Question 1

The number of values of z C , satisfying the equations |z-(4+8i)|= 10 and |z-(3+5i)|+|z-(5+11i)|=4 5 , is:

Accepted Answer

The first equation |z-(4+8i)|= 10 represents a circle with center C(4, 8) and radius r = 10 . The second equation |z-(3+5i)|+|z-(5+11i)|=4 5 represents an ellipse with foci at S_1(3, 5) and S_2(5, 11), and length of the major axis 2a = 4 5 . The distance between the foci is 2ae = (5-3)^2 + (11-5)^2 = 4+36 = 40 = 2 10 . The center of the ellipse is the midpoint of the line segment joining the foci, which is ( 3+5 2 , 5+11 2 ) = (4, 8). This is the same as the center of the circle. The semi-major

Question 2

Let S = \ z C : z^2 + 6 \,iz - 3 = 0\ . Then _ z S z^8 is equal to :

Accepted Answer

Given equation is z^2 + 6 iz - 3 = 0. Rearranging the terms, we get z^2 - 3 = - 6 iz. Squaring both sides, we obtain: (z^2 - 3)^2 = (- 6 iz)^2 z^4 - 6z^2 + 9 = -6z^2 z^4 + 9 = 0 z^4 = -9 Squaring again, we get: (z^4)^2 = (-9)^2 z^8 = 81 Since the given equation is a quadratic in z, it has two roots, say z_1 and z_2. The discriminant of the quadratic equation is = ( 6 i)^2 - 4(1)(-3) = -6 + 12 = 6 0, which means the roots are distinct. Thus, the set S contains two distinct elements z_1 and z_2. F

Question 3

Let the set of all values of k R such that the equation z( z + 2 + i) + k(2 + 3i) = 0, z C , has at least one solution, be the interval [ , ]. Then 9( + ) is equal to:

Accepted Answer

Let z = x + iy, then z = x - iy. Substituting z into the given equation: (x + iy)(x - iy + 2 + i) + k(2 + 3i) = 0 x^2 + y^2 + 2x + ix + 2iy - y + 2k + 3ki = 0 Separating the real and imaginary parts, we get: Real part: x^2 + y^2 + 2x - y + 2k = 0 Imaginary part: x + 2y + 3k = 0 x = -2y - 3k Substituting x into the real part equation: (-2y - 3k)^2 + y^2 + 2(-2y - 3k) - y + 2k = 0 4y^2 + 12ky + 9k^2 + y^2 - 4y - 6k - y + 2k = 0 5y^2 + (12k - 5)y + 9k^2 - 4k = 0 For the equation to have at least on

Question 4

Let z_1, z_2 C be the distinct solutions of the equation z^2 + 4z - (1 + 12i) = 0. Then |z_1|^2 + |z_2|^2 is equal to :

Accepted Answer

Given equation is z^2 + 4z - (1 + 12i) = 0 Let the roots be z_1 and z_2. Sum of roots: z_1 + z_2 = -4 Product of roots: z_1 z_2 = -(1 + 12i) We know that (z_1 - z_2)^2 = (z_1 + z_2)^2 - 4z_1 z_2 (z_1 - z_2)^2 = (-4)^2 - 4(-(1 + 12i)) = 16 + 4 + 48i = 20 + 48i Taking modulus on both sides: |z_1 - z_2|^2 = |20 + 48i| = 20^2 + 48^2 = 400 + 2304 = 2704 = 52 Using the parallelogram law for complex numbers: |z_1 + z_2|^2 + |z_1 - z_2|^2 = 2(|z_1|^2 + |z_2|^2) |-4|^2 + 52 = 2(|z_1|^2 + |z_2|^2) 16 + 52

Question 5

Let S=\ z C : z^2+4z+16=0\ . Then _ z S |z+ 3 i|^2 is equal to:

Accepted Answer

Given the equation z^2 + 4z + 16 = 0, we find its roots using the quadratic formula: z = -4 16 - 64 2 = -4 -48 2 = -2 2 3 i The set S contains two complex numbers: z_1 = -2 + 2 3 i and z_2 = -2 - 2 3 i. We need to evaluate the sum _ z S |z + 3 i|^2, which is |z_1 + 3 i|^2 + |z_2 + 3 i|^2. For z_1 = -2 + 2 3 i: |z_1 + 3 i|^2 = |-2 + 2 3 i + 3 i|^2 = |-2 + 3 3 i|^2 |-2 + 3 3 i|^2 = (-2)^2 + (3 3 )^2 = 4 + 27 = 31 For z_2 = -2 - 2 3 i: |z_2 + 3 i|^2 = |-2 - 2 3 i + 3 i|^2 = |-2 - 3 i|^2 |-2 - 3 i|^

