Quadratic Equations (1) PDF Past Paper

44 JEE-Mathematics 12. If one root of the equation x2 + px + 12 = 0 is 4, while the equation x2 + px + q = 0 has equal roots, then the value of ‘q’ is- (A) 3 (B) 12 (C) 49/4 (D) 4 13. The sum of the values of m for which...

44 JEE-Mathematics 12. If one root of the equation x2 + px + 12 = 0 is 4, while the equation x2 + px + q = 0 has equal roots, then the value of ‘q’ is- (A) 3 (B) 12 (C) 49/4 (D) 4 13. The sum of the values of m for which the quadratic polynomial P(x) = x2 + (m + 5)x + (5m + 1) is a perfect square (m Î R) is (A) 3 (B) 7 (C) 8 (D) 10 14. 2 3 2 If equations x – 5x + 5 = 0 and x + ax + bx + 5 = 0 have common root, then value of a + b (a, b Î Q) is - (A) 4 (B) –4 (C) 0 (D) can't find 15. If the equation ax2 + bx + c = 0 has distinct real roots, both negative, then- (A) a,b,c must be of same sign (B) a,b must be of opposite sign (C) a,c must be of opposite sign (D) a,b must be of same sign and opposite to sign of c 16. If P(x) = x2 – (2 – p)x + p – 2 assumes both positive and negative value, then the complete set of values of 'p' is- (A) (–¥,2) (B) (6,¥) (C) (2,6) (D) (–¥,2) È (6, ¥) 17. If value of a for which the sum of the squares of the roots of the equation x2 – (a – 2)x – a – 1=0 assume the least value is- (A) 2 (B) 3 (C) 0 (D) 1 18. If x2 + 2ax + 10 – 3a > 0 for all x Î R, then (A) – 5 < a < 2 (B) a < – 5 (C) a > 5 (D) 2 < a < 5 19. If the expression y = 8x – x2 – 15 is negative, then x lies in the interval- (A) (3,5) (B) (5,50) (C) (3,¥) (D) (–¥,3) È (5,¥) 20. Let ƒ(x) = ax2 + bx + 8 (a,b Î R) be a quadratic polynomial whose graph is symmetric about the line x = 2. If minimum value of ƒ(x) is 6, then the value of 2a – b is- (A) 0 (B) 1 (C) 2 (D) 3 21. Let g(x) = x2 – (b + 1)x + (b – 1), where b is a real parameter. The largest natural number b satisfying g(x) > –2 " x Î R, is - (A) 1 (B) 2 (C) 3 (D) 4 22. y = x – 6x + 5, x Î [2,4], then- 2 (A) least value of y is –3 (B) least value of y is 3 (C) greatest value of y is 4 (D) greatest value of y is –3 16x 2 - 12x + 9 23. Range of the expression : ( x Î R ) is- 16x 2 + 12x + 9 é1 ù æ 1ù (A) ê ,3ú (B) ç -¥, ú (C) [3, ¥ ) (D) R ë3 û è 3û  Quadratic equation 45 24. If the roots of equation (4p – p2 – 5)x2 – (2p – 1)x + 3p = 0 lie on either side of unity then the number of integral values of p is- (A) 4 (B) 2 (C) 3 (D) 1 25. If exactly one root of the equation 2kx2 – 4kx + 2k – 1 = 0 lies in [0,1), then complete range of k is- (A) (–¥,0] (B) (–¥,0) (C) (0,¥) (D) [0,¥) 26. If " p Î R one root of the equation x2 + 2px + q2 – p2 – 6 =0 is less than 1 and other root is greater than 1, then range of q is - (A) ( -¥, -2 ) (B) (–2, 2) ( (C) - 5, 5 ) (D) ( 2, ¥ ) 27. Let ƒ(x) = 2x2 + px + 1 is given. If ƒ(x) is negative integer for only one real value of x, then product of all possible values of p is - (A) –3 (B) –16 (C) 5 (D) –7 28. 