, Î R, then prove tht the roots of the eqution x  + x  + x  = 0 re lwys rel nd nnot hve roots if = =. 7. If, re the roots of x + px + = 0 nd, d re the roots of x + qx + = 0. Show tht q  p = (  ) (  ) ( + d) ( + d). 8. Find ll vlues of the prmeter ' ' suh tht the roots, of the eqution x + 6 x + = 0 stisfy the inequlity + <. 9. If,, g re the roots of the eqution x 3 + px + qx + r = 0, then find the vlue of æ ö ç  è g ø æ ö ç  è g ø æ ö ç g . è ø Arride lerning Online Elerning Ademy A479 Indr Vihr, Kot Rjsthn Pge No. #
14 æ 0. If, nd g re roots of x 3 + x 7 = 0, then find the vlue of å ö ç +. è ø. Find the vlue of '' so tht x x + = 0 nd x 4x + = 0 hve ommon root.. If x + px + q = 0 nd x + qx + p = 0, (p ¹ q) hve ommon root, show tht + p + q = 0 ; show tht their other roots re the roots of the eqution x + x + pq = The equtions x  x + = 0 & x 3  px + qx = 0, where ¹ 0, q ¹ 0 hve one ommon root & the seond eqution hs two equl roots. Prove tht (q + ) = p. 4. If the equtions x + x + = 0, x + x + 5 = 0 & x + ( + ) x + 36 = 0 hve ommon positive root, then find, nd the roots of the equtions. 5. Drw the grph of the following expressions : (i) y = x + 4x + 3 (ii) y = 9x + 6x + (iii) y = x + x 6. If x e rel, then find the rnge of the following rtionl expressions : (i) y = x x + x + + (ii) y = x x  x x Solve for rel vlues of 'x' : (i) x 3 x 3 (5 + 6) + (56) = 0 (ii) x x 3 = 0, 0 8. If,, re non zero, unequl rtionl numers then prove tht the roots of the eqution ( )x + 3 x + x 6 + = 0 re rtionl. 9. Find ll the vlues of 'K' for whih one root of the eqution x²  (K + ) x + K² + K  8 = 0, exeeds & the other root is smller thn. 0. If & re the two distint roots of x² + (K  3) x + 9 = 0, then find the vlues of K suh tht, Î ( 6, ).. If p, q, r, s Î R nd pr = (q + s), then show tht tlest one of the equtions x + px + q = 0, x + rx + s = 0 hs rel roots.. Find ll vlues of for whih tlest one of the roots of the eqution x ( 3) x + = 0 is greter thn. 3. If x is root of x + x + = 0, x is root of  x + x + = 0 where 0 < x < x, show tht the eqution x + x + = 0 hs root x 3 stisfying 0 < x < x 3 < x. 4. Otin rel solutions of the simultneous equtions xy + 3 y²  x + 4 y  7 = 0, xy + y²  x  y + = 0. Arride lerning Online Elerning Ademy A479 Indr Vihr, Kot Rjsthn Pge No. # 3
15 PART  I : IITJEE PROBLEMS (PREVIOUS YEARS) * Mrked Questions re hving more thn one orret option.. The set of ll rel numers x for whih x  x + + x > 0, is [IITJEE 00] (A) (,) (, )  U (B) (, ) U(, ) , U (, (D) (, ) (C) ( ) ). If x + ( ) x + ( ) = 0 where, Î R then find the vlues of for whih eqution hs unequl rel roots for ll vlues of. [IITJEE 003] x 5x 3. Find the rnge of vlues of t for whih sint =  + p p, t Î, [IITJEE 005] 3x x 4. In qudrti eqution x + x + = 0, if, re roots of eqution, D = 4 nd +, +, re in G. P. then [IITJEE 005] (A) D ¹ 0 (B) D = 0 (C) D = 0 (D) D = 0 5. If roots of the eqution x 0x d = 0 re, nd those of x 0x = 0 re, d