Tiêu đề THEORY OF EQUATIONS.pdf ✅

THEORY OF EQUATIONS WORKSHEET 01 CUQ 1. The equation whose roots are 1,3, −2 is a) x 3 + 2x 2 − 5x + 6 = 0 b) x 3 − 2x 4 + 5x − 6 = 0 c) x 3 − 2x 2 − 5x + 6 = 0 d) x 3 − 2x 2 − 5x − 6 = 0 2. The equations whose roots are −2,3 ± √5 is a) x 3 − 4x 4 − 8x + 8 = 0 b) x 3 + 4x 2 + 8x + 8 = 0 c) x 3 − 4x 2 − 8x − 8 = 0 d) x 3 + 4x 2 + 8x + 8 = 0 3. The quotient we get when we divide x 3 − 3x 2 − x + 3 with (x + a) is a) x 4 + 4x − 3 b) x 2 + 3x + 4 c) x 2 − 4x + 3 d) x 2 + 4x + 3 4. The value of K so that 3x 4 + 4x 3 + 2x 2 + 10x + k is divisible by x + 2 is a) 2 b) -2 c) 4 d) -4 5. The equation whose roots are 1, −1,3 is a) x 3 + 3x 2 − x + 3 = 0 b) x 3 − 3x 2 + x + 3 = 0 c) x 3 + 3x 2 + x + 3 = 0 d) x 3 − 3x 2 − x + 3 = 0 6. The roots of x 3 − 3x 2 + 2x = 0 are a) 0,0,2 b) 0,1,2 c) −21, −1,1 d) 1, ±i 7. The roots of x 3 + 3x 2 − 11x + 6 = 0 are a) −1, ±√2 b) 0,1,2 c) −2, −1,1 d) 1, ±i 8. The roots of 2x 3 − 3x 2 − 11x + 6 = 0 are a) 11 2 , 1/2, −1/5 b) −1/2,1/3,1/5 c) 1 2 , 1/2, −6 d) −2,1/2,3 9. If 1,1, α are the roots of x 3 − 6x 2 + 9x − 4 = 0, then the value of α is a) 4 b) −3/2 c) 7/4 d) -5 10. If x 3 − x 2 + 33x + 5 = 0, then the values of s1, s2, s3 respectively are a) 1,33, −5 b) −1,33,5 c) 1, −33, −5 d) −1, −33,5 JEE MAIN LEVEL - 1 SINGLE CORRECT CHOICE TYPE 1. The number of Changes of sign in the equa- tion x 7 + 5x 6 − x 3 + 7x + 2 = 0 is a) 3 b) 2 c) 4 d) 1 2. The number of positive roots of x 9 + 4x 8 − 2x 3 + 7x + 8 = 0 is a) atmost one b) atleast one c) atmost two imaginary d) atleast three 3. The number of Imaginary roots of x 7 + 5x 6 − x 3 + 7x + 2 = 0 is a) at most three b) atleast two c) atleast three d) at least two 4. The equation whose roots are 3 ± √2, 2 ± 3i is a) x 4 − 6x 3 + 8x 2 + 2x − 1 = 0 b) x 4 − 9x 3 + 27x 2 − 33x + 14 = 0 c) x 4 − 10x 3 + 44x 2 − 106x + 91 = 0 d) x 4 − 6x 3 + 4x 2 + 54x + −117 = 0 5. The equation whose roots are 2 + √3, 2 − √3, 1 + 2i, 1 − 2i is a) x 4 − 7x 3 − 25x 2 − 43x + 40 = 0 b) x 4 − 7x 3 + 25x 2 + 43x − 40 = 0 c) x 4 + 6x 3 + 14x 2 − 22x + 5 = 0 d) x 4 − 6x 3 + 14x 2 − 22x + 5 = 0 6. The biquadratic equation, two of whose roots are 1 + i, 1 − √2 is a) x 4 − 4x 3 + 5x 2 − 2x − 2 = 0
