Complex Numbers PYQs Set 1 (PDF)

Set 1 of 14 · Questions 1-30 of 417

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Complex Numbers · Set 1 (Q1-30)
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1. $The conjugate of a complex number is \frac{1}{i - 1} . Then, that complex number is$ (1) $- \frac{1}{i - 1}$ (2) $\frac{1}{i + 1}$ (3) $- \frac{1}{i + 1}$ (4) $\frac{1}{i - 1}$ 2. $If α is a complex number such that α^{2} - α + 1 = 0, then α^{2011} =$ (1) $1$ (2) $- α$ (3) $α^{2}$ (4) $α$ 3. $Let z_{0} be a root of a quadratic equation, x^{2} + x + 1 = 0 . If z = 3 + 6 i z_{0}^{81} - 3 i z_{0}^{93}, then arg z is equal to:$ (1) $\frac{π}{4}$ (2) $\frac{π}{6}$ (3) $\frac{π}{3}$ (4) $0$ 4. $Let z be the complex number satisfying \| z - 5 \| \leq 3 and having maximum positive principal argument. Then 34 (\frac{5 z - 12}{5 iz + 16})^{2} is equal to:$ (1) $26$ (2) $12$ (3) $20$ (4) $16$ 5. $One of the 15^{th} roots of -1 is$ (1) $cis0$ (2) $cis \frac{14 π}{15}$ (3) $cis \frac{13 π}{15}$ (4) $cis \frac{8 π}{15}$ 6. $Let z_{1} and z_{2} be two roots of the equation z^{2} + az + b = 0, z being complex. Further, assume that the origin, z_{1} and z_{2} form an equilateral tringle. Then$ (1) $a^{2} = b$ (2) $a^{2} = 2 b$ (3) $a^{2} = 3 b$ (4) $a^{2} = 4 b$ 7. $If z_{1} = 1 - 2 i; z_{2} = 1 + i and z_{3} = 3 + 4 i, then (\frac{1}{z _{1}} + \frac{3}{z _{2}}) \frac{z _{3}}{z _{2}} =$ (1) $13 - 6 i$ (2) $13 - 3 i$ (3) $6 - \frac{13}{2} i$ (4) $\frac{13}{2} - 3 i$ 8. $Let z be a complex number such that ∣ z ∣ + z = 3 + i (where i = - 1). Then ∣ z ∣ is equal to:$ (1) $\frac{34}{3}$ (2) $\frac{5}{3}$ (3) $\frac{41}{4}$ (4) $\frac{5}{4}$ 9. $If z_{1} = 2 + 3 i, z_{2} = 4 - 5 i and z_{3} are are three points in the Argand plane such that 5 z_{1} + xz_{2} + yz_{3} = 0 (x, y \in R) and z_{3} is the midpoint of the segment joining the points z_{1} and z_{2} then x + y =$ (1) $-5$ (2) $0$ (3) $4$ (4) $-1$ 10. $If α, β and γ are the roots of the equation x^{3} - 8 x + 8 = 0, then \sum α^{2} and \sum \frac{1}{αβ} are respectively$ (1) $0 and -16$ (2) $16 and 8$ (3) $-16 and 0$ (4) $16 and 0$ 11. $\frac{( sin \frac{π}{8} + icos \frac{π}{8} ) ^{8}}{( sin \frac{π}{8} - icos \frac{π}{8} ) ^{8}} is equal to$ (1) $i$ (2) $- i$ (3) $1$ (4) $2$ 12. $If z - \frac{4}{z} = 2, then the maximum value of ∣ z ∣ is equal to:$ (1) $5 + 1$ (2) $2$ (3) $2 + 2$ (4) $3 + 1$ 13. $The minimum value of \| z - 1 \| + \| z - 5 \| is$ (1) $5$ (2) $4$ (3) $3$ (4) $2$ 14. $If ω is a complex cube root of unity, then sin ((ω^{10} + ω^{23}) π - \frac{π}{4}) =$ (1) $1/ 2$ (2) $1/2$ (3) $1$ (4) $3 /2$ 15. $For n \in N, let S_{n} = {z \in C : ∣ z - 3 + 2 i ∣ = \frac{n}{4}} and T_{n} = {z \in C : ∣ z - 2 + 3 i ∣ = \frac{1}{n}} . Then the number of elements in the set {n \in N : S_{n} \cap T_{n} = φ} is:$ (1) $0$ (2) $2$ (3) $3$ (4) $Infinite$ 16. $Let z \in C be such that ∣ z ∣ < 1 . If ω = \frac{5 + 3 z}{5 ( 1 - z )}, then:$ (1) $5 Re (ω) > 4$ (2) $4 Im (ω) > 5$ (3) $5 Re (ω) > 1$ (4) $5 Im (ω) < 1$ 17. $The area (in sq. units) of the triangle whose vertices are the points represented by the complex numbers 0, z, z e^{i