jfpr% ekuo /kez iz.ksrk ln~xq# Jh j.knksm+nklth egkjkt

Transcription

1 Phone : (0755) , WhatsApp QUADRATIC EQUATIONS PART OF 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 iz.ksrk ln~xq# Jh j.knksm+nklth egkjkt QUADRATIC EQUATIONS Some questions (Assertion Reason type) are given elow. Each question contains Statement (Assertion) and Statement (Reason). Each question has choices (A), (B), (C) and (D) out of which ONLY ONE is correct. So select the correct choice : Choices are : (A) Statement is True, Statement is True; Statement is a correct explanation for Statement. (B) Statement is True, Statement is True; Statement is NOT a correct explanation for Statement. (C) Statement is True, Statement is False. (D) Statement is False, Statement is True.. Statement-: If x R, x + x + 5 is positive. Statement-: If < 0, ax + x + c, a have same sign x R.. Statement-: If is a root of x x = 0, then will e the other root. Statement-: Irrational roots of a quadratic equation with rational coefficients always occur in conjugate pair.. Statement-: The roots of the equation x + i x + = 0 are always conjugate pair. Statement-: Imaginary roots of a quadratic equation with real coefficients always occur in conjugate pair.. Consider the equation (a a + ) x + (a 5a + 6)x + a = 0 Statement : If a =, then aove equation is true for all real x. Statement : If a =, then aove equation will have two real and distinct roots. 5. Consider the equation (a + )x + (a ) x = a Statement : Roots of aove equation are rational if 'a' is rational and not equal to. Statement : Roots of aove equation are rational for all rational values of 'a'. 6. Let f(x) = x = x + (a + ) x + 5 Statement : f(x) is positive for same < x < and for all ar Statement : f(x) is always positive for all xr and for same real 'a'. 7. Consider f(x) = (x + x + ) a (x + ) a (x + x + ) = 0 Statement : Numer of values of 'a' for which f(x) = 0 will e an identity in x is. Statement : a = the only value for which f(x) = 0 will represent an identity. 8. Let a,, c e real such that ax + x + c = 0 and x + x + = 0 have a common root Statement : a = = c Statement : Two quadratic equations with real coefficients can not have only one imaginary root common. 9. Statement : The numer of values of a for which (a a + ) x + (a 5a + ) x + a = 0 is an identity in x is. Statement : If ax + x + c = 0 is an identity in x then a = = c = Let a (, 0). Statement : ax x + < 0 for all x R Statement : If roots of ax + x + c = 0,, c R are imaginary then signs of ax + x + c and a are same for all x R.. Let a,, c R, a 0. Statement : Difference of the roots of the equation ax + x + c = 0 = Difference of the roots of the equation ax + x c = 0 Statement : The two quadratic equations over reals have the same difference of roots if product of the coefficient of the two equations are the same.. Statement : If the roots of x 5 0x + Px + Qx + Rx + S = 0 are in G.P. and sum of their reciprocal is 0, then S. Statement : x. x. x.x.x 5 = S, where x, x, x, x, x 5 are the roots of given equation.. Statement : If 0 < <, then the equation (x sin ) (x cos ) = 0 has oth roots in (sin, cos ) 8 of

