Complex Numbers PYQs Set 11 (PDF)

Set 11 of 14 · Questions 301-330 of 417

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Complex Numbers · Set 11 (Q301-330)
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301. $If the point (x, y) satisfies the equation \frac{x + i ( x - 2 )}{3 + i} - i = \frac{2 y + i ( 1 - 3 y )}{i - 3}, then x + y =$ (1) $4$ (2) $2$ (3) $0$ (4) $-2$ 302. $Let S = {z \in C : ∣ z - 1∣ = 1 and (2 - 1) (z + z ˉ) - i (z - z ˉ) = 22} . Let z_{1}, z_{2} \in S be such that ∣ z_{1} ∣ = max_{z \in S} ∣ z ∣ and ∣ z_{2} ∣ = min_{z \in S} ∣ z ∣ . Then (2 z_{1} - z_{2})^{2} equals:$ (1) $1$ (2) $4$ (3) $3$ (4) $2$ 303. $\sum_{k = 1}^{6} (sin \frac{2 πk}{7} - icos \frac{2 πk}{7}) =$ (1) $-1$ (2) $0$ (3) $i$ (4) $- i$ 304. $The area of the polygon, whose vertices are the non-real roots of the equation z ˉ = i z^{2} :$ (1) $\frac{3 3}{4}$ (2) $\frac{3 3}{2}$ (3) $\frac{3}{2}$ (4) $\frac{3}{4}$ 305. $Let O be the origin, the point A be z_{1} = 3 + 22 i, the point B (z_{2}) be such that 3 ∣ z_{2} ∣ = ∣ z_{1} ∣ and ar g (z_{2}) = ar g (z_{1}) + \frac{π}{6} . Then$ (1) $area of triangle ABO is \frac{11}{3}$ (2) $ABO is an obtuse angled isosceles triangle$ (3) $area of triangle ABO is \frac{11}{4}$ (4) $ABO is a scalene triangle$ 306. $Let z_{1} and z_{2} be two distinct complex numbers and let z = (1 - t) z_{1} + t z_{2} for some real number t with 0 < t < 1 . If Arg (w) denotes the principal argument of a nonzero complex number w, then$ (1) $∣ z - z_{1} ∣ + ∣ z - z_{2} ∣ = ∣ z_{1} - z_{2} ∣$ (2) $Arg (z - z_{1}) = Arg (z - z_{2})$ (3) $z - z_{1} z_{2} - z_{1} z ˉ - z_{1} ˉ z_{2} ˉ - z_{1} ˉ = 0$ (4) $Arg (z - z_{1}) = Arg (z_{2} - z_{1})$ 307. $Let a complex number be w = 1 - 3 i . Let another complex number z be such that \| zw \| = 1 and arg (z) - arg (w) = \frac{π}{2} . Then the area of the triangle with vertices origin, z and w is equal to:$ (1) $4$ (2) $\frac{1}{2}$ (3) $\frac{1}{4}$ (4) $2$ 308. $If z is a complex number such that \frac{z - i}{z - 1} is purely imaginary, then the minimum value of ∣ z - (3 + 3 i) ∣ is:$ (1) $22 - 1$ (2) $32$ (3) $62$ (4) $22$ 309. $Let z_{1} and z_{2} be two complex numbers satisfying (z_{1}) = 9 and (z_{2} - 3 - 4 i) = 4 . Then the minimum value of (z_{1} - z_{2}) is:$ (1) $0$ (2) $1$ (3) $2$ (4) $2$ 310. $If the four complex numbers z, z, z - 2 Re (z) and z - 2 Re (z) represent the vertices of a square of side 4 units in the Argand plane, then \| z \| is equal to:$ (1) $42$ (2) $2$ (3) $22$ (4) $4$ 311. $If P is the point in the Argand diagram corresponding to the complex number 3 + i and if OPQ is an isosceles right angled triangle, right angled at ' O', then Q represents the complex number$ (1) $3 - i or 1 - i 3$ (2) $- 1 \pm i 3$ (3) $- 1 + i 3 or 1 - i 3$ (4) $1 \pm i 3$ 312. $Let w = \frac{3 + i}{2} and P = (w^{n} : n = 1, 2, 3, \dots) . Further H_{1} = (z \in C : Rez > \frac{1}{2}) and H_{2} = (z \in C : Rez < \frac{- 1}{2}), where C is the set of all complex numbers. If z_{1} \in P \cap H_{1}, z_{2} \in P \cap H_{2}, and O represents the origin, then ∠ z_{1} O z_{2} = (JEE Advanced 2013)$ (1) $π /2$ (2) $π /6$ (3) $2 π /3$ (4) $5 π /6$ 313. $If \|z + 4\| \leq 3, then the maximum value of \|z + 1\| is$ (1) $4$ (2) $10$ (3) $6$ (4) $0$ 314. $If z is a complex number such that ∣ z ∣ \geq 1, then the minimum value of z + \frac{1}{2} (3 + 4 i) is:$ (1) $\frac{5}{2}$ (2) $2$ (3) $3$ (4) $0$ 315. $Let z_{1} and z_{2} be two distinct complex numbers and let z = (1 - t) z_{1} + t z_{2} for some real number t with 0 < t < 1 . If arg (w) denotes the principal argument of a nonzero complex number w, then (IIT-JEE, 2010)$ (1) $∣ z - z_{1} ∣ + ∣ z - z_{2} ∣ = ∣ z_{1} - z_{2} ∣$ (2) $∣ z - z_{1} ∣ = ∣ z - z_{2} ∣$ (3) $z - z_{1} z_{2} - z_{1} z ˉ - z ˉ_{1} z ˉ_{2} - z ˉ_{1} = 0$ (4) $ar g (z - z_{1}) = ar g (z_{2} - z_{1})$ 316. $For z \in C if the minimum value of (∣ z - 32 ∣ + ∣ z - p 2 i ∣) is 52, then a value of p is_____________$ (1) $3$ (2) $\frac{7}{2}$ (3) $4$ (4) $\frac{9}{2}$ 317. $Let w = \frac{3 + i}{2} and P = (w_{n} : n = 1, 2, 3, \dots .) . Further H_{1} = (z \in C : Re z > \frac{1}{2}) and H_{2} = (z \in C : Re z < - \frac{1}{2}), , where C is the set of all complex numbers. If Z_{1} \in P \cap H_{1}, Z_{2} \in P \cap H_{2} and O represents the origin, then ∠ z_{1} O z_{2} =$ (1) $\frac{π}{2}$ (2) $\frac{π}{6}$ (3) $\frac{2 π}{3}$ (4) $\frac{5 π}{6}$ 318. $If z, z ˉ, - z, - z ˉ forms a rectangle of area 23 square units, then one such z is$ (1) $\frac{1}{2} + 3 i$ (2) $\frac{5 + 3 i}{4}$ (3) $\frac{3}{2} + \frac{3 i}{2}$ (4) $\frac{3 + 11 i}{2}$ 319. $The number of points of intersection of \|z-(4 + 3i)\| = 2 and \|z\| + \|z-4\| = 6, z \in C is :$ (1) $0$ (2) $1$ (3) $2$ (4) $3$ 320. $The number of complex numbers z, satisfying ∣ z ∣ = 1 and \frac{z}{z ˉ} + \frac{z ˉ}{z} = 1, is:$ (1) $4$ (2) $8$ (3) $10$ (4) $6$ 321. $If z a complex number such that ∣ z ∣ \geq 2, then the minimum value of z + \frac{1}{2}$ (1) $is strictly greater than 5/2$ (2) $is strictly greater than 3/2 but less than 5/2$ (3) $is equal to 5/2$ (4) $lie in the interval (1, 2)$ 322. $If at least one value of the complex number z = x + iy satisfy the condition \| z + 2 \| = a^{2} - 3 a + 2 and the inequality \| z + i 2 \| < a^{2} then minimum value of a is$ 323. $Let z be those complex number which satisfy ∣ z + 5∣ \leq 4 and z (1 + i) + z ˉ (1 - i) \geq - 10, i = - 1 . If the maximum value of ∣ z + 1 ∣^{2} is α + β 2, then the value of (α + β) is$ 324. $If z is any complex number satisfying \| z - 3 - 2 i \| \leq 2, then the minimum value of \| 2 z - 6 + 5 i \| is____. (IIT-JEE, 2011)$ 325. $If for the complex numbers z satisfying \|z - 2 - 2i\| \leq 1, the maximum value of \|3iz + 6\| is attained at a + ib, then a + b is equal to.$ 326. $If z is any complex number satisfying \|z - 3 - 2i\| \leq 2, then the minimum value of \|2z - 6 + 5i\| is$ 327. $The sum of the square of the modulus of the elements in the set {z = a + ib : a, b \in Z, z \in C, \| z - 1 \| \leq 1, \| z - 5 \| \leq \| z - 5 i \|} is_____________$ 328. $A point z moves in the complex plane such that ar g (\frac{z - 2}{z + 2}) = \frac{π}{4}, then the minimum value of ∣ z - 92 - 2 i ∣^{2} is equal to ____.$ 329. $Let P = {z \in C : ∣ z + 2 - 3 i ∣ \leq 1} and Q = {z \in C : z (1 + i) + z ˉ (1 - i) \leq - 8} . Let in P \cap Q, ∣ z - 3 + 2 i ∣ be maximum and minimum at z_{1} and z_{2} respectively. If ∣ z_{1} ∣^{2} + 2∣ z_{2} ∣^{2} = α + β 2, where α, β are integers, then α + β equals$ 330. $Let S = {z \in C : ∣ z - 2∣ \leq 1, z (1 + i) + z ˉ (1 - i) \leq 2} . Let ∣ z - 4 i ∣ attains minimum and maximum values, respectively, at z_{1} \in S and z_{2} \in S . If 5 (∣ z_{1} ∣^{2} + ∣ z_{2} ∣^{2}) = α + β 5, where α and β are integers, then the value of α + β is equal to_____.$