3 2 Let r1, r2, r3 be roots of equation x – 2x + 4x + 5074 = 0, then the value of (r1 + 2)(r2 + 2)(r3 + 2) is (A) 5050 (B) –5050 (C) –5066 (D) –5068 29. Let a, b, c are roots of equation x3 + 8x + 1 = 0, then the value of bc ac ab + + is equal to (8b + 1)(8c + 1) (8a + 1)(8c + 1) (8a + 1)(8b + 1) (A) 0 (B) –8 (C) –16 (D) 16 30. Let ƒ(x) = x3 + x + 1 and P(x) be a cubic polynomial such that P(0) = –1 and the roots of P(x) = 0 are the squares of the roots of ƒ(x) = 0, then value of P(9) is - (A) 98 (B) 899 (C) 80 (D) 898 EXERCISE (O-2) 1. If a and b are the roots of the equation x 2 – x + 1 = 0, then a2009 + b2009 = (A) –2 (B) –1 (C) 1 (D) 2 2 2. The sum of all the real values of x satisfying the equation 2( x -1)(x + 5x - 50) = 1 is : (A) 16 (B) –5 (C) –4 (D) 14 3. The number of integral values of m for which the equation (1 + m2)x2 – 2(1 + 3m)x + (1 + 8m) = 0 has no real root is : (A) infinitely many (B) 2 (C) 3 (D) 1 4. Sum of all distinct integral value(s) of a such that equation x2 – ax + a + 1 = 0 has integral roots, is equal to- (A) 2 (B) 4 (C) 3 (D) None of these 5. 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 46 JEE-Mathematics 6. If one root of the equation x2 + px + q = 0 is the square of the other, then (A) p3 + q2 – q(3p + 1) = 0 (B) p3 + q2 + q(1 + 3p) = 0 (C) p3 + q2 + q(3p – 1) = 0 (D) p3 + q2 + q(1 – 3p) = 0 a 7. Let a, b, be the roots of the equation x2 – px + r = 0 and , 2b be the roots of the equation x2 – qx + r = 2 0. Then the value of r is - 2 2 2 2 (A) (p – q)(2q – p) (B) (q – p)(2p – q) (C) (q – 2p)(2q – p) (D) (2p – q)(2q – p) 9 9 9 9 8. Let p, q Î ¤. If 2 - 3 is a root of the quadratic equation, x2 + px + q = 0, then : (A) q2 + 4p + 14 = 0 (B) p2 – 4q – 12 = 0 (C) q2 – 4p – 16 = 0 (D) p2 – 4q + 12 = 0 æ pö 9. If a and b are the roots of the quadratic equation, x2 + xsinq –2sinq = 0, q Î ç 0, ÷ , then è 2ø a12 + b12 (a -12 ) + b-12 ( a - b ) 24 is equal to : 26 212 212 212 (A) ( sin q + 8 )12 (B) ( sin q - 8 )6 (C) ( sin q - 4 )12 (D) ( sin q + 8 )12 10. If f(x) is a quadratic expression such that f(1) + f(2) = 0, and –1 is a root of f(x) = 0, then the other root of f(x) = 0 is :- 5 8 8 5 (A) - (B) (C) - (D) 8 5 5 8 11. If the equations x2 + bx – 1 = 0 and x2+x + b = 0 have a common root different from –1, then 'b' is equal to :- (A) 2 (B) 3i (C) 3i (D) 2 12. Let p(x) be a quadratic polynomial such that p(0) = 1. If p(x) leaves remainder 4 when divided by x – 1 and it leaves remainder 6 when divided by x + 1; then : (A) p(2) = 19 (B) p(–2) = 19 (C) p(–2) = 11 (D) p(2) = 11 13. If a,b are the roots of the equation ax + bx + c = 0, then the roots of the equation 2 a(2x + 1)2 + b(2x + 1) (x – 1) + c(x – 1)2 = 0 are 2a + 1 2b + 1 2a - 1 2b - 1 a +1 b +1 2a + 3 2b + 3 (A) , (B) , (C) , (D) , a -1 b -1 a +1 b +1 a-2 b-2 a -1 b -1 14. Let a, b, c be the sides of a triangle. No two of them are equal and l Î R. If the roots of the equation x2 + 2(a + b + c)x + 3l(ab + bc + ca) = 0 are real then - 4 5 1 5 æ 4 5ö (A) l < (B) l > (C) l Î æç , ö÷ (D) l Î çè , ÷ø 3 3 è 3 3ø 3 3 15. The equations ax + bx + c = 0 and bx + cx + a = 0, where b – 4ac ¹ 0 have a common root, then 2 2 2 a3 + b3 + c3 is equal to (a ¹ 0) (A) 3abc (B) abc (C) 0 (D) 1  Quadratic equation 47 16. If a & b(a < b ) , are the roots of the equation, x2 + bx + c = 0, where c < 0 < b , then - (A) 0 < a < b (B) a < 0 < b < |a | (C) a < b < 0 (D) a < 0 < a < b 17. 