2 Phone : (0755) , WhatsApp QUADRATIC EQUATIONS PART OF Statement : If f and f possess opposite signs then there exist at least one solution of the equation f(x) = 0 in open interval (a, ).. Statement : If a / then < < p where, are roots of equation x + ax + a = 0 Statement : Roots of quadratic equation are rational if discriminant is perfect square. 5. Statement- : The numer of real roots of x + x + = 0 is zero. Statement- : xr, x Statement-: If all real values of x otained from the equation x (a ) x + (a ) = 0 are non-positive, then a (, 5] Statement-: If ax + x + c is non-positive for all real values of x, then ac must e ve or zero and a must e ve. 7. Statement-: If a,, c, d R such that a < < c < d, then the equation (x a) (x c) + (x ) (x d) = 0 are real and distinct. Statement-: If f(x) = 0 is a polynomial equation and a, are two real numers such that f f < 0 has at least one real root. x x 8. Statement-: f(x) = 0 xr x x 5 Statement-: ax + x + c > 0 xr if a > 0 and ac < Statement-: If a + + c = 0 then ax + x + c = 0 must have as a root of the equation Statement-: If a + + c = 0 then ax + x + c = 0 has roots of opposite sign. 0. Statement-: ax + x + C = 0 is a quadratic equation with real coefficients, if + is one root then other root can e any other real numer. Statement-: If P q is a real root of a quadratic equation, then P - q is other root only when the coefficients of equation are rational. Statement-: If px + qx + r = 0 is a quadratic equation (p, q, rr) such that its roots are, & p + q + r < 0, p q + r < 0 & r > 0, then [] + [] =, where [] denotes G.I.F. Statement-: If for any two real numers a &, function f(x) is such that f.f < 0 f(x) has at least one real root lying etween (a, ). Statement-: If x is a root of a quadratic equation then another root of this equation must e x Statement-: If ax + x + c = 0, a,, c Q, having irrational roots then they are in conjugate pairs.. Statement-: If roots of the quadratic equation ax + x + c = 0 are distinct natural numer then oth roots of the equation cx + x + a = 0 cannot e natural numers. Statement-: If, e the roots of ax + x + c = 0 then, are the roots of cx + x + a = 0.. Statement-: The (x p) (x r) + (x q) (x s) = 0 where p < q < r < s has non real roots if > 0. Statement-: The equation (p, q, r R) x + qx + r = 0 has non-real roots if q pr < Statement-: One is always one root of the equation (l m)x + (m n) x + (n l ) = 0, where l, m, nr. Statement-: If a + + c = 0 in the equation ax + x + c = 0, then is the one root. 6. Statement-: If (a ) x + (a a + ) x + (a 7a + 0) = 0 is an identity, then the value of a is. Statement-: If a = = 0 then ax + x + c = 0 is an identity. 7. Statement-: x + x + > 0 x R Statement-: ax + x + c > 0 x R if ac < 0 and a > Statement-: Maximum value of is x x / Statement-: Minimum value of ax + x + c (a > 0) occurs at x. a 9. Statement-: If quadratic equation ax + x = 0 have non-real roots then a < 0 Statement-: For the quadratic expression f(x) = ax + x + c if ac < 0 then f(x) = 0 have non real roots. 0. Statement-: Roots of equation x 5 0x + Px + Qx + Rx + S = 0 are in G.P. and sum of their reciprocal is equal to 0 then s =. Statement-: If x, x, x, x are roots of equation ax + x + cx + dx + e = 0 (a 0) x + x + x + x = /a c xx a d e xx x xx xx a a 9 of

3 Phone : (0755) , WhatsApp QUADRATIC EQUATIONS PART OF. Statement-: The real values of a form which the quadratic equation x (a + 8a ) + a a = 0. Possesses roots of opposite signs are given y 0 < a <. Statement-: Disc 0 and 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 a Other root = = rational for all rational a. a (C) is correct option. 6. Here f(x) is a downward paraola D = (a + ) + 0 > 0 From the graph clearly st () is true ut st () is false - 7. f(x) = 0 represents an identity if a a 6 = 0 a =, a a 6 = 0 a =, a a = 0 a =, a a =0 a =, a = is the only values. Ans.: D 8. (A) x + x + = 0 D = < 0 x + x + = 0 and ax + x + c = 0 have oth the roots common a = = c. 9. (A) (a a + ) x + (a 5a + 6) x + a = 0 Clearly only for a =, it is an identify. 0. Statement II is true as if ax + x + c = 0 has imaginary roots, then for no real x, ax + x + c is zero, meaning therey ax + x + c is always of one sign. Further lim ax x c = signum. x statement I is false, ecause roots of ax x + = 0 are real for any a (-, 0) and hence ax x + takes zero, positive and negative values. Hence (d) is the correct answer.. Statement I is true, as Difference of the roots of a quadratic equation is always D, D eing the discriminant of the quadratic equation and the two given equations have the same discriminant. Statement II is false as if two quadratic equations over reals have the same product of the coefficients, their discriminents need not e same. Hence is the correct answer.. Roots of the equation x 5 0x + px + qx + rx + s = 0 are in G.P., let roots e a, ar, ar, ar, ar a + ar + ar + ar + ar = 0... (i) and a ar ar ar ar 0... (ii) from (i) and (ii); ar =... (iii) Now, - S = product of roots = a 5 r 0 = (ar ) 5 =. s. Hence is the correct answer.. Let, f(x) = (x sin ) (x cos ) then, f(sin ) = - < 0; f(cos ) = - < 0 Also as 0 < < ; sin < cos There-fore equation f(x) = 0 has one root in (-, sin ) and other in (cos, ) Hence is the correct answer. 0 of

