M_Ch-01_Complex Numbers.pdf

Chapter 1 Complex Numbers Only One Option Correct Type Questions 1. 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) d...

Chapter 1 Complex Numbers Only One Option Correct Type Questions 1. 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 [IIT-JEE-2007 (Paper-1)] (A) 3eiπ/ 4 + 4i (B) (3 – 4i )eiπ/ 4 (C) (4 + 3i)eiπ/ 4 (D) (3 + 4i )eiπ/ 4 z 2. If |z| = 1 and z ≠ ± 1, then all the values of lie on [IIT-JEE-2007 (Paper-2)] 1− z2 (A) A line not passing through the origin (B) | z | = 2 (C) The x-axis (D) The y-axis 3. A particle P starts from the point z0 = 1 + 2i , where i = − 1. It moves first horizontally away from origin by 5 units and then vertically away from origin by 3 units to reach a point z1. From z1 the particle moves 2 π units in the direction of the vector iˆ + ˆj and then it moves through an angle in anticlockwise direction on 2 a circle with centre at origin, to reach a point z2. The point z2 is given by [IIT-JEE-2008 (Paper-2)] (A) 6 + 7i (B) –7 + 6i (C) 7 + 6i (D) – 6 + 7i 4. Let z = x + iy be a complex number where x and y are integers. Then the area of the rectangle whose vertices are the roots of the equation zz 3 + zz 3 = 350 is [IIT-JEE-2009 (Paper-1)] (A) 48 (B) 32 (C) 40 (D) 80 15 5. Let z = cosθ + i sinθ. Then the value of  Im(z m =1 2 m −1 ) at θ = 2° is [IIT-JEE-2009 (Paper-1)] 1 1 (A) (B) sin 2° 3 sin 2° 1 1 (C) (D) 2 sin 2° 4 sin 2° Corporate Office : Aakash Tower, 8, Pusa Road, New Delhi-110005 Ph.011-47623456 MATHEMATICS ARCHIVE - JEE (Advanced) 6. Let z be a complex number such that the imaginary part of z is non-zero and a = z2 + z +1 is real. Then a cannot take the value [IIT-JEE-2012 (Paper-1)] 1 (A) –1 (B) 3 1 3 (C) (D) 2 4 1 7. Let complex numbers α and lie on circles (x – x0)2 + (y – y0)2 = r2 and (x – x0)2 + (y – y0)2 = 4r2, α 2 respectively. If z0 = x0 + iy 0 satisfies the equation 2 z0 = r 2 + 2, then α = [JEE (Adv)-2013 (Paper-1)] 1 1 (A) (B) 2 2 1 1 (C) (D) 7 3 8. Let S be the set of all complex numbers z satisfying z − 2 + i ≥ 5. If the complex number z0 is such that 1  1  4 − z0 − z0 is the maximum of the set  : z ∈ S  , then the principal argument of is z0 − 1  z − 1  z0 − z0 + 2i [JEE (Adv)-2019 (Paper-1)] π π (A) (B) 2 4 π 3π (C) − (D) 2 4 9. Let θ1, θ2,...., θ10 be positive valued angles (in radian) such that θ1 + θ2 +... + θ10 = 2π. Define the complex iθ iθ numbers z1 = e 1 , zk = zk – 1e k for k = 2, 3,..., 10, where i = –1. Consider the statement P and Q given below: [JEE (Adv)-2021 (Paper-1)] P : z2 – z1 + z3 – z2 +... + z10 – z9 + z1 – z10 ≤ 2π Q : z22 – z12 + z32 – z22 +... + z10 2 – z92 + z12 – z10 2 ≤ 4π Then, (A) P is TRUE and Q is FALSE (B) Q is TRUE and P is FALSE (C) Both P and Q are TRUE (D) Both P and Q are FALSE One or More Option(s) Correct Type Questions 10. Let z1 and z2 be two distinct complex numbers and let z = (1 – t) z1 + tz2 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 (Paper-1)] (A) |z – z1| + |z – z2| = |z1 – z2| (B) Arg (z – z1) = Arg (z – z2) z − z1 z − z1 (C) =0 (D) Arg (z – z1) = Arg (z2 – z1) z2 − z1 z2 − z1 11. Let ω be a complex cube root of unity with ω ≠ 1 and P = [pij] be a n × n matrix with pij = ωi+j. Then P2 ≠ 0, when n = [JEE (Adv)-2013 (Paper-2)] (A) 57 (B) 55 (C) 58 (D) 56 Corporate Office : Aakash Tower, 8, Pusa Road, New Delhi-110005 Ph.011-47623456 ARCHIVE - JEE (Advanced) MATHEMATICS 3 +i  1  − 1 12. Let ω = and P = {ωn : n = 1, 2, 3, …}. Further H1 = z ∈ C : Re z >  and H2 = z ∈ C : Re z <  , where C is 2  2  2  the set of all complex numbers. If z1 ∈ P ∩ H1, z2 ∈ P ∩ H2 and O represents the origin, then ∠z1Oz2 = [JEE (Adv)-2013 (Paper-2)] π π (A) (B) 2 6 2π 5π (C) (D) 3 6  1  13. Let a, b ∈ » and a2 + b2 ≠ 0. Suppose S = z ∈ » : z = , t ∈ », t ≠ 0  , where i = −1. If z = x + iy and z  a + ibt  ∈ S, then (x, y) lies on [JEE (Adv)-2016 (Paper-2)] 1  1  (A) The circle with radius and centre  , 0  for a > 0, b ≠ 0 2a  2a  1  1  (B) The circle with radius − and centre  − , 0  for a < 0, b ≠ 0 2a  2a  (C) The x-axis for a ≠ 0, b = 0 (D) The y-axis for a = 0, b ≠ 0 14. Let a, b, x and y be real numbers such that a – b = 1 and y ≠ 0. If the complex number z = x + iy satisfies  az + b  Im   = y , then which of the following is(are) possible value(s) of x? [JEE (Adv)-2017 (Paper-1)]  z +1  (A) −1 + 1 − y 2 (B) 1 − 1 + y 2 (C) 1 + 1 + y 2 (D) −1 − 1 − y 2 15. For a non-zero complex number z, let arg(z) denote the principal argument with – π < arg(z) ≤ π. Then, which of the following statement(s) is (are) FALSE? [JEE (Adv)-2018 (Paper-1)] π (A) arg(–1 – i) = , where i = −1 4 (B) The function f : » → (–π, π], defined by f(t) = arg(–1 + it) for all t ∈ » , is continuous at all points of » , where i = −1 z  (C) For any two non-zero complex number z1 and z2, arg  1  − arg( z1 ) + arg( z2 ) is an integer multiple of 2π  z2  (D) For any three given distinct complex numbers z1, z2 and z3, the locus of the point z satisfying the condition  ( z − z1 )( z2 − z3 )  arg  = π, lies on a straight line  ( z − z3 )( z2 − z1 )    16. Let s, t, r be non-zero complex numbers and L be the set of solutions z = x + iy ( x, y ∈ », i = −1) of the equation sz + tz + r = 0, where z = x − iy. Then, which of the following statement(s) is (are) TRUE? [JEE (Adv)-2018 (Paper-2)] (A) If L has exactly one element, then |s| ≠ |t| (B) If |s| = |t|, then L has infinitely many elements (C) The number of elements in L ∩ {z : |z – 1 + i| = 5} is at most 2 (D) If L has more than one element, then L has infinitely many elements Corporate Office : Aakash Tower, 8, Pusa Road, New Delhi-110005 Ph.011-47623456 MATHEMATICS ARCHIVE - JEE (Advanced) 17. Let S be the set of all complex numbers z satisfying |z2 + z + 1| = 1. Then which of the following statements is/are TRUE? [JEE (Adv)-2020 (Paper-1)] 1 1 (A) z + ≤ for all z ∈ S (B) z ≤ 2 for all z ∈ S 2 2 1 1 (C) z + ≥ for all z ∈ S (D) The set S has exactly four elements 2 2 18. For any complex number w = c + id, let arg(w) ∈ (–π, π], where i = –1. Let α and β be real numbers such z + α π that for all complex numbers z = x + iy satisfying arg  z + β  = 4 , the ordered pair (x, y) lies on the circle   x2 + y2 + 5x – 3y + 4 = 0 [JEE (Adv)-2021 (Paper-1)] Then which of the following statements is (are) TRUE? (A) α = –1 (B) αβ = 4 (C) αβ = – 4 (D) β = 4 19. Let z denote the complex conjugate of a complex number z. If z is a non-zero complex number for which 2 1 both real and imaginary parts of ( z ) + are integers, then which of the following is/are possible value(s) z2 of |z|? [JEE (Adv)-2022 (Paper-2)] 1/4 1/4  43 + 3 205   7 + 33  (A)   (B)    2   4  1/4 1/4  9 + 65   7 + 13  (C)   (D)    4   6  Linked Comprehension Type Questions Paragraph for Q.Nos. 19 to 21 Let A, B, C be three sets of complex numbers as defined below A = {z ; Imz ≥ 1} B = {z : |z – 2 – i| = 3} C = {z : Re((1 − i )z ) = 2 }. [IIT-JEE-2008 (Paper-1)] Choose the correct answer : 20. The number of elements in the set A ∩ B ∩ C is (A) 0 (B) 1 (C) 2 (D) ∞ 21. Let z be any point in A ∩ B ∩ C. The | z + 1 – i |2 + | z – 5 – i |2 lies between (A) 25 and 29 (B) 30 and 34 (C) 35 and 39 (D) 40 and 44 Corporate Office : Aakash Tower, 8, Pusa Road, New Delhi-110005 Ph.011-47623456 ARCHIVE - JEE (Advanced) MATHEMATICS 22. Let z be