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 ) & the other in ( b, +¥) 18. The graph of y = ax2 + bx + c is shown. Which of the following does NOT hold good? (A) ab2c3 > 0 y (B) ab3c2 < 0 (C) ab3c5 > 0 x O (D) b2 > 4ac 19. The sum of the solutions of the equation x -2 + x ( ) x - 4 + 2 = 0, (x > 0) is equal to : (A) 4 (B) 9 (C) 10 (D) 12 20. The set of all real numbers x for which x2 – |x + 2| + x > 0, is (A) (–¥, –2) U (2, ¥) (B) (–¥, – 2 ) U ( 2 , ¥) (C) (–¥, –1) U (1, ¥) (D) ( 2 , ¥) 21. If a,b are roots of equation 3x2 – 9x – l = 0, lÎ R such that 1 < a < 3 & 3 < b < 5 then l lies in (A) (–6,30) (B) (–6,0) (C) (0,30) (D) Null set 2 22. The set of values of 'a' for which ƒ(x) = ax + 2x(1 – a) – 4 is negative for exactly three integral values of x, is- (A) (0,2) (B) (0,1] (C) [1,2) (D) [2,¥) 23. If a,b,g,d are the roots of equation x – bx + 3 = 0, then the equation whose solutions are 4 a+b+ g a+b+d a+ g+d b+ g +d , , , is- d2 g2 b2 a2 (A) 3x4 – bx3 – 1 = 0 (B) 3x4 – bx3 + 1 = 0 (C) 3x4 + bx3 – 1 = 0 (D) 3x4 + bx3 + 1 = 0 [MATCHING COLUMN TYPE] 2 24. Let f(x) = x2 - 6x + 5 x - 5x + 6 Match the expressions/statements in Column I with expressions/statements in Column II. Coloumn-I Coloumn-II (A) If –1 < x < 1, then f(x) satisfies (p) 0 < f(x) < 1 (B) If 1 < x < 2, then f(x) satisfies (q) f(x) < 0 (C) If 3 < x < 5, then f(x) satisfies (r) f(x) > 0 (D) If x > 5, then f(x) satisfies (s) f(x) < 1 48 JEE-Mathematics EXERCISE (S-1) 1. a , b are the roots of the equation K (x2 – x) + x + 5 = 0. If K1 & K2 are the two values of K for which the roots a, b are connected by the relation (a/b) + (b/a) = 4/5. Find the value of (K1/K2) + (K2/K1). 2. Let the quadratic equation x2 + 3x – k = 0 has roots a, b and x2 + 3x – 10 = 0 has roots c, d such that modulus of difference of the roots of the first equation is equal to twice the modulus of the difference of the roots of the second equation. If the value of 'k' can be expressed as rational number in the lowest form as m n then find the value of (m + n). 3. If a, b are the roots of ax2 + bx + c = 0, (a ¹ 0) and a + d, b + d are the roots of Ax2 + Bx + C = 0, b2 - 4ac B2 - 4AC (A ¹ 0) for some constant d, then prove that, =. a2 A2 4. If the quadratic equations, x2 + bx + c = 0 and bx2 + cx + 1 = 0 have a common root then prove that either b + c + 1 = 0 or b2 + c2 + 1 = b c + b + c. 5. Find the value of m for which the quadratic equations x2 – 11x + m = 0 and x2 – 14x + 2m = 0 may have common root. 6. Let a, b be arbitrary real numbers. Find the smallest natural number 'b' for which the equation x2 + 2(a + b)x + (a – b + 8) = 0 has unequal real roots for all a Î R. ax 2 + 2(a + 1) x + 9a + 4 7. Find the range of values of a, such that f (x) = is always negative. x 2 - 8x + 32 8. Consider the quadratic polynomial f(x) = x2 – 4ax + 5a2 – 6a (a) Find the smallest positive integral value of 'a' for which f(x) positive for every real x. (b) Find the largest distance between the roots of the equation f(x) = 0 (c) Find the set of values of 'a' for which range of f(x) is [–8, ¥) 2x 2 + 2x + 3 9. We call 'p' a good number if the inequality £ p is satisfied for any real x. Find the x2 + x +1 smallest integral good number. 