4 Phone : (0755) , WhatsApp QUADRATIC EQUATIONS PART OF sin cos Hence (d) is the correct answer.. (B) x ax a = 0 g() < 0 a > / 5. equation can e written as ( x ) (a ) x (a ) = 0 x = & x = a 6. (A) Let f(x) = (x a) (x c) + (x ) (x d) Then f = (a ) (a d) > 0 Since x 0 and x = a [ x is non positive] f = ( a) ( c) < 0 0 < a < a 5 f(d) = (d a) (d ) > 0 i.e., a (, 5] Hence a root of f(x) = 0 lies etween a & and another Hence ans. (B). root lies etween ( & d). 7. x + x + > 0 x R a = > 0 ac = = - < 0 x + x + 5 > 0 x R a = > 0 ac = 0 = -6 < 0 x x So > 0 xr a is correct x x 5 9. (A) If the coefficients of quadratic equation are not rational then root may e and. Hence the roots of the given equation are real and distinct. 8. ax + x + c = 0 Put x = a + + c = 0 which is given So clearly is the root of the equation Nothing can e said aout the sign of the roots. c is correct. 0. (D) R is oviously true. So test the statement let f(x) = (x p) (x r) + (x q) (x s) = 0 Then f(p) = (p q) (p s) f(r) = (r q) (r s) If > 0 then f(p) > 0, f(r) < 0 There is a root etween p & r Thus statement- is false.. (A) Both Statement- and Statement- are true and Statement- is the correct explanation of Statement-.. (C) Clearly Statement- is true ut Statement- is false. ax + x + c = 0 is an identity when a = = c = 0.. (A) for x + x + a > 0 and D < 0. (A) x x + x 5. The roots of the given equation will e of opposite signs. If they are real and their product is negative D 0 and product of root is < 0 a a (a 8a ) 8(a a) 0 and 0 a a < 0 0 < a <. Ans.. If x...to infinity, then x = 5 5 Que. from Compt. Exams 5 (d) None of these. For the equation x x 6 0, the roots are [EAMCET 988, 9] One and only one real numer Real with sum one Real with sum zero (d) Real with product zero. If ax x c 0, then x = [MP PET 995] of

