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DPP for SOLUTION OF QUADRATIC EQUATIONS AND NATURE OF ROOTS MATH Chapter: QUADRATIC EQUATIONS AND INEQUATIONS Topic: SOLUTION OF QUADRATIC EQUATIONS AND NATURE OF ROOTS Date: 04 Jun 2025 Type: DPP 1. Two students while solving a quadratic equation in , one copied the constant term incorrectly and got the roots and . The other copied the constant term and coefficient of correctly as and respectively. The correct roots are (1) (2) (3) (4) 2. The value of for which the quadratic equation, has real and equal roots are (1) (2) (3) (4) None of these 3. If the roots of the given equation are real, then (1) (2) (3) (4) 4. Let be a root of the equation . The value of is equal to (1) (2) (3) (4) 5. The sum of the squares of the roots of and the squares of the roots of , is (1) (2) (3) (4) 6. If and are the roots of the equation , then is equal to (1) (2) (3) (4) 7. If , then (1) (2) (3) (4) 8. The quadratic equation with real coefficients has purely imaginary roots. Then the equation has (1) only purely imaginary roots (2) all real roots (3) two real and two purely imaginary roots (4) neither real nor purely imaginary roots x 3 2 x 2 −6 1 3, −2 −3, 2 −6, −1 6, −1 k kx 2 + 1 =kx + 3x − 11x 2 −11, −3 5, 7 5, −7 (cos p − 1)x 2 + (cos p)x + sin p = 0 p ∈ (−π, 0) p ∈ (− π 2 , π 2 ) p ∈ (0, π) p ∈ (0, 2π) r x 2 + 2x + 6 = 0 (r + 2)(r + 3)(r + 4)(r + 5) 51 −51 −126 126 |x − 2| 2 + |x − 2| − 2 = 0 x 2 − 2|x − 3| − 5 = 0 26 36 30 24 α β 375x 2 − 25x − 2 = 0 limn→∞ n∑ r=1 α r + limn→∞ n∑ r=1 β r 1 12 29 358 7 116 21 346 x 2 + y 2 = 25, xy = 12 x = {3, 4} {3, −3} {3, 4, −3, −4} {−3, −3} p(x) = 0 p(p(x)) = 0
9. If the roots of the equation be imaginary, then for all real values of the expression is : (1) (2) (3) (4) 10. The expression has the positive value if (1) (2) (3) (4) DPP for SOLUTION OF QUADRATIC INEQUATIONS AND NEWTON FORMULA AND MISCELLANEOUS EQUATIONS MATH Chapter: QUADRATIC EQUATIONS AND INEQUATIONS Topic: SOLUTION OF QUADRATIC INEQUATIONS AND NEWTON FORMULA AND MISCELLANEOUS EQUATIONS Date: 04 Jun 2025 Type: DPP 1. The set of all for which the equation has exactly one real root is: (1) (2) (3) (4) 2. Suppose that and are positive number with . The value of equals (1) (2) (3) (4) 3. Let and be the roots of the equation If then : (1) (2) (3) (4) 4. If , then equals (1) (2) (3) (4) 5. Exact set of values of for which is having four real solutions is (1) (2) (3) (4) 6. If are roots of the equation and for each posi- tive integer , then the value of is equal to (1) (2) (3) (4) bx 2 + cx + a = 0 x, 3b 2x 2 + 6bcx + 2c 2 > 4ab < 4ab > −4ab < −4ab x 2 + 2bx + c b 2 − 4c > 0 b 2 − 4c < 0 c 2 < b b 2 < c a ∈ R x|x − 1| + |x + 2| + a = 0 (−6, −3) (−∞,∞) (−6,∞) (−∞, −3) x y xy = 1 9 ; x (y + 1) = 7 9 ; y (x + 1) = 5 18 (x + 1)(y + 1) 8 9 16 9 10 9 35 18 α β 5x 2 + 6x − 2 = 0. Sn = α n + β n , n = 1, 2, 3... 5S6 + 6S5 = 2S4 5S6 + 6S5 + 2S4 = 0 6S6 + 5S5 + 2S4 = 0 6S6 + 5S5 = 2S4 log(3x−1)(x − 2) = log(9x2−6x+1) (2x 2 − 10x − 2) x 9 − √15 3 + √15 2 + √5 6 − √5 a x 3 (x + 1) = 2(x + a)(x + 2a) [−1, 2] [−3, 7] [−2, 4] [− 1 8 , 1 2 ] α, β x 2 + 5√2x + 10 = 0, α > β Pn = α n − β n n ( P17P20+5√2P11P19 P18P19+5√2P 2 18 ) . . . . 4 3 2 1
