Numerical Analysis and Computer Programming – Mathematics Optional PYQs (2008–2026)

Here are all 114 Numerical Analysis and Computer Programming questions asked in UPSC Mathematics Optional Paper II from 2008 to 2026, arranged under the 7 topics of the official syllabus. Within each topic the latest questions come first.

Each question ends with a code such as [2024/10m/1a] – the year, the marks and the question number in the paper.

6.1] Solution of equations

28 questions

1. By using the Newton-Raphson iterative formula, establish the formula $x_{i+1}=\frac{1}{3}(2x_i+\frac{N}{x_i^2})$ to find the cube root of $N$. Use this formula, if established, to find the cube root of 63 correct to four decimal places with the initial approximation 3.9. [2026/10m/5b]

2. Solve the following system of linear equations by Gauss-Seidel method : $\begin{array}{cc}\hfill 10x+2y+z\hfill & \hfill =9\hfill \\ \hfill 2x+20y-2z\hfill & \hfill =-44\hfill \\ \hfill -2x+3y+10z\hfill & \hfill =22\hfill \end{array}$ [2025/10m/5b]

3. Solve the following system of linear equations by Gauss-Jordan method : $\begin{array}{cc}\hfill 2x+3y-z\hfill & \hfill =5\hfill \\ \hfill 4x+4y-3z\hfill & \hfill =3\hfill \\ \hfill 2x-3y+2z\hfill & \hfill =2\hfill \end{array}$ [2024/10m/5b]

4. Solve the system of linear equations $\begin{array}{cc}\hfill 7x_1-x_2+2x_3\hfill & \hfill =11\hfill \\ \hfill 2x_1+8x_2-x_3\hfill & \hfill =9\hfill \\ \hfill x_1-2x_2+9x_3\hfill & \hfill =7\hfill \end{array}$ correct up to 4 significant figures by the Gauss-Seidel iterative method. Take initially guessed solution as $x_1=x_2=x_3=0$. [2023/15m/6b]

5. Compute a root of the equation ${log}_{10}(2x+1)-x^2+3=0$, in the interval $[0,3]$, by Regula-Falsi method, correct to 6 decimal places. [2023/15m/8b]

6. Solve, by Gauss elimination method, the system of equations $\begin{array}{cc}\hfill 2x+2y+4z\hfill & \hfill =18\hfill \\ \hfill x+3y+2z\hfill & \hfill =13\hfill \\ \hfill 3x+y+3z\hfill & \hfill =14\hfill \end{array}$ [2022/10m/5b]

7. Find a positive root of the equation $3x=1+cosx$ by a numerical technique using initial values $0,\frac{\pi }{2}$; and further improve the result using Newton-Raphson method correct to 8 significant figures. [2021/10m/5b]

8. Solve the system of equations $\begin{array}{cc}\hfill 3x_1+9x_2-2x_3\hfill & \hfill =11\hfill \\ \hfill 4x_1+2x_2+13x_3\hfill & \hfill =24\hfill \\ \hfill 4x_1-2x_2+x_3\hfill & \hfill =-8\hfill \end{array}$ correct up to 4 significant figures by using Gauss-Seidel method after verifying whether the method is applicable in your transformed form of the system. [2021/15m/7b]

9. Show that the equation : $f\left(x\right)=cos\frac{\pi (x+1)}{8}+0.148x-0.9062=0$ has one root in the interval $(-1,0)$ and one in $(0,1)$. Calculate the negative root correct to four decimal places using Newton-Raphson method. [2020/10m/5b]

10. For the solution of the system of equations : $\begin{array}{cc}\hfill 4x+y+2z\hfill & \hfill =4\hfill \\ \hfill 3x+5y+z\hfill & \hfill =7\hfill \\ \hfill x+y+3z\hfill & \hfill =3\hfill \end{array}$ set up the Gauss-Seidel iterative scheme and iterate three times starting with the initial vector $X^{\left(0\right)}=0$. Also find the exact solutions and compare with the iterated solutions. [2020/15m/6b]

11. Apply Newton-Raphson method, to find a real root of transcendental equation $x{log}_{10}x=1.2$, correct to three decimal places. [2019/10m/5b]

12. Apply Gauss-Seidel iteration method to solve the following system of equations : $\begin{array}{cc}\hfill 2x+y-2z\hfill & \hfill =17,\hfill \\ \hfill 3x+20y-z\hfill & \hfill =-18,\hfill \\ \hfill 2x-3y+20z\hfill & \hfill =25,\hfill \end{array}$ correct to three decimal places. [2019/15m/7b]

