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QUADRATIC EQUATIONS WORKSHEET 01 CUQ 1. The roots of 15x 2 + 14x − 8 = 0 are a) −2 5 or 4 3 b) −4 3 or 2 5 c) −4 5 or 2 3 d) 4 5 or −2 3 2. If one root of the equation ix2 − 2(1 + i)x + (2 − i) = 0 is 2 − i, then the other root is a) −i b) 2 + i c) i d) −2 − i 3. The roots of 3x 2 + 4x − 7 = 0 are a) rational and equal b) rational and not equal c) irrational d) imaginary 4. The value of k, so that 16x 2 + kx + 1 = 0 has equal roots a) +8 b) -8 c) \pm 8 d) \pm 4 5. The discriminant of x 2 + 3x + 7 = 0 a) -18 b) 1 c) -19 d) 9 6. The quadratic equation whose sum and product of the roots is -1 and 1 4 is a) 4x 2 − 4x + 1 = 0 b) x 2 + x + 4 = 0 c) 4x 2 + x + 1 = 0 d) 4x 2 + 4x + 1 = 0 7. If 2 + 3i is a root of equation x 2 + ax + b = 0, then (a, b) is a) (4, −1c) b) (−4,1c) c) (−4, −1c) d) (4,1c) 8. If α, β are roots of x 2 + kx + 2 = 0 and α + β = 1, then k is a) 2 b) 3 c) -1 d) -4 JEE MAIN LEVEL - 1 1. The roots of √3x 2 + 10x − 8√3 = 0 are a) 2√3, 4√3 b) −2√3, 4√3 c) 2 √3 , −4√3 d) −2 √3 , −4 √3 2. If α, β are roots of ax2 + bx + c = 0, then 1 α2 + 1 β2 is a) b 2−2ac a2 b) −b c c) b 2−2ac c 2 d) b 2−2ac ac 3. The condition that one root of ax2 + bx + c = 0 may be triple of the other is a) b 2 = 3ac b) 2b 2 = 9ac c) 2b 2 = 3ac d) 3b 2 = 16ac 4. If one root is equal to the square of other root of the equation x 2 + x − k = 0, then k is a) 0 b) 1 c) 2 d) -1 5. If α, β are the roots of 3x 2 − 5x + 7 = 0, then α 2 + β 2 is a) 5 3 b) 7 3 c) −17 9 d) 35 9 6. If α, β are the roots of ax2 − ax + b = 0, then α β + β α is a) 1
b) -1 c) a b + 2 d) a b − 2 7. If α, β are the roots of x 2 − p(x + a) + c = 0, then (1 + α)(1 + β) is a) 1 b) 1 − c c) c d) 1 + c 8. The condition that the roots of ax2 + bx + c = 0 may be in the ratio m: n a) mnb2 = ac(m + n) 2 b) mnb = ac(m + n) 2 c) mn2 a = bc(m + n) 2 d) mnc2 = ab(m + n) 2 JEE MAIN LEVEL - 2 9. If one root of px 2 − 14x + 8 = 0 is six times the other, then p is a) 1 b) 2 c) 3 d) 4 10. If α, β are roots of ax2 + bx + c = 0, then αβ 2 + α 2β + αβ is a) ac−bc a 2 b) bc−ac a 2 c) ac−bc b2 d) bc−ac b2 11. If α, β are the roots of ax2 + bx + c = 0, then α 3+β 3 α−3+β−3 is a) c 2 a2 b) c 3 a3 c) 3abc−b 3 c 3 d) b 2−2ac ac 12. If α, β are the roots of x 2 − px + q = 0, then α −3+β −3 α3+β3 is a) q 2 b) −q 2 c) 1/q 3 d) q 3 JEE MATN LEVEL -3 13. The condition that the roots of the equation ax2 + bx + c = 0 may differ by 5 is a) b 2 − 25a 2 = 4ac b) b 3 − 5a 2 = 4ac c) b 2 + 15a 2 = 4ac d) None 14. If one root of x 2 + px + q = 0 may be the square of the other , then p 3 + q 2 is a) q(3p + a) b) −q(3p + a) c) q(3p − a) d) q(1 − 3p) INTEGER ANSWER TYPE: 15. If the sum of the squares of the roots of x 2 + px − 3 = 0 is 10 , then value of p > 0 is JEE MATN LEVEL -4 16. The equation whose roots are greater by 1 than those x 2 − 5x + 6 = 0 is a) x 2 − 7x + 12 = 0 b) x 2 + 7x − 12 = 0 c) x 2 − 7x − 12 = 0 d) x 2 + 7x + 12 = 0 17. The equations whose roots are decreased by 1 , than those of 2x 2 − 3x + 1 = 0 is a) 2x 2 + x = 0 b) 2x 2 − x = 0 c) 2x 2 + x + 1 = 0 d) 2x 2 + x − 1 = 0 18. The equation whose roots are multiplied by 3 of those of 2x 2 − 5x + 6 = 0 is a) 2x 2 + 15x + 54 = 0 b) 2x 2 − 15x + 54 = 0 c) 2x 2 − 15x − 54 = 0 d) 2x 2 + 15x − 54 = 0 JEE MAIN LEVEL - 5 19. If α, β are the roots of ax2 + bx + c = 0 and α + h, β + h are the roots of px 2 + qx + r = 0, then h is a) ( b a − q p ) b) 1 2 ( b a − q p ) c) − 1 2 ( a b − p q ) d) 1 2 ( b a − p q ) 20. If α, β are the roots of 2x 2 + x + 3 = 0, then the equation whose roots are 1−α 1+α , 1−β 1+β is a) 2x 2 + x + 3 = 0
