HSC Science Class 12 - Tamil Nadu Board of Secondary Education Question Bank Solutions for Mathematics

If z = x + iy is a complex number such that Im `((2z + 1)/("i"z + 1))` = 0, show that the locus of z is 2x² + 2y² + x – 2y = 0

Obtain the Cartesian form of the locus of z = x + iy in the following cases:

[Re(iz)]² = 3

Obtain the Cartesian form of the locus of z = x + iy in the following cases:

Im[(1 – i)z + 1] = 0

Obtain the Cartesian form of the locus of z = x + iy in the following cases:

|z + i| = |z – 1|

Obtain the Cartesian form of the locus of z = x + iy in the following cases:

`bar(z) = z^-1`

Show that the following equations represent a circle, and, find its centre and radius.

|z – 2 – i| = 3

Show that the following equations represent a circle, and, find its centre and radius.

|2z + 2 – 4i| = 2

Show that the following equations represent a circle, and, find its centre and radius.

|3z – 6 + 12i| = 8 

Obtain the Cartesian equation for the locus of z = x + iy in the following cases:

|z – 4| = 16

Obtain the Cartesian equation for the locus of z = x + iy in the following cases:

|z – 4|² – |z – 1|² = 16

Choose the correct alternative:

If  `|"z" - 3/2|`, then the least value of |z| is

Choose the correct alternative:

If |z| = 1, then the value of  `(1 + "z")/(1 + "z")` is

Choose the correct alternative:

If |z₁| = 1,|z₂| = 2, |z₃| = 3 and |9z₁z₂ + 4z₁z₃ + z₂z₃| = 12, then the value of |z₁ + z₂ + z₃| is

Choose the correct alternative:

If z = x + iy is a complex number such that |z + 2| = |z – 2|, then the locus of z is

Choose the correct alternative:

The principal argument of the complex number `((1 + "i" sqrt(3))^2)/(4"i"(1 - "i" sqrt(3))` is

Solve the cubic equation: 2x³ – x² – 18x + 9 = 0 if sum of two of its roots vanishes

Solve the equation 9x³ – 36x² + 44x – 16 = 0 if the roots form an arithmetic progression

Solve the equation 3x³ – 26x² + 52x – 24 = 0 if its roots form a geometric progression

Determine k and solve the equation 2x³ – 6x² + 3x + k = 0 if one of its roots is twice the sum of the other two roots

Find all zeros of the polynomial x⁶ – 3x⁵ – 5x⁴ + 22x³ – 39x² – 39x + 135, if it is known that 1 + 2i and `sqrt(3)` are two of its zeros