Title COMPLEX NUMBERS AND QUADRATIC EQUATIONS Q-9.pdf ✅

CLASS : XIth SUBJECT : MATHS DATE : DPP NO. : 9 1. If α1,α2 and β1,β2 are the roots of the equation ax 2 +bx + c = 0 and px 2 +qx + r = 0 respectively and system of equations α1y + α2z = 0 and β1y + β2z = 0 has a non-zero solution, then a) a 2 qc = p 2 br b)b 2 = pr = q 2ac c) c 2 = ar = r 2pb d)None of these 2. If 1, ω, ω 2 are the cube roots of unity, then (1 ― ω + ω 2 )(1 ― ω 2 + ω 4 )(1 ― ω 4 + ω 8 )(1 ― ω 8 + ω 16)... upto 2n factors is a) 2n b)2 2n c) 1 d) ― 2 2n 3. If α and β are different complex numbers with |β| = 1, then | β ― α 1 ― αβ | is a) 0 b)3/2 c) 1/2 d)1 4. In a right-angled triangle, the sides are a, b and c, with c as hypotenuse, and c ― b ≠ 1, c + b ≠ 1 . Then the value of (logc+b a + logc―b a)/(2logc+b a × logc―b a) will be a) 2 b) ―1 c) 1 2 d)1 5. The set of real values of x for which 10x 2 + 17x ― 34 x 2 + 2x ― 3 < 8, is a) ( ―5/2, 2) b) ( ― 3, ― 5/2) ∪ (1, 2) c) ( ― 3, 1)d) None of these 6. If ( 1 + cos φ + i sin φ 1 + cos φ ― i sin φ ) = u + iv, where u and v all real numbers, then u is a) ncos φ b)cos nφ c) cos ( nφ 2 ) d)sin ( nφ 2 ) 7. The number of real roots of the equation 2 x 4 +5 x 2 +3 = 0, is a) 4 b)1 c) 0 d)3 8. If α and β are the roots of x 2 ―2x + 4 = 0, then the value of α 6 + β 6 is a) 32 b)64 c) 128 d)256 9. If |z + 4| ≤ 3, then the greatest and the least value of |z +1| are a) 6, ― 6 b)6, 0 c) 7, 2 d)0, ― 1 Topic :- COMPLEX NUMBERS AND QUADRATIC EQUATIONS Solutions
10. If P,P′ represent the complex number z1 and its additive inverse respectively, then the equation of the circle with PP′ as a diameter is a) z z1 = z1 z b)zz = z1z1 = 0 c) zz1 + zz1 = 0 d)None of these 11. If x + 1 is a factor of x 4 + (p ― 3)x 3 ― (3p ― 5)x 2 + (2p ― 9)x + 6, then the value of p is a) ―4 b)0 c) 4 d)2 12. If A = {x:f(x) = 0} and B = {x:g(x) = 0}, then A ∩ B will be the set of roots of the equation a){f(x)} 2 + {g(x)} 2 = 0 b) f(x) g(x) c) g(x) f(x) d)None of these 13. If α and β are the roots of the equation x 2 +px + q = 0 and if the sum (α + β)x ― α 2 + β 2 2 .x 2 + α 3 + β 3 3 .x 3 ―... exists then it is equal to a)log(x 2 + px + q) b)log(x 2 ― px + q) c) log(1 + px + qx 2 ) d)log(1 ― px + qx 2 ) 14. Let z be a complex number satisfying | z ― 5i| ≤ 1 such that amp (z) is minimum. Then z is equal to a) 2 6 5 + 24i 5 b) 24 5 + 2 6i 5 c) 2 6 5 ― 24i 5 d)None of these 15. If α and β are the roots of x 2 +px + 1 = 0 and γ and δ are the roots of x 2 +qx + 1 = 0, then the value of (α ― γ)(β ― γ)(α + δ)(β + δ), is a) p 2 ― q 2 b)q 2 ― p 2 c) p 2 d)q 2 16. For two complex numbers z1, z2 the relation |z1 + z2 | = |z1 | +|z2| holds, if a) arg(z1) = arg(z2) b)arg(z1 ) + arg(z2 ) = π 2 c) z1z2 = 1 d)|z1 | = |z2| 17. If ω is a complex cube root of unity, then sin {(ω 10 + ω 23π ― π 4 )} is equal to a) 1 2 b) 1 2 c) 1 d) 3 2 18. If the equation x 3 ―3x + a = 0 has distinct roots between 0 and 1, then the value of a is a) 2 b)1/2 c) 3 d)None of these 19. If α,β are roots of the equation 375x 2 ―25x ― 2 = 0 and Sn = a n + β n , then lim n→∞ ∑ n r=1 Sr is equal to a) 7/116 b)1/12 c) 29/348 d)None of these 20. If y = tan x cot 3x,x ∈ R, then a) 1 3 < y < 1 b) 1 3 ≤ y ≤ 1 c) 1 3 ≤ y ≤ 3 d)None of these