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Complex Numbers

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  1. 151.For a natural number n, let _n=19^n-12^n.
  2. 152.Let a,b R be such that the equation ax^2-2bx+15=0 has a repeated root .
  3. 153.The sum of the cubes of all the roots of the equation x^4-3x^3-2x^2+3x+1=0 is ____ Answer:…
  4. 154.The sum of all the roots of the equation e^2x-11e^x-45e^-x+812=0 is _e p.
  5. 155.Let , be the roots of the equation x^2-4 x+5=0 and , be the roots of the equation x^2-(3√2…
  6. 156.The number of real solutions of the equation e^4x+4e^3x-58e^2x+4e^x+1=0 is _____ Answer: 2
  7. 157.Let f(x) be a quadratic polynomial such that f(-2)+f(3)=0.
  8. 158.Let be a root of the equation 1+x^2+x^4=0.
  9. 159.Let x,y>0.
  10. 160.Let p and q be two real numbers such that p+q=3 and p^4+q^4=369.
  11. 161.For n N, let S_n=z C: |z-3+2i|= n4 and T_n=z C: |z-2+3i|=1n.
  12. 162.For z C, if the minimum value of (|z-3√2|+|z-p√2,i|) is 5√2, then a value of p is
  13. 163.Let O be the origin and A be the point z_1=1+2i.
  14. 164.If z=x+iy satisfies |z|-2=0 and |z-i|-|z+5i|=0, then
  15. 165.Let the minimum value v_0 of v=|z|^2+|z-3|^2+|z-6i|^2, z C, be attained at z=z_0.
  16. 166.Let S=z C: z^2+ z=0.
  17. 167.Let S be the set of all (,), <,<2, for which the complex number 1-i1+2i is purely imaginar…
  18. 168.Let S_1=z_1 C: |z_1-3|=12 and S_2=z_2 C: |z_2-|z_2+1||=|z_2+|z_2-1||.
  19. 169.Let z=a+ib, b0, be a complex number satisfying z^2= z2^1-|z|.
  20. 170.If z=2+3i, then z^5+ z^5 is equal to:
  21. 171.If z0 is a complex number such that |z-1z|=2, then the maximum value of |z| is:
  22. 172.Let S=z=x+iy: |z-1+i||z|, |z|<2, |z+i|=|z-1|.
  23. 173.If the numbers appearing on two throws of a fair six-faced die are and , then the probabil…
  24. 174.Let a,b be two non-zero real numbers.
  25. 175.If for some p,q,r R, not all having the same sign, one of the roots of the equation (p^2+q…
  26. 176.The number of distinct real roots of the equation x^5(x^3-x^2-x+1)+x(3x^3-4x^2-2x+4)-1=0 i…
  27. 177.The minimum value of the sum of the squares of the roots of x^2+(3-a)x+1=2a is:
  28. 178.Let the abscissae of the two points P and Q on a circle be the roots of x^2-4x-6=0 and the…
  29. 179.If , are the roots of the equation (after simplifying the given logarithmic expression, wh…
  30. 180.The sum of all real values of x for which 3x^2-9x+17x^2+3x+10=5x^2-7x+193x^2+5x+12 is equa…
  31. 181.Let , be the roots of the equation x^2-√2x+√6=0, and let 1^2+1,1^2+1 be the roots of the e…
  32. 182.Let f(x)=ax^2+bx+c be such that f(1)=3, f(-2)= and f(3)=4.
  33. 183.Let , (>) be the roots of the quadratic equation x^2-x-4=0.
  34. 184.Let S=x[-6,3]--2,2: |x+3|-1|x|-20 and T=x Z: x^2-7|x|+90.
  35. 185.Let i=-1.
  36. 186.Let z be those complex numbers which satisfy |z+5|4 and z(1+i)+ z(1-i)-10, i=-1.
  37. 187.Let the lines (2-i)z=(2+i) z and (2+i)z+(i-2) z-4i=0 (here i^2=-1) be normal to a circle C…
  38. 188.Let and be two real numbers such that +=1 and =-1.
  39. 189.The number of solutions of the equation _4(x-1)=_2(x-3) is Answer: 1
  40. 190.Let and be the roots of x^2-6x-2=0.
  41. 191.If , R are such that 1-2i (here i^2=-1) is a root of z^2+ z+=0, then (-) is equal to
  42. 192.The integer 'k', for which the inequality x^2-2(3k-1)x+8k^2-7>0 is valid for every x R, is
  43. 193.The sum of the 162th power of the roots of the equation x^3-2x^2+2x-1=0 is Answer: 3
  44. 194.Let a,b,c be in an arithmetic progression.
  45. 195.The number of real roots of the equation (x+1)^2+|x-5|=274 is Answer: 2
  46. 196.If the least and largest real values of , for which the equation z+|z-1|+2i=0 (z C, i=-1) …
  47. 197.Let p and q be two positive numbers such that p+q=2 and p^4+q^4=272.
  48. 198.If the equation a|z|^2+ z+ z+d=0 represents a circle, where a,d are real constants, then w…
  49. 199.Let z_1,z_2 be the roots of the equation z^2+az+12=0 and z_1,z_2 form an equilateral trian…
  50. 200.Let a complex number be w=1-√3 i.