13. Write down the basic algorithm for solving the equation : $x{e}^x-1=0$ by bisection method, correct to 4 decimal places. [2018/10m/5e]

14. Explain the main steps of the Gauss-Jordan method and apply this method to find the inverse of the matrix $\left[\begin{array}{ccc}\hfill 2\hfill & \hfill 6\hfill & \hfill 6\hfill \\ \hfill 2\hfill & \hfill 8\hfill & \hfill 6\hfill \\ \hfill 2\hfill & \hfill 6\hfill & \hfill 8\hfill \end{array}\right].$ [2017/10m/5b]

15. Write an algorithm in the form of a flow chart for Newton-Raphson method. Describe the cases of failure of this method. [2017/15m/8b]

16. Find the solution of the system $\begin{array}{cc}\hfill 10x_1-2x_2-x_3-x_4\hfill & \hfill =3\hfill \\ \hfill -2x_1+10x_2-x_3-x_4\hfill & \hfill =15\hfill \\ \hfill -x_1-x_2+10x_3-2x_4\hfill & \hfill =27\hfill \\ \hfill -x_1-x_2-2x_3+10x_4\hfill & \hfill =-9\hfill \end{array}$ using Gauss-Seidel method (make four iterations). [2015/15m/8b]

17. Apply Newton-Raphson method to determine a root of the equation $cosx-x{e}^x=0$ correct up to four decimal places. [2014/10m/5b]

18. Solve the system of equations $\begin{array}{cc}\hfill 2x_1-x_2\hfill & \hfill =7\hfill \\ \hfill -x_1+2x_2-x_3\hfill & \hfill =1\hfill \\ \hfill -x_2+2x_3\hfill & \hfill =1\hfill \end{array}$ using Gauss-Seidel iteration method (Perform three iterations). [2014/15m/6b]

19. Develop an algorithm for Newton — Raphson method to solve $f\left(x\right)=0$ starting with initial iterate $x_0$, $n$ be the number of iterations allowed, eps be the prescribed relative error and delta be the prescribed lower bound for $f\prime \left(x\right)$. [2013/20m/7a]

20. Use Newton-Raphson method to find the real root of the equation $3x=cosx+1$ correct to four decimal places. [2012/12m/5b]

21. Solve the following system of simultaneous equations, using Gauss-Seidel iterative method : $\begin{array}{cc}\hfill 3x+20y-z\hfill & \hfill =-18\hfill \\ \hfill 20x+y-2z\hfill & \hfill =17\hfill \\ \hfill 2x-3y+20z\hfill & \hfill =25.\hfill \end{array}$ [2012/20m/6c]

22. Find the positive root of the equation $10x{e}^{-x^2}-1=0$ correct up to 6 decimal places by using Newton-Raphson method. Carry out computations only for three iterations. [2010/12m/5c]

23. Given the system of equations $\begin{array}{cc}\hfill 2x+3y\hfill & \hfill =1\hfill \\ \hfill 2x+4y+z\hfill & \hfill =2\hfill \\ \hfill 2y+6z+Aw\hfill & \hfill =4\hfill \\ \hfill 4z+Bw\hfill & \hfill =C\hfill \end{array}$ State the solvability and uniqueness conditions for the system. Give the solution when it exists. [2010/20m/7a]

24. The equation $x^2+ax+b=0$ has two real roots $\alpha$ and $\beta$. Show that the iterative method given by : $x_{k+1}=-(a{x}_k+b)/x_k,k=0,1,2,\dots \dots$ is convergent near $x=\alpha$, if $\left|\alpha \right|>\left|\beta \right|$. [2009/6m/5c(i)]

25. Develop an algorithm for Regula-Falsi method to find a root of $f\left(x\right)=0$ starting with two initial iterates $x_0$ and $x_1$ to the root such that sign $\left(f\left(x_0\right)\right)\ne$ sign $\left(f\left(x_1\right)\right)$. Take $n$ as the maximum number of iterations allowed and eps be the prescribed error. [2009/30m/7a]

26. Find the smallest positive root of equation $x{e}^x-cosx=0$ using regula-falsi method. Do three iterations. [2008/12m/5c]

27. Apply Gauss-Seidel method to calculate $x$, $y$, $z$ from the system: $\begin{array}{cc}\hfill -x-y+6z\hfill & \hfill =42\hfill \\ \hfill 6x-y-z\hfill & \hfill =11.33\hfill \\ \hfill -x+6y-z\hfill & \hfill =32\hfill \end{array}$ with initial values $(4.67,7.62,9.05)$. Carry out computations for two iterations. [2008/15m/7a(ii)]

