Chapter 1. Quadratic Equations

Transcription

1 1 Chpter Eqution: An eqution is sttement of equlity etween two expression for prticulr vlues of the vrile. For exmple 5x + 6, x is the vrile (unknown) The equtions cn e divided into the following two kinds: Conditionl Eqution: It is n eqution in which two lgeric expressions re equl for prticulr vlue/s of the vrile e.g., ) x is true only for x / ) x + x 6 0 is true only for x, - Note: for simplicity conditionl eqution is clled n eqution. Identity: It is n eqution which holds good for ll vlue of the vrile e.g; ) ( + ) x x + x is n identity nd its two sides re equl for ll vlues of x. ) (x + ) (x + 4) x + 7x + 1 is lso n identity which is true for ll vlues of x. For convenience, the symol shll e used oth for eqution nd identity. 1. Degree of n Eqution: The degree of n eqution is the highest sum of powers of the vriles in one of the term of the eqution. For exmple x st degree eqution in single vrile x + 7y 8 1 st degree eqution in two vriles x 7x nd degree eqution in single vrile xy 7x + y nd degree eqution in two vriles x x + 7x rd degree eqution in single vrile x y + xy + x rd degree eqution in two vriles 1. Polynomil Eqution of Degree n: An eqution of the form n x n + n-1 x n x + x + 1 x (1) Where n is non-negtive integer nd n, n-1, ,,, 1, 0 re rel constnts, is clled polynomil eqution of degree n. Note tht the degree of the eqution in the single vrile is the highest power of x which pper in the eqution. Thus x 4 + x x 4 + x + x + x + 1 0, x 4 0 re ll fourth-degree polynomil equtions. By the techniques of higher mthemtics, it my e shown tht nth degree eqution of the form (1) hs exctly n solutions (roots). These roots my e rel, complex or mixture of oth. Further it my e shown tht if such n eqution hs complex roots, they occur in pirs of conjugtes complex numers. In other words it cnnot hve n odd numer of complex roots. A numer of the roots my e equl. Thus ll four roots of x 4 0 re equl which re zero, nd the four roots of x 4 x Comprise two pirs of equl roots (1, 1, -1, -1).

2 1.4 Liner nd Cuic Eqution: The eqution of first degree is clled liner eqution. For exmple, i) x (in single vrile) ii) x + y 4 (in two vriles) The eqution of third degree is clled cuic eqution. For exmple, i) x + x + 1 x (in single vrile) ii) 9x + 5x + x 0 (in single vrile) iii) x y + xy + y 8 (in two vriles) 1.5 Qudrtic Eqution: The eqution of second degree is clled qudrtic eqution. The word qudrtic comes from the Ltin for squre, since the highest power of the unknown tht ppers in the eqution is squre. For exmple x x (in single vrile) xy x + y 9 (in two vrile) Stndrd form of qudrtic eqution The stndrd form of the qudrtic eqution is x + x + c 0, where, nd c re constnts with 0. If 0 then this eqution is clled complete qudrtic eqution in x. If 0 then it is clled pure or incomplete qudrtic eqution in x. For exmple, 5 x + 6 x + 0 is complete qudrtic eqution is x. nd x 4 0 is pure or incomplete qudrtic eqution. 1.6 Roots of the Eqution: The vlue of the vrile which stisfies the eqution is clled the root of the eqution. A qudrtic eqution hs two roots nd hence there will e two vlues of the vrile which stisfy the qudrtic eqution. For exmple the roots of x + x 6 0 re nd Methods of Solving Qudrtic Eqution: There re three methods for solving qudrtic eqution: i) By fctoriztion ii) By completing the squre iii) By using qudrtic formul i) Solution y Fctoriztion: Method: Step I: Write the eqution in stndrd form. Step II: Fctorize the qudrtic eqution on the left hnd side if possile. Step III: The left hnd side will e the product of two liner fctors. Then equte ech of the liner fctor to zero nd solve for vlues of x. These vlues of x give the solution of the eqution. Exmple 1: Solve the eqution x + 5x x + 5x Write in stndrd form x + 5x 0

