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**Q.1. If where z = x + iy, then the point (x, y) lies on a (2020)(1) Circle whose centre is at (2) Straight line whose slope is -2/3.(3) Straight line whose slope is 3/2.(4) Circle whose diameter is âˆš5/2.Ans. **(4)

We have

...(1)

Therefore,

Center and radius

Hence, the diameter of circle is âˆš5/2.

(1) Ï€ - tan

(2) Ï€ - tan

(3) - tan

(4) - tan

Ans.

We have

Given that the number is real, then

4 sinÎ¸ + 3 cosÎ¸ = 0 â‡’ tan Î¸ = -3/4

Hence, the argument of sinÎ¸ + i cosÎ¸ is

(1)

(2) âˆš10

(3) âˆš7

(4) âˆš8

Ans.

Let the complex number be z = x + iy. So,

|x|+|y| = 4 (1)

Now,

Minimum value of

Maximum value of |z| = 4

So, |z| âˆˆ (2âˆš2, 4).

Hence, the value of |z| cannot be âˆš7.**Q.4. Let Then the sum of the elements in A is: (2019)(1) 5Ï€/6(2) Ï€(3) 3Ï€/4(4) 2Ï€/3Ans.** (4)

Since, z is purely imaginary, then

Now, the sum of elements in A =

(1) Ï€/4

(2) Ï€/6

(3) Ï€/3

(4) 0

Ans.

âˆµ z

Ans.

(1) I(z) = 0

(2) R(z) > 0 and I(z) > 0

(3) R(z) < 0 and I(z) > 0

(4) R(z) = -(3)

Ans.

(1) 91

(2) -85

(3) 85

(4) -91

Ans.

Then |z| is equal to: (2019)

(1)

(2) 5/3

(3)

(4) 5/4

Ans.

Since, |z| + z = 3 + i

Let z = a + ib, then

Compare real and imaginary coefficients on both sides

Then,

(1) 2

(2) 1

(3) 1/2

(4)

Ans.

âˆµ t is purely imaginary number.

(2) âˆš2

(3) 1

(4) 2

Ans.

|Z

z

z

So, minimum value of |z

(1) 0

(2) 1

(3) (-1 + 2i)

(4) -1

Ans.

where Ï‰ is imaginary cube root of unity.

(1) straight line whose slope is 1.

Ans.

Let zâˆˆS then

Since, z is a complex number and let z = x + iy

Then, (by rationalisation)

Then compare both sides

...(1)

...(2)

Now squaring and adding equations (1) and (2)

(1) 5 Re (Ï‰) > 4

(2) 4 Im (Ï‰) > 5

(3) 5 Re (Ï‰) >1

(4) 5 Im (Ï‰) < 1

Ans.

(1)

(2)

(3)

(4)

Ans.

Then, from equation (1),

Now, square on both side; we get

Ans.

(1) a circle of radius 1/2.

(2) the line through the origin with slope 1.

(3) a circle of radius 1.

(4) the line through the origin with slope -1.

Ans.

Given equation is, |z - 1| = |z - i|

Hence, locus is straight line with slope 1.

(1) n = 20 and Re(z) = -10

(2) n = 40 and Re(z) = 10

(3) n = 40 and Re(z) = -10

(4) n = 20 and Re(z) = 10

Ans.

On comparing real and imaginary parts,

Hence, Re(z) = -10

(1) -1

(2) 0

(3) 1

(4) 2

Ans.

(1) An empty set

(2)

(3) equal to R

(4) {0}

Ans.

If Re(z) â‰ 1

then, Î± = 0

(1) 2

(2) 5

(3) 6

(4) 3

Ans.

Also,

âˆ´ Least positive integer n is 3.

Q.22. Let z âˆˆ C, the set of complex numbers. Then the equation, 2|z + 3i| â€“ |z â€“ i| = 0 represents a circle with radius (2017)

(1) a cirlce with radius 8/3

(2) an ellipse with length of minor axis 16/9

(3) an ellipse with length of major axis 16/3

(4) a circle with diameter 10/3

Ans.

(1) 1

(2) 2

(3) 3/4

(4) 1/2

Ans.

(1) Ï€/3

(2) Ï€/6

(3) sin

(4) sin

Ans.

To be purely imaginary if

(1) 2 + 2i

(2) â€“ 2 â€“ 2i

(3) 1 + i

(4) â€“ 1 â€“ i

Ans.

By rotation theorem

Ans.

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