28. Draw a flow chart for solving equation $F\left(x\right)=0$ correct to five decimal places by Newton-Raphson method. [2008/30m/7b]

6.2] Interpolation

13 questions

1. Find the unique polynomial of degree 2 or less which fits the following data : $\begin{array}{cccc}\hfill x\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 3\hfill \\ \hfill f\left(x\right)\hfill & \hfill 1\hfill & \hfill 3\hfill & \hfill 55\hfill \end{array}$ Also obtain the bound on the truncation error. [2025/15m/7b]

2. Using Newton’s forward difference formula for interpolation, estimate the value of $f\left(2.5\right)$ from the following data : $\begin{array}{ccccccc}\hfill x\hfill & \hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill & \hfill 4\hfill & \hfill 5\hfill & \hfill 6\hfill \\ \hfill f\left(x\right)\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 8\hfill & \hfill 27\hfill & \hfill 64\hfill & \hfill 125\hfill \end{array}$ [2024/15m/8b]

3. Derive Newton’s backward difference interpolation formula and also do error analysis. [2021/15m/8b]

4. Write the three point Lagrangian interpolating polynomial relative to the points $x_0$, $x_0+\epsilon$ and $x_1$. Then by taking the limit $\epsilon \to 0$, establish the relation $f\left(x\right)=\frac{(x_1-x)(x+x_1-2x_0)}{{(x_1-x_0)}^2}f\left(x_0\right)+\frac{(x-x_0)(x_1-x)}{(x_1-x_0)}f\prime \left(x_0\right)+\frac{{(x-x_0)}^2}{(x_1-x_0)}f\left(x_1\right)+E\left(x\right)$ where $E\left(x\right)=\frac{1}{6}{(x-x_0)}^2(x-x_1)f‴\left(\xi \right)$ is the error function and $min.(x_0,x_0+\epsilon ,x_1)<\xi <max.(x_0,x_0+\epsilon ,x_1)$ [2020/15m/8b]

5. Using Newton’s forward difference formula find the lowest degree polynomial $u_x$ when it is given that $u_1=1$, $u_2=9$, $u_3=25$, $u_4=55$ and $u_5=105$. [2018/10m/5b]

6. For given equidistant values $u_{-1}$, $u_0$, $u_1$ and $u_2$, a value is interpolated by Lagrange’s formula. Show that it may be written in the form $u_x=y{u}_0+x{u}_1+\frac{y(y^2-1)}{3!}{\Delta }^2{u}_{-1}+\frac{x(x^2-1)}{3!}{\Delta }^2{u}_0,$ where $x+y=1$. [2017/15m/6b]

7. Let $f\left(x\right)=e^{2x}cos3x$, for $x\in [0,1]$. Estimate the value of $f\left(0.5\right)$ using Lagrange interpolating polynomial of degree 3 over the nodes $x=0$, $x=0.3$, $x=0.6$ and $x=1$. Also, compute the error bound over the interval $[0,1]$ and the actual error $E\left(0.5\right)$. [2016/20m/6c]

8. Find the Lagrange interpolating polynomial that fits the following data : $\begin{array}{ccccc}\hfill x\hfill & \hfill -1\hfill & \hfill 2\hfill & \hfill 3\hfill & \hfill 4\hfill \\ \hfill f\left(x\right)\hfill & \hfill -1\hfill & \hfill 11\hfill & \hfill 31\hfill & \hfill 69\hfill \end{array}$ Find $f\left(1.5\right)$. [2015/20m/6c]

9. In an examination, the number of students who obtained marks between certain limits were given in the following table : $\begin{array}{cccccc}\hfill \mathrm{\text{Marks}}\hfill & \hfill 30-40\hfill & \hfill 40-50\hfill & \hfill 50-60\hfill & \hfill 60-70\hfill & \hfill 70-80\hfill \\ \hfill \mathrm{\text{No. of Students}}\hfill & \hfill 31\hfill & \hfill 42\hfill & \hfill 51\hfill & \hfill 35\hfill & \hfill 31\hfill \end{array}$ Using Newton forward interpolation formula, find the number of students whose marks lie between 45 and 50. [2013/10m/5c]