7. If , then (1) (2) (3) or (4) None of these 8. A man standing on a railway platform noticed that a train took to cross the platform (this means the time elapsed from the moment the engine enters the platform till the last compart- ment leaves the platform) which is long, and that it took to pass him. Assuming that the train was moving with uniform speed, what is the length of the train in meters? (1) (2) (3) (4) 9. Let and be the roots of the equation If then which one of the following statements is not true? (1) (2) (3) (4) 10. Let be the largest real root and be the smallest real root of the polynomial equation Then is (1) (2) (3) (4) DPP for CONDITION FOR COMMON ROOTS MATH Chapter: QUADRATIC EQUATIONS AND INEQUATIONS Topic: CONDITION FOR COMMON ROOTS Date: 04 Jun 2025 Type: DPP 1. The value of for which the equations and have a common root is (1) (2) (3) (4) 2. If a root of the equation is , while the roots of the equation are same, then the value of will be (1) (2) (3) (4) None of these 3. The number of ordered pairs of integers such that and the equations and have a common real root is (1) (2) (3) (4) |x − 2| + |x − 3| = 7 x = 6 −1 6 −1 21 s 88 m 9s 55 60 66 72 α β x 2 − x − 1 = 0. pk = (α) k + (β) k , k ≥ 1, (p1 + p2 + p3 + p4 + p5) = 26 p5 = 11 p3 = p5 − p4 p5 = p2 ⋅ p3 a b x 6 − 6x 5 + 15x 4 − 20x 3 + 15x 2 − 6x + 1 = 0 a 2+b 2 a+b+1 1 2 2 3 5 4 13 7 ‵a ′ x 2 − 3x + a = 0 x 2 + ax − 3 = 0 3 1 −2 2 x 2 + px + 12 = 0 4 x 2 + px + q = 0 q 4 4/49 49/4 (a, b) 1 ≤ a, b ≤ 2021 x 2 − ax + b = 0 x 3 − a 2 + bx + a − b = 0 2017 2018 2019 2021
4. Let be in . If and are the roots of the equation, and and are the roots of the equation, then is equal to (1) (2) (3) (4) 5. If the equation and have common root, then is equal to (1) (2) (3) (4) 6. If a root of the equations and is common, then its value will be (where and ) (1) (2) (3) or (4) None of these 7. Let and and have a common root then Statement : Equation of other roots is Statement : (1) Statement is true, Statement is true; Statement is not the correct explanation of Statement . (2) Statement is false, Statement is true. (3) Statement is true, Statement is false. (4) Statement is true, Statement is true;Statement is the correct explanation of Statement . 8. If are in , then the equations and have a common root if are in (1) (2) (3) (4) None of these 9. If equations and have a common root, then equals (1) (2) (3) (4) 10. The quadritic equations and have one root in common. The other roots of the first and second equations are integers in the ratio Then the common root is (1) (2) (3) (4) DPP for RELATION BETWEEN ROOTS AND COEFFICIENTS MATH λ ≠ 0 R α β x 2 − x + 2λ = 0 α γ 3x 2 − 10x + 27λ = 0 βγ λ 36 27 9 18 2ax 2 − 3bx + 4c = 0 3x 2 − 4x + 5 = 0 ( a+b c ) (a, b, c ∈ R) 2 34 5 34 15 17 15 x 2 + px + q = 0 x 2 + αx + β = 0 p ≠ α q ≠ β q−β α−p pβ−αq q−β q−β α−p pβ−αq q−β a ≠ b, c ≠ 0 x 2 + ax + bc = 0 x 2 + bx + ac = 0 −1 x 2 + cx + ab = 0 −2 a + b + c = 0 −1 −2 −2 −1 −1 −2 −1 −2 −1 −2 −2 −1 a, b, c G. P. ax 2 + 2bx + c = 0 dx 2 + 2ex + f = 0 d a , e b , f c A. P. G. P. H. P. ax 2 + bx + c = 0 (a, b, c ∈ R, a ≠ 0) 2x 2 + 3x + 4 = 0 a : b : c 1 : 2 : 3 2 : 3 : 4 4 : 3 : 2 3 : 2 : 1 x 2 − 6x + a = 0 x 2 − cx + 6 = 0 4 : 3. 1 4 3 2