JEE Main Complex Numbers Practice Questions With Solutions

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JEE Main Complex Numbers Practice Questions with Solutions

Mathematics is a critical section of the JEE Main entrance exam. One of the most important chapters covered under the mathematics section is Complex Numbers. The JEE Main Complex Numbers Practice Questions with Solutions are developed for the JEE Main aspirants to practise various problems from this chapter and obtain high scores in the Mathematics section of the JEE Main entrance test. This section tests the analytical and problem-solving skills of the JEE Main aspirants; therefore, the JEE Main Complex Numbers practice tests should be attempted on a regular basis. These practice tests usually consist of 15 to 20 questions based on the Complex Numbers chapter. The total questions asked from the Complex Number chapter is usually 1 or 2; however, the number of questions asked is not fixed and may vary each year.

The JEE Main Complex Numbers practice tests also contain some of the questions asked from this chapter in the previous years’ JEE Main examination. The important topics covered under Complex Number, from which questions are framed in the JEE Main exam, are Basic Operations, Polar and Exponential Forms, Solving Quadratic Equations, Argand Diagram & Geometric Interpretation, Square Roots and Properties, De Moivre’s Theorem, Locus and Equations, and Application-Based Problems. The JEE Main Complex Numbers Practice Questions with Solutions consist of many problems on each of these topics, making it easier for students to learn the basic concepts of Complex Numbers well and solve problems effectively. Each question has solutions along with it for candidates to understand the steps involved in solving a problem. Regular practice using these practice tests will improve the students’ accuracy and speed for solving questions from Complex Numbers in the Mathematics section of the JEE Main exam. It will also build confidence in the students, which is important for success in the Joint Entrance Exam.

JEE Main Mathematics Complex Numbers Practice Questions

Chemistry Physics

Question 1.

Let $z$ be a complex number such that the real part of $\frac{z - 2 i}{z + 2 i}$ is zero. Then, the maximum value of $| z - (6 + 8 i) |$ is equal to

\infty

Question 2.

The sum of all possible values of $θ \in [- π, 2 π]$ , for which $\frac{1 + i \cos θ}{1 - 2 i \cos θ}$ is purely imaginary, is equal to :

4 π

3 π

2 π

5 π

Question 3.

Let $z$ be a complex number such that $| z + 2 | = 1$ and $lm (\frac{z + 1}{z + 2}) = \frac{1}{5}$ . Then the value of $| Re (\overset{―}{z + 2}) |$ is

\frac{2 \sqrt{6}}{5}

\frac{24}{5}

\frac{\sqrt{6}}{5}

\frac{1 + \sqrt{6}}{5}

Question 4.

If the set $R = {(a, b) : a + 5 b = 42, a, b \in N}$ has $m$ elements and $\sum_{n = 1}^{m} (1 - i^{n!}) = x + i y$ , where $i = \sqrt{- 1}$ , then the value of $m + x + y$ is

Question 5.

If $z_{1}, z_{2}$ are two distinct complex number such that $| \frac{z_{1} - 2 z_{2}}{\frac{1}{2} - z_{1} {\bar{z}}_{2}} | = 2$ , then

either

z_{1}

lies on a circle of radius

\frac{1}{2}

z_{2}

lies on a circle of radius 1.

z_{1}

lies on a circle of radius

\frac{1}{2}

and

z_{2}

lies on a circle of radius 1.

either

z_{1}

lies on a circle of radius 1 or

z_{2}

lies on a circle of radius

\frac{1}{2}

both

z_{1}

and

z_{2}

lie on the same circle.

Question 6.

Let $S_{1} = {z \in C : | z | \leq 5}, S_{2} = {z \in C : Im (\frac{z + 1 - \sqrt{3} i}{1 - \sqrt{3} i}) \geq 0}$ and $S_{3} = {z \in C : Re (z) \geq 0}$ . Then the area of the region $S_{1} \cap S_{2} \cap S_{3}$ is :

\frac{125 π}{24}

\frac{125 π}{6}

\frac{125 π}{12}

\frac{125 π}{4}

Question 7.

Consider the following two statements :

Statement I: For any two non-zero complex numbers $z_{1}, z_{2}, (| z_{1} | + | z_{2} |) | \frac{z_{1}}{| z_{1} |} + \frac{z_{2}}{| z_{2} |} | \leq 2 (| z_{1} | + | z_{2} |), and$

Statement II : If $x, y, z$ are three distinct complex numbers and $a, b, c$ are three positive real numbers such that $\frac{a}{| y - z |} = \frac{b}{| z - x |} = \frac{c}{| x - y |}$ , then $\frac{a^{2}}{y - z} + \frac{b^{2}}{z - x} + \frac{c^{2}}{x - y} = 1$ .