10. Find $\frac{dy}{dx}$ at $x=0.1$ from the following data : $\begin{array}{ccccc}\hfill x\hfill & \hfill 0.1\hfill & \hfill 0.2\hfill & \hfill 0.3\hfill & \hfill 0.4\hfill \\ \hfill y\hfill & \hfill 0.9975\hfill & \hfill 0.9900\hfill & \hfill 0.9776\hfill & \hfill 0.9604\hfill \end{array}$ [2012/20m/7a]

11. Draw a flow chart for Lagrange’s interpolation formula. [2011/20m/7c]

12. Using Lagrange interpolation formula, calculate the value of $f\left(3\right)$ from the following table of values of $x$ and $f\left(x\right)$ : $\begin{array}{ccccccc}\hfill x\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 2\hfill & \hfill 4\hfill & \hfill 5\hfill & \hfill 6\hfill \\ \hfill f\left(x\right)\hfill & \hfill 1\hfill & \hfill 14\hfill & \hfill 15\hfill & \hfill 5\hfill & \hfill 6\hfill & \hfill 19\hfill \end{array}$ [2009/15m/7b]

13. The following values of the function $f\left(x\right)=sinx+cosx$ are given: $\begin{array}{cccc}\hfill x\hfill & \hfill {10}^{\circ }\hfill & \hfill {20}^{\circ }\hfill & \hfill {30}^{\circ }\hfill \\ \hfill f\left(x\right)\hfill & \hfill 1.1585\hfill & \hfill 1.2817\hfill & \hfill 1.3360\hfill \end{array}$ Construct the quadratic interpolating polynomial that fits the data. Hence calculate $f\left(\frac{\pi }{12}\right)$. Compare with exact value. [2008/15m/7a(i)]

6.3] Numerical integration

16 questions

1. Evaluate the integral ${\int }_0^{0.8}f\left(x\right)dx$, where $f\left(x\right)=0.2+25x-200x^2+675x^3-900x^4+400x^5$ by using single application of Simpson’s $\frac{3}{8}$ rule. [2026/15m/7b]

2. Find the constant $p$ and error term for the quadrature formula ${\int }_{x_0}^{x_1}f\left(x\right)dx=\frac{h}{2}(f_0+f_1)+p{h}^2(f_0\prime -f_1\prime )$ where $x_0+h=x_1$, $f_0=f\left(x_0\right)$, $f_1=f\left(x_1\right)$ and prime ($\prime$) represents derivative with respect to $x$. Hence deduce the composite rule for integrating ${\int }_a^{b}f\left(x\right)dx,a=x_0<x_1<\cdots <x_N=b$ [2025/15m/8b]

3. Integrate $f\left(x\right)=5x^3-3x^2+2x+1$ from $x=-2$ to $x=4$ using (i) Simpson’s $\frac{3}{8}$ rule with width $h=1$, and (ii) Trapezoidal rule with width $h=1$. [2024/15m/7b]

4. The velocity of a train which starts from rest is given by the following table, the time being reckoned in minutes from the start and the velocity in km/hour : $\begin{array}{ccccccccccc}\hfill t\left(\mathrm{\text{minutes}}\right)\hfill & \hfill 2\hfill & \hfill 4\hfill & \hfill 6\hfill & \hfill 8\hfill & \hfill 10\hfill & \hfill 12\hfill & \hfill 14\hfill & \hfill 16\hfill & \hfill 18\hfill & \hfill 20\hfill \\ \hfill v\left(\mathrm{\text{km/hour}}\right)\hfill & \hfill 16\hfill & \hfill 28.8\hfill & \hfill 40\hfill & \hfill 46.4\hfill & \hfill 51.2\hfill & \hfill 32\hfill & \hfill 17.6\hfill & \hfill 8\hfill & \hfill 3.2\hfill & \hfill 0\hfill \end{array}$ Using Simpson’s $\frac{1}{3}$rd rule, estimate approximately in km the total distance run in 20 minutes. [2022/15m/7b]

5. Find a quadrature formula ${\int }_0^{1}f\left(x\right)\frac{dx}{\sqrt{x(1-x)}}={\alpha }_{1}f\left(0\right)+{\alpha }_{2}f\left(\frac{1}{2}\right)+{\alpha }_{3}f\left(1\right)$ which is exact for polynomials of highest possible degree. Then use the formula to evaluate ${\int }_0^1\frac{dx}{\sqrt{x-x^3}}$ (correct up to three decimal places). [2020/20m/7b]

6. Draw a flow chart and write a basic algorithm (in FORTRAN/C/C${}^{++}$) for evaluating $y={\int }_0^6\frac{dx}{1+x^2}$ using Trapezoidal rule. [2019/10m/5e]

