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4. QUADRATIC EQUATIONS AND LINEAR INEQUALITIES CLASS XI MATHEMATICS VOLUME - I JEE 132 1. If a,b,c are positive real numbers such that a+b+c=1 then the least value of ( )( )( ) ( )( )( ) 1 1 1 1 1 1 abc abc is:[1.1] A) 4 B) 8 C) 3 D) 2 2. If a,b are the roots of the equation ax2 +bx+c=0 and Sn =an + bn then aSn+1+bSn cSn-1 = n ≥ 2 [1.1] A) 0 B) a+ b+ c C) (a + b+ c) n D) n2 abc. 3. A group of students decided to buy a Alarm Clock priced between Rs. 170 to Rs 195. But at the last moment, two students backed out of the decision so that the remaining students had to pay 1 Rupee more than they had planned. If the students paid equal shares, the price of the Alarm Clock is [1.1] A) 190 B) 196 C) 180 D) 171. 4. The equation x x x 2 1 1 2 1 , has [1.1] A) No real root B) One real root C) Two equal roots D) Infinitely many roots 5. The roots of the equation x x 2 2 3 3 0 are [1.1] A) Real and equal B) Rational and equal C) Irrational and equal D) Irrational and unequal 1. The number of real roots of equation [1.1] (x-1)2 +(x - 2) 2 + (x - 3) 2 = 0 is A) 2 B) 1 C) 0 D) 3 2. The number of real solutions of the equation |x2 + 4x + 3| + 2x + 5 = 0 are [1.1] A) 1 B) 2 C) 3 D) 4 3. The equation (a + 2)x2 + (a - 3)x = 2a - 1, a ≠ -2 has roots rational for [1.1] A) All rational values of a except a = -2 B) All real values of a except a = -2 C) Rational values of a > -(1/2) D) None of these 4. Let a and b be the roots of x2 – 6x -2=0 with a >β. If an = an -βn for n ≥ 1, then the value of a10 – 2a8 /2a9 is [1.1] A) 1 B) 2 C) 3 D) 4 5. If both the roots of (2a-4) 9x – (2a-3)3x + 1=0 are non- negative, then [1.1] A) 0 < a< 2 B) 2 < a < 5 /2 C) a < 5/2 D) a > 3 6. If a, b, c, d are four consecutive terms of an increasing AP then the roots of the equation(x - a)(x - c) + 2(x - b)(x - d) = 0 are [1.1] A) Real and distinct B) Nonreal complex C) Real and equal D) Integers 7. For what values of k will the equation x2 - 2(1 + 3k)x + 7(3 + 2k) = 0 have equal roots [1.1] A) 1, –10/9 B) 2, –10/9 C) 3, –10/9 D) 4, –10/9 8. The range of a value of ‘a’ for which all the roots of the equation (a-1)(1+x+x2 )2 = (a+1)(1+x2 +x4 )are imaginary is [1.1] A) (-∞,-2] B) (2, ∞) C) (-2,2) D) [2, ∞) 9. Both the roots of given equation (x - a)(x - b)+(x-b) (x - c) + (x - c)(x - a) = 0 are always [1.1] A) Positive B) Negative C) Real D) Imaginary 1. If 31+x + 31-x = 10, then the values of are [1.1] A) 1, -1 B) 1, 0 C) 1, 2 D) -1, -2 2. The roots of the equation x x 2 2 3 3 0 are [1.1] A) Rational and equal B) Rational and not equal C) Irrational D) Imaginary 3. If ‘3’ is root of x2 + kx - 24 = 0 then it is also root of [1.1] A) x2 + 5x + k = 0 B) x2 + kx + 24 = 0 C) x2 - kx + 6 = 0 D) x2 - 5x + k = 0 4. The number of real solutions of the equation sin (ex ) =5x +5-x is [1.1] A) 0 B) 1 C) 2 D) Infinitely many 5. For the equation |x|2 +|x|- 6 = 0 the roots are [1.1] A) One and only one real number B) Real with sum one C) Real with sum zero D) Real with product zero 6. Product of real roots of the equation t 2 x2 +|x|+9=0 [1.1] A) Is always positive B) Is always negative C) Does not exist D) 9/t2 7. If the product of the roots of the equation 5x2 - 4x +2+k (4x2 – 2x-1)= 0 is 2 then k = [1.1] A) -8/9 B) 8/9 C) 4/9 D) -4/9 8. The real 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 lease value is [1.1] A) 0 B) 1 C) 2 D) 3 9. The real values of a for which the quadratic equation 3x2 + 2(a2 +1)x + (a2 – 3a+2) = 0 possesses roots of opposite signs lie in [1.1] A) (-∞, 1) B) (-∞, 0) C) (1, 2) D) 3 2 ,2