then the vlue of d is (,, nd d re distint numers) [IITJEE 006] 6. Let,, e the sides of tringle. No two of them re equl nd l Î R. If the roots of the eqution x + ( + + )x + 3l ( + + ) = 0 re F rel, then [IITJEE 006] (A) l< 4 (B) l> 5 (C) lî 3 3 H G 5I K J F 3, 3 (D) lîhg 4 5 I, K J Let, e the roots of the eqution x px + r = 0 nd, e the roots of the eqution x qx + r = 0. Then the vlue of r is [IITJEE 007] (A) 9 (p q) (q p) (B) 9 L NM (q p) (p q) O QP (C) 9 (q p) (q p) (D) 9 (p q) (q p) 8. Let p nd q e rel numers suh tht p ¹ 0, p 3 ¹ q nd p 3 ¹  q. If nd re nonzero omplex numers stisfying + = p nd = q, then qudrti eqution hving nd s its roots is :[IITJEE 00] (A) (p 3 + q) x (p 3 + q)x + (p 3 + q) = 0 (B) (p 3 + q) x (p 3 q)x + (p 3 + q) = 0 (C) (p 3. q) x. (5p 3. q)x + (p 3. q) = 0 (D) (p 3 q) x (5p 3 + q)x + (p 3 q) = 0 9. Let nd e the roots of x 6x = 0, with >. If n = n n for n ³, then the vlue of is (A) (B) (C) 3 (D) 4 [IITJEE 0] 0. A vlue of for whih the equtions [IITJEE 0] x + x = 0 ; x + x + = 0, hve one root in ommon is (A) (B)  i 3 (C) i 5 (D) Arride lerning Online Elerning Ademy A479 Indr Vihr, Kot Rjsthn Pge No. # 4
16 PART  II : AIEEE PROBLEMS (PREVIOUS YEARS). If ¹ ut = 5 3, = 5 3, then the eqution hving the roots nd is. [AIEEE00] () 3x + 9x + 3 = 0 () 3x 9x + 3 = 0 (3) 3x 9x 3 = 0 (4) x 6x + = 0. The vlue of for whih one root of the qudrti eqution ( 5 + 3)x + (3 )x + = 0 is twie s lrge s the other, is : [AIEEE003] () 3 () 3 (3) 3 (4) If ( p) is root of qudrti eqution x + px + ( p) = 0, then its roots re : [AIEEE004] () 0, (), (3) 0, (4), 4. If one root of the eqution x + px + = 0 is 4, while the eqution x + px + q = 0 hs equl roots, then the vlue of q is : [AIEEE004] () 49/4 () (3) 3 (4) 4 p æ P ö æ Q ö 5. In tringle PQR, ÐR =. If tn ç nd tn ç re the roots of x + x + = 0, ¹ 0, then : [AIEEE005] è ø è ø () = +. () =. (3) = +. (4) = The vlue of '' for whih the sum of the squres of the roots of the eqution x ( ) x = 0 ssume the lest vlue is  [AIEEE 005] () () 0 (3) 3 (4) 7. If oth the roots of the qudrti eqution x kx + k +k 5 = 0 re less thn 5, then 'k' lies in the intervl [AIEEE 005] () (5, 6) () (6, ) (3) (,4) (4) [4, 5] 8. If the roots of the qudrti eqution x + px + q = 0 re tn 30 nd tn 5 respetively, then the vlue of + q p is : [AIEEE006] () 3 () 0 (3) (4) 9. All the vlues of 'm' for whih oth roots of the eqution x mx + m = 0 re greter thn ut less thn 4 lie in the intervl : [AIEEE006] () m > 3 () < m < 3 (3) < m < 4 (4) < m < 0 0. If 'x' is rel, the mximum vlue of 3x 3x + 9x + 7 is  [AIEEE 006] + 9x () 4 () (3) 7 (4) 4. If the differene etween the roots of the eqution x + x + = 0 is less thn 5, then the set of possile vlues of '' is () ( 3, 3) () ( 3, ) (3) (3, ) (4) (, 3) [AIEEE007]. The qudrti equtions x 6x + = 0 nd x x + 6 = 0 hve one root in ommon. The other roots of the first nd seond equtions re integers in the rtio 4 : 3. Then the ommon root is [AIEEE 008] () 4 () 3 (3) (4) 3. How mny rel solution does the eqution x 7 + 4x 5 + 6x x 560 = 0 hs? [AIEEE 008] () () 3 (3) 5 (4) 7 4. If the eqution x + x + 3 = 0 nd x + x + = 0,,, Î R, hve ommons root, then : : is : [JEE Mins_03] () : :3 () 3 : : (3) : 3 : (4) 3 : : Arride lerning Online Elerning Ademy A479 Indr Vihr, Kot Rjsthn Pge No. # 5
