# JEE Main

Explore popular questions from Complex Numbers and Quadratic Equations for JEE Main. This collection covers Complex Numbers and Quadratic Equations previous year JEE Main questions hand picked by experienced teachers.

## Mathematics

Correct Marks 4

Incorrectly Marks -1

Q 1. If {tex} \left( \frac { 1 + i } { 1 - i } \right) ^ { x } = 1 , {/tex} then

A

{tex} x = 2 n + 1 , {/tex} where {tex} n {/tex} is any positive integer

{tex} x = 4 n , {/tex} where {tex} n {/tex} is any positive integer

C

{tex} x = 2 n , {/tex} where {tex} n {/tex} is any positive integer

D

{tex} x = 4 n + 1 , {/tex} where {tex} n {/tex} is any positive integer

##### Explanation

Correct Marks 4

Incorrectly Marks -1

Q 2. Let {tex} z _ { 1 } {/tex} and {tex} z _ { 2 } {/tex} be the roots of the equation {tex} z ^ { 2 } + a z + b = 0 , z {/tex} being complex number. Further, assume that the origin, {tex} z _ { 1 } {/tex} and {tex} z _ { 2 } {/tex} form an equilateral triangle, then

A

{tex} a ^ { 2 } = 4 b {/tex}

B

{tex} a ^ { 2 } = b {/tex}

C

{tex} a ^ { 2 } = 2 b {/tex}

{tex} a ^ { 2 } = 3 b {/tex}

##### Explanation

Correct Marks 4

Incorrectly Marks -1

Q 3. If {tex} z , \omega {/tex} are two non-zero complex numbers such that {tex} | z \omega | = 1 {/tex} and {tex} \arg ( z ) - \arg ( \omega ) = \pi / 2 , {/tex} then {tex} \overline { z } \omega {/tex} is equal to

A

{tex}-1{/tex}

B

{tex}1{/tex}

{tex} -i {/tex}

D

{tex} i {/tex}

##### Explanation

Correct Marks 4

Incorrectly Marks -1

Q 4. If {tex} \left| z ^ { 2 } - 1 \right| = | z | ^ { 2 } + 1 , {/tex} then {tex} z {/tex} lies on

A

the real axis

the imaginary axis

C

a circle

D

an ellipse

##### Explanation

Correct Marks 4

Incorrectly Marks -1

Q 5. If the cube roots of unity are 1 ,{tex}\omega{/tex}, {tex} \omega ^ { 2 } , {/tex} then the roots of the equation {tex} ( x - 1 ) ^ { 3 } + 8 = 0 {/tex} are

A

{tex} - 1,1 + 2 \omega , 1 + 2 \omega ^ { 2 } {/tex}

{tex} - 1,1 - 2 \omega , 1 - 2 \omega ^ { 2 } {/tex}

C

{tex} - 1 , - 1 , - 1 {/tex}

D

{tex} - 1 , - 1 + 2 \omega, - 1 , - 2 \omega ^ { 2 } {/tex}

##### Explanation

Correct Marks 4

Incorrectly Marks -1

Q 6. If {tex} z _ { 1 } {/tex} and {tex} z _ { 2 } {/tex} are two non-zero complex numbers such that {tex} \left| z _ { 1 } + z _ { 2 } \right| = \left| z _ { 1 } \right| + \left| z _ { 2 } \right| , {/tex} then {tex} \arg \left( z _ { 1 } \right) - \arg \left( z _ { 2 } \right) {/tex} is equal to

A

{tex} - \frac { \pi } { 2 } {/tex}

{tex}0{/tex}

C

{tex} - \pi {/tex}

D

{tex} \frac { \pi } { 2 } {/tex}

##### Explanation

Correct Marks 4

Incorrectly Marks -1

Q 7. If {tex} \omega = \frac { z } { z - \frac { 1 } { 3 } i } {/tex} and {tex} | \omega | = 1 , {/tex} then {tex} z {/tex} lies on

A

a parabola

a straight line

C

a circle

D

an ellipse

##### Explanation

Correct Marks 4

Incorrectly Marks -1

Q 8. If {tex} z ^ { 2 } + z + 1 = 0 , {/tex} where {tex} z {/tex} is a complex number, then the value of {tex} \left( z + \frac { 1 } { z } \right) ^ { 2 } + \left( z ^ { 2 } + \frac { 1 } { z ^ { 2 } } \right) ^ { 2 } + \left( z ^ { 3 } + \frac { 1 } { z ^ { 3 } } \right) ^ { 2 } + \cdots + \left( z ^ { 6 } + \frac { 1 } { z ^ { 6 } } \right) ^ { 2 } {/tex} is

A

18

B

54

C

6

12

##### Explanation

Correct Marks 4

Incorrectly Marks -1

Q 9. If {tex} | z + 4 | \leq 3 , {/tex} then the maximum value of {tex} | z + 1 | {/tex} is

A

0

B

4

C

10

6

##### Explanation

Correct Marks 4

Incorrectly Marks -1

Q 10. The conjugate of a complex number is {tex} \frac { 1 } { i - 1 } {/tex} . Then that complex number is

A

{tex} \frac { - 1 } { i - 1 } {/tex}

B

{tex} \frac { 1 } { i + 1 } {/tex}

{tex} \frac { - 1 } { i + 1 } {/tex}

D

{tex} \frac { 1 } { i - 1 } {/tex}

##### Explanation

Correct Marks 4

Incorrectly Marks -1

Q 11. If {tex} \left| z - \frac { 4 } { z } \right| = 2 , {/tex} then the maximum value of {tex} | z | {/tex} is equal to

