Complex Numbers PYQs Set 12 (PDF)

Set 12 of 14 · Questions 331-360 of 417

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Complex Numbers · Set 12 (Q331-360)
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331. $Let S_{1}, S_{2} and S_{3} be three sets defined as S_{1} = {z \in C : \| z - 1 \| \leq 2} S_{2} = {z \in C : Re ((1 - i) z) \geq 1} S_{3} = {z \in C : Im (z) \leq 1} Then the set S_{1} \cap S_{2} \cap S_{3}$ (1) $is a singleton$ (2) $has exactly two elements$ (3) $has infinitely many elements$ (4) $has exactly three elements$ 332. $Let S = z : 3 ⩽ \| 2z-3 (1+i) \| ⩽ 7 be a set of complex numbers. Then min_{z \in S} ((z + \frac{1}{2} (5 + 3 i))) is equal to :$ (1) $2$ (2) $\frac{5}{2}$ (3) $\frac{3}{2}$ (4) $\frac{1}{2}$ 333. $Let S = {z = x + iy : \|z -1 + i\| > \|z\|, \|z\| < 2, \|z + i\| = \|z -1\|}. Then the set of all values of x, for which w = 2x + iy \in S for some y \in R, is$ (1) $(- 2, \frac{1}{2 2})$ (2) $(- \frac{1}{2}, \frac{1}{4})$ (3) $(2, \frac{1}{2})$ (4) $(- \frac{1}{2}, \frac{1}{2 2})$ 334. $If S = z \in C : \| z-i \| = \| z+i \| = \| z-1 \|, then, n (S) is:$ (1) $1$ (2) $0$ (3) $3$ (4) $2$ 335. $Let A = {z \in C : 1 \leq \| z - (1 + i) \| \leq 2} and B = z \in A : \| z- (1-i) \| =1 . Then, B :$ (1) $is an empty set$ (2) $contains exactly two elements$ (3) $contains exactly three elements$ (4) $is an infinite set$ 336. $For a non-zero complex number z, let arg (z) denote the principal argument with - π < arg (z) \leq π . Then, which of the following statement(s) is (are) FALSE? (JEE Advanced 2018)$ (1) $arg(-1-i)=π4, where i = - 1$ (2) $The function f : R \to (- π, π], defined by f (t) = arg (- 1 + it) for all t \in R, is continuous at all points of R, where i = - 1$ (3) $For any two non-zero complex numbers z1 and z2, argz1z2-argz1+argz2 is an integer multiple of 2 π$ (4) $For any three given distinct complex numbers z_{1}, z_{2} and z3, the locus of the point z satisfying the condition argz-z1z2-z3z-z3z2-z1=π, lies on a straight line$ 337. $A man walks a distance of 3 units from the origin towards the north-east (N 45° E) direction. From there, he walks a distance of 4 units towards the north-west (N 45° W) direction to reach a point P . Then the position of P in the Argand plane is$ (1) $3 e^{iπ /4} + 4 i$ (2) $(3 - 4 i) e^{iπ /4}$ (3) $(4 + 3 i) e^{iπ /4}$ (4) $(3 + 4 i) e^{iπ /4}$ 338. $The area (in sq. units) of the region S = {z \in C : ∣ z - 1∣ \leq 2; (z + z ˉ) + i (z - z ˉ) \leq 2, Im (z) \geq 0} is$ (1) $\frac{7 π}{3}$ (2) $\frac{7 π}{4}$ (3) $\frac{17 π}{8}$ (4) $\frac{3 π}{2}$ 339. $Let ∣ z_{1} - 8 - 2 i ∣ \leq 1 and ∣ z_{2} - 2 + 6 i ∣ \leq 2, z_{1}, z_{2} \in C . Then the minimum value of ∣ z_{1} - z_{2} ∣ is:$ (1) $13$ (2) $10$ (3) $3$ (4) $7$ 340. $Let a circle C in complex plane pass through the points z_{1} = 3 + 4i, z2 = 4 + 3i and z3 = 5i. If z (\neq = z_{1}) is a point on C such that the line through z and z_{l} is perpendicular to the line through z2 and z3, then arg(z) is equal to:$ (1) $tan^{- 1} (\frac{2}{5}) - π$ (2) $tan^{- 1} (\frac{24}{7}) - π$ (3) $tan^{- 1} (3) - π$ (4) $tan^{- 1} (\frac{3}{4}) - π$ 341. $Let A = {z \in C : \frac{z + 1}{z - 1} < 1} and B = {z \in C : ar g (\frac{z - 1}{z + 1}) = \frac{2 π}{3}} . Then A \cap B is:$ (1) $a portion of a circle centered at (0, - \frac{1}{3}) that lies in the second and third quadrants only$ (2) $a portion of a circle centered at (0, - \frac{1}{3}) that lies in the second quadrant only$ (3) $an empty set$ (4) $a portion of a circle of radius \frac{2}{3} that lies in the third quadrant only$ 342. $Let the curve z (1 + i) + z ˉ (1 - i) = 4, z \in C, divide the region ∣ z - 3∣ \leq 1 into two parts of areas α and β . Then ∣ α - β ∣ equals:$ (1) $1 + \frac{π}{2}$ (2) $1 + \frac{π}{3}$ (3) $1 + \frac{π}{6}$ (4) $1 + \frac{π}{4}$ 343. $Let A = z \in C : \| z-2 \| ⩽ 4 and B = z \in C : \| z-2 \| + \| z+2 \| =5 . Then the max ((z_{1} - z_{2}) : z_{1} \in A) and (z_{2} \in B) is :$ (1) $\frac{17}{2}$ (2) $8$ (3) $9$ (4) $\frac{15}{2}$ 