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MANOJ CHAUHAN CHAUHA N SIR(IIT SIR(IIT-DELHI) EX. SR. SR. FA FA CUL CULTY TY (BA (BANSAL NSAL CLA CLASSES) SSES)
EXERCISE–III
Q.1
Find the values of 0 < /2, satisfying the following equation, cos cos cos ( ) = 1/8. [JEE '99, 6] P R If tan and tan Q are the roots of the equation 2 2 2 ax2 + bx + c = 0 (a 0) then : [JEE '99, 2+2]
Q.2(i) In a triangle PQR, (A) a + b = c (C) a + c = b
(B) b + c = a (D) b = c
(ii) If the roots of the equation x2 2ax + a2 + a 3 = 0 are real & less than 3 then (A) a < 2 (B) 2 a 3 (C) 3 < a 4 (D) a > 4 Q.3
If , are the roots of the equation, (x a)(x b) + c = 0, find the roots of the equation, (x ) (x ) = c. [REE 2000 (Mains), 3]
Q.4(a) For the equation, 3x2 + px + 3 = 0 , p > 0 if one of the roots is square of the other, then p is equal to: (A) 1/3 (B) 1 (C) 3 (D) 2/3 (b) If & ( < ), are the roots of the equation, x2 + bx + c = 0, where c < 0 < b, then (A) 0 < < (B) < 0 < < (C) < < 0 (D) < 0 < < (c) If b > a , then the equation, (x a) (x b) 1 = 0, has : (A) both roots in [a, b] (B) both roots in ( a) (C) both roots in [b ) (D) one root in ( a) & other in (b, + ) [JEE 2000 Screening, 1 + 1 + 1 out of 35] (d) If , are the roots of ax2 + bx + c = 0, (a 0) and + , + , are the roots of, Ax2 + Bx + C = 0, (A 0) for some constant , then prove that, b 2 4ac B 2 4AC = . a2 A2
[JEE 2000, Mains, 4 out of 100]
Q.5
Let a, b, c be real numbers with a 0 and let be the roots of the equation ax2 + bx + c = 0. Express the roots of a3x2 + abcx + c3 = 0 in terms of [JEE 2001, Mains, 5 out of 100]
Q.6
The set of all real numbers x for which x2 – |x + 2| + x > 0, is (A) (– , –2) (2, ) (B) (–, – 2 ) ( 2 , ) (C) (– , –1)
Q.7
(1, )
(D) ( 2 , )
[JEE 2002 (screening), 3]
If x2 + (a – b)x + (1 – a – b) = 0 where a, b R then find the values of ‘a’ for which equation has unequal real roots for all values of ‘b’. [JEE 2003, Mains-4 out of 60]
Q.8(a) If one root of the equation x2 + px + q = 0 is the square of the other, then (A) p3 + q 2 – q(3p + 1) = 0 (B) p3 + q 2 + q(1 + 3p) = 0 (C) p3 + q 2 + q(3p – 1) = 0 (D) p3 + q 2 + q(1 – 3p) = 0 ETOOS Academy Pvt. Ltd. : F-106, Road No. 2, Indraprastha Industrial Area, End of Evergreen Motors
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(b) If x2 + 2ax + 10 – 3a > 0 for all x R, then (A) – 5 < a < 2 (B) a < – 5
Q.9
[JEE 2004 (Screening)] (D) 2 < a < 5
(C) a > 5
1 2 x 5x 2 Find the range of values of t for which 2 sin t = , t 3x 2 2 x 1
, 2 2 . [JEE 2005(Mains), 2]
Q.10(a) Let a, b, c be the sides of a triangle. No two of them are equal and R. If the roots of the equation x2 + 2(a + b + c)x + 3(ab + bc + ca) = 0 are real, then (A)
4 3
(B)
5
3
1 5 (C) , 3 3
4 5 (D) , 3 3 [JEE 2006, 3]
(b) If roots of the equation x 2 10cx 11d 0 are a, b and those of x2 – 10ax – 11b = 0 are c, d, then find the value of a + b + c + d. (a, b, c and d are distinct numbers) [JEE 2006, 6] Q.11(a) Let , be the roots of the equation x2 – px + r = 0 and x2 – qx + r = 0. Then the value of 'r' is 2 (A) (p–q)(2q – p) 9