α} (0 < α < π) is$ (1) $\frac{1}{2} \| z \|^{2}$ (2) $\frac{1}{2} \| z \|^{2} sinα$ (3) $\frac{1}{2} \| z \|^{2} sinαcosα$ (4) $\frac{1}{2} \| z \|^{2} cosα$ 18. $If ω represents a complex cube root of unity, then (1 + \frac{1}{ω}) (1 + \frac{1}{ω ^{2}}) + (2 + \frac{1}{ω}) (2 + \frac{1}{ω ^{2}}) + \dots + (n + \frac{1}{ω}) (n + \frac{1}{ω ^{2}}) =$ (1) $\frac{n ( n ^{2} + 1 )}{3}$ (2) $\frac{n ( n ^{2} + 2 )}{3}$ (3) $\frac{n ( n ^{2} - 2 )}{3}$ (4) $\frac{n ^{2} ( n - 1 )}{6}$ 19. $ω is a complex cube root of unity. Match the items of List-I to the items of List-II. List - I (A) ω^{1010} + ω^{2020} (B) (1 + ω - ω^{2}) (1 - ω + ω^{2}) (C) (2 + ω^{2} + ω^{4})^{5} (D) (3 + 5 ω + 3 ω^{2})^{3} List - II (i) 0 (ii) 1 (iii) -1 (iv) 4 (v) 8 The correct match is$ (1) $A - (iii), B - (iv), C - (i), D - (v)$ (2) $A - (i), B - (iv), C - (ii), D - (v)$ (3) $A - (iii), B - (iv), C - (ii), D - (v)$ (4) $A - (iii), B - (i), C - (i), D - (iv)$ 20. $The locus of z = x + iy, such that Im (\frac{z - 3 i}{iz + 4}) = 0 is$ (1) $x^{2} - y^{2} + 7 y - 12 = 0$ (2) $x^{2} + y^{2} - 7 y + 12 = 0$ (3) $x^{2} + y^{2} - 7 y + 12 = 0 and (x,y)\neq(0,4)$ (4) $x^{2} - y^{2} + 7 y - 12 = 0 and (x,y)\neq(0,4)$ 21. $If the roots of the equation (z - 4)^{3} = 8 i are a - 2 i, b + i and c + i, then abc is equal to$ (1) $133$ (2) $413$ (3) $213$ (4) $53$ 22. $If \frac{1 - 10 icos θ}{1 - 10 3 isin θ} is purely real then one of the values of θ is$ (1) $\frac{π}{6}$ (2) $\frac{π}{4}$ (3) $\frac{π}{3}$ (4) $\frac{π}{2}$ 23. $Geometrically, the set {z \in C : \| z - 2 - 2 i \| \leq 1} represents..........$ (1) $a closed circular disc with center at (- 2, - 2) and with radius 1$ (2) $a closed circular disc with centre at (2, 2) and with radius 1$ (3) $a closed circular disc with center at (1, 1) and with radius 0.5$ (4) $a closed circular disc with centre at (- 1, - 1) and with radius 0.5$ 24. $If x = - 5 + 2 - 4, then the value of the expression x^{4} + 9 x^{3} + 35 x^{2} - x + 4 is A .Find - A .$ 25. $Let ω be the complex number cos \frac{2 π}{3} + i sin \frac{2 π}{3} . Then the number of distinct complex numbers z satisfying z + 1 ω ω^{2} ω z + ω^{2} 1 ω^{2} 1 z + ω = 0 is equal to ____ . (IIT-JEE, 2010)$ 26. $Sum of squares of modulus of all the complex numbers z satisfying z ˉ = i z^{2} + z^{2} - z is equal to$ 27. $If z_{1}, z_{2}, z_{3} are complex numbers such that ∣ z_{1} ∣ = ∣ z_{2} ∣ = ∣ z_{3} ∣ = \frac{1}{z _{1}} + \frac{1}{z _{2}} + \frac{1}{z _{3}} = 1 then ∣ z_{1} + z_{2} + z_{3} ∣ is$ 28. $Let z = a + ib, b \neq = 0 be complex numbers satisfying z^{2} = z . 2^{1 - \| z \|} . Then the least value of n \in N, such that z^{n} = (z + l)^{n}, is equal to_____$ 29. $If z_{1} = 10 + 6 i, z_{2} = 4 + 6 i and z is a complex number such that amp (\frac{z - z _{1}}{z - z _{2}}) = \frac{π}{4} then the value of \| z - 7 - 9 i \| is equal to N . Find N^{2} .$ 30. $Let α, β be the roots of the equation x^{2} - 6 x + 3 = 0 such that Im (α) > Im (β) . Let a, b be integers not divisible by 3 and n be a natural number such that \frac{α ^{99}}{β} + α^{98} = 3^{n} (a + ib), i = - 1 . Then n + a + b is equal to_______$