Between the above two statements,

both Statement I and Statement II are incorrect.

Statement I is correct but Statement II is incorrect.

Statement I is incorrect but Statement II is correct.

both Statement I and Statement II are correct.

Question 8.

The area (in sq. units) of the region $S = {z \in C : | z - 1 | \leq 2; (z + \bar{z}) + i (z - \bar{z}) \leq 2, lm (z) \geq 0}$ is

\frac{7 π}{4}

\frac{3 π}{2}

\frac{7 π}{3}

\frac{17 π}{8}

Question 9.

Let $α$ and $β$ be the sum and the product of all the non-zero solutions of the equation $(\bar{z})^{2} + | z | = 0, z \in C$ . Then $4 (α^{2} + β^{2})$ is equal to :

Question 10.

z

is a complex number such that

| z | ⩽ 1

, then the minimum value of

| z + \frac{1}{2} (3 + 4 i) |

is :

\frac{5}{2}

\frac{3}{2}

Question 1.

Let

S = | z \in C : | z - 1 ∣= 1

and

(\sqrt{2} - 1) (z + \bar{z}) - i (z - \bar{z}) = 2 \sqrt{2} ∣

. Let

z_{1}, z_{2} \in S

be such that

| z_{1} | = max_{z \in s} | z |

and

| z_{2} | = min_{z \in S} | z |

. Then

{| \sqrt{2} z_{1} - z_{2} |}^{2}

equals :

Question 2.

Let $z_{1}$ and $z_{2}$ be two complex numbers 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 -

15 \sqrt{15}

30 \sqrt{3}

25 \sqrt{3}

Question 3.

If $z$ is a complex number, then the number of common roots of the equations $z^{1985} + z^{100} + 1 = 0$ and $z^{3} + 2 z^{2} + 2 z + 1 = 0$ , is equal to

Question 4.

If $z = x + i y, x y \neq 0$ , satisfies the equation $z^{2} + i \bar{z} = 0$ , then $| z^{2} |$ is equal to :

\frac{1}{4}

Question 5.

Let $r$ and $θ$ respectively be the modulus and amplitude of the complex number $z = 2 - i (2 \tan \frac{5 π}{8})$ , then $(r, θ)$ is equal to

(2 \sec \frac{11 π}{8}, \frac{11 π}{8})

(2 \sec \frac{3 π}{8}, \frac{3 π}{8})

(2 \sec \frac{5 π}{8}, \frac{3 π}{8})

(2 \sec \frac{3 π}{8}, \frac{5 π}{8})

Question 6.

If $z = \frac{1}{2} - 2 i$ is such that $| z + 1 | = α z + β (1 + i), i = \sqrt{- 1}$ and $α, β \in R$ , then $α + β$ is equal to

-

-

Question 7.

S = {z \in C : | z - i | = | z + i | = | z - 1 |}

, then,

n (S)

is :

Question 8.

If the set

{Re (\frac{z - \bar{z} + z \bar{z}}{2 - 3 z + 5 \bar{z}}) : z \in C, Re (z) = 3}

is equal to

the interval

(α, β]

, then

24 (β - α)

is equal to :

Question 9.

Let $S = {z \in C : \bar{z} = i (z^{2} + Re (\bar{z}))}$ . Then $\sum_{z \in S} | z |^{2}$ is equal to :

\frac{7}{2}

\frac{5}{2}

Question 10.

Let $C$ be the circle in the complex plane with centre $z_{0} = \frac{1}{2} (1 + 3 i)$ and radius $r = 1$ . Let $z_{1} = 1 + i$ and the complex number $z_{2}$ be outside the circle $C$ such that $| z_{1} - z_{0} | | z_{2} - z_{0} | = 1$ . If $z_{0}, z_{1}$ and $z_{2}$ are collinear, then the smaller value of ${| z_{2} |}^{2}$ is equal to :

\frac{3}{2}

\frac{5}{2}

\frac{13}{2}

\frac{7}{2}

JEE Main Complex Numbers Practice Questions With Solutions

JEE Main Complex Numbers Practice Questions with Solutions

JEE Main Mathematics Complex Numbers Practice Questions

Question 1.

Question 2.

Question 3.

Question 4.

Question 5.

Question 6.

Question 7.

Question 8.

Question 9.

Question 10.

Question 1.

Question 2.

Question 3.

Question 4.

Question 5.

Question 6.

Question 7.

Question 8.

Question 9.

Question 10.

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