Complex Numbers PYQs Set 5 (PDF)

Set 5 of 14 · Questions 121-150 of 417

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Complex Numbers · Set 5 (Q121-150)
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121. $If the locus of z \in C, such that Re (\frac{z - 1}{2 z + i}) + Re (\frac{z - 1}{2 z - i}) = 2, is a circle of radius r and center (a, b), then \frac{15 ab}{r ^{2}} is equal to:$ (1) $16$ (2) $24$ (3) $12$ (4) $18$ 122. $The real root of the equation x^{3} - 6 x + 9 = 0 is$ (1) $6$ (2) $-3$ (3) $-6$ (4) $-9$ 123. $S ≐ z \in C/ \| z-1+i \| =1 represents$ (1) $a circle with center (- 1, 1) and radius 1 unit$ (2) $a circle with centre (1, 2) and radius 5 units$ (3) $a circle with centre (1, - 1) and radius 1 unit$ (4) $an ellipse with centre (1, - 1)$ 124. $If x + iy = \frac{3 + i}{1 + 3 i}, then (x^{2} + y^{2})^{2} =$ (1) $0$ (2) $1$ (3) $2$ (4) $3$ 125. $If 1, ω, ω^{2} are the cube roots of unity, k is positive integer and (1 - ω + ω^{2})^{3 k} + (1 - ω^{2} + ω)^{3 k} = (1 - ω + ω^{2})^{3 k + 1} + (1 + ω - ω^{2})^{3 k + 1}, then k =$ (1) $r, r, \in N$ (2) $2 r + 1, r \in N$ (3) $4 r + 1, r \in N$ (4) $3 r, r \in N$ 126. $If \frac{3 + isinθ}{4 - icosθ}, θ \in [0, 2 π] is a real number, then an argument of sinθ + icosθ is:$ (1) $- tan^{- 1} \frac{3}{4}$ (2) $π - tan^{- 1} \frac{3}{4}$ (3) $π - tan^{- 1} \frac{4}{3}$ (4) $tan^{- 1} \frac{4}{3}$ 127. $The number of complex roots of the equation x^{11} - x^{7} + x^{4} - 1 = 0 whosearguments lie in the first quadrant is$ (1) $2$ (2) $3$ (3) $7$ (4) $9$ 128. $One of the value of (- 32 i)^{\frac{2}{5}} is$ (1) $4 cis \frac{2 π}{5}$ (2) $4 cis \frac{3 π}{5}$ (3) $4 cis \frac{4 π}{5}$ (4) $4 cis \frac{6 π}{5}$ 129. $For a \in C, let A = {z \in C : Re(a + z) > Im(a + z)} and B = {z \in C : Re(a + z) < Im(a + z)}. Then among the two statements: (S1): If Re(A), Im(A) > 0, then the set A contains all the real numbers (S2): If Re(A), Im(A) < 0, then the set B contains all the real numbers,$ (1) $Only (S2) is true$ (2) $only (SI) is true$ (3) $Both are true$ (4) $Both are false$ 130. $Let the product of ω_{1} = (8 + i) sinθ + (7 + 4 i) cos θ and ω_{2} = (1 + 8 i) sin θ + (4 + 7 i) cosθ be α + iβ, i = - 1 . Let p and q be the maximum. and the minimum values of α + β respectively. Then p + q is equal to:$ (1) $130$ (2) $150$ (3) $160$ (4) $140$ 131. $The number of complex numbers z satisfying z ˉ = i z^{2} is$ (1) $3$ (2) $4$ (3) $2$ (4) $5$ 132. $The number of complex numbers z such that \| z - 1\| = \| z + 1\| = \|z - i\| equals$ (1) $1$ (2) $2$ (3) $¥$ (4) $0$ 133. $If α and β are the roots of x^{2} - ax + b^{2} = 0, then α^{2} + β^{2} is equal to$ (1) $2 a^{2} - b^{2}$ (2) $a^{2} + b^{2}$ (3) $a^{2} - 2 b^{2}$ (4) $a^{2} - b^{2}$ 134. $Real part of (cos 4 + isin 4 + 1)^{2020} is$ (1) $2^{2020} cos^{2020} 2 cos 2020$ (2) $2^{2020} cos^{2020} 2 cos 4040$ (3) $2^{1020} cos^{2020} 2 cos 4040$ (4) $2^{2020} cos^{2020} 1 cos 2020$ 135. $If