any point in A ∩ B ∩ C and let w be any point satisfying | w – 2 – i | < 3. Then, | z | – | w | + 3 lies between (A) –6 and 3 (B) –3 and 6 (C) –6 and 6 (D) –3 and 9 Paragraph for Q.Nos. 22 and 23 Let S = S1 ∩ S2 ∩ S3, where S1 = {z ∈ » :| z | < 4} ,   z − 1 + 3i   S2 = z ∈ » : Im   > 0  and   1 − 3i   S3 = {z ∈ » : Re z > 0}. [JEE (Adv)-2013 (Paper-2)] Choose the correct answer : 23. Area of S = 10π 20π (A) (B) 3 3 16π 32π (C) (D) 3 3 24. min | 1 − 3i − z | = z ∈S 2− 3 2+ 3 (A) (B) 2 2 3− 3 3+ 3 (C) (D) 2 2 Matrix-Match / Matrix Based Type Questions 25. Match the statements in Column-I with those in Column-II. [IIT-JEE-2010 (Paper-2)] [Note: Here z takes values in the complex plane and Im z and Re z denote, respectively, the imaginary part and the real part of z.] Column I Column II 4 (A) The set of points z satisfying (p) An ellipse with eccentricity 5 | z – i | z || = | z + i | z || is contained in or equal to (B) The set of points z satisfying (q) The set of points z satisfying Im z = 0 | z + 4 | + | z – 4 | = 10 is contained in or equal to (C) If | w | = 2, then the set of points (r) The set of points z satisfying | Im z | ≤ 1 1 z =w − is contained in or equal to w (D) If | w | = 1, then the set of points (s) The set of points z satisfying | Re z | ≤ 2 1 z =w + is contained in or equal to w (t) The set of points z satisfying | z | ≤ 3 Corporate Office : Aakash Tower, 8, Pusa Road, New Delhi-110005 Ph.011-47623456 MATHEMATICS ARCHIVE - JEE (Advanced)  2k π   2k π  26. Let zk = cos   + i sin   ; k = 1, 2,..., 9. [JEE (Adv)-2014 (Paper-2)]  10   10  List-I List-II P. For each zk there exists a zj such that 1. True zk. zj = 1 Q. There exists a k ∈ {1, 2,..., 9} such that 2. False z1.z = zk has no solution z in the set of complex numbers | 1 − z1 | | 1 − z2 |... | 1 − z9 | R. equals 3. 1 10 9 2k π S. 1 −  cos  10  equals k =1 4. 2 P Q R S (A) 1 2 4 3 (B) 2 1 3 4 (C) 1 2 3 4 (D) 2 1 4 3 Integer / Numerical Value Type Questions 2π 2π 27. Let ω be the complex number cos + i sin. Then the number of distinct complex numbers z satisfying 3 3 z +1 ω ω2 ω z + ω2 1 = 0 is equal to [IIT-JEE-2010 (Paper-1)] 2 ω 1 z+ω 28. If z is any complex number satisfying | z – 3 – 2i | ≤ 2, then the minimum value of | 2z – 6 + 5i | is [IIT-JEE-2011 (Paper-1)] iπ 29. Let ω = e 3 , and a, b, c, x, y, z be non-zero complex number such that a+b+c=x a + bω + c ω2 = y a + bω2 = c ω | x |2 + | y |2 + | z |2 Then the value of is [IIT-JEE-2011 (Paper-2)] | a |2 + | b |2 + | c |2  kπ   kπ  30. For any integer k, let α k = cos   + i sin   , where i = −1. The value of the expression  7   7  12  | αk +1 − αk | k =1 3 is [JEE (Adv)-2015 (Paper-2)]  | α 4k −1 − α 4k −2 | k =1 Corporate Office : Aakash Tower, 8, Pusa Road, New Delhi-110005 Ph.011-47623456 ARCHIVE - JEE (Advanced) MATHEMATICS 31. Let ω ≠ 1 be a cube root of unity. Then the minimum of the set {|a + bω + cω2|2 : a, b, c distinct non-zero integers} equals ____. [JEE (Adv)-2019 (Paper-1)] 32. For a complex number z, let Re(z) denote the real part of z. Let S be the set of all complex numbers z satisfying z4 – |z|4 = 4iz2, where i = −1. Then the minimum possible value of |z1 – z2|2, where z1, z2 ∈ S with Re(z1) > 0 and Re(z2) < 0, is _____ [JEE (Adv)-2020 (Paper-2)] 2 + 3z + 4z 2 33. Let z be a complex number with non-zero imaginary part. If is a real number, then the value of 2 − 3z + 4z 2 |z|2 is _________. [JEE (Adv)-2022 (Paper-1)] 34. Let z 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 − z 2 = i ( z + z 2 ) is _________. [JEE (Adv)-2022 (Paper-1)] ‰ ‰ ‰ Corporate Office : Aakash Tower, 8, Pusa Road, New Delhi-110005 Ph.011-47623456