3 Fctorize the left hnd side x + 6x x 0 x(x + ) -1(x + ) 0 (x 1) (x + ) 0 Equte ech of the liner fctor to zero. x 1 0 or x + 0 x 1 or x - x 1 x 1, - re the roots of the Eqution. Solution Set { 1, -} Exmple : Solve the eqution 6x 5x 4 6x 5x 4 6x 5x 4 0 6x 8x + x 4 0 x (x 4) +1 (x 4) 0 (x + 1) (x 4) 0 Either x or x 4 0 Which gives x 1 which gives x 4 1 x x Required Solution Set, ii) Solution of qudrtic eqution y Completing the Squre Method: Step I: Write the qudrtic eqution is stndrd form. Step II: Divide oth sides of the eqution y the co-efficient of x if it is not lredy 1. Step III: Shift the constnt term to the R.H.S. Step IV: Add the squre of one-hlf of the co-efficient of x to oth side. Step V: Write the L.H.S s complete squre nd simplify the R.H.S. Step VI: Tke the squre root on oth sides nd solve for x. Exmple : Solve the eqution x 15 4x y completing the squre. x 15 4x Step I Write in stndrd form: x + 4x 15 0 Step II Dividing y to oth sides: x + 4 x 5 0

4 4 Step III Shift constnt term to R.H.S: x + 4 x 5 Step IV Step V: : Step VI: Adding the squre of one hlf of the co-efficient of x. i.e., 4 on oth sides: 6 x + 4 x Write the L.H.S. s complete squre nd simplify the R.H.S. 4 x x Tking squre root of oth sides nd Solve for x x x x x + 6, 4 7 x x - x x, x x, x Hence, the solution set Exmple 4: Solve the eqution 5 -, x x + x x 0 Dividing oth sides y, we hve x x + y completing the squre.

5 5 sides. x x x - x Adding the squre of one hlf of the co-efficient of x i.e., x x - + x 4 x x 4 Tking squre root on oth sides x x x x x + - x x 4 x x x x Solution Set, iii) Derivtion of Qudrtic formul Consider the stndrd form of qudrtic eqution x + x + c 0. Solve this eqution y completing the squre. x + x + c 0 Dividing oth sides y on oth Tke the constnt term to the R.H.S x + x - c

6 6 To complete the squre on L.H.S. dd to oth sides. X + c x c x c x + 4 Tking squre root of oth sides x - - 4c 4 formul. x - - 4c which is clled the Qudrtic Where, co-efficient of x, coefficient of x, c constnt term Actully, the Qudrtic formul is the generl solution of the qudrtic eqution x + x + c c c Note:, re lso clled roots of the qudrtic eqution Method: To solve the qudrtic eqution y Using Qudrtic formul: Step I: Write the Qudrtic Eqution in Stndrd form. Step II: By compring this eqution with stndrd form x + x + c 0 to identify the vlues of,, c. Step III: Putting these vlues of,, c in Qudrtic formul - - 4c x nd solve for x. Exmple 5: Solve the eqution x + 5x x + 5x x + 5x 0 Composing with the stndrd form x + x + c 0, we hve, 5, c -. Putting these vlues in Qudrtic formul

7 7 x - - 4c -5 (5) - 4()(-) () x x x x x x x - Sol. Set Exmple 6: Solve the eqution 1, 15x x 0 y using Qudrtic formul: 15x x 0 Compring this eqution with Generl Qudrtic Eqution Here, 15, -, c - Putting these vlues in Qudrtic formul x x x x - - 4c -(-) (-) - 4(15)(- ) (15) -(-) x 0 10 x 0 x Sol. Set, 5

8 Exmple 7: Solve the eqution y using Qudrtic formul. x - 5 x x - 5 x - 1 Multiplying throughout y (x 5)(x 1), we get (x 1) + 5(x 5) (x 5) (x 1) x x 5 8x 4x x 6x x 9x Compring this eqution with Generl Qudrtic Eqution Here,, -9, c 9 Putting these vlues in the Qudrtic formul x ± 4c ( 9) ( 9) 4()(9) () x 9 4 x 9 4 x 1 4 x 6 4 x x Sol. Set, Exercise 1.1 Q.1. Solve the following equtions y fctoriztion. (i). x + 7x 8 (ii). x + 7x (iii). x x x 6 (iv). x (1 x) (v). (x + ) (x + 1) 1 (vi). 1 5 x 5 x 1