4. QUADRATIC EQUATIONS AND LINEAR INEQUALITIES CLASS XI MATHEMATICS VOLUME - I JEE 134 1. If the roots of ax2 + bx + c = 0 and px2 + qx+ r=0 differ by the same quantity, then b ac q pr 2 2 4 4 [2.1] A) p a 2 B) c p 2 C) a p 2 D) p c 2 2. If ax2 + bx + c = 0 and bx2 + cx + a = 0 have a common root and a ≠ 0 then abc abc 3 3 3 [2.1] A) 1 B) 2 C) 3 D) 9 3. The value of λ in order that the equations 2x2 + 5λx + 2 = 0 and 4x2 + 8λx + 3 = 0 have a common root is given by. [2.1] A) 1 B) -1 C) ±1 D) 3 4. If x2 - hx - 21 = 0, x2 – 3hx + 35 = 0 (h > 0) have a common root, then the value of h is [2.1] A) ±8 B) ±4 C) 4 D) 2 5. Let A,G and H be the A.M, G.M and H.M of two positive numbers a and b. The quadratic equation whose roots are A and H is [2.1] A) Ax A G x AG 2 2 2 2 0 B) Ax A H x AH 2 2 2 2 0 C) Hx H G x HG 2 2 2 2 0 D) Gx H G x GH 2 2 2 2 0 6. The roots of the equation ax2 + bx + c = 0, a ∈ R+ are two consecutive odd positive integers. Then [2.1] A) |b| ≤ 4a B) |b| ≥ 4a C) |b| ≥ 2a D) |b| ≤ a 7. If the difference between the roots of the equation x2 + ax+1= 0 is less than 5 , then the set of possible values of a is [2.1] A) (-3, 3) B) (-3, ∞) C) (3, ∞) D) (-∞, -3) 8. If the sum of two roots of the equation x3 – p x2 + qx – r = 0 is zero, then [2.1] A) pq = r B) qr = p C) pr = q D) pqr = 1 9. If α and β are the roots of x2 - 2x + 4 = 0 then the value of α6 + β6 is [2.1] A) 32 B) 64 C) 128 D) 256 10 If the roots of the equation x3 – px2 + qx – r = 0 are in A.P., then [2.1] A) 2p3 = 9 pq – 27 r B) 2q3 = 9 pq – 27 r C) p3 = 9 pq – 27 r D) 2p3 = 9 pq + 27 r SESSION-3 MAXIMUM AND MINIMUM VALUES, CONDITIONS FOR NUMBER K, LOCATION OF ROOTS. 3.1 MAXIMUM AND MINIMUM VALUES, CONDITIONS FOR NUMBER K, LOCATION OF ROOTS. Signs of ‘a’ and ax2 + bx + c : If the equation ax2 + bx + c = 0 has non real roots (∆ < 0) then ‘a’ and ax2 + bx + c will have same sign ∀ x ∈ R. If the equation ax2 + bx + c = 0 has equal roots then ‘a’ and ax2 + bx + c will have same sign x R b 2a . If the equation ax2 + bx + c = 0 has real roots α and β(∆ > 0, α > β) then (i) α < x < β ⇔ a and ax2 + bx + c will have opposite sign. (ii) x < α or x > β⇔ a and ax2 + bx + c will have same sign. Maximum And Minimum Values of ax2 +bx+ c; a,b,c ∈ R : (i) If a > 0 then f (x) = ax2 + bx + c has absolute minimum at x = -(b/2a) and the minimum value is 4 4 2 ac b a − (ii) If a < 0 then f (x) = ax2 + bx + c has absolute maximum at x = -(b/2a) and the maximum value is 4 4 2 ac b a − (iii) If f x ax bx c ax bx c 2 2 (or) f x ax bx c ax bx c 2 2 , (b2 - 4ac < 0), then the minimum and maximum values of f (x) are given by f c a (iv) If x is real then the maximum and minimum values of a x b x c x x c a c b c , , are a c b c 2 and a c b c 2 Locating the roots of Quadratic Equation under given conditions: Location of real roots of the equation f(x) = ax2 + bx + c = 0 (∆ = 0) whose roots are α, β (α < β ) with respect to one real number K. If both roots are greater than k then α + β > 2k and af (k) > 0 If both roots are less than k, then α + β < 2k and a f (k) > 0 If k lies between the roots then a f (k) < 0 with respect to two real numbers k1 , k2 . α, β If k1 < α < β < k2 then i) af (k1 ) > 0, ii) af (k2 ) > 0, iii) 2k1 < α <β < 2k2 If k1 <α <β < k2 then i) af (k1 ) < 0, ii) af (k2 ) < 0 If k1 <α< k2 < β then i) af (k1 ) > 0 and ii) af (k2 ) < 0 Position of roots of a quadratic equation: Let f(x) = ax2 + bx + c, a ≠ 0, b, c ∈ R and α, β be the roots of f (x) = 0 . Suppose k, k1 ,k2 ∈ R and k1 < k2 . Then, the following holds good.