QUADRATIC EQUATION Contents Topi Pge No. Theory 0-04 Exerise - 05-09 Exerise - 09-3 Exerise - 3 4-5 Exerise - 4 6 Answer Key 7-8 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 E-LEARNING ACADEMY A-479 indr Vihr, Kot Rjsthn 34005 Contt No. 8033545007
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 & Co-effiients: (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 E-lerning Ademy A-479 Indr Vihr, Kot Rjsthn 34005 Pge No. #
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 o-effiient 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) = 0. 5. 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 o-ordinte 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 y-xis t point (0, ). * the x-o-ordinte of point of intersetion of prol with x-xis re the rel roots of the qudrti eqution f (x) = 0. Hene the prol my or my not interset the x-xis 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 E-lerning Ademy A-479 Indr Vihr, Kot Rjsthn 34005 Pge No. #
(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 y-o-ordinte of point on prol 0 y = x + x + whose x-o-ordinte is x. Hene if the point lies ove the x-xis for some x = x, 0 0 then f (x 0 ) > 0 nd vie-vers. We get six different positions of the grph with respet to x-xis 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 E-lerning Ademy A-479 Indr Vihr, Kot Rjsthn 34005 Pge No. # 3
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 ) < 0. 0. 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 3 0 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 E-lerning Ademy A-479 Indr Vihr, Kot Rjsthn 34005 Pge No. # 4
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) A-3. A-4. A-5. A-6. 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 = 0 5 - -3 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 A-7. The vlue of 8 + 8 + 8 + 8... is (A) 0 (B) 6 (C) 8 (D) none of these Setion (B) : Nture of Roots nd Common Roots B-. B-. B-3. 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 E-lerning Ademy A-479 Indr Vihr, Kot Rjsthn 34005 Pge No. # 5
B-4. 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 B-5. 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 x-xis 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) C-3. 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 + C-4. If 'x' is rel nd k =, then : x + x + (A) 3 k 3 (B) k ³ 5 (C) k 0 (D) none C-5. Let, nd e rel numers suh tht 4 + + = 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 3 > 0, then the intervl in whih '' lies is (A) < 5 (B) 5 < < (C) > 5 (D) < < 5 Arride lerning Online E-lerning Ademy A-479 Indr Vihr, Kot Rjsthn 34005 Pge No. # 6
D-3. 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) (, ) D-4. 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 D-5. 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 D-6. 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) (, ) D-7. D-8. The rel vlues of '' for whih the qudrti eqution x - ( 3 + 8 - ) x + - 4 = 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) E-3. 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),, E-4. 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 E-lerning Ademy A-479 Indr Vihr, Kot Rjsthn 34005 Pge No. # 7
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 = 0. 4. 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 Column-I Column-II (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 E-lerning Ademy A-479 Indr Vihr, Kot Rjsthn 34005 Pge No. # 8
8. Mth the following Column Column-I Column-II (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 E-lerning Ademy A-479 Indr Vihr, Kot Rjsthn 34005 Pge No. # 9
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) 3 + 6. 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 E-lerning Ademy A-479 Indr Vihr, Kot Rjsthn 34005 Pge No. # 0
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 E-lerning Ademy A-479 Indr Vihr, Kot Rjsthn 34005 Pge No. #
7.* If, re non-zero 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 + ) + = 0. 5. 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 E-lerning Ademy A-479 Indr Vihr, Kot Rjsthn 34005 Pge No. #
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 = 0. 3. 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 + 9 + x + 9 7. Solve for rel vlues of 'x' : (i) x -3 x -3 (5 + 6) + (5-6) = 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 E-lerning Ademy A-479 Indr Vihr, Kot Rjsthn 34005 Pge No. # 3
PART - I : IIT-JEE 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 [IIT-JEE 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. [IIT-JEE 003] x 5x 3. Find the rnge of vlues of t for whih sint = - + p p, t Î-, [IIT-JEE 005] 3x -x- 4. In qudrti eqution x + x + = 0, if, re roots of eqution, D = 4 nd +, +, 3 + 3 re in G. P. then [IIT-JEE 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) [IIT-JEE 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 [IIT-JEE 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 3 3 7. 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- [IIT-JEE 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 3 + 3 = q, then qudrti eqution hving nd s its roots is :[IIT-JEE 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 0-8 9 is (A) (B) (C) 3 (D) 4 [IIT-JEE 0] 0. A vlue of for whih the equtions [IIT-JEE 0] x + x = 0 ; x + x + = 0, hve one root in ommon is (A) (B) - i 3 (C) i 5 (D) Arride lerning Online E-lerning Ademy A-479 Indr Vihr, Kot Rjsthn 34005 Pge No. # 4
PART - II : AIEEE PROBLEMS (PREVIOUS YEARS). If ¹ ut = 5 3, = 5 3, then the eqution hving the roots nd is. [AIEEE-00] () 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 : [AIEEE-003] () 3 () 3 (3) 3 (4) 3. 3. If ( p) is root of qudrti eqution x + px + ( p) = 0, then its roots re : [AIEEE-004] () 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 : [AIEEE-004] () 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) = +. 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 - [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 : [AIEEE-006] () 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 : [AIEEE-006] () 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 + 7 7 () 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) [AIEEE-007]. 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 3 + 30x 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 E-lerning Ademy A-479 Indr Vihr, Kot Rjsthn 34005 Pge No. # 5
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 - 4 4 x + = 0 will e- is- (A) 0 (B) (C) (D) infinite 0. Sum of roots of the eqution (x + 3) 4 x + 3 + 3 = 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 E-lerning Ademy A-479 Indr Vihr, Kot Rjsthn 34005 Pge No. # 6
Exerise # PART - I A-. (B) A-. (C) A-3. (A) A-4. (C) A-5. (C) A-6. (C) A-7. (D) B-. (A) B-. (C) B-3. (B) B-4. (B) B-5. (A) C-. (B) C-. (B) C-3. (C) C-4. (A) C-5. (A) D-. (B) D-. (B) D-3. (B) D-4. (B) D-5. (B) D-6. (D) D-7. (B) D-8. (B) E-. (A) E-. (C) E-3. (B) E-4. (B) PART-II. (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 = 0. 3. 8, 3 5. 3 ± 8. (-, 0) È (9/, ) 9. r 3 0. 3. = 0, 4 4. = - 7, = - 8 ; roots (3, 4), (3, 5), (3, ) Arride lerning Online E-lerning Ademy A-479 Indr Vihr, Kot Rjsthn 34005 Pge No. # 7
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 < 6.75. 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) 5. 0 0 0 6. (A) 7. (D) 8. (B) 9. (C) 0. (B) PART-II. (). () 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 E-lerning Ademy A-479 Indr Vihr, Kot Rjsthn 34005 Pge No. # 8