Question 6

Let z be a complex number such that |z+2| = |z-2| and ( z+3 z-i ) = 4 . Then |z|^2 is equal to:

Accepted Answer

Given |z+2| = |z-2|, the point z lies on the perpendicular bisector of the line segment joining (-2, 0) and (2, 0). This means z lies on the imaginary axis. Let z = iy, where y R . We are given ( z+3 z-i ) = 4 . Substituting z = iy, we get: iy+3 iy-i = 3+iy i(y-1) = -i(3+iy) y-1 = y - 3i y-1 = y y-1 - i 3 y-1 For a complex number X + iY to have an argument of 4 , its real and imaginary parts must be equal and strictly positive. Therefore: y y-1 = -3 y-1 > 0 Since y 1, equating the numerators giv

Question 7

Let the circles C_1 : |z| = r and C_2 : |z - 3 - 4i| = 5, z C , be such that C_2 lies within C_1. If z_1 moves on C_1, z_2 moves on C_2 and |z_1 - z_2| = 2, then |z_1 - z_2| is equal to:

Accepted Answer

The center of circle C_1 is O_1(0,0) and its radius is r_1 = r. The center of circle C_2 is O_2(3,4) and its radius is r_2 = 5. The distance between the centers of the two circles is d = 3^2 + 4^2 = 5. Since C_2 lies entirely within C_1, the minimum distance between a point on C_1 and a point on C_2 is given by r_1 - (d + r_2). We are given that |z_1 - z_2| = 2. r - (5 + 5) = 2 r - 10 = 2 r = 12 The maximum distance between a point on C_1 and a point on C_2 is given by r_1 + d + r_2. |z_1 - z_2|

Question 8

Let x and y be real numbers such that 50 ( 2x 1+3i - y 1-2i ) = 31 + 17i, i = -1 . Then the value of 10(x - 3y) is :

Accepted Answer

Given the equation: 50 ( 2x 1+3i - y 1-2i ) = 31 + 17i Rationalizing the denominators inside the parentheses: 2x 1+3i = 2x(1-3i) (1+3i)(1-3i) = 2x(1-3i) 1^2 + 3^2 = 2x(1-3i) 10 = x(1-3i) 5 y 1-2i = y(1+2i) (1-2i)(1+2i) = y(1+2i) 1^2 + 2^2 = y(1+2i) 5 Substituting these back into the given equation: 50 ( x(1-3i) 5 - y(1+2i) 5 ) = 31 + 17i 10(x(1-3i) - y(1+2i)) = 31 + 17i 10(x - y) - 10i(3x + 2y) = 31 + 17i Equating the real and imaginary parts on both sides: 10(x - y) = 31 10x - 10y = 31 -10(3x +

Question 9

Let A=\ z C :|z-2| 4\ and B=\ z C :|z-2|+|z+2|=5\ . Then the max \ |z_ 1 -z_ 2 |: z_ 1 ~A . and .z_ 2 ~B \ is :

Accepted Answer

|z - 2| 4 (x - 2)^2 + y^2 16 |z - 2| + |z + 2| = 5 x^2 a^2 + y^2 b^2 = 1 4x^2 25 + 4y^2 9 = 1 Maximum value of |z_1 - z_2| = 6 + 5 2 = 17 2

Question 10

Let z be a complex number such that |z-6|=5 and |z+2-6 i|=5. Then the value of z^ 3 +3 z^ 2 -15 z+141 is equal to

Accepted Answer

From |z - 6| = 5, point z lies on circle centered at (6, 0) with radius 5. From |z + 2 - 6i| = 5, point lies on circle centered at (-2, 6) with radius 5. Let z = x + iy. First condition gives: x^2 + y^2 = 12x - 11. Second gives: x^2 + y^2 = -4x + 12y - 15. Setting equal: 16x - 12y + 4 = 0, so x = 3y - 1 4 . Substituting into (x-6)^2 + y^2 = 25: (3y - 25)^2 + 16y^2 = 400, which simplifies to 25y^2 - 150y + 225 = 0, giving y = 3 and x = 2. Thus z = 2 + 3i, so z^2 = -5 + 12i and z^3 = -46 + 9i. z^3

