Other / Removed from syllabus

The secant method is used to find the root of an equation $f(x) = 0$. It is started from two distinct estimates $x_a$ and $x_b$ for the root. It is an iterative procedure involving linear interpolation to a root. The iteration stops if $f(x_b)$ is very small and then $x_b$ is the solution. The procedure is given below. Observe that there is an expression which is missing and is marked by ?. Which is the suitable expression that is to be put in place of ? so that it follows all steps of the secant method?

Secant
Initialize: xa, xb, ε, N     // ε = convergence indicator
                              // N = maximum no. of iterations
    fb = f(xb)
    i = 0
    while (i < N and |fb| > ε) do
        i = i + 1             // update counter
        xt = ?                // missing expression for
                              // intermediate value
        xa = xb               // reset xa
        xb = xt               // reset xb
        fb = f(xb)            // function value at new xb
    end while
    if |fb| > ε then          // loop is terminated with i=N
        write "Non-convergence"
    else
        write "return xb"
    end if

The velocity $v$ (in kilometer/minute) of a motorbike which starts from rest, is given at fixed intervals of time $t$ (in minutes) as follows: $$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline t & 2 & 4 & 6 & 8 & 10 & 12 & 14 & 16 & 18 & 20 \\ \hline v & 10 & 18 & 25 & 29 & 32 & 20 & 11 & 5 & 2 & 0 \\ \hline \end{array}$$ The approximate distance (in kilometers) rounded to two places of decimals covered in 20 minutes using Simpson's 1/3rd rule is _____.

Function $f$ is known at the following points: $$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|} \hline x & 0 & 0.3 & 0.6 & 0.9 & 1.2 & 1.5 & 1.8 & 2.1 & 2.4 & 2.7 & 3.0 \\ \hline f(x) & 0 & 0.09 & 0.36 & 0.81 & 1.44 & 2.25 & 3.24 & 4.41 & 5.76 & 7.29 & 9.00 \\ \hline \end{array}$$ The value of $\int_0^3 f(x)\,dx$ computed using the trapezoidal rule is

The bisection method is applied to compute a zero of the function $f(x) = x^4 - x^3 - x^2 - 4$ in the interval [1,9]. The method converges to a solution after _____ iterations.