7. $\begin{array}{ccccccccccc}\hfill \mathrm{\text{Time (Minutes)}}\hfill & \hfill 2\hfill & \hfill 4\hfill & \hfill 6\hfill & \hfill 8\hfill & \hfill 10\hfill & \hfill 12\hfill & \hfill 14\hfill & \hfill 16\hfill & \hfill 18\hfill & \hfill 20\hfill \\ \hfill \mathrm{\text{Speed (Km/h)}}\hfill & \hfill 10\hfill & \hfill 18\hfill & \hfill 25\hfill & \hfill 29\hfill & \hfill 32\hfill & \hfill 20\hfill & \hfill 11\hfill & \hfill 5\hfill & \hfill 2\hfill & \hfill 8.5\hfill \end{array}$ Starting from rest in the beginning, the speed (in Km/h) of a train at different times (in minutes) is given by the above table : Using Simpson’s $\frac{1}{3}$rd rule, find the approximate distance travelled (in Km) in 20 minutes from the beginning. [2018/10m/5d]

8. Find the values of the constants $a$, $b$, $c$ such that the quadrature formula ${\int }_o^{h}f\left(x\right)dx=h[af\left(o\right)+bf\left(\frac{h}{3}\right)+cf\left(h\right)]$ is exact for polynomials of as high degree as possible, and hence find the order of the truncation error. [2018/15m/7b]

9. Derive the formula ${\int }_a^{b}ydx=\frac{3h}{8}[(y_0+y_n)+3(y_1+y_2+y_4+y_5+\cdots +y_{n-1})+2(y_3+y_6+\cdots +y_{n-3})].$ Is there any restriction on $n$? State that condition. What is the error bound in the case of Simpson’s $\frac{3}{8}$ rule? [2017/20m/7b]

10. For an integral ${\int }_{-1}^{1}f\left(x\right)dx$, show that the two-point Gauss quadrature rule is given by ${\int }_{-1}^{1}f\left(x\right)dx=f\left(\frac{1}{\sqrt{3}}\right)+f(-\frac{1}{\sqrt{3}})$. Using this rule, estimate ${\int }_2^{4}2x{e}^{x}dx$. [2016/15m/7c]

11. Use five subintervals to integrate ${\int }_0^1\frac{dx}{1+x^2}$ using trapezoidal rule. [2014/10m/5c]

12. Draw a flowchart for Simpson’s one-third rule. [2014/15m/7b]

13. The velocity of a train which starts from rest is given in the following table. The time is in minutes and velocity is in km/hour. $\begin{array}{ccccccccccc}\hfill t\hfill & \hfill 2\hfill & \hfill 4\hfill & \hfill 6\hfill & \hfill 8\hfill & \hfill 10\hfill & \hfill 12\hfill & \hfill 14\hfill & \hfill 16\hfill & \hfill 18\hfill & \hfill 20\hfill \\ \hfill v\hfill & \hfill 16\hfill & \hfill 28.8\hfill & \hfill 40\hfill & \hfill 46.4\hfill & \hfill 51.2\hfill & \hfill 32.0\hfill & \hfill 17.6\hfill & \hfill 8\hfill & \hfill 3.2\hfill & \hfill 0\hfill \end{array}$ Estimate approximately the total distance run in 30 minutes by using composite Simpson’s $\frac{1}{3}$ rule. [2013/15m/7c]

14. Calculate ${\int }_2^{10}\frac{dx}{1+x}$ (upto 3 places of decimal) by dividing the range into 8 equal parts by Simpson’s $\frac{1}{3}$rd Rule. [2011/12m/5c]

15. A solid of revolution is formed by rotating about the $x$-axis, the area between the $x$-axis, the line $x=0$ and $x=1$ and a curve through the points with the following co-ordinates : $\begin{array}{cccccc}\hfill x\hfill & \hfill .00\hfill & \hfill .25\hfill & \hfill .50\hfill & \hfill .75\hfill & \hfill 1\hfill \\ \hfill y\hfill & \hfill 1\hfill & \hfill .9896\hfill & \hfill .9589\hfill & \hfill .9089\hfill & \hfill .8415\hfill \end{array}$ Find the volume of the solid. [2011/20m/7a]

16. Find the value of the integral ${\int }_1^5{log}_{10}xdx$ by using Simpson’s $\frac{1}{3}$-rule correct up to 4 decimal places. Take 8 subintervals in your computation. [2010/20m/7b]

