The steady-state temperature distribution in a square plate ABCD is governed by the 2-dimensional Laplace equation. The side AB is kept at a temperature of 100°C and the other three sides are kept at a temperature of 0°C. Ignoring the effect of discontinuities in the boundary conditions at the corners, the steady-state temperature at the center of the plate is obtained as T₀°C. Due to symmetry, the steady-state temperature at the center will be same (T₀°C), when any one side of the square is kept at a temperature of 100°C and the remaining three sides are kept at a temperature of 0°C. Using the principle of superposition, the value of T₀ is _________ (rounded off to two decimal places).

The solution of the differential equation

$\rm \frac{d^3y}{dx^3}-5.5\frac{d^2y}{dx^2}+9.5\frac{dy}{dx}-5y=0$

is expressed as 𝑦 = 𝐶₁𝑒^2.5𝑥 + 𝐶₂𝑒^𝛼𝑥 + 𝐶₃𝑒^𝛽𝑥 , where 𝐶₁, 𝐶₂, 𝐶₃, 𝛼, and 𝛽 are constants, with α and β being distinct and not equal to 2.5. Which of the following options is correct for the values of 𝛼 and 𝛽?

Two vectors [2 1 0 3]^𝑇 and [1 0 1 2]^𝑇 belong to the null space of a 4 × 4 matrix of rank 2. Which one of the following vectors also belongs to the null space?

Cholesky decomposition is carried out on the following square matrix [𝐴]. 

$\rm [A]=\begin{bmatrix}8&-5\\\ -5&a_{22}\end{bmatrix}$

Let 𝑙_ij and 𝑎ij be the (i, j)^th elements of matrices [𝐿] and [𝐴], respectively. If the element 𝑙₂₂ of the decomposed lower triangular matrix [𝐿] is 1.968, what is the value (rounded off to the nearest integer) of the element 𝑎₂₂?