Complex Numbers PYQs Set 7 (PDF)

Set 7 of 14 · Questions 181-210 of 417

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Complex Numbers · Set 7 (Q181-210)
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181. $The sum of the products of the non-conjugate roots of i^{1/4} taken two at a time is$ (1) $2$ (2) $0$ (3) $-1$ (4) $-2$ 182. $If (3 + i)^{100} = 2^{99} (p + iq), then p and q are roots of the equation:$ (1) $x^{2} - (3 - 1) x - 3 = 0$ (2) $x^{2} + (3 + 1) x + 3 = 0$ (3) $x^{2} + (3 - 1) x - 3 = 0$ (4) $x^{2} - (3 + 1) x + 3 = 0$ 183. $Let z = x + i y be a complex number where x and y are integers. Then the area of the rectangle whose vertices are the roots of the equation z z ˉ^{3} + z ˉ z^{3} = 350 is$ (1) $48$ (2) $32$ (3) $40$ (4) $80$ 184. $The radius of the circle represented by (1 + i) (1 + 3 i) (1 + 7 i) = x + iy is (i = - 1) .$ (1) $1000$ (2) $1010$ (3) $10000$ (4) $100$ 185. $If α is a non-real root of x^{7} = 1, then α (1 + α) (1 + α^{2} + α^{4}) =$ (1) $1$ (2) $2$ (3) $-1$ (4) $-2$ 186. $Let z \in C with Im (z) = 10 and its satisfies \frac{2 z - n}{2 z + n} = 2 i - 1, i = - 1 for some natural number n, then$ (1) $n = 20 and Re (z) = - 10$ (2) $n = 40 and Re (z) = - 10$ (3) $n = 40 and Re (z) = 10$ (4) $n = 20 and Re (z) = 10$ 187. $If z_{1} = 2 - 3 i and the roots of the equation z^{3} + b z^{2} + cz + d = 0 are i, z_{1} and z_{1}, then b + c + d =$ (1) $13$ (2) $7$ (3) $9 - 10 i$ (4) $10 - 10 i$ 188. $If ω is a complex cube root of unity, then \sum_{r = 1}^{9} r (r + 1 - ω) (r + 1 - ω^{2}) =$ (1) $5025$ (2) $4020$ (3) $2016$ (4) $3015$ 189. $If (x + iy) = (\frac{1 + i}{1 - i})^{3} - (\frac{1 - i}{1 + i})^{3}, then the true statement among the following is$ (1) $x < y$ (2) $x > y$ (3) $x \neq = 0$ (4) $x = y$ 190. $If z = x + iy, then the equation \| z + 1 \| = \| z - 1 \| represents$ (1) $a parabola$ (2) $x -axis$ (3) $y -axis$ (4) $a circle$ 191. $If Z_{1} and Z_{2} are complex numbers such that ∣ Z_{1} + Z_{2} ∣ = ∣ Z_{1} ∣ + ∣ Z_{2} ∣ then the difference in the amplitudes of Z_{1} and Z_{2} is$ (1) $\frac{π}{4}$ (2) $\frac{π}{3}$ (3) $\frac{π}{2}$ (4) $0$ 192. $If \frac{z - i}{z + 2 i} = 1, ∣ z ∣ = \frac{5}{2} then value of ∣ z + 3 i ∣ is:$ (1) $\frac{7}{2}$ (2) $10$ (3) $5$ (4) $3$ 193. $If w = \frac{z}{z - \frac{1}{3} i} and \|w\| = 1, then z lies on$ (1) $a parabola$ (2) $a straight line$ (3) $a circle$ (4) $an ellipse.$ 194. $If 1, ω, ω^{2} are the cube roots of unity then (1 + ω) (1 + ω^{2}) (1 + ω^{4}) (1 + ω^{8}) is equal to$ (1) $1$ (2) $0$ (3) $ω^{2}$ (4) $ω$ 195. $If x is real, then the interval in which no value of the expression \frac{2 ( x ^{2} + 2 x - 11 )}{2 x - 5} lies, is$ (1) $(2, 5)$ (2) $(3, 6)$ (3) $(3, 4)$ (4) $(6, 8)$ 196. $A (z_{1}) and B (z_{2}) are two points in the argand plane. Then, the locus of the complex number z satisfying arg (\frac{z - z _{1}}{z - z _{2}}) = 0 or π, is$ (1) $the circle with AB as a diameter$ (2) $the ellipse with A, B as extremities of the major axis$ (3) $the perpendicular bisector of AB$ (4) $the straight line passing through the points A and B$ 197. $The