A

{tex} \sqrt { 3 } + 1 {/tex}

{tex} \sqrt { 5 } + 1 {/tex}

C

{tex}2{/tex}

D

{tex} 2 + \sqrt { 2 } {/tex}

##### Explanation

Correct Marks 4

Incorrectly Marks -1

Q 12. The number of complex number z such that {tex} | z - 1 | = | z + 1 | = | z - i | {/tex} equals

A

{tex} \infty {/tex}

B

0

C

1

2

##### Explanation

Correct Marks 4

Incorrectly Marks -1

Q 13. If {tex} \alpha {/tex} and {tex} \beta {/tex} are the roots of the equation {tex} x ^ { 2 } - x + 1 = 0 {/tex} , then {tex} \alpha ^ { 2009 } + \beta ^ { 2009 } = ? {/tex}

A

2

B

-2

C

-1

1

##### Explanation

Correct Marks 4

Incorrectly Marks -1

Q 14. If {tex} \omega ( \neq 1 ) {/tex} is a cube root of unity, and {tex} ( 1 + \omega ) ^ { 7 } = A + B \omega . {/tex} Then {tex} ( A , B ) {/tex} equals

{tex} ( 1,1 ) {/tex}

B

{tex} ( 1,0 ) {/tex}

C

{tex} ( - 1,1 ) {/tex}

D

{tex} ( 0,1 ) {/tex}

##### Explanation

Correct Marks 4

Incorrectly Marks -1

Q 15. Let {tex} \alpha , \beta {/tex} be real and {tex} z {/tex} be a complex number. If {tex} z ^ { 2 } + \alpha z + \beta = 0 {/tex} has two distinct roots on the line {tex} R z = 1 {/tex} , then it is necessary that

A

{tex} \beta \in ( - 1,0 ) {/tex}

B

{tex} | \beta | = 1 {/tex}

{tex} \beta \in ( 1 , \infty ) {/tex}

D

{tex} \beta \in ( 0,1 ) {/tex}

##### Explanation

Correct Marks 4

Incorrectly Marks -1

Q 16. If {tex} z {/tex} is a complex number of unit modulus and argument {tex} \theta {/tex} , then {tex} \arg \left( \frac { 1 + z } { 1 + \overline { z } } \right) {/tex} equals

A

{tex}0{/tex}

B

{tex} \frac { \pi } { 2 } - \theta {/tex}

{tex}\theta{/tex}

D

{tex} \pi - \theta {/tex}

##### Explanation

Correct Marks 4

Incorrectly Marks -1

Q 17. If , but , then the equation whose roots are and is

A

C

D

None of these

##### Explanation

, . α, β are roots of . Therefore ,

Correct Marks 4

Incorrectly Marks -1

Q 18. If one root of the equation is the square of the other, then

A

B

C

##### Explanation

Let α and be the roots then ,
Now

Correct Marks 4

Incorrectly Marks -1

Q 19. If the roots of the equation x2 - 2ax + a2 + a - 3 = 0 are less than 3 then

a < 2

B

2 ≤ a ≤ 3

C

3 < a ≤ 4

D

a > 4

##### Explanation

Since roots are less than a real number, roots must be real
⇒ 4a2 - 4(a2 + a - 3) ≥ 0 ⇒ a ≤ 3. ... (1)
Let f(x) = x2 - 2ax + a2 + a - 3. Since 3 lie outside the roots,
f(3) > 0 ⇒ a < 2 or a > 3. . . . .(2)
From (1), (2) we have a < 2

Correct Marks 4

Incorrectly Marks -1

Q 20. Sum of the real roots of the equation x2 + 5|x| + 6 = 0

A

Equals to 5

B

Equals to 10

C

Equals to -5

Does not exit

##### Explanation

Since x2, 5|x| and 6 are positive so x2 + 5|x| + 6 = 0 does not have any real root. Therefore sum does not exist

Correct Marks 4

Incorrectly Marks -1

Q 21. If = 7 then x =

A

B

log7(3/4)

D

log7(4/3)

##### Explanation

x = log(3/4) 7 = =

Correct Marks 4

Incorrectly Marks -1

Q 22. The value of 'a' for which the sum of the squares of the roots of the equation x2 - (a - 2)x - a - 1 = 0 assumes the least value is :

A

0

1

C

2

D

3

##### Explanation

Here α+β=a-2,αβ=-(a-1), Now,α22=(α+β)2- 2αβ
= (a- 2)2 + 2 (a +1) = a2 - 2a + 6 = (a-1)2 + 5
So α2 + β2 will be maximum at a = 1.

Correct Marks 4

Incorrectly Marks -1

Q 23. If a, b, c are in G.P. then the equation ax2 + 2bx + c = 0 and dx2 + 2ex + f = 0 have a common root if d/a , e/b, f/c are in :

AP

B

GP

C

HP

D

None of these

##### Explanation

Here b = (As a, b, c → GP)
So, ax2 + 2 + c = 0 ⇒ (x+)2 = 0
or x = -/, Now, substituting in eqn (2);
dc/a - 2e/ + f = 0 ⇒ d/a - 2e/ + f/c = 0
⇒ d/a -2e/b + f/c = 0

Correct Marks 4

Incorrectly Marks -1

Q 24. If the roots of the equation = are such that α + β = 0, then the value of λ is-

B

c

C

D

##### Explanation

(λ + 1) (x2 - bx) = (λ - 1) (ax - c)
⇒ λx2 + (-bλ - b - aλ + a) x + cλ - c = 0
Since α + β = 0, we have
= 0
⇒ -bλ - b -aλ + a = 0 ⇒ λ = .

Correct Marks 4

Incorrectly Marks -1

Q 25. The equation x + = 1 + has-

No real root

B

One real root

C

Two equal roots

D

Infinitely

##### Explanation

We have x + = 1 +
⇒ x = 1 provided x = 1 ≠ 0 i.e., x ≠ 1.
The given equation has no solution.