2 , 2 be the roots of the equation [JEE 2007, 3+6]
2 2 2 (B) (q – p)(2p – q) (C) (q – 2p)(2q – p) (D) (2p–q)(2q – p) 9 9 9
MATCH THE COLUMN:
x 2 6x 5 (b) Let f (x) = 2 x 5x 6 Match the expressions / statements in Column I with expressions / statements in Column II. (A) (B) (C) (D)
Q.12
Column I
Column II
If – 1 < x < 1, then f (x) satisfies If 1 < x < 2, the f (x) satisfies If 3 < x < 5, then f (x) satisfies If x > 5, then f (x) satisfies
(P) (Q) (R) (S)
0 < f (x) < 1 f (x) < 0 f (x) > 0 f (x) < 1
ASSERTION & REASON: Let a, b, c, p, q be real numbers. Suppose , are the roots of the equation x2 + 2 px + q = 0 and
, 1 are the roots of the equation ax2 + 2bx + c = 0, where 2 {–1, 0, 1} STATEMENT-1 : ( p2 – q)(b2 – ac) 0 and STATEMENT-2 : b pa
c qa or (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1 (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True [JEE 2008, 3 (–1)]
Q.13
The smallest value of k, for which both the roots of the equation, x2 – 8kx + 16(k 2 – k + 1) = 0 are real, distinct and have values at least 4, is [JEE 2009, 4 (–1)]
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0, p3 q, p3 q. If and are nonzero complex numbers satisfying p and q, then a quadratic equation having and as its roots
Q.14 Let p and q be real numbers such that p
is (A) (p3 + q) x2 (p3 + 2q) x + (p3 + q) = 0
(B) (p3 + q) x2 (p3 2q) x + (p3 + q) = 0
(B) (p3 q) x2 (5p3 2q) x + (p3 q) = 0 (D) (p3 q) x2 (5p3 + 2q) x + (p3 q) = 0 [JEE 2010] Q.15 A value of b for which the equations x2 + bx 1 = 0 x2 + x + b = 0, have one root in common is (A) 2 (B) i 3 Q.16
(C) i 5
Let and be the roots of x2 6x 2 = 0, with If an = a10 2a8 is : 2a9
(A) 1
(D) 2
n n for n 1, then the value of [JEE 2011]
(B) 2
(C) 3
(D) 4
Q. 17 If the equations x2 + 2x + 3 = 0 and ax2 +bx + c = 0, a, b, c R, have a common root, then a : b : c is : [IIT Mains - 2013] (A) 3 : 1 : 2 (B) 1 : 2 : 3 (C) 3 : 2 : 1 (D) 1 : 3 : 2 Q.18 The real number k for which the equation, 2x3 + 3x + k = 0 has two distinct real roots in [0, 1] (A) does not exist (B) lies between 1 and 2 (C) lies between 2 and 3 (D) lies between 1 and 0 [IIT Mains - 2013] Q. 19 The number of values of k, for which the system of equations : (k + 1) x + 8y = 4k kx + (k + 3)y = 3k 1 has no solution, is : (A) 3 (B) infinite (C) 1
[IIT Mains - 2013] (D) 2 1
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Q. 20 Let and be the roots of equation px2 + qx + r = 0, p 0. If p, q, r in A.P. and 4 , then the value of | | is [IIT Mains - 2014] 2 13 9 2 17 (C) 9
(A)
61 9 34 (D) 9
(B)
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Anwer Key 1. 4. 6. 9. 11. 12. 15. 18.
/ 3,
2. 5. 7.
(a) C, (b) B, (c) D B , 3 , 10. 2 10 10 2 (a) D, (b) (A) P, R, S ; (B) Q, S ; 13. A 16. B 19. A
(i) A, (ii) A 2 and a>1
3.
(a, b)
2 or 2 and 2 8.
(a) D ; (b) A
14. 17. 20.
B B A
(a) A, (b) 1210 (C) Q, S ; (D) P, R, S k=2 C C
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