smallest positive integral value of ' n' such that (\frac{1 + sin \frac{π}{8} + icos \frac{π}{8}}{1 + sin \frac{π}{8} - icos \frac{π}{8}})^{n} is purely imaginary is,$ (1) $2$ (2) $8$ (3) $4$ (4) $3$ 198. $In Argand's plane, the point corresponding to \frac{( 1 - i 3 ) ( 1 + i )}{( 3 + i )} lies in$ (1) $quadrant I$ (2) $quadrant II$ (3) $quadrant III$ (4) $quadrant IV$ 199. $Let O be the origin and A be the point z_{1} = 1 + 2i. If B is the point z_{2}, Re(z_{2}) < 0, such that OAB is a right angled isosceles triangle with OB as hypotenuse, then which of the following is NOT true?$ (1) $arg z_{2} = π - tan^{- 1} 3$ (2) $arg (z_{1} - 2 z_{2}) = - tan^{- 1} \frac{4}{3}$ (3) $(z_{2}) = 10$ (4) $(2 z_{1} - z_{2}) = 5$ 200. $The locus of the variable point z = x + iy whose amplitude is always equal to θ, is$ (1) $x^{2} + y^{2} = tan^{2} θ$ (2) $y = xtan θ$ (3) $\frac{x ^{2}}{sin ^{2} θ} + \frac{y ^{2}}{cos ^{2} θ} = 1$ (4) $\frac{x ^{2}}{sin ^{2} θ} - \frac{y ^{2}}{cos ^{2} θ} = 1$ 201. $If α, β \in C are the distinct roots, of the equation x^{2} - x + 1 = 0, then α^{101} + β^{107} is equal to:$ (1) $1$ (2) $2$ (3) $-1$ (4) $0$ 202. $Let a, b \in R and a^{2} + b^{2} \neq = 0 . Suppose S = (z \in C : z = \frac{1}{a + ibt}, t \in R, t \neq = 0), where i = - 1 . If z = x + iy and z \in S, then (x, y) lies on (JEE Advanced 2016)$ (1) $the circle with radius \frac{1}{2 a} and center (\frac{1}{2 a}, 0) for a > 0, b \neq = 0$ (2) $the circle with radius - \frac{1}{2 a} and center (- \frac{1}{2 a}, 0) for a<0,b\neq0$ (3) $the x-axis for a \neq = 0, b = 0$ (4) $the y -axis for a = 0, b \neq = 0$ 203. $Let a complex number z, \| z \| \neq = 1,satisfy log_{\frac{1}{2}} (\frac{\| z \| + 11}{( \| z \| - 1 ) ^{2}}) \leq 2 Then, the largest value of \| z \| is equal to$ (1) $8$ (2) $7$ (3) $6$ (4) $5$ 204. $Area of the triangle formed by the complex numbers z, iz and z + iz in the Argand diagram as vertices is$ (1) $\frac{1}{2} \cdot ∣ z ∣^{2}$ (2) $\frac{1}{2} \cdot z^{2}$ (3) $z^{2}$ (4) $\| z \|^{2}$ 205. $If z = 2 + 3 i, then (z ˉ)^{5} + (z)^{5} is equal to:$ (1) $244$ (2) $224$ (3) $245$ (4) $265$ 206. $Let z = \frac{- 1 + 3 i}{2}, where i = - 1 and r, s \in {1,2,3}. Let P = ((- z)^{r} z^{2 s} z^{2 s} z^{r}) and I be the identity matrix of order 2. Then the total number of ordered pairs (r, s) for which p^{2} = - I is$ 207. $Let z_{1} and z_{2} be n^{th} roots of unity which are ends of a line segment that subtend a right angle at the origin. Then n must be of the form 4 k + r where r < 4 and r >= 0 . Find r$ 208. $Given that the equation z^{2} + (p + iq) z + r + is = 0, where p, q, r, s are real and non-zero has a real root, then pqs = s^{n} + q^{m} r . Find n + m .$ 209. $The value of the expression 1. (2 - ω) (2 - ω^{2}) + 2 \cdot (3 - ω) (3 - ω^{2}) + \dots \dots . \dots + (n - 1) \cdot (n - ω) (n - ω^{2}) where ω is an imaginary cube root of unity, is \frac{1}{A} (n - 1) n (n^{2} + Bn + C) find A + B + C$ 210. $Let A = (\frac{1967 + 1686 isinθ}{7 - 3 icosθ} : θ \in R) . If A contains exactly one positive integer n, then find the value of n . [2023 Adv.]$