17 BOARD PATTERN QUESTIONS Solve eh of the following questions ( to 6):. x + x + = 0. x + 3x + 5 = 0 3. x + x + = 0 4. x + + = 0 x 5. 7x 0x + = 0 6. x 8x + 0 = 0 7. The roots of the eqution x x + = 0 re (A) rel nd different (C) rel nd equl (B) imginry nd different (D) rtionl nd different 8. If the roots of the eqution x + x + = 0 e rel nd different, then the roots of the eqution x (A) rtionl (B) irrtionl (C) rel (D) imginry 9. The numer of rel solutions of x x  = 4 x x + = 0 will e is (A) 0 (B) (C) (D) infinite 0. Sum of roots of the eqution (x + 3) 4 x = 0 is (A) 4 (B) (C) (D) 4. If, re roots of the eqution x 35 x + = 0, then the vlue of ( 35) 3. ( 35) 3 is equl to (A) (B) 8 (C) 64 (D) None of these. If, re roots of the eqution x 5x + 6 = 0 then the eqution whose roots re + 3 nd + 3 is (A) x x + 30 = 0 (B) (x 3) 5 (x 3) + 6 = 0 (C) Both (A) nd (B) (D) None of these 3. If, re the root of qudrti eqution x 3x + 5 = 0 then the eqution whose roots re ( 3 + 7) nd ( 3 + 7) is (A) x + 4x + = 0 (B) x 4x + 4 = 0 (C) x 4x = 0 (D) x + x + 3 = 0 4. The minimum vlue of the expression 4x + x + is (A) /4 (B) / (C) 3/4 (D) Arride lerning Online Elerning Ademy A479 Indr Vihr, Kot Rjsthn Pge No. # 6
18 Exerise # PART  I A. (B) A. (C) A3. (A) A4. (C) A5. (C) A6. (C) A7. (D) B. (A) B. (C) B3. (B) B4. (B) B5. (A) C. (B) C. (B) C3. (C) C4. (A) C5. (A) D. (B) D. (B) D3. (B) D4. (B) D5. (B) D6. (D) D7. (B) D8. (B) E. (A) E. (C) E3. (B) E4. (B) PARTII. (A). (A) 3. (D) 4. (C) 5. (B) 6. (D) 7. (A) Q ; (B) R ; (C) PQ ; (D) P 8. (A) R ; (B) Q ; (C) S ; (D) P 9. (A) 0. (D) Exerise # PART  I. (A). (A) 3. (A) 4. (D) 5. (B) 6. (B) 7. (D) 8. (A) 9. (B) 0. (B). (A). (A) 3. (B) 4. (C) 5. (D) 6. (C) 7. (A) 8. (C) 9. (D) 0. (B).* (BC).* (BC) 3.* (ABCD)4.* (CD) 5.* (ABD) 6.* (BD) 7.* (BCD) 8. (A) PART  II. (i) x + ( + ) x + ( + ) = 0 (ii) x + ( 4 ) x + + ( ) = 0 ( r + ). 3x 9x + 3 = , ± 8. (, 0) È (9/, ) 9. r = 0, 4 4. =  7, =  8 ; roots (3, 4), (3, 5), (3, ) Arride lerning Online Elerning Ademy A479 Indr Vihr, Kot Rjsthn Pge No. # 7
19 5. (i) (ii) (iii) é 3ù 6. (i) ê, ú ë û (ii) é ù ê, ú ë û 7. (i) x = ±, ± (ii) x = ( ), x = ( 6 ) 9. K Î (, 3) 0. 6 < K < Î [9, ) 4. x Î R if y =, x = if y =  3 Exerise # 3 L NM PART  I O QP ÈL N M. (B). > 3.  p  p 3, p, p 4. (C) (A) 7. (D) 8. (B) 9. (C) 0. (B) PARTII. (). () 3. (3) 4. () 5. (3) 6. () 7. (3) 8. () 9. () 0. (). (). (3) 3. () 4. () O Q P Exerise # 4 7. (A) 8. (D) 9. (A) 0. (C). (C). (C) 3. (D) 4. (C) Arride lerning Online Elerning Ademy A479 Indr Vihr, Kot Rjsthn Pge No. # 8
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