9 (vii). (viii). x - 1 x + x + + x x (ix). x + ( c) x c 0 (x). ( + )x +( + + c) x + ( + c) 0 x + x + (xi). + (xii). x - 1 x - 1 x - 1 x - Q.. Solve the following equtions y the method of completing the squre. (i). x 6x (ii). x 10x (iii). (x )(x + ) (x + 11) (iv). x + ( + )x + 0 (v). x x (vi) x - 5 x (vii). x 5x (viii). x x + 0 Q. Solve the following equtions y using qudrtic formul. (i). x x 9 0 (ii). (x + 1) x + 14 (iii) (iv). ( ) x+1 x+ x+ x (v). x (m n)x (m n) 0 (vi) mx + (1 + m)x (vii) x ( )x 6 0 (viii). x ( )x 0 (ix) x x + 1 x + x + 1 x + x + Q.4 The sum of numer nd its squre is 56. Find the numer. Q.5 A projectile is fired verticlly into the ir. The distnce (in meter) ove the ground s function of time (in seconds) is given y s t 16 t. When will the projectile hit the ground? Q.6 The hypotenuse of right tringle is 18 meters. If one side is 4 meters longer thn the other side, wht is the length of the shorter side? Answers 1.1 Q.1. (i). {1, -8} (ii). (iv). { (v). 4 1, 1, (iii). {, } (vi). { }

10 10 (vii). 1, (viii). {-, -} (ix). (x). (xi). (xii). Q.. (i). {, 4} (ii). {, 16 } (iii). { 1 11 (iv). {, } (v). {, 1 } (vi). {-1, 5} } (vii). {, } (viii) { ( + ), ( )} Q.. (i). { 5, } (ii).{1 } (iii). { (iv). { 5, 1 } (v). { m n, (m n) } (vi) (vii). {, } (viii). {, } (ix) 11 1 } 6 1 1, m 6 6, Q.4. 7, - 8 Q seconds Q m 1.8 Clssifiction of Numers 1. The Set N of Nturl Numers: Whose elements re the counting, or nturl numers: N {1,,, }. The Set Z of Integers: Whose elements re the positive nd negtive whole numers nd zero: Z { , -, -1, 0, 1,, }. Whose elements re ll those numers tht cn e represented s the quotient of two integers, where 0. Among the elements of Q re such numers s,,,. In symol Q, Z, 0 Equivlently, rtionl numers re numers with terminting or repeting deciml representtion, such s 1.15, 1.5, , 0.

11 4. 11 Whose elements re the numers with deciml representtions tht re nonterminting nd non-repeting. Among the elements of this set re such numers s, 7,. An irrtionl numer cnnot e represented in the form, where, Z. In symols, Q {irrtionl numers} 5. The Set R of Rel Numers: Which is the set of ll rtionl nd irrtionl numers: R {x x Q Q'} 6. The set I of Imginry Numers: Whose numers cn e represented in the form x + yi, where x nd y re rel numers, I {x + yi x, y R, y 0, i -1} If x 0, then the imginry numer is clled pure imginry numer. An imginry numer is defined s, numer whose squre is negtive i.e, 1, -, The set C of Complex Numers: Whose memers cn e represented in the form x + y i, where x nd y rel numers nd i -1 : C {x + yi x, y R, i -1} With this fmilir identifiction, the foregoing sets of numers re relted s indicted in Fig. 1. Nturl numers Imginry numers Integers Zero Complex numers Rtionl numers Negtive of nturl numers Rel numers Non-integers Irrtionl numers Fig. 1 Hence, it is cler tht N Z Q R C 1.9 Nture of the roots of the Eqution x + x + c 0 The two roots of the Qudrtic eqution x + x + c 0 re: x ± 4c

