Complex Numbers PYQs Set 10 (PDF)

Set 10 of 14 · Questions 271-300 of 417

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Complex Numbers · Set 10 (Q271-300)
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271. $In the Argand plane, the values of Z satisfying the equation \| z - 1 \| = \| i (z + 1) \| lie on$ (1) $the Y - axis$ (2) $a parabola$ (3) $a hyperbola$ (4) $the X - axis$ 272. $Let z \in C be such that \frac{z ^{2} + 3 i}{z - 2 + i} = 2 + 3 i . Then the sum of all possible values of z^{2} is:$ (1) $- 19 + 2 i$ (2) $- 19 - 2 i$ (3) $19 - 2 i$ (4) $19 + 2 i$ 273. $If z = cos α + i sin α; 0 < α < \frac{π}{4}, then \frac{1 + z ^{4}}{1 - z ^{3}} =$ (1) $\frac{cos 2 α}{sin \frac{3}{2} α}$ (2) $\frac{cosα}{sin \frac{3}{2} α}$ (3) $\frac{cos 2 α}{sin \frac{α}{2}}$ (4) $\frac{cosα}{sin \frac{α}{2}}$ 274. $Let z = x + i y be a complex number where x and y are integers. Then the area of the rectangle whose vertices are the roots of the equation z z ˉ^{3} + z ˉ z^{3} = 350 is (IIT-JEE, 2009)$ (1) $48$ (2) $32$ (3) $40$ (4) $80$ 275. $Let A = (θ \in [0, 2 π] : 1 + 10 Re (\frac{2 cos θ + isin θ}{cos θ - 3 isin θ}) = 0) . Then \sum θ^{2} is equal to$ (1) $\frac{21}{4} π^{2}$ (2) $6 π^{2}$ (3) $\frac{27}{4} π^{2}$ (4) $8 π^{2}$ 276. $The value of (\frac{1 + sin \frac{2 π}{9} + icos \frac{2 π}{9}}{1 + sin \frac{2 π}{9} - icos \frac{2 π}{9}})^{3} is:$ (1) $- \frac{1}{2} (1 - i 3)$ (2) $\frac{1}{2} (1 - i 3)$ (3) $\frac{1}{2} (3 - i)$ (4) $- \frac{1}{2} (3 - i)$ 277. $If (\frac{1 + i}{1 - i})^{m} = 1, then the least positive integral value of m is$ (1) $4$ (2) $2$ (3) $1$ (4) $3$ 278. $Let α and β be the sum and the product of all the non-zero solutions of the equation (z ˉ)^{2} + ∣ z ∣ = 0, z \in C . Then 4 (α^{2} + β^{2}) is equal to:$ (1) $6$ (2) $8$ (3) $2$ (4) $4$ 279. $If ω is a root of the equation x + \frac{1}{x} + 1 = 0, then 136 1 + ω 4 + 3 ω 9 + 6 ω 1 + ω + ω^{2} 5 + 4 ω + 3 ω^{2} 11 + 9 ω + 6 ω^{2} is equal to$ (1) $1$ (2) $-1$ (3) $0$ (4) $1 + ω$ 280. $If the conjugate of (x + iy) (1 - 2 i) is (1 + i), then$ (1) $x + iy = 1 - i$ (2) $x + i y = \frac{1 - i}{1 - 2 i}$ (3) $x - iy = \frac{1 - i}{1 + 2 i}$ (4) $x - iy = \frac{1 - i}{1 + i}$ 281. $A complex number z is said to be unimodular if ∣ z ∣ = 1 . Suppose z_{1} and z_{2} are complex numbers such that \frac{z _{1} - 2 z _{2}}{2 - z _{1} z ˉ _{2}} is unimodular and z_{2} is not unimodular. Then the point z_{1} lies on a:$ (1) $straight line parallel to x -axis$ (2) $straight line parallel to y -axis$ (3) $circle of radius 2$ (4) $circle of radius 2$ 282. $The complex number \frac{( - 3 + 3 i ) ( 1 - i )}{( 3 + 3 i ) ( i ) ( 3 + 3 i )} when represented in the Argand diagram is$ (1) $in the second quadrant$ (2) $in the first quadrant$ (3) $on the Y -axis (imaginary axis)$ (4) $on the X -axis (real axis)$ 283. $Let z be a complex number such that \| z - 6 \| = 5 and \| z + 2 - 6 i \| = 5 . Then the value of z^{3} + 3 z^{2} - 15 z + 141 is equal to$ (1) $50$ (2) $61$ (3) $37$ (4) $42$ 284. $The real part of (1 - cos θ + isin θ)^{- 1} is$ (1) $\frac{1}{2}$ (2) $\frac{1}{1 + cos θ}$ (3) $tan \frac{θ}{2}$ (4) $cot \frac{θ}{2}$ 285. $If i = - 1 then 1 + i^{2} + i^{4} + i^{6} + \dots + i^{2024} =$ (1) $i$ (2) $- i$ (3) $1$ (4) $-1$ 286. $Let z_{1} and z_{2} be two complex number such that