6.4] Numerical solution of ODEs

9 questions

1. Consider the initial value problem : $\frac{dy}{dx}=1+\frac{y}{x}$, $y\left(1\right)=1$, $h=1$ (Step-size) (i) Apply the classical fourth order Runge-Kutta method to approximate the solution $y\left(2\right)$. (ii) Use Euler’s method to find $y\left(2\right)$ and $y\left(3\right)$. [2026/15m/8b]

2. Given $\frac{dy}{dx}=\frac{y^2-x}{y^2+x}$ with initial condition $y=1$ at $x=0$. Find the value of $y$ for $x=0.4$ by Euler’s method, correct to 4 decimal places, taking step length $h=0.1$. [2023/10m/5b]

3. Using Runge-Kutta method of fourth order, solve the differential equation $\frac{dy}{dx}=x+y^2$ with $y\left(0\right)=1$, at $x=0.2$. Use four decimal places for calculation and step length 0.1. [2022/15m/8b]

4. Using Runge-Kutta method of fourth order, solve $\frac{dy}{dx}=\frac{y^2-x^2}{y^2+x^2}$ with $y\left(0\right)=1$ at $x=0.2$. Use four decimal places for calculation and step length 0.2. [2019/10m/5d]

5. Solve the initial value problem $\frac{dy}{dx}=x(y-x)$, $y\left(2\right)=3$ in the interval $[2,2.4]$ using the Runge-Kutta fourth-order method with step size $h=0.2$. [2015/15m/7b]

6. Use Runge-Kutta formula of fourth order to find the value of $y$ at $x=0.8$, where $\frac{dy}{dx}=\sqrt{x+y}$, $y\left(0.4\right)=0.41$. Take the step length $h=0.2$. [2014/20m/6c]

7. Use Euler’s method with step size $h=0.15$ to compute the approximate value of $y\left(0.6\right)$, correct up to five decimal places from the initial value problem $\begin{array}{cc}\hfill y\prime \hfill & \hfill =x(y+x)-2\hfill \\ \hfill y\left(0\right)\hfill & \hfill =2\hfill \end{array}$ [2013/15m/7b]

8. Provide a computer algorithm to solve an ordinary differential equation $\frac{dy}{dx}=f(x,y)$ in the interval $[a,b]$ for $n$ number of discrete points, where the initial value is $y\left(a\right)=\alpha$, using Euler’s method. [2012/12m/5c]

9. Find the value of $y\left(1.2\right)$ using Runge-Kutta fourth order method with step size $h=0.2$ from the initial value problem : $y\prime =xy,y\left(1\right)=2.$ [2009/15m/7c]

6.5] Number systems and representation

12 questions

1. Find the decimal equivalent of the following floating point 10-bit numbers with 5 bits as fractional part : 1010111011 and 0100101010. [2026/8m/6b(i)]

2. Evaluate ${\left(11001.1011\right)}_{10}+{\left(33.24\right)}_8+{\left(101101\right)}_2$ and convert the final result in hexadecimal system. [2026/7m/6b(ii)]

3. (i) Convert the number ${\left(3479\right)}_{10}$ into binary system and the number ${\left(7AE.9F\right)}_{16}$ into decimal system. (ii) Determine the truth table for the Boolean function $F(x,y,z)=(x+y+z\prime )(x\prime +y\prime )$ Also derive the full disjunctive normal form of $F(x,y,z)$ from the truth table. [2025/10m/5c]

4. (i) Determine the decimal equivalent in sign magnitude form of ${\left(8D\right)}_{16}$ and ${\left(FF\right)}_{16}$. (ii) Determine the decimal equivalent of ${\left(9B2.1A\right)}_{16}$. [2024/10m/5c]

5. Evaluate, using the binary arithmetic, the following numbers in their given system : (i) ${\left(634.235\right)}_8-{\left(132.223\right)}_8$ (ii) ${\left(7AB.432\right)}_{16}-{\left(5CA.D61\right)}_{16}$ [2023/10m/5c]

6. (i) Convert the number ${\left(1093.21875\right)}_{10}$ into octal and the number ${\left(1693.0628\right)}_{10}$ into hexadecimal systems. (ii) Express the Boolean function $F(x,y,z)=xy+x\prime z$ in a product of maxterms form. [2022/10m/5c]

7. (i) Convert ${\left(3798.3875\right)}_{10}$ into octal and hexadecimal equivalents. (ii) Obtain the principal conjunctive normal form of $(\neg P\to R)\wedge (Q\rightleftarrows P)$. [2021/10m/5c]