12 1 The expression 4c which pper under rdicl sign is clled the Discriminnt (Disc.) of the qudrtic eqution. i.e., Disc 4c The expression 4c discrimintes the nture of the roots, whether they re rel, rtionl, irrtionl or imginry. There re three possiilities. (i) 4c < 0 (ii) 4c 0 (iii) 4c > 0 (i) If 4c < 0, then roots will e imginry nd unequl. (ii) If 4c 0, then roots will e rel, equl nd rtionl. (This mens the left hnd side of the eqution is perfect squre). (iii) If 4c > 0, then two cses rises: () 4c is perfect squre, the roots re rel, rtionl nd unequl. (This men the eqution cn e solved y the fctoriztion). () 4c is not perfect squre, then roots re rel, irrtionl nd unequl. Exmple 1: Find the nture of the roots of the given eqution 9x + 6x x + 6x Here 9, 6, c 1 Therefore, Discriminnt 4c (6) 4(9) (1) Becuse 4c 0 roots re equl, rel nd rtionl. Exmple : Find the nture of the roots of the Eqution x 1x x 1x Here, -1, c 9 Discriminnt 4c (-1) -4() (9) Disc 4c 61 which is positive Hence the roots re rel, unequl nd irrtionl. Exmple : For wht vlue of K the roots of Kx + 4x + (K ) 0 re equl. Kx + 4x + (K ) 0 Here K, 4, c K Disc 4c (4) 4(K)(K ) 16 4K + 1K

13 1 The roots re equl if 4c 0 i.e. 16 4K + 1K 0 4K K 4 0 K 4K + K 4 0 K(K 4) + 1(K 4) 0 Or K 4, 1 Hence roots will e equl if K 4, 1 Exmple 4: Show tht the roots of the eqution ( + )x ( + + c)x + c 0 re rel ( + )x ( + + c)x + c 0 Here, ( + ), ( + + c), c c Discriminnt 4c [ ( + + c)] 4[( + ) c] 4( + + c + + c + c) 8(c + c) 4( + + c + + c + c c 4( + + c + ) c ) 4[( + + )+ c ] 4[(+ ) + c ] Since ech term is positive, hence Disc > 0 Hence, the roots re rel. Exmple 5: For wht vlue of K the roots of eqution x + 5x + k 0 will e rtionl. x + 5x + k 0 Here,, 5, c k The roots of the eqution re rtionl if Disc 4c 0 So, 5 4()k 0 5 8k 0 k 5 Ans 8 Exercise 1. Q1. Find the nture of the roots of the following equtions (i) x + x (ii) 6x 7x +5 (iii) x + 7x 0 (iv) Q. For wht vlue of K the roots of the given equtions re equl. (i) x + (K + 1)x + 4K (ii) x + (K )x 8k 0 (iii) (K + 6)x + 6x + K 0 (iv) (K + )x Kx + K 1 0 Q. Show tht the roots of the equtions (i) (mx + c) + x will e equl if c + m

14 14 (ii) (mx + c) 4x will e equl if c m (iii) x + (mx + c) hs equl roots if c (1 + m ). Q4. If the roots of (c )x ( c)x + ( c) 0 re equl then prove tht + + c c Q5. Show tht the roots of the following equtions re rel (i) x 1 ( m+ )x + 0 m (ii) x x + + c (iii)( 4c)x + 4( + c)x 4 0 Q6. Show tht the roots of the following equtions re rtionl (i) ( c)x + (c )x + c( ) 0 (ii) ( + )x + ( + + c)x + ( + c) 0 (iii) ( + )x x ) 0 (iv) p x - (p q) x q 0 Q7. For wht vlue of K the eqution (4 k) x + (k+) x + 8k will e perfect squre. (Hint : The eqution will e perfect squre if Disc. 4c 0 ) Answers 1. Q1. (i) Rel, rtionl, unequl (ii) unequl, rel nd rtionl (iii) ir-rtionl, unequl, rel (iv) Rel, unequl, ir-rtionl -11 Q. (i) 1, (ii) - (iii) 1, - (iv) 9 Q7. 0, 1.10 Sum nd Product of the Roots (Reltion etween the roots nd Co-efficient of x + x + c 0) The roots of the eqution x + x + c 0 re ± 4c 4c Sum of roots Add the two roots + 4c 4c

