HSC Arts Class 12 - Tamil Nadu Board of Secondary Education Question Bank Solutions

Test for consistency and if possible, solve the following systems of equations by rank method:

2x – y + z = 2, 6x – 3y + 3z = 6, 4x – 2y + 2z = 4

Find the value of k for which the equations kx – 2y + z = 1, x – 2ky + z = -2, x – 2y + kz = 1 have no solution

Find the value of k for which the equations kx – 2y + z = 1, x – 2ky + z = -2, x – 2y + kz = 1 have unique solution

Find the value of k for which the equations kx – 2y + z = 1, x – 2ky + z = -2, x – 2y + kz = 1 have infinitely many solution

Investigate the values of λ and µ the system of linear equations 2x + 3y + 5z = 9, 7x + 3y – 5z = 8, 2x + 3y + λz = µ, have no solution

Investigate the values of λ and µ the system of linear equations 2x + 3y + 5z = 9, 7x + 3y – 5z = 8, 2x + 3y + λz = µ, have a unique solution

Investigate the values of λ and µ the system of linear equations 2x + 3y + 5z = 9, 7x + 3y – 5z = 8, 2x + 3y + λz = µ, have an infinite number of solutions

Solve the following system of homogenous equations:

3x + 2y + 7z = 0, 4x – 3y – 2z = 0, 5x + 9y + 23z = 0

Solve the following system of homogenous equations:

2x + 3y – z = 0, x – y – 2z = 0, 3x + y + 3z = 0

Determine the values of λ for which the following system of equations x + y + 3z = 0; 4x + 3y + λz = 0, 2x + y + 2z = 0 has a unique solution

Determine the values of λ for which the following system of equations x + y + 3z = 0; 4x + 3y + λz = 0, 2x + y + 2z = 0 has a non-trivial solution

By using Gaussian elimination method, balance the chemical reaction equation:

\[\ce{C2H + O2 -> H2O + CO2}\]

Find the modulus of the following complex numbers

`(2"i")/(3 + 4"i")`

Find the modulus of the following complex numbers

`(2 - "i")/(1 + "i") + (1 - 2"i")/(1 - "i")`

Find the modulus of the following complex numbers

(1 – i)¹⁰

Find the modulus of the following complex numbers

2i(3 – 4i)(4 – 3i)

For any two complex numbers z₁ and z₂, such that |z₁| = |z₂| = 1 and z₁ z₂ ≠ -1, then show that `(z_1 + z_2)/(1 + z_1 z_2)` is real number

Which one of the points 10 – 8i, 11 + 6i is closest to 1 + i

If |z| = 3, show that 7 ≤ |z + 6 – 8i| ≤ 13

If |z| = 1, show that 2 ≤ |z² – 3| ≤ 4