8. Find the equivalent numbers given in a specified number to the system mentioned against them : (i) Integer 524 in binary system. (ii) $101010110101.101101011$ to octal system. (iii) decimal number 5280 to hexadecimal system. (iv) Find the unknown number ${\left(1101.101\right)}_8\to \left(?\right)10$. [2019/15m/6b]

9. Find the equivalent of numbers given in a specified number system to the system mentioned against them. (i) ${\left(111011.101\right)}_2$ to decimal system (ii) ${\left(1000111110000.00101100\right)}_2$ to hexadecimal system (iii) ${\left(C4F2\right)}_{16}$ to decimal system (iv) ${\left(418\right)}_{10}$ to binary system. [2018/15m/6b]

10. Convert the following decimal numbers to equivalent binary and hexadecimal numbers : (i) $4096$ (ii) $0.4375$ (iii) $2048.0625$ [2016/10m/5d]

11. (i) Compute ${\left(3205\right)}_{10}$ to the base 8. (ii) Let $A$ be an arbitrary but fixed Boolean algebra with operations $\wedge$, $\vee$ and $\prime$ and the zero and the unit element denoted by 0 and 1 respectively. Let $x,y,z,\dots \dots$ be elements of $A$. If $x,y\in A$ be such that $x\wedge y=0$ and $x\vee y=1$ then prove that $y=x\prime$. [2011/12m/5d]

12. Find the decimal equivalent of ${\left(357.32\right)}_8$. [2009/6m/5d(ii)]

6.6] Computer systems, logic gates and Boolean algebra

25 questions

1. Consider the logic circuit $L$ : [Figure: logic circuit $L$ with inputs $A$, $B$, $C$; the first AND gate takes $A$, $B$, $C$; the second AND gate takes $A$, $\bar{B}$ (through a NOT gate) and $C$; the third AND gate takes $\bar{A}$ (through a NOT gate) and $B$; the outputs of the three AND gates feed an OR gate whose output is $Y$] (i) Express $Y$ as a Boolean expression. (ii) Write 8-bit special sequences for $A$, $B$ and $C$. (iii) Find the truth table of $L$ using 8-bit special sequences. [2026/10m/5c]

2. (i) Convert the number ${\left(3479\right)}_{10}$ into binary system and the number ${\left(7AE.9F\right)}_{16}$ into decimal system. (ii) Determine the truth table for the Boolean function $F(x,y,z)=(x+y+z\prime )(x\prime +y\prime )$ Also derive the full disjunctive normal form of $F(x,y,z)$ from the truth table. [2025/10m/5c]

3. Simplify the Boolean function $F(x,y,z)=xyz+x\prime yz+xy\prime z+xyz\prime$ and draw the corresponding GATE network. [2025/15m/6b]

4. Draw the logical circuit for the Boolean expression $Y=AB\bar{C}+B\bar{C}+\bar{A}B$. Also, obtain the output $Y$ (truth table) for the three input bit sequences : $A=10001111$, $B=00111100$, $C=11000100$ [2024/15m/6b]

5. (i) Find the conjunctive normal form (CNF) of the following Boolean function : $f(x,y,z,t)=x\cdot y\cdot z+\bar{x}\cdot y\cdot (t+\bar{z})$ \(ii\) Express the Boolean function $f(x,y,z)=x+\overline{(\bar{x}\cdot \bar{y}+\bar{x}\cdot z)}+z$ in disjunctive normal form (DNF) and construct the truth table for the function. [2023/15m/7a]

6. (i) Convert the number ${\left(1093.21875\right)}_{10}$ into octal and the number ${\left(1693.0628\right)}_{10}$ into hexadecimal systems. (ii) Express the Boolean function $F(x,y,z)=xy+x\prime z$ in a product of maxterms form. [2022/10m/5c]

7. Find a combinatorial circuit corresponding to the Boolean function $f(x,y,z)=[x\cdot (\bar{y}+z)]+y$ and write the input/output table for the circuit. [2022/15m/6b]

8. (i) Convert ${\left(3798.3875\right)}_{10}$ into octal and hexadecimal equivalents. (ii) Obtain the principal conjunctive normal form of $(\neg P\to R)\wedge (Q\rightleftarrows P)$. [2021/10m/5c]

9. Obtain the Boolean function $F(x,y,z)$ based on the table given below. Then simplify $F(x,y,z)$ and draw the corresponding GATE network : $\begin{array}{cccc}\hfill x\hfill & \hfill y\hfill & \hfill z\hfill & \hfill F(x,y,z)\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}$ [2021/15m/6b]

