Complex Numbers and Quadratic Equations

Let z = (1 + i)(1 + 2i)(1 + 3i) (1 + ni), where i = -1. If |z|2 = 44200, then n is equal to .

Let = -1 + i 32 and = -1 - i 32, i = -1. If (7 - 7 + 9 )20 + (9 + 7 - 7 )20 + (-7 + 9 + 7 )20 + (14 + 7 + 7 )20 = m10, then m is

For t > -1, let t and t be the roots of the equation ((t + 2)1/7 - 1 )x2 + ((t + 2)1/6 - 1 )x + ((t + 2)1/21 - 1 ) = 0 If t -1+ t = a and t -1+ t = b, then 72(a + b)2 is equal to .

Let A = \z C : |z - 2 - i| = 3\, B = \z C : Re(z - iz) = 2\ and S = A B. Then z S |z|2 is equal to .

If is a root of the equation x2+x+1=0 and k=1n ( k+ 1 k )2=20, then n is equal to .

If the set of all a R - \1\, for which the roots of the equation (1 - a)x2 + 2(a - 3)x + 9 = 0 are positive is (- , - ] [ , ), then 2 + + is equal to .

Let integers a, b [-3,3] be such that a+b 0. Then the number of all possible ordered pairs (a, b), for which | z-az+b |=1 and vmatrixz+1& & 2\ &z+ 2&1\ 2&1&z+ vmatrix=1, z C, where and 2 are the roots of x2+x+1=0, is equal to .

If the equation a(b-c)x2+b(c-a)x+c(a-b)=0 has equal roots, where a+c=15 and b= 365, then a2+c2 is equal to .

Let , be the roots of the equation x2-ax-b=0 with Im( )<Im( ). Let Pn= n- n. If P3=-5 7 i, P4=-3 7 i, P5=11 7 i and P6=45 7 i, then | 4+ 4 | is equal to .

The sum of the square of the modulus of the elements in the set \z = a + ib : a, b Z, z C, |z - 1| 1, |z - 5| |z - 5i|\ is .

The number of distinct real roots of the equation |x + 1||x + 3| - 4|x + 2| + 5 = 0, is

Let x1, x2, x3, x4 be the solution of the equation 4x4 + 8x3 - 17x2 - 12x + 9 = 0 and (4 + x12 ) (4 + x22 ) (4 + x32 ) (4 + x42 ) = 12516m. Then the value of m is .

Let , be roots of x2 + 2x - 8 = 0. If Un = n + n, then U10 + 2 U92U8 is equal to .

The number of distinct real roots of the equation |x||x + 2| - 5|x + 1| - 1 = 0 is .

The number of real solutions of the equation x|x + 5| + 2|x + 7| - 2 = 0 is .

Let S = \ 2 2 : ( 4 + 4 )x2 + ( 2 )x + ( 6 + 6 ) = 0 has real roots\. If and be the smallest and largest elements of the set S, respectively, then 3 (( - 2)2 + ( - 1)2 ) equals .

Let P=\z C: |z+2-3i| 1\ and Q=\z C: z(1+i)+ z(1-i) -8\. Let in P , |z-3+2i| be maximum and minimum at z1 and z2 respectively. If |z1|2+2|z2|2= + 2, where , are integers, then + equals .

If denotes the number of solutions of |1-i|x=2x and = ( |z| (z) ), where z= 4(1+i)4 [ 1- i +i+ -i1+ i ], i= -1, then the distance of the point ( , ) from the line 4x-3y=7 is .

Let a, b, c be the lengths of three sides of a triangle satisfying the condition (a2+b2 )x2-2b(a+c)x+ (b2+c2 )=0. If the set of all possible values of x is the interval ( , ), then 12 ( 2+ 2 ) is equal to .

Let , N be roots of the equation x2-70x+ =0, where 2, 3 N. If assumes the minimum possible value, then ( -1+ -1)( +35)| - | is equal to :

The number of real solutions of the equation x (x2+3|x|+5|x-1|+6|x-2| )=0 is .

Let , be the roots of the equation x2-x+2=0 with Im( )>Im( ). Then 6+ 4+ 4-5 2 is equal to .

Let , be the roots of the equation x2- 6x+3=0 such that Im( )>Im( ). Let a, b be integers not divisible by 3 and n be a natural number such that 99 + 98=3n(a+ib), i= -1. Then n+a+b is equal to .

If satisfies the equation x2+x+1=0 and (1+ )7=A+B +C 2, A, B, C 0, then 5(3A-2B-C) is equal to .

Let the complex numbers and 1 lie on the circles |z-z0 |2=4 and |z-z0 |2=16 respectively, where z0=1+i. Then, the value of 100| |2 is .