5 Phone : (0755) , WhatsApp QUADRATIC EQUATIONS PART OF ac a c ac a ac (d) None of these. If the equations x x 5 0 and x x 0 have a common root, then 0 0, (d), [RPET 989] 5. If the equation x x 0 has equal roots and one root of the equation x x 0 is, then (, ) = (, ) (,) (, ) (d) (, ) x x 6. If x is real and k, then [MNR 99; RPET 997] x x k k 5 k 0 (d) None of these 7. If a c d, then the roots of the equation ( x a)( x c) ( x )( x d) 0 are [IIT 98] Real and distinct Real and equal Imaginary (d) None of these 8. If the roots of the equation qx px q 0 where p, q are real, e complex, then the roots of the equation x qx p 0 are Real and unequal Real and equal Imaginary (d) None of these 9. The values of 'a' for which ( a ) x ( a ) x is positive for any x are [UPSEAT 00] a a a (d) a or a x x m 0. If the roots of equation are equal ut opposite in sign, then the value of m will e ax c m [RPET 988, 00; MP PET 996, 00; P. CET 000] a a a a (d) a a a a. The coefficient of x in the equation x px q 0 was taken as 7 in place of, its roots were found to e and 5, The roots of the original equation are [IIT 977, 79], 0, 0 5, 8 (d) None of these. If one root of the equation ax x c 0 e n times the other root, then na c( n ) n ac( n ) nc a( n ) (d) None of these. If one root of the quadratic equation ax x c 0 is equal to the n th power of the other root, then the value of n ( n n ac ) n ( a c) [IIT 98] n (d) n. If sin, cos are the roots of the equation ax x c 0, then [MP PET 99] a ac 0 ( a c) c a ac 0 (d) a ac 0 5. If oth the roots of the quadratic equation x kx k k 5 0 are less than 5, then k lies in the interval [AIEEE 005] (, ) [, 5] (5, 6] (d) (6, ) 6. If the roots of the equations x x c 0 and x cx 0 differ y the same quantity, then c is equal to [BIT Ranchi 969; MP PET 99] 0 (d) 7. If the product of roots of the equation log k x kx e 0 is 7, then its roots will real when [IIT 98] k k k (d) None of these 8. If a root of the given equation a ( c) x ( c a) x c( a ) 0 is, then the other will e [RPET 986] a( c) ( c a) c( a ) (d) None of these ( c a) a( c) a( c) 9. In a triangle ABC the value of A is given y 5 cos A 0, then the equation whose roots are sin A and tan A will e [Roorkee 97] 5x 8x 6 0 5x 8x 6 0 5x 8 x 6 0 (d) 5x 8x If one root of the equation ax x c 0 the square of the other, then a( c ) cx, where X is a ( a ) a (d) None of these of

6 Phone : (0755) , WhatsApp QUADRATIC EQUATIONS PART OF. If 8, are the roots of x ax 0 and, are the roots of x x 0, then the roots of x ax 0 are 8, 9, 8, (d) 9, [EAMCET 987] x. The set of values of x which satisfy 5x x 8 and, x (, ) (,) (,) (,) (d) (,) n n is [EAMCET 989]. If, are the roots of x ax 0 and if Vn, then [RPET 995; Karnataka CET 000; P. CET 00] V n avn Vn V n avn avn V n avn Vn (d) Vn avn Vn 7. The value of c for which, where and are the roots of x 7x c 0, is 0 6 (d) 5. For what value of the sum of the squares of the roots of x ( ) x ( ) 0 is minimum [AMU 999] / / (d) / 6. The product of all real roots of the equation x x 6 0 is [Roorkee 000] (d) 6 7. For the equation x px 0, p 0 if one of the root is square of the other, then p is equal to [IIT Screening 000] (d) 8. If, e the roots of x px q 0 and h, h are 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 quadratic equation whose roots are a and where a and are the roots of x x 0, then [Kerala (Engg.) 00], q p, q 5 p 5 p, q (d) None of these 0. The value of a for which one root of the quadratic equation ( a 5a ) x (a ) x 0 is twice as large as the other, is [AIEEE 00]. If a,, c are in G.P., then the equations ax x c 0 and dx ex f 0 have a common root if [IIT 985; P. CET 000; DCE 000] (d) d e f,, are in a c A.P. G.P. H.P. (d) None of these. The value of a for which the equations x x a 0 and x ax 0 have a common root is [P. CET 999] (d). If ( x ) is a factor of x ( p ) x (p 5) x ( p 7) x 6, then p [IIT 975] (d) None of these. The roots of the equation x x 57x 8x 5 0, If one of them is i 6, are i 6, i 6, i 6, (d) None of these 5. The values of a for which x (a ) x a( a ) 0 may have one root less than a and other root greater than a are given y 0 [UPSEAT 00] a a 0 a 0 (d) a 0 or a ANSWER KEY(Que. from Compt. Exams) a c c c 5 a 6 a 7 a 8 a 9 d 0 a a 5 a 6 d 7 8 c 9 0 d c c 5 c 6 a 7 c 8 c 9 d 0 a a d a c 5 d 6 d 7 8 c 9 a 0 a of