b) 2x 2 − x + 3 = 0 c) 2x 2 + x − 3 = 0 d) 2x 2 − x − 3 = 0 21. If α, β be the roots of equation x 2 + x + 1 = 0, then the equation whose roots are α 19 , β 7 is a) x 2 − x − 1 = 0 b) x 2 − x + 1 = 0 c) x 2 + x − 1 = 0 d) x 2 + x + 1 = 0 JEE ADVANCED LEVEL -1 MULTIPLE CORRECT CHOICE TYPE : 22. The roots of (x + a)(x + b) − 8k = (k − b) 2 are real and equal. Where a, b, c ∈ R then a) a + b = 0 b) a = b c) k = −2 d) k = 0 REASONING TYPE: 23. Statement-I: The quadratic equations whose roots are −√3, √3 is x 2 − 3 = 0. Statement-II: The quadratic equation whose roots are α, β is x 2 − (α + β)x + αβ = 0. a) Both the statements are true b) Both the statements are false c) Statement I is true and Statement II is false d) Statement I is false and Statement II is true COMPREHENSION TYPE While solving an equation of the form x = √a + √a + √a + ⋯ ... we can replace x = √a + √a + √a + ⋯ ... with ' x '. The given equations can be modified as x = √a + x. Using this idea answer the following ques- tions. 24. If x = 2 + √2 + √2 + √2 + ⋯ ., then x = ? a) 1 b) 4 c) 2 d) 5 25. If x = √6 + √6 + √6 + √6 + ⋯ ..., then x = a) -3 b) -2 c) 3 d) 11 26. If x = √3 − √3 − √3 − √3 ... ... ., then the value of x is a) −1 + √ 11 2 b) −1 − √ 11 2 c) −1−√13 2 d) −1+√13 2 MATRIX MATCH TYPE: 27. If α, β are roots x 2 + x + 2 = 0, then Column - I Column - II a) α + β p) 0 b) αβ q) 2 c) α 2 + β 2 r) 1 d) α 2 + β 2 + αβ s) -3 t) -1 INTEGER ANSWER TYPE: 28. If the product of the roots of 5x 2 − 4x + 38 + k(−4x 2 − 2x − 1) = 0 is -5 , then the value of k is JEE ADVANCEDLEVEL-2&3 MULTIPLE CORRECT CHOICE TYPE 29. The equation whose roots are numerically equal but opposite sign of the roots of 3x 2 − 5x − 7 = 0 is ax 2 + bx + c = 0, then a, b, c values are a) 3 b) 5 c) -7 d) 7 MATRIX MATCH TYPE: 30. If α, β are roots of ax2 + bx + c = 0, then Column - I Column - II
a) α β + β α p) c 2 a 2 b) α 2+β 2 α−2+β−2 q) c 5(3abc−b 3) a8 c) α 3 + β 3 r) b 2−2ac ac d) α 5β 8 + α 8β 5 s) 3abc−b 3 a3 t) b/a JEE ADVANCED LEVEL-4&5 COMPREHENSION TYPE If α, β are roots of ax2 + bx + c = 0, then α + β = −b/a, αβ = c/a. 31. If α, β are the roots of ax 2 + bx + c = 0, then the value of 1 aα+b + 1 aβ+b is a) b ac b) c ab c) a bc d) abc 32. If the roots of the equation x 2 + 5x + 16 = 0 are α, β and the roots of the equation x 2 + px + q = 0 are α 2 + β 2 , αβ 2 , then a) p = 1, q = −36 b) p = −1, q = −56 c) p = 1, q = 56 d) p = −1, q = 56 33. If p, q ∈ {1,2,3,4}, then the number of equa- tions of the form px2 + qx + 1 = 0 having real roots is a) 15 b) 9 c) 7 d) 8 WORKSHEET 02 CUQ 1. The condition that the equations a1x 2 + b1x + c1 = 0; a2x 2 + b2x + c2 = 0 may have common root is a) (c1a2 − c2a1 ) 2 = (a1b2 − a2b1 )(b1c2 − b2c1 ) b) (a1b2 − a2b1 ) 2 = (b1c2 − b2c1 )(c1a2 − c2a1 ) c) (b1c2 − b2c1 ) 2 = (c1a2 − c2a1 )(a1b2 − a2b1 ) d) (c1a2 + c2a1 ) 2 = (a1b2 − a2b1 )(b1c2 − b2c1 ) 2. If a1x 2 + b1x + c1 = 0 and a2x 2 + b2x + c2 = 0 have a common root, then the com- mon root is a) c1a2−c2a1 a1b2−a2b1 b) a2b2−a2b1 c1a2−c2a1 c) b1c2−b2c1 c1a2−c2a1 d) c1a2−c2a1 b1c2−b2c1 3. If x 2 − 5x + 6 = 0 and 2x 2 − 10x + k = 0 have both the roots common, then k = a) 6 b) 12 c) -5 d) -10 4. If a > 0, then the expression ax 2 + bx + c is positive for all values of ' x ' provided a) b 2 − 4ac > 0 b) b 2 − 4ac < 0 c) b 2 − 4ac = 0 d) b 2 − ac < 0 5. If 4 < x < 8, then the value of 12x − x 2 − 32 is a) zero b) positive c) negative d) Not determined 6. If x ∈ R, then the value of x 2 − 6x + 10 is a) zero b) positive c) negative d) not determined 7. The maximum value of 10x − 5x 2 − 1 is a) -1 b) 1 c) 3 d) 4 8. The extreme value of x 2 − 5x + 6 is a) 1 4 b) −1 4 c) 1 2 d) −1 2 9. The minimum value of x 2 − 8x + 17 is a) 17