z_{1} + z_{2} = 5 and z_{1}^{3} + z_{2}^{3} = 20 + 15 i . Then ∣ z_{1}^{4} + z_{2}^{4} ∣ equals-$ (1) $303$ (2) $75$ (3) $1515$ (4) $253$ 287. $The values of θ, for which \frac{3 + 2 isin θ}{1 - 2 isin θ} is real are$ (1) $θ = nπ + π /3 for n \in Z$ (2) $θ = nπ + π /4 for n \in Z$ (3) $θ = nπ + π /2 for n \in Z$ (4) $θ = nπ for n\inZ$ 288. $The modulus of amplitude of (1 + i 3)^{8} are respectively$ (1) $256 and π /3$ (2) $256 and 2 π /3$ (3) $2 and 2 π /3$ (4) $256 and 8 π /3$ 289. $The common roots of the equations z^{3} + 2 z^{2} + 2 z + 1 = 0, z^{2014} + z^{2015} + 1 = 0 are$ (1) $ω, ω^{2}$ (2) $1, ω, ω^{2}$ (3) $- 1, ω, ω^{2}$ (4) $- ω, - ω^{2}$ 290. $Let z = x + iy and a point P represent z in the Argand plane. If the real part of \frac{z - 1}{z + i} is 1 , then a point that lies on the locus of P is$ (1) $(2016, 2017)$ (2) $(- 2016, 2017)$ (3) $(- 2016, - 2017)$ (4) $(2016, - 2017)$ 291. $If cis α is the common value of (- 1)^{1/4} and (- i)^{1/2} then tan α =$ (1) $-1$ (2) $1$ (3) $3$ (4) $\frac{1}{3}$ 292. $If z = \frac{( 3 + i ) ^{3} ( 3 i + 4 ) ^{2}}{( 8 + 6 i ) ^{2}}, then \| z \| is equal to$ (1) $1$ (2) $3$ (3) $0$ (4) $2$ 293. $If 1, ω and ω^{2} are the cube roots of unity, then (a + b + c) (a + bω + c ω^{2}) (a + b ω^{2} + cω) =$ (1) $a^{3} + b^{3} + c^{3}$ (2) $a^{3} + b^{3} + c^{3} - 3 abc$ (3) $(a + b + c)^{3} - 3 abc$ (4) $a^{3} + b^{3} + c^{3} + 3 abc$ 294. $If Re (\frac{z - 1}{2 z + i}) = 1 Where z = x + iy then the point (x, y) lies on a:$ (1) $Straight line with slope 2$ (2) $Straight line with slope - \frac{1}{2}$ (3) $circle with diameter \frac{5}{2}$ (4) $circle with diameter \frac{1}{2}$ 295. $Let s, t, r be non-zero complex numbers and L be the set of solutions z = x + i y (x, y \in R, i = - 1) of the equation sz + t z ˉ + r = 0, where z ˉ = x - i y . Then, which of the following statement(s) is (are) TRUE? (JEE Advanced 2018)$ (1) $If L has exactly one element, then \| s \| \neq = \| t \|$ (2) $If \| s \| = \| t \|, then L has infinitely many elements$ (3) $The number of elements in L \cap {z : \| z - 1 + i \| = 5} is at most 2$ (4) $If L has more than one element, then L has infinitely many elements$ 296. $If z^{2} + z + 1 = 0, where z is complex number, then the value of (z + \frac{1}{z})^{2} + (z^{2} + \frac{1}{z ^{2}})^{2} + (z^{3} + \frac{1}{z ^{3}})^{2} +.......+ (z^{6} + \frac{1}{z ^{6}})^{2} is:$ (1) $54$ (2) $6$ (3) $12$ (4) $18$ 297. $Let z = (\frac{3}{2} + \frac{i}{2})^{5} + (\frac{3}{2} - \frac{i}{2})^{5} . If R(z) and I(z) respectively denote the real and imaginary parts of z, then:$ (1) $I(z) = 0$ (2) $R (z) > 0 and I (z) > 0$ (3) $R (z) < 0 and I (z) > 0$ (4) $R (z) = - (3)$ 298. $If α denotes the number of solutions of \| 1 - i \|^{x} = 2^{x} and β = (\frac{\| z \|}{arg ( z )}), where z = \frac{π}{4} (1 + i)^{4} (\frac{1 - π i}{π + i} + \frac{π - i}{1 + π i}), i = - 1, then the distance of the point (α, β) from the line 4 x - 3 y = 7 is______$ 299. $Let z = \frac{1 - i 3}{2}, i = - 1 . Then the value of 21 + (z + \frac{1}{z})^{3} + (z^{2} + \frac{1}{z ^{2}})^{3} + (z^{3} + \frac{1}{z ^{3}})^{3} + ... + (z^{21} + \frac{1}{z ^{21}})^{3} is.$ 300. $If z_{1}, z_{2} and z_{3}, z_{4} are two pairs of conjugate complex numbers, then arg (\frac{z _{1}}{z _{4}}) + arg (\frac{z _{2}}{z _{3}}) equals N degree. Find N.$