Q 1: An ordinary differential equation (ODE), \frac{dy}{dx}=2y , with an initial condition y(0) = 1, has the analytical solution y=e^{2x} . Using the Runge-Kutta second-order method, numerically integrate the ODE to calculate y at x = 0.5 using a step size of h = 0.5. If the relative percentage error is defined as,
\varepsilon=\left|\frac{y_{analytical}-y_{numerical}}{y_{analytical}}\right|\times100then the value of ε at x = 0.5 is ___________
Q 2: The function cos (x) is approximated using the Taylor series around x = 0 as cos (x) ≈ 1 + a x + b x2 + c x3 + d x4. The values of a, b, c, and d are
Q 3: For the function given below, the correct statements is/are
f\left(x\right)=\begin{pmatrix}-x&x<0\\x^2&x\geq0\end{pmatrix}Q 4: A, B, C, and D are vectors of length 4.
A=\begin{bmatrix}a_1&a_2&a_3&a_4\end{bmatrix} , B=\begin{bmatrix}b_1&b_2&b_3&b_4\end{bmatrix} , C=\begin{bmatrix}c_1&c_2&c_3&c_4\end{bmatrix} , and D=\begin{bmatrix}d_1&d_2&d_3&d_4\end{bmatrix}
It is known that B is not a scalar multiple of A. Also, C is linearly independent of A and B. Further, D = 3 A + 2 B + C. The rank of the matrix \begin{bmatrix}a_1&a_2&a_3&a_4\\b_1&b_2&b_3&b_4\\c_1&c_2&c_3&c_4\\d_1&d_2&d_3&d_4\end{bmatrix} is ______.
Q 5: Let A be a square matrix of size n×n (n>1). The elements of A = {aij} are given by a_{ij}=\begin{Bmatrix}i\times j,&if\;i\geq j\\0,&if\;i<j\end{Bmatrix} . The determinant of A is
Q 6: To solve an algebraic equation f(x)= 0, an iterative scheme of the type x_{(n+1)}=g(x_n) is proposed, where
g(x)=x-\frac{f(x)}{f'(x)}At the solution x = s, g'(s) = 0, g”(s) ≠ 0. The order of convergence for this iterative scheme near the solution is ____.
Q 7: The probability distribution function of a random variable X is shown in the following figure.
From this distribution, random samples with sample size n = 68 are taken. If X̅, i.e. \sigma_{\overline X} is ____________ (round off to 3 decimal places).
Q 8: For the ordinary differential equation
\frac{d^3y}{dt^3}+6\frac{d^2y}{dt^2}+11\frac{dy}{dt}+6y=1With initial conditions y(0)=y'(0)=y''(0)=y'''(0)=0 , the value of \lim\limits_{t\rightarrow\infty}y(t) =____________ (round off to 3 decimal places).