15 15 + 4c 4c Co-efficient of x Hence, sum of roots Co-efficient of x Product of roots + 4c 4c x 4c 4 4c 4 c Constnt term i.e. product of roots Co-efficient of x Exmple 1: Find the sum nd the Product of the roots in the Eqution x + 4 7x x + 4 7x x 7x Here, -7, c Sum of the roots Product of roots c 4 Exmple : Find the vlue of K if sum of roots of (k 1)x + (4K 1)x + (K + ) 0 is 5 (k 1)x + (4K 1)x + (K + ) 0 Here (k 1), 4K 1, c K +

16 16 Sum of roots 5 (4K - 1) 5 Sum of roots (K - 1) 5 (K 1) (4K 1) 10K 5 8K + 10K + 8K K 7 K 7 18 Exmple : If one root of 4x x + K 0 is times the other, find the vlue of K. Given Eqution is 4x x + K 0 Let one root e, then other will e. Sum of roots ( ) Product of roots () K 4 K 4 K 1 Putting the vlue of we hve 16 K x

17 Exercise Q1. Without solving, find the sum nd the product of the roots of the following equtions. (i) x x (ii) y + 5y 1 0 (iii) x 9 0 (iv) x + 4 7x (v) 5x + x 7 0 Q. Find the vlue of k, given tht (i) The product of the roots of the eqution (k + 1)x + (4k + )x + (k 1) 0 is 7 (ii) The sum of the roots of the eqution x + k x will e equl to the product of its roots. (iii) The sum of the roots of the eqution 4 x + k x is. Q. (i)if the difference of the roots of x 7x + k 4 0 is 5, find the vlue of k nd the roots. (ii) If the difference of the roots of 6x x + c 0 is 5 6, find the vlue of k nd the roots. Q4. If, β re the roots of x + x + c 0 find the vlue of (i) + β (ii) 1 1 (iii) (iv) (v) Q5. If p, q re the roots of x 6x + 0 find the vlue of (p + q ) pq (p + q ) pq (p + q) Q6. The roots of the eqution px + qx + q 0 re nd β, (v) Prove tht Q7. Find the condition tht one root of the eqution px + qx + r 0 is squre of the other. Q8. Find the vlue of k given tht if one root of 9x 15x + k 0 exceeds the other y. Also find the roots. Q9. If, β re the roots of the eqution px + qx + r 0 then find the vlues of (i) + β (ii) ( β) (iii) β + β Q1.(i) 1, 1 (ii) Answers , (iii) 0, - 9 (iv) 7, 1 7 (v), 5 5

18 Q.(i) (ii) (iii) Q.(i) K 10, roots 6, 1 (ii), ; c 1 c c c 4c Q4. (i) (ii) (iii) (iv) (v) c c c c Q Q7. Pr (p + r)+q pqr Q8. K - 14, roots re 7, Q9. (i) q pr p (ii) q 4pr p (iii) 1.11 Formtion of Qudrtic Eqution from the given roots : Let, e the roots of the Eqution x + x + c 0 The sum of roots (I) c Product of roots. (II) Or The eqution is x + x + c 0 Divide this eqution y c x x+ 0 c x x+ 0 From I nd II this eqution ecomes x - ( + β ) x + β 0 Or x (Sum of roots) x + Product of roots 0 Or x (S) x + (P) 0 is the required eqution, where S nd P Alternte method:- Let, e the roots of the eqution x + x + c 0 i.e., x nd x β x - 0 nd x - β 0 (x - ) ( x - β ) 0 x - x - β x + β 0

19 x - ( + β ) x + β 0 19 Or x (Sum of roots) x + Product of roots 0 Or x S x + P 0 is the required eqution, where S nd P Exmple 4: Form qudrtic Eqution whose roots re 5, 5 Roots of the required Eqution re 5 nd 5 Therefore S Sum of roots 5 5 S 0 P Product of roots ( 5)( 5) 9 (5) P 45 Required eqution is x (Sum of roots) x + (Product of roots) 0 Or x Sx + P 0 x 0(x) + ( 45) 0 x x 45 0 Exmple 5: If, re the roots of the eqution x + x + c 0, find the eqution whose roots re,. Becuse, re the roots of the Eqution x + x + c 0 The sum of roots Product of roots Roots of the required eqution re, Therefore, S sum of roots of required eqution + ( ) ( ) c