344. $The equation arg (\frac{z - 1}{z + 1}) = \frac{π}{4} represents a circle with:$ (1) $center at (0, -1) and radius 2$ (2) $center at (0,1) and radius 2$ (3) $center at (0,0) and radius 2$ (4) $centre at (0,1) and radius 2$ 345. $If z = x + iy represents a point P in the argànd plane, then the area of the region represented by the inequality 2 < \| z - (1 + i) \| < 3 is$ (1) $49 π$ (2) $36 π$ (3) $25 π$ (4) $5 π$ 346. $If S = (z \in C : \frac{z - i}{z + 2 i} \in R), then:$ (1) $S contains exactly two elements$ (2) $S contains only one element$ (3) $S is a circle in the complex plane$ (4) $S is a straight line in the complex plane$ 347. $The area of the triangle with vertices A (z), B (iz) and C (z + iz) is:$ (1) $1$ (2) $\frac{1}{2} \| z \|^{2}$ (3) $\frac{1}{2}$ (4) $\frac{1}{2} \| z + iz \|^{2}$ 348. $Let the complex numbers α and (\frac{1}{α}) lie on circles (x - x_{0})^{2} + (y - y_{0})^{2} = r^{2} and (x - x_{0})^{2} + (y - y_{0})^{2} = 4 r^{2} respectively. If z_{0} = x_{0} + i y_{0} satisfies the equation 2 (z_{0})^{2} = r^{2} + 2, then \| α \| =$ (1) $\frac{1}{2}$ (2) $\frac{1}{2}$ (3) $\frac{1}{7}$ (4) $\frac{1}{3}$ 349. $Let S be the set of all complex numbers z satisfying ∣ z - 2 + i ∣ \geq 5 . If the complex number z_{0} is such that \frac{1}{∣ z _{0} - 1∣} is the maximum of the set {\frac{1}{∣ z - 1∣} : z \in S}, then the principal argument of \frac{4 - z _{0} - z ˉ _{0}}{z _{0} - z ˉ _{0} + 2 i} is (JEE Advanced 2019)$ (1) $\frac{π}{4}$ (2) $\frac{3 π}{4}$ (3) $- \frac{π}{2}$ (4) $\frac{π}{2}$ 350. $If z be a complex number satisfying \| Re (z) \| + \| Im (z) \| = 4 then \| z \| cannot be:$ (1) $10$ (2) $8$ (3) $7$ (4) $\frac{17}{2}$ 351. $Let C be the set of all complex numbers. Let S_{1} = {z \in C : ∣ z - 3 - 2 i ∣^{2} = 8}, S_{2} = {z \in C : Re (z) \geq 5} and S_{3} = {z \in C : ∣ z - z ˉ ∣ \geq 8} . Then the number of elements in S_{1} \cap S_{2} \cap S_{3} is equal to$ (1) $1$ (2) $0$ (3) $2$ (4) $Infinite$ 352. $Let C be the set of all complex numbers. Let S_{1} = {z \in C : ∣ z - 2∣ \leq 1} and S_{2} = {z \in C : z (1 + i) + z ˉ (1 - i) \geq 4} . Then, the maximum value of z - \frac{5}{2}^{2} for z \in S_{1} \cap S_{2} is equal to:$ (1) $\frac{3 + 2 2}{4}$ (2) $\frac{5 + 2 2}{2}$ (3) $\frac{3 + 2 2}{2}$ (4) $\frac{5 + 2 2}{4}$ 353. $Let S_{1} = {z \in C : \| z \| \leq 5}, S_{2} = (z \in C : Im (\frac{z + 1 - 3 i}{1 - 3 i}) \geq 0) and S_{3} = {z \in C : Re (z) \geq 0} . Then the area of the region S_{1} \cap S_{2} \cap S_{3} is :$ (1) $\frac{125 π}{12}$ (2) $\frac{125 π}{4}$ (3) $\frac{125 π}{24}$ (4) $\frac{125 π}{6}$ 354. $Let \frac{z ˉ - i}{2 z ˉ + i} = \frac{1}{3}, z \in C, be the equation of a circle with center at C . If the area of the triangle, whose vertices are at the points (0, 0), C and (α, 0) is 11 square units, then α^{2} equals:$ (1) $50$ (2) $100$ (3) $\frac{81}{25}$ (4) $\frac{121}{25}$ 355. $Let z be a complex number such that \| z \| = 1 . If \frac{2 + k ^{2} z}{k + z} = kz, k \in R, then the maximum distance of k + i k^{2} from the circle \| z - (1 + 2 i) \| = 1 is:$ (1) $5 + 1$ (2) $3$ (3) $3 + 1$ (4) $2$ 356. $The number of elements in the set {z = a + ib C : a, b Z and 1< \|z-3 + 2i\| < 4} is _______ .$ 357. $Let z and w be two complex numbers such that w = z z ˉ - 2 z + 2, \frac{z + i}{z - 3 i} = 1 and Re (w) has minimum value. Then, the minimum value of n \in N for which w^{n} is real, is equal to$ 358. $Let z1 and z2 be two complex numbers such that arg (z_{1} - z_{2}) = \frac{π}{4} and z_{1}, z_{2} satisfy the equation \|z - 3\| = Re(z). Then the imaginary part of z1 + z2 is equal to.$ 359. $Let z ˉ denote the complex conjugate of a complex number z and let i = - 1 . In the set of complex numbers, the number of distinct roots of the equation z ˉ - z^{2} = i (z ˉ + z^{2}) is [2022 Adv.]$ 360. $Let ω be the complex number cos \frac{2 π}{3} + i sin \frac{2 π}{3} . Then the number of distinct complex numbers z satisfying z + 1 ω ω^{2} ω z + ω^{2} 1 ω^{2} 1 z + ω = 0 is equal to$