10. Number of integral values of 'a' for which 2x2 – 2ax + a2 – a – 6 = 0 has roots of opposite sign is 11. Find all values of p for which the roots of the equation (p – 3)x2 – 2px + 5p = 0 are real and positive. 12. Find all the values of the parameter 'a' for which both roots of the quadratic equation x2 – ax + 2 = 0 belong to the interval (0, 3). 13. At what values of 'a' do all the zeroes of the function f(x) = (a - 2)x2 + 2ax + a + 3 lie on the interval (- 2, 1)? 14. If both the roots of the quadratic equation x2 – 2kx + k2 + k - 5 = 0 are less than 5, then find k. 15. When y2 + my + 2 is divided by (y – 1) then the quotient is f (y) and the remainder is R1. When y2 + my + 2 is divided by (y + 1) then quotient is g (y) and the remainder is R2. If R1 = R2 then find the value of m.  Quadratic equation 49 EXERCISE (S-2) 1. Let P (x) = x2 + bx + c, where b and c are integer. If P(x) is a factor of both x4 + 6x2 + 25 and 3x4 + 4x2 + 28x + 5, find the value of P(1). 2. Find the complete set of real values of ‘a’ for which both roots of the quadratic equation 2 ( a2 – 6a + 5) x2 – a + 2a x + (6a – a2 – 8) = 0 lie on either side of the origin. a b 3. Let a, b and g are the roots of the cubic x3 – 3x2 + 1 = 0. Find a cubic whose roots are , a-2 b-2 g and. Hence or otherwise find the value of (a – 2)(b – 2)(g – 2). g -2 4. Find the product of the real roots of the equation, x2 + 18x + 30 = 2 x 2 + 18x + 45 5. If the roots of x2 - ax + b = 0 are real & differ by a quantity which is less than c (c > 0), prove that b lies between (1/4) (a2 - c2) & (1/4)a2. 6. Suppose a, b, c Î ¢ such that greatest common divisor of x2 + ax + b and x2 + bx + c is (x + 1) and the least common multiple of x2 + ax + b and x2 + bx + c is (x3 – 4x2 + x + 6). Find the value of (a + b + c). 7. If roots of the equation (x – a) (x – 4 + b) + (x – 2 + a) (x + 2 – b) = 0 are p and q then find the absolute value of the sum of the roots of the equations 2(x – p) (x – q) – (x – a) (x – 4 + b) = 0 and 2(x – p) (x – q) – (x – 2 + a) (x + 2 – b) = 0. 8. Let P(x) = 4x2 + 6x + 4 and Q(y) = 4y2 – 12y + 25. Find the unique pair of real numbers (x, y) that satisfy P(x) · Q(y) = 28. a+ b 9. If largest real root of the equation x4 – 4x3 + 5x2 – 4x + 1 = 0 can be expressed as (where c does not contain any perfect square and a, b, c are natural numbers in lowest form), then the value of a + b – c is 10. Find the values of ‘a’ for which -3 < [(x2 + ax - 2)/(x2 + x + 1)] < 2 is valid for all real x. 11. Find values of k for which the equation x2 + (1 – 2k) |x| + (k2 – 1) = 0 has : (a) No solution (b) One real solution (c) Two real solutions (d) Three real solutions (e) Four real solutions 50 JEE-Mathematics EXERCISE (JM) 1. Let for a ¹ a1 ¹ 0, f(x) = ax2 + bx + c, g(x) = a1x2 + b1x + c1 and p(x) = f(x) – g(x). If p(x) = 0 only for x = –1 and p(–2) = 2, then the value of p(2) is: [AIEEE-2011] (1) 18 (2) 3 (3) 9 (4) 6 2. Sachin and Rahul attempted to solve a quadratic equation. Sachin made a mistake in writing down the constant term and ended up in roots (4, 3). Rahul made a mistake in writing down coefficient of x to get roots (3, 2). The correct roots of equation are: [AIEEE-2011] (1) –4, –3 (2) 6, 1 (3) 4, 3 (4) –6, –1 3. If the equations x2 + 2x + 3 = 0 and ax2 + bx + c = 0, a, b, c Î R, have a common root, then a : b : c is : [JEE-MAIN-2013] (1) 1 : 2 : 3 (2) 3 : 2 : 1 (3) 1 : 3 : 2 (4) 3 : 1 : 2 4. Let a and b be the roots of equation x2 – 6x – 2 = 0. If an = an – bn, for n ³ 1, then the value of a10 - 2a 8 2a 9 is equal to : [JEE-MAIN-2015] (1) 3 (2) – 3 (3) 6 (4) – 6 x 2 + 4x - 60 5. The sum of all real values of x satisfying the equation ( x 2 - 5x + 5) = 1 is :- [JEE-MAIN-2016] (1) 5 (2) 3 (3) –4 (4) 6 6. Let a and b be two roots of the equation x2 + 2x + 2 = 0, then a15 + b15 is equal to : [JEE(Main)-2019] (1) 512 (2) –512 (3) –256 (4) 256 7. The number of all possible positive integral values of a for which the roots of the quadratic equation, 6x2–11x+a = 0 are rational numbers is : [JEE(Main)-2019] (1) 2 (2) 5 (3) 3 (4) 4 8. Consider the quadratic equation (c–5)x2–2cx + (c–4) = 0, c¹5. Let S be the set of all integral values of c for which one root of the equation lies in the interval (0,2) and its other root lies in the interval (2,3). Then the number of elements in S is : [JEE(Main)-2019] (1) 11 (2) 18 (3) 10 (4) 12 9. If l be the ratio of the roots of the quadratic equation in x, 3m x +m(m–4)x+2 = 0, then the least 2 2 1 value of m for which l + = 1 , is : [JEE(Main)-2019] l (1) 2 - 3 (2) 4 - 3 2 (3) -2 + 2 (4) 4 - 2 3 10. Let a and b be the roots of the equation x2 – x – 1 = 0. If pk = (a)k + (b)k, k ³ 1, then which one of the following statements is not true ? [JEE (Main)-2020] (1) (p1 + p2 + p3 + p4 + p5) = 26 (2) p5 = 11 (3) p3 = p5 – p4 (4) p5 = p2 · p3 33 11. The least positive value of 'a' for which the equation 2x2 + (a – 10)x + = 2a has real roots is 2 [JEE(Main)-2020] 12. If A = {x Î R : |x| < 2} and B = {x Î R : |x – 2| ³ 3}; then : (1) A È B = R – (2, 5) (2) A Ç B = (–2, –1) (3) B – A = R – (–2, 5) (4) A – B = [–1, 2) [JEE(Main)-2020]  Quadratic equation 51 EXERCISE (JA) 1. The smallest value of k, for which both the roots of the equation, x2 – 8kx + 16(k2 – k + 1) = 0 are real, distinct and have values at least 4, is [JEE 2009, 4 (–1)] 2. Let p and q be real numbers such that p ¹ 0, p3 ¹ q and p3 ¹ –q. If a and b are nonzero complex a b numbers satisfying a + b = – p and a3 + b3 = q, then a quadratic equation having b and as a its roots is [JEE 2010, 3] 3 2 3 3 3 2 3 3 (A) (p + q)x – (p + 2q)x + (p + q) = 0 (B) (p + q)x – (p – 2q)x + (p + q) = 0 3 2 3 3 (C) (p – q)x – (5p – 2q)x + (p – q) = 0 (D) (p3 – q)x2 – (5p3 + 2q)x + (p3 – q) = 0 3. Let a and b be the roots of x2 – 6x – 2 = 0, with a > b. If an = an – bn for n ³ 1, then the value a10 - 2a 8 of is [JEE 2011] 2a 9 (A) 1 (B) 2 (C) 3 (D) 4 4. A value of b for which the equations x2 + bx – 1 = 0 x2 + x + b = 0, have one root in common is - [JEE 2011] (A) - 2 (B) -i 3 (C) i 5 (D) 2 5. Let S be the set of all non-zero numbers a such that the quadratic equation ax2 – x + a = 0 has two distinct real roots x1 and x2 satisfying the inequality |x1 – x2| < 1. Which of the following intervals is(are) a subset(s) of S ? [JEE 2015, 4M, –0M] æ 1 1 ö æ 1 ö (C) æç 0, 1 ö (A) ç - 2 , - ÷ (B) ç - , 0÷ ÷ (D) æç 1 , 1 ö÷ è 5ø è 5 ø è 5ø è 5 2 ø PARAGRAPH Let p,q be integers and let a,b be the roots of the equation, x2 – x – 1 = 0, where a ¹ b. For n = 0,1,2,...., let an = pan + qbn. FACT : If a and b are rational numbers and a + b 5 = 0 , then a = 0 = b. 6. If a4 = 28, then p + 2q = (A) 14 (B) 7 (C) 12 (D) 21 [JEE(Advanced)-2017, 3(–1)] 7. a12 = (A) 2a11 + a10 (B) a11 – a10 (C) a11 + a10 (D) a11 + 2a10 [JEE(Advanced)-2017, 3(–1)] 52 JEE-Mathematics ANSWERS Do yourself-1 1 (a) –1, –2; (b) 4 ; (c) 1 ± 2 ; 1 7 2 a, ; 3. 4. 3, – 1 a 3 5 5. 28x2 – 20x + 1 = 0 6. a1 = –2, a2 = 1 7. a1 = 2, a2 = 9/2 8. p = ±7 9. k = ±3 5 10. p=0 11. a = 2 12. x13 + x 32 = 3pq - p3 a(a 2 -18a + 9) 215 13. x13 + x32 = 14. 27 27 Do yourself-2 1. (a) imaginary; (b) real & distinct ; (c) real & coincident -22 3. a = 20 ± 6 5 4. k1 = , k2 = 2 5. a=4 3 æ1 ö 6. a = ±2 7. k=3 9. For all m Î ç , + ¥ ÷ è 4 ø æ 1ö æ1 ö 10. For all c Î [2, 4) 11. k = 13 12. For all m Î ç -¥, - ÷ È ç , ¥ ÷ è 2ø è2 ø æ 1ö 13. For all m Î ç -¥, - ÷ È (1, + ¥ ) 14. b = –4, c = 1; è 7ø 15. 7 16. –15 17. (a) c = 0; (b) c = 1; (c) b ® negative, c ® negative 18. (ii), (iii) and (iv) 19. (A) b 2 - 2ac (B) ( bc 4 3ac - b 2 ) (C) ( b 2 b 2 - 4ac ) a2 a 7 2 2 ac 1 1 27a 3 + 36a 20. + = - 21. a1 = 3/2, a2 = 3 22. a = –4 x13 x 32 8 3 {2, 18} for a = 6, ìí , üý for a = - 2 18 6 23. a1= - , a2 = 6. 24. 2 î19 19 þ 19 125 27 25. a1 = - , a2 = 26. a1 = 1/2, a2 = 1 27. p1 = 0, q1 = 0, p2 = 1, q2 = –2 8 8  Quadratic equation 53 Do yourself-3 1. (a) c2y2 + y(2ac – b2 ) + a2 =0; (b) acx2 – bx + 1 = 0; (c) acx2 + (a + c)bx + (a + c)2 = 0 2. x2 – 2 (p2 – 2q)x + p2(p2 – 4q) = 0 3. x2 – p(p4 – 5p2q + 5q2)x + p2q2(p2 – 4q)(p2 – q) = 0 Do yourself-4 9 1. 1 3. b= ,c=5 4. c = 0, c = 6 2 6. a = –2 7. m = –2 Do yourself-5 æ 1ö 1. xÎR 2. xÎR 3. x Î ç -¥, ÷ È ( 3, ¥ ) è 2ø æ1 ö 4. x Î ( -6, – 3) È ç , 2 ÷ - {1} È (9, ¥) 5. [3,7] 6. xΡ è2 ø æ1 ö 7. x Î (–¥, –3) È (–2,3) 8. ç 2, 3÷ 9. (2, 3) è ø 10 (–¥, –2) È æç -2, - 1 ö÷ È (1, ¥ ) è 2ø Do yourself-6 1. x Î R – (0,1] 2. (1/2, 2] 3. (3, +¥) æ 1 5ù 4. [1, 2) È (2, +¥) 5. [20/9, 4) È (5, +¥) 6. ç , ú è 2 2û 7. [0, 2] 8. x Î [–18, 2) Do yourself-7 1. (–¥,–10] È [10,¥) 2. x Î [0,10] 3. x Î [–2,6] é 7 ù é 3ù 4. ê - 2 ,1ú È ê2, 2 ú ë û ë û 5. (- 2, 2 ) 6. f æ 3 3ö æ 7ö 7. ç- , ÷ 8. f 9. ç 1, ÷ è 2 2ø è 3ø æ 6 10 ö 10. [–1,3] 11. ( -¥, -2 ) È ( 2, ¥ ) 12. ç- , ÷ è 7 7 ø æ 7ö æ7 ö 13. x Î R 14. ç -¥, - ÷ È ç , ¥ ÷ è 2ø è2 ø 54 JEE-Mathematics Do yourself-8 1. (a) 1 (b) –1 2. (1) (i) a < 0 (ii) b < 0 (iii) c < 0 (iv) D > 0 (v) a + b < 0 (vi) ab > 0 (2) (i) a < 0 (ii) b > 0 (iii) c = 0 (iv) D > 0 (v) a + b > 0 (vi) ab = 0 (3) (i) a < 0 (ii) b = 0 (iii) c = 0 (iv) D = 0 (v) a + b = 0 (vi) ab = 0 3. Third quadrant 4. (a) a > 9/16 (b) a < –2 5. c