20 0 S c c c c c x c P Product of roots of required eqution. P 1 Required eqution is: x Sx + P 0 c x x c cx ( c)x + c 0 Exercise 1.4 Q1. Form qudrtic equtions with the following given numers s its roots. (i), - (ii) +i, i (iii) +, - (iv) + 5, 5 (v) i, 4 5 i Q. Find the qudrtic eqution with roots (i)equl numericlly ut opposite in sign to those of the roots of the eqution x + 5x 7 0 (ii)twice the roots of the eqution 5x + x + 0 (iii)exceeding y thn those of the roots of 4x + 5x Q. Form the qudrtic eqution whose roots re less y 1 thn those of x 4x 1 0 Q4. Form the qudrtic eqution whose roots re the squre of the roots of the eqution x x 5 0 Q5. Find the eqution whose roots re reciprocl of the roots of the eqution px qx + r 0 Q6. If, re the roots of the eqution x 4x + 0 find the eqution whose roots re (i) α, β (ii) α, β 1 1 (iii), (iv) α +, β +

21 1 Q7. If α, β re the roots of x + x + c 0 form n eqution whose roots re (i), (ii) (iii), Answers 1.4 Q1. (i) x + x 6 0 (ii) x 6x (iii) x 4x (iv) x + 6x (v) x - 5 x Q. (i) x 5x 7 0 (ii) 5x - 6x (iii) 4x 11x Q. x + x 0 Q4. 4x 9x Q5. rx qx + p 0 Q6. (i) x 1x (ii) x 40x (iii) x 1x (iv) x 8x Q7. (i) cx ( c)x + c 0 (ii) cx + ( c)x + c 0 (iii) cx (c )x + ( + c) 0 Summry Qudrtic Eqution: An eqution of the form x + x + c 0, 0, where,, c R nd x is vrile, is clled qudrtic eqution. If, re its roots then + 4c 4c, Nture of Roots: (i) If 4c > 0 the roots re rel nd distinct. (ii) If 4c 0 the roots re rel nd equl. (iii) If 4c < 0 the roots re imginry. (iv) If 4c is perfect squre, roots will e rtionl, otherwise irrtionl. Reltion etween Roots nd Co-efficients If nd e the roots of the eqution x + x + c 0 - Then sum of roots Product of roots c Formtion of Eqution If nd e the roots of the eqution x + x + c 0 then we hve x (sum of roots)x + (product of roots) 0

22 Short Questions Write Short nswers of the following questions: Solve the following qudrtic equtions y fctoriztion Q.1 x + 7x Q. x x Q. x(x + 7) (x 1) ( x + 4) Q4. 6x 5x 4 Q5. x + 5x Q6. x + x 1 Q7. m x + (1 + m ) x Solve the following equtions y completing the squre: Q8. x x Q9. x +1x Q10. x + 5x 6 0 Q11. x 6x Solve the following equtions y qudrtic formul : Q1. 4x +7x 1 0 Q1. 9 x x 8 0 Q14. X x 18 0 Q15. X x x 6 Q16. x 5x 0 Q x + 8 x Q18 Define discriminnt Discuss the nture of the roots of the eqution:

23 Q19 x 7x + 0 Q0. x 5x 0 Q1. x + x Q. x x + 0 Q. 9x + 6x Q4. x 1x For wht vlue of K the roots of the following equtions re equl: Q5 Kx + 4x + 0 Q6. x + 5x + K 0 Q7 Prove tht the roots of the eqution ( + ) x x - 0 re rtionl Q8 Write reltion etween the roots nd the coefficients of the qudrtic eqution x + x + c 0 Q.9 If the sum of the roots of 4x + k x 7 0 is, Find the vlue of k. Q.0 Find the vlue of K if the sum of the roots of eqution (k 1)x + (4k 1) x + (K + ) 0 is 5/ Find the sum nd product of the roots of following equtions: Q1 7x -5x Q. x 9 0 Q. 9x + 6x Q4. For wht vlue of k the sum of roots of eqution x + kx my e equl to the product of roots? Q5. If α, β re the roots of x px p c 0 then prove tht (1+ α) (1+ β) 1 c Write the qudrtic eqution for the following equtions whose roots re : Q.6 -, -

