HPAS Maths Optional PYQs: Numerical Methods & Computer Programming
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On that page, you will find year-wise and question-wise solutions.
Based on the provided excerpts from the HPAS Maths Optional Question Papers (2014–2024), the questions related to Numerical Analysis and Computer Programming/Algorithms are compiled below, categorized by year.
HPAS 2024 Numerical Analysis and Computer Programming Questions
Question 1(d): Show that the Newton-Raphson process has a quadratic convergence.
Question 6(b): Using Euler’s method, solve the initial value problem \(\frac{dy}{dt} = 1 – t + 4y\) with \(y(0)=1\), in the interval \(0 \le t \le 0.5\) with step size \(h=0.1\). If the exact solution is given, compute the error and the percentage error.
Question 7(b): Write the algorithm for the Bisection method for finding a real root of \(f(x)=0\) which lies in \([a, b]\). Further, develop a simple program in C language for finding a real root of the equation \(x^3 – 2x – 1 = 0\) using the Bisection method.
HPAS 2023 Numerical Analysis and Computer Programming Questions
Question 7(b): By applying Gauss’s quadrature formula, compute the integral \(\int_{5}^{12} \frac{1}{x} dx\) and find the error.
Question 7(c): Show that the rate of convergence of the Newton-Raphson method is quadratic, and determine a root of the equation \(x^{10}-1=0\) with the initial point \(x_0 = 0.5\).
Question 8(a): Let the polynomial \(\phi(x)\) be of the form \(\phi(x) = \sum_{i=0}^{n} L_i(x)y_i\) (Lagrangian interpolation). Show that \(\sum_{i=0}^{n} L_i(x) = 1\).
Question 8(b): Using the 4th order Runge-Kutta method, solve the differential equation \(\frac{dy}{dx} = -xy^2\) with \(y(0)=1\). Taking a step size of \(h=0.2\), determine \(y(0.4)\).
HPAS 2021 Numerical Analysis and Computer Programming Questions
Question 5(b): Find the root of the equation \(x \sin x + \cos x = 0\) by using the Newton-Raphson method.
Question 8(a): Determine the maximum error in evaluating the integral \(\int_{0}^{\pi/2} \cos x \,dx\) by both the Trapezoidal and Simpson’s rules using four subintervals.
Question 8(b): Apply Euler’s modified method to find the value of \(y\) at \(x = 0.1\) correct to five decimal places, given \(\frac{dy}{dx} = x^2 + y\) with the initial condition \(y(0) = 0.94\).
HPAS 2020 Numerical Analysis and Computer Programming Questions
Question 1(d): Find the value of \(y(0.5)\) for the initial value problem \(\frac{dy}{dx}=y\), \(y(0)=1\), using Euler’s method with a step size of \(h=0.1\).
Question 5(b): Obtain the interpolating polynomial by Newton’s divided difference formula for the given data. Also, find the value of \(f(3.5)\).
Question 7(c): Find a real root of the equation \(xe^x – \cos x = 0\) using the Regula-Falsi method.
Question 8(a): Evaluate the integral \(I = \int_{0}^{1} \frac{dx}{1+x}\) correct to three decimal places by using the trapezoidal rule and Simpson’s one-third rule, taking \(h=0.25\).
Question 8(c): Write a flow chart in C language for the Runge-Kutta method of the fourth order.
HPAS 2019 Numerical Analysis and Computer Programming Questions
Question 1(d): Obtain the smallest positive root of the equation \(x \log_{10}x – 1.2 = 0\), correct to 2 decimal places, using the Newton-Raphson method.
Question 5(a): Solve the following system of equations by the Gauss-Seidel method, correct to 2 decimal places.
Question 5(b): From the following table, obtain \(f(2.07)\) using the best formula (Interpolation).
Question 8(a): What do the following explain in the C Language: (i) ++nc, (ii) else if (condition), (iii) tolower(c), (iv) Pointer.
HPAS 2018 Numerical Analysis and Computer Programming Questions
Question 8(a): Write an algorithm and draw a flow chart for integrating \(\int_{a}^{b} f(x) dx\) by the Trapezoidal rule, taking a step size \(h\).
Question 8(b): Write an algorithm and draw a flow chart for finding the value of \(y\) at \(x=x_n\) for the differential equation \(\frac{dy}{dx}=f(x,y)\), taking a step size \(h\), when the initial values of \(x\) and \(y\) are given, by Euler’s method.
HPAS 2017 Numerical Analysis and Computer Programming Questions
Question 8(b): Find a root of the equation \(x \log_{10}x – 1.2 = 0\) correct to four places of decimals (numerical root finding).
HPAS 2016 Numerical Analysis and Computer Programming Questions
Question 7(a): By using the Newton-Raphson method, find a root of the equation: \(x \sin x+\cos x=0\).
Question 7(b): Determine the cubic polynomial which takes the values \(y(0)=1\), \(y(1)=0\), \(y(2)=1\), \(y(3)=10\) (Interpolation).
Question 8(a): Draw a flowchart to find the roots of a quadratic equation \(ax^{2}+bx+c=0\).
HPAS 2015 Numerical Analysis and Computer Programming Questions
Question 1(e): Using method of false position (Regula-Falsi), find the real root of the equation: \(x^{3}-2x-5=0\).
Question 1(f): Draw a flow chart to find the largest number from two numbers.
Question 8(a): Find the polynomial of the lowest possible degree which assumes the values 3, 12, 15, -21; when \(x\) has values 3, 2, 1, -1 respectively (Interpolation).
Question 8(b): Using Euler’s method with step-size 0.1, find the value of \(y(0.5)\) from the differential equation: \(\frac{dy}{dx}=x^{2}+y^{2}, y(0)=0\).
HPAS 2014 Numerical Analysis and Computer Programming Questions
Question 1(f): Draw a flow chart to print all even numbers between 1 and 50.
Question 8(a): Find the roots of the quadratic equation: \(x^{2}-5x+2=0\), correct to four decimal places by the Newton-Raphson method.
Question 8(b): Evaluate \(\int_{0}^{1}\frac{dx}{1+x^{2}}\) using Simpson’s \(\frac{1}{3}\) and \(\frac{3}{8}\) rule. Hence obtain the approximate value of \(\pi\) in each case.