IIT-JEE 2007 | Complex Numbers Question 1 | Mathematics

1

IIT-JEE 2007

MCQ (Single Correct Answer)

+3

-0.75

If $$\left| z \right|\, =1\,and\,z\, \ne \, \pm \,1,$$ then all the values of $${z \over {1 - {z^2}}}$$ lie on

A

a line not passing through the origin

B

$$\left| z \right|\, = \,\sqrt 2 $$

C

the x-axis

D

the y-axis

2

IIT-JEE 2007

MCQ (Single Correct Answer)

+3

-0.75

A man walks a distance of 3 units from the origin towards the north-east ($$N\,{45^ \circ E }$$) direction. From there, he walks a distance of 4 units towards the north-west $$\left( {N\,{{45}^ \circ }\,W} \right)$$ direction to reach a point P. Then the position of P in the Argand plane is

A

$$3{e^{i\pi /4}} + 4i$$

B

$$\left( {3 - 4i} \right){e^{i\pi /4}}$$

C

$$\left( {4 + 3i} \right){e^{i\pi /4}}$$

D

$$\left( {3 + 4i} \right){e^{i\pi /4}}$$

3

JEE Advanced 2026 Paper 1 Online

MCQ (Single Correct Answer)

+4

-1

Match each entry in List-I to the correct entry in List-II and choose the correct option.

List-I	List-II
(P) If $\alpha$ and $\beta$ are the distinct roots of the equation $x^2 + x + 1 = 0$, then the quadratic equation with roots $\frac{1}{(\alpha+1)^{2026}}$ and $\frac{1}{(\beta+1)^{2026}}$ is	(1) $x^2 + x + 1 = 0$
(Q) If $\alpha$ and $\beta$ are the distinct roots of the equation $x^2 + x + 1 = 0$, then the quadratic equation with roots $\frac{1}{(\alpha+1)^{2027}}$ and $\frac{1}{(\beta+1)^{2027}}$ is	(2) $x^2 - x + 1 = 0$
(R) If $\gamma$ and $\delta$ are the distinct roots of the equation $x^2 - x + 1 = 0$, then the value of $\frac{1}{(\gamma-1)^{2026}} + \frac{1}{(\delta-1)^{2026}}$ is	(3) $x^2 + x - 1 = 0$
(S) If $p$ and $r$ are the distinct roots of the equation $x^2 + x - 1 = 0$, then the value of $\frac{1}{(p+1)^3} + \frac{1}{(r+1)^3}$ is	(4) $-1$
	(5) $-4$

A

(P) $\rightarrow$ (1), (Q) $\rightarrow$ (2), (R) $\rightarrow$ (5), (S) $\rightarrow$ (4)

B

(P) $\rightarrow$ (3), (Q) $\rightarrow$ (1), (R) $\rightarrow$ (4), (S) $\rightarrow$ (5)

C

(P) $\rightarrow$ (1), (Q) $\rightarrow$ (2), (R) $\rightarrow$ (4), (S) $\rightarrow$ (5)

D

(P) $\rightarrow$ (2), (Q) $\rightarrow$ (3), (R) $\rightarrow$ (5), (S) $\rightarrow$ (4)

4

JEE Advanced 2023 Paper 1 Online

MCQ (Single Correct Answer)

+3

-1

Let $z$ be a complex number satisfying $|z|^3+2 z^2+4 \bar{z}-8=0$, where $\bar{z}$ denotes the complex conjugate of $z$. Let the imaginary part of $z$ be nonzero.

Match each entry in List-I to the correct entries in List-II.

List - I	List - II
(P) $\|z\|^2$ is equal to	(1) 12
(Q) $\|z-\bar{z}\|^2$ is equal to	(2) 4
(R) $\|z\|^2+\|z+\bar{z}\|^2$ is equal to	(3) 8
(S) $\|z+1\|^2$ is equal to	(4) 10
	(5) 7

The correct option is:

A

$$ (P) \rightarrow(1) \quad(Q) \rightarrow(3) \quad(R) \rightarrow(5) \quad(S) \rightarrow(4) $$

B

$$ (P) \rightarrow(2) \quad(Q) \rightarrow(1) \quad(R) \rightarrow(3) \quad(S) \rightarrow(5) $$

C

$$ (P) \rightarrow(2) \quad(Q) \rightarrow(4) \quad(R) \rightarrow(5) \quad(S) \rightarrow(1) $$

D

$$ (P) \rightarrow(2) \quad(Q) \rightarrow(3) \quad(R) \rightarrow(5) \quad(S) \rightarrow(4) $$

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