Question 11

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

Accepted Answer

z = (1 + i)(1 + 2i)(1 + 3i) (1 + ni) where i = -1 |z|^2 = _ k=1 ^ n |1 + ki|^2 = _ k=1 ^ n (1 + k^2) Computing: (1+1)(1+4)(1+9)(1+16)(1+25) = 2 5 10 17 26 = 10 10 17 26 = 100 442 = 44200 Therefore n = 5

Question 12

If z= 3 2 + i 2 , i= -1 , then (z^ 201 -i )^ 8 is equal to

Accepted Answer

z = 3 2 + i 2 = e^ i /6 . z^ 201 = e^ i 201 /6 = e^ i 33.5 = e^ i(33 + /2) . Since e^ i 33 = e^ i = -1, we get z^ 201 = -1 e^ i /2 = -i. z^ 201 - i = -i - i = -2i. (z^ 201 - i)^8 = (-2i)^8 = 2^8 i^8 = 256 1 = 256.

Question 13

Let S=\ z: 3 |2 z-3(1+i)| 7\ be a set of complex numbers. Then _ z S | (z+ 1 2 (5+3 i) ) | is equal to :

Accepted Answer

3 2 | z - 3 2 (1+i) | 7 2 _ z S | z - ( -5 2 - 3 2 i ) | = PB PB = PC - 7 2 5 - 7 2 3 2

Question 14

Let S= \ z C : 4 z^ 2 + z =0 \ . Then _ z S |z|^ 2 is equal to:

Accepted Answer

Let z=x+iy. From 4z^2+ z =0, separating real and imaginary parts: Real: 4x^2-4y^2+x=0, Imaginary: y(8x-1)=0. Case 1: y=0 x(4x+1)=0 z=0 or z=- 1 4 . Case 2: x= 1 8 1 16 -4y^2+ 1 8 =0 y^2= 3 64 . So |z|^2= 1 64 + 3 64 = 1 16 . |z|^2=0+ 1 16 + 1 16 + 1 16 = 3 16

Question 15

Let = -1+i 3 2 and = -1-i 3 2 , i= -1 . If (7-7 +9 )^ 20 +(9+7 -7 )^ 20 +(-7+9 +7 )^ 20 +(14+7 +7 )^ 20 =m^ 10 , then m is \_\_\_\_\_

Accepted Answer

= , = ^2 (cube roots of unity). Let z_1 = 7-7 +9 ^2, z_2 = 9+7 -7 ^2, z_3 = -7+9 +7 ^2. Calculating: z_1 = 6-8i 3 , z_2 = 9+7i 3 , z_3 = -15+i 3 . All have |z_k| = 228 . Verify: z_2 = z_1 , z_3 = z_1 ^2. z_1^ 20 +z_2^ 20 +z_3^ 20 = z_1^ 20 (1+ ^ 20 + ^ 40 ) = z_1^ 20 (1+ ^2+ ) = 0. Fourth term: 14+7 +7 ^2 = 14+7(-1) = 7. So 7^ 20 = 49^ 10 . Total = 0 + 49^ 10 = m^ 10 m = 49.

Question 16

Let z be the complex number satisfying |z-5| 3 and having maximum positive principal argument. Then 34 | 5 z-12 5 i z+16 |^ 2 is equal to :

Accepted Answer

The region |z - 5| 3 is a disk centered at (5, 0) with radius 3. Maximum positive principal argument from origin occurs at the tangent point from origin to the circle. = 3 5 , = 4 5 . The tangent point is the foot of perpendicular from (5,0) to the line 3x - 4y = 0, giving z = 16 5 + 12 5 i. 5z - 12 = 4 + 12i |5z - 12|^2 = 160 5iz + 16 = 4 + 16i |5iz + 16|^2 = 272 34 | 5z-12 5iz+16 |^2 = 34 160 272 = 34 10 17 = 20.

Question 17

If x^ 2 +x+1=0, then the value of (x+ 1 x )^ 4 + (x^ 2 + 1 x^ 2 )^ 4 + (x^ 3 + 1 x^ 3 )^ 4 + + (x^ 25 + 1 x^ 25 )^ 4 is:

Accepted Answer

Given x^2 + x + 1 = 0, so x = (primitive cube root of unity) with x^3 = 1. From the equation: x + 1 x = -1. Since x^3 = 1, value of x^n + x^ -n has period 3: n 0 3 : x^n + x^ -n = 2 n 1, 2 3 : x^n + x^ -n = -1 Fourth powers: (2)^4 = 16, (-1)^4 = 1. For n = 1 to 25: 8 multiples of 3 contribute 8 16 = 128, remaining 17 terms contribute 17 1 = 17. Total = 128 + 17 = 145.

Complex Number — Previous Year Questions & Solutions