24 Q7. ἰ, - ἰ 4 Q8. - +, - Q.9 Form the qudrtic eqution whose roots re equl numericlly ut opposite in sign to those of x 7x 6 0 If α, β re the roots of the eqution x 4 x + 0 find eqution whose roots re: Q40. 1 α, 1 β Q41 -, - β Answers Q1. {-, - 4} Q {- 1, } Q {, -} Q4 {4/, - ½} Q5 {1, -6} Q6 {-1, ½} Q7 { -1, - 1/m} Q8 {-9, 1} Q9 {-11, 5} Q10 {1, 6} Q11 {, 4} Q1. {1, -6} Q1 { {-8/9, 1} Q14 Q15 {6, -} Q16 {, } Q17 {, -1/} Q18 {-1/4 } Q19. Roots re rtionl, rel nd unequl Q0 Roots re irrtionl, rel nd unequl Q1 Roots re imginry Q Roots re equl nd rel Q Roots re equl nd rel Q4 Roots re unequl, rel nd irrtionl Q5. K 4/ Q6. K 5 Q9 K -1 Q0. K 7/18 Q1 S 5/7, P 4/7 Q S 0, P - 9 Q S -/, 1/9 Q4 K - 5 Q6 x +5 x Q7 x + 0 Q8 x + 4x Q9 x + 7 x 0 Q40 x 4 x Q41 x + 4 x + 0

25 5 Ojective Type Questions Q1. Ech question hs four possile nswers.choose the correct nswer nd encircle it. 1. The stndrd form of qudrtic eqution is: () x + x 0 () x 0 (c) x + x + c 0 (d) x + c 0. The roots of the eqution x + 4x 1 0 re: () (7, ) () ( 7, ) (c) ( 7, ) (d) (7, ). To mke x 5x complete squre we should dd: () 5 () (c) (d) The fctors of x 7x re: () (x 4)(x + ) () (x 4)(x ) (c) (x + 4)(x + ) (d) (x + 4)(x ) The qudrtic formul is: () 4c () +4c (c) 4c +4c (d) A second degree eqution is known s: () Liner () Qudrtic (c) Cuic (e) None of these Fctors of x 1 re: () (x 1)(x x 1) () (x 1)(x + x + 1) (c) (x 1)(x + x 1) (d) (x 1)(x x + 1) To mke 49x + 5x complete squre we must dd: () 5 14 (c) 5 7 (d) lx + mx + n 0 will e pure qudrtic eqution if: () l 0 () m 0 (c) n 0 (d) Both l, m If the discrimnnt 4c is negtive, the roots re: () Rel () Rtionl (c) Irrtionl (d) Imginry 11. If the discriminnt 4c is perfect squre, its roots will e: () Imginry () Rtionl (c) Equl (d) Irrtionl 1. The product of roots of x x 5 0 is: () 14 5

26 () (c) 5 () 5 (d) The sum of roots of x x 5 0 is: () () (c) (d) 14. If nd 5 re the roots of the eqution, then the equtions is: () x + x () x x 10 0 (c) x + x 10 0 (d) x 5x If ± re the roots of the eqution, then the eqution is: () x 0 () x 9 0 (c) x + 0 (d) x If S is the sum nd P is the product of roots, then eqution is: () x + Sx + P 0 () x + Sx P 0 (c) x Sx + P 0 (d) x Sx P Roots of the eqution x + x 1 0 re: () Equl () Irrtionl (c) Imginry (d) Rtionl 18. If the discriminnt of n eqution is zero, then the roots will e: () Imginry () Rel (c) Equl (d) Irrtionl 19. Sum of the roots of x x + c 0 is: () c c () (c) (d) 0. Product of roots of x + x c 0 is: () c c () (c) (d) Answers 1. c c d c c c 19. d 0.

27 7