ω is an imaginary cube root of unity, then (1 + ω - ω^{2})^{7} equals-$ (1) $128 ω$ (2) $- 128 ω$ (3) $128 ω^{2}$ (4) $- 128 ω^{2}$ 136. $If the Cartesian co-ordinates of a point are (\frac{- 5 3}{2}, \frac{5}{2}), then its polar co-ordinates are$ (1) $(5, \frac{5 π}{6})$ (2) $(5, \frac{2 π}{3})$ (3) $(5, \frac{11 π}{18})$ (4) $(5, \frac{13 π}{18})$ 137. $If z = \frac{3}{2} + \frac{i}{2}, i = - 1, then (z^{201} - i)^{8} is equal to$ (1) $1$ (2) $256$ (3) $-1$ (4) $0$ 138. $If z^{2} + z + 1 = 0, where z is a complex number, then (z + \frac{1}{z})^{3} + (z^{4} + \frac{1}{z ^{4}})^{3} is equal to$ (1) $1$ (2) $0$ (3) $-1$ (4) $-2$ 139. $If α, β, γ are the roots of the equation x^{3} + 4 x + 2 = 0, then α^{3} + β^{3} + γ^{3} =$ (1) $-6$ (2) $2$ (3) $6$ (4) $-2$ 140. $Let z \in C and i = - 1, if a, b, c \in (0, 1) be such that a^{2} + b^{2} + c^{2} = 1 and b + ic = (1 + a) z, then \frac{1 + iz}{1 - iz} =$ (1) $\frac{a + ib}{1 + c}$ (2) $\frac{a - ib}{1 + c}$ (3) $\frac{a - ib}{1 - c}$ (4) $\frac{a + ib}{1 - c}$ 141. $If 3 x + i (4 x - y) = 6 - i where x and y are real numbers, then the values of x and y are respectively,$ (1) $3,9$ (2) $2,9$ (3) $2,4$ (4) $3,4$ 142. $Let z be a complex number such that \frac{z - 2 i}{z + i} = 2, z \neq = - i . then z lies on the circle of radius 2 and centre$ (1) $(2, 0)$ (2) $(0, 0)$ (3) $(0, 2)$ (4) $(0, - 2)$ 143. $Let z = x + iy be a non-zero complex number such that z^{2} = i \| z \|^{2} where i = - 1 then z lies onthe:$ (1) $line, y = x$ (2) $imaginary axis$ (3) $real axis$ (4) $line, y = - x$ 144. $If a, b \in R and i = - 1, then the number of ordered pairs of real numbers (a, b) satisfying the condition (a + bi)^{3} = a - bi is$ (1) $3$ (2) $2$ (3) $4$ (4) $5$ 145. $Let a, b, x and y be real numbers such that a - b = 1 and y ¹ 0. If the complex number z = x + iy satisfies Im (\frac{az + b}{z + 1}) = y, then which of the following is(are) possible value(s) of x?$ (1) $1 - 1 + y^{2}$ (2) $- 1- 1 - y^{2}$ (3) $1 + 1 + y^{2}$ (4) $- 1 + 1 - y^{2}$ 146. $Let w = z z ˉ + k_{1} z + k_{2} i z + λ (1 + i), k_{1}, k_{2} \in R . Let Re(w) = 0 be the circle C of radius 1 in the first quadrant touching the line y = 1 and the y-axis. If the curve Im(w) = 0 intersects C at A and B, then 30 (A B)^{2} is equal to________.$ 147. $The number of real values of a satisfying the equation a^{2} - 2 asin x + 1 = 0 is$ 148. $Let z_{1}, z_{2} be the roots of the equation z^{2} + az + 12 = 0 and Z_{1}, Z_{2} form an equilateral triangle with origin. Then, the value of \|a\| is$ 149. $Let integers a, b \in [- 3, 3] be such that a + b \neq = 0 . Then the number of all possible ordered pairs (a, b), for which \frac{z - a}{z + b} = 1 and z + 1 ω ω^{2} ω z + ω^{2} 1 ω^{2} 1 z + ω = 1, z \in C, where ω and ω^{2} are the roots of x^{2} + x + 1 = 0, is equal to .$ 150. $ω is an imaginary cube root of unity. If (1 + ω^{2})^{m} = (1 + ω^{4})^{m} then least positive integral value of m is$