b) x 4 − 4x 3 − 5x 2 + 2x + 2 = 0 c) x 4 + 4x 3 − 5x 2 + 2x − 2 = 0 d) x 4 + 4x 3 + 5x 2 − 2x + 2 = 0 7. If 3x 4 − 27x 3 + 36x 2 − 5 = 0, then s1 + s2 = a) 3 b) 21 c) -21 d) -3 8. If 1,1, α with roots of the equation x 3 − 6x 2 + x − 1 = 0 then α is a) -4 b) 4 c) -6 d) 6 9. If α, β, γ with roots of the equation 2x 3 − x 2 + x − 1 = 0, then α 2 + β 2 + γ 2 = a) 5/4 b) 3/4 c) −5/4 d) −3/4 10. If one root of x 3 + 2x 2 + 3x + k = 0 is the sum of the other two roots, then k = a) 0 b) 1 c) 2 d) 3 JEE MAIN LEVEL - 2 11. If α1, α2, α3, α4 are roots of the equation 7x 4 − 2x 3 + 4x + 11 = 0, then ∑α1α2α3 = a) −4 7 b) 4 7 c) 2/7 d) −2/7 12. If α, β, γ are the roots of the equation 2x 3 − 5x 2 + 3x − 1 = 0, then 1 αβ + 1 βγ + 1 γα = a) -4 b) -5 c) 5 d) -4 JEE MAIN LEVEL - 3 13. If α, β, γ are the roots of x 3 + ax + b = 0, then (α + β) −1 + (β + γ) −1 + (γ + α) −1 = a) a b b) −a b c) a 2 b2 d) −a 2 b2 14. If the product of two of the roots x 3 − kx 2 + 5x + 3 = 0 is -1 , then k = a) 2 b) 3 c) 4 d) 5 15. If the sum of two of the roots at x 3 + px 2 + qx + r = 0 is zero, then pq = a) −r b) r c) 2r d) −2r JEE MAIN LEVEL - 4 16. The sum of two roots of the equation x 3 − 3x 2 + kx + 48 = 0 is zero then K = a) 16 b) -16 c) 24 d) -24 17. If the product of two roots of x 3 + 3x 2 − 10x + k = 0 is 8 , then K = a) -10 b) 24 c) -24 d) 10 18. If α, β, γ are the roots of x 3 + px 2 + qx + r = 0, then (β + γ − 3α)(γ + α − 3β)(α + β − 3γ) = a) 3P 3 + 16pq + 64r b) 3P 3 − 16pq + 64r c) 3P 3 − 16pq d) 3P 3 + 16pq 19. If α, β, γ are the roots of the equation x 3 + px 2 + qx + r = 0 such that α + β = 0, then a) pr = q b) pr + q = 0 c) pq + r = 0 d) pq = r
20. If α, β, γ are the roots of x 3 + 2x 2 + 3x + 4 = 0, then ∑α 2β 2 = a) 7 b) -14 c) 14 d) -7 JEE MAIN LEVEL - 5 21. The polynomial of degree 4 having real co- efficients with three of its roots as 2 ± √3 and 1 ± 2i is a) x 4 − 6x 3 − 14x 2 + 22 + 5 = 0 b) x 4 − 6x 3 − 19x + 22 − 5 = 0 c) x 4 − 6x 3 − 19x 2 − 22 + 5 = 0 d) x 4 − 6x 3 + 14x 2 − 22x + 5 = 0 22. The sum of fifth power of the roots of the equation x 4 − 7x 2 + 4x − 3 = 0 is a) 99 b) -140 c) -99 d) 140 JEE ADVANCED LEVEL - 1 MULTI CORRECT CHOICE TYPE : 23. If one root of ax 4 + bx 3 + cx 2 + dx + e = 0, where a,b,c, d,e are rational numbers, is √2 + √3, then the other roots are a) √2 − √3 b) −√2 + √3 c) −√2 − √3 d) √2 + √3 COMPREHENSION TYPE Let α, β, γ are the roots of x 3 + px 2 + qx + r = 0, then 24. The value of (α + β)(β + γ)(γ + α) is a) r − pq b) p − rq c) q - r p d) pq − r 25. The value of ∑ 1 α is a) q r b) − p r c) − r q d) − q r 26. The value of ∑α 2 is a) q 2 − 2p b) p 2 − 2q c) 1 − p 2q d) 1 − pq 2 MATRIX MATCH TYPE: 27. If α, β, γ are the roots of the equation f(x) = x 3 − 10x 2 + 7x + 8 = 0, then Column-I Column-II a) α + β + γ p) − 43 4 b) α 2 + β 2 + γ 2 q) − 7 8 c) 1 α + 1 β + 1 γ r) 86 d) α βγ + β γα + γ αβ s) 0 t) 10 INTEGER ANSWER TYPE 28. If x 4 − 8x 3 + 23x 2 − 28x + 12 = 0, then s3 = 20 + x then x is JEE ADVANCED LEVEL-2&3 MULTIPLE CORRECT CHOICE TYPE 29. If α, β, γ are the roots of x 3 + x 2 + x + 1 = 0, then α 3 + β 3 + γ 3 = a) (−a) 2019 b) -1 c) cos 180∘ d) sin 270∘ COMPREHENSION TYPE If x 4 − 16x 3 + 86x 2 − 176x + 105 = 0, then an- swer the following questions. 30. S1 + S2 = a) 108 b) 106 c) 104 d) 102 31. S1 + S2 − S3 = a) -70 b) -72
c) -74 d) -76 32. S1 + S2 + S3 + S4 = a) 300 b) 383 c) 283 d) 183 INTEGER ANSWER TYPE 33. If α, β, 1 are roots of x 3 − 2x 2 − 5x + 6 = 0, then α + β is 5 − k, then the value of k = JEE ADVANCED LEVEL-485 MULTIPLE CORRECT CHOICE TYPE 34. If α is real root and β, γ are the complex roots of the equation x 3 + 3x 2 + 3x + 28 = 0, then 2α + 2β + 3γ = a) -5 b) 5cos 180∘ c) 5sin 270∘ d) 5log 5 1 5 MATRIX MATCH TYPE: 35. If α, β, γ are the roots of the equation x 3 − px 2 − qx − r = 0, then match the following Column-I Column-II a) 1 α + 1 β + 1 γ = p) p 2−2q r 2 b) 1 αβ + 1 βγ + 1 γα = q) q 2−2pr r 2 c) 1 α2 + 1 β2 + 1 γ2 = r) p r d) 1 α2β2 + 1 β2γ2 + 1 γ2α2 = s) − q r WORKSHEET 02 CUQ 1. The equation whose roots are reciprocals of the roots of 6x 6 + 25x 5 + 316x 4 − 31x 2 + 25x − 6 = 0 is a) x 3 − 6x 2 + 7x + 2 = 0 b) 4x 3 − 2x 3 + 6x 2 − 3x − 1 = 0 c) 6x 6 − 25x 5 + 31x 4 − 31x 2 + 25x − 6 = 0 d) x 4 + 3x 3 + 4x 2 − 28x − 16 = 0 2. If the roots of ax 3 + bx 2 + cx + d = 0 are in A.P., then the roots of dx 3 + cx 2 + bx + a = 0 are in a) A.P b) G.P c) H.P d) none 3. If the roots of ax 3 + bx 2 + cx + d = 0 are in H.P, then the roots of dx 3 − cx 2 + bx − a = 0 are in a) A.P b) G.P c) H.P d) none 4. If the roots of ax 3 + bx 2 + cx + d = 0 are in G.P, then the roots of dx 3 − cx 2 + bx − a = 0 are in a) A.P b) G.P c) H.P d) none 5. The equation whose roots are diminish by 1 than those of x 3 + 2x 2 + 3x + 4 = 0 a) x 3 + 5x 2 + 10x + 10 = 0 b) x 5 + 8x 2 + 20x + 16 = 0 c) x 4 − 8x 2 + 16x − 9 = 0 d) x 3 + x 2 − x + 1 = 0 6. If the roots of 2x 3 − 3x 2 + kx + 6 = 0 are in A.P, then K = a) 3 b) -5 c) 7 d) -11 7. If the roots of 4x 3 − 12x 2 + 11x + k = 0 are in A.P, then K = a) -3 b) 1 c) 2 d) 3 8. If α, β, γ are the roots of x 3 − 10x 2 + 6x − 8 = 0, then α 2 + β 2 + γ 2 = a) -88 b) 88 c) -7 d) 1 JEE MAIN LEVEL - 1 SINGLE CORRECT CHOICE TYPE 1. If α, β, γ are the roots of 2x 3 − 5x 2 − 7x + 8 = 0, then the equation whose roots are