10. Let $g(w,x,y,z)=(w+x+y)(x+\bar{y}+z)(w+\bar{y})$ be a Boolean function. Obtain the conjunctive normal form for $g(w,x,y,z)$. Also express $g(w,x,y,z)$ as a product of maxterms. [2020/10m/5c]

11. Given the Boolean expression $X=AB+ABC+A\bar{B}\bar{C}+A\bar{C}$ (i) Draw the logical diagram for the expression. (ii) Minimize the expression. (iii) Draw the logical diagram for the reduced expression. [2019/15m/8a]

12. Simplify the boolean expression : $(a+b)\cdot (\bar{b}+c)+b\cdot (\bar{a}+\bar{c})$ by using the laws of boolean algebra. From its truth table write it in minterm normal form. [2018/15m/8a]

13. Write the Boolean expression $z(y+z)(x+y+z)$ in its simplest form using Boolean postulate rules. Mention the rules used during simplification. Verify your result by constructing the truth table for the given expression and for its simplest form. [2017/10m/5c]

14. Let $A$, $B$, $C$ be Boolean variables, $\bar{A}$ denote complement of $A$, $A+B$ is an expression for $A$ OR $B$ and $A\cdot B$ is an expression for $A$ AND $B$. Then simplify the following expression and draw a block diagram of the simplified expression, using AND and OR gates. $A\cdot (A+B+C)\cdot (\bar{A}+B+C)\cdot (A+\bar{B}+C)\cdot (A+B+\bar{C}).$ [2016/15m/8c]

15. Find the principal (or canonical) disjunctive normal form in three variables $p$, $q$, $r$ for the Boolean expression $((p\wedge q)\to r)\vee ((p\wedge q)\to -r)$. Is the given Boolean expression a contradiction or a tautology? [2015/10m/5c]

16. Use only AND and OR logic gates to construct a logic circuit for the Boolean expression $z=xy+uv$. [2014/10m/5d]

17. For any Boolean variables $x$ and $y$, show that $x+xy=x$. [2014/15m/8b]

18. (i) Compute ${\left(3205\right)}_{10}$ to the base 8. (ii) Let $A$ be an arbitrary but fixed Boolean algebra with operations $\wedge$, $\vee$ and $\prime$ and the zero and the unit element denoted by 0 and 1 respectively. Let $x,y,z,\dots \dots$ be elements of $A$. If $x,y\in A$ be such that $x\wedge y=0$ and $x\vee y=1$ then prove that $y=x\prime$. [2011/12m/5d]

19. Find the logic circuit that represents the following Boolean function. Find also an equivalent simpler circuit : $\begin{array}{cccc}\hfill x\hfill & \hfill y\hfill & \hfill z\hfill & \hfill f(x,y,z)\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}$ [2011/20m/7b]

20. (i) Suppose a computer spends 60 per cent of its time handling a particular type of computation when running a given program and its manufacturers make a change that improves its performance on that type of computation by a factor of 10. If the program takes 100 sec to execute, what will its execution time be after the change ? (ii) If $A\oplus B=AB\prime +A\prime B$, find the value of $x\oplus y\oplus z$. [2010/12m/5d]

21. (i) Find the hexadecimal equivalent of the decimal number ${\left(587632\right)}_{10}$ (ii) For the given set of data points $(x_1,f\left(x_1\right)),(x_2,f\left(x_2\right)),\dots \dots ,(x_n,f\left(x_n\right))$ write an algorithm to find the value of $f\left(x\right)$ by using Lagrange’s interpolation formula. (iii) Using Boolean algebra, simplify the following expressions (i) $a+a\prime b+a\prime b\prime c+a\prime b\prime c\prime d+\dots \dots$ (ii) $x\prime y\prime z+yz+xz$ where $x\prime$ represents the complement of $x$. [2010/20m/7c]

22. Find the values of two valued boolean variables $A$, $B$, $C$, $D$ by solving the following simultaneous equations : $\begin{array}{cc}\hfill \bar{A}+AB\hfill & \hfill =0\hfill \\ \hfill AB\hfill & \hfill =AC\hfill \\ \hfill AB+A\bar{C}+CD\hfill & \hfill =\bar{C}D\hfill \end{array}$ where $\bar{x}$ denotes the complement of $x$. [2009/6m/5c(ii)]

23. Realize the following expression by using NAND gates only : $g=(\bar{a}+\bar{b}+c)\bar{d}(\bar{a}+e)f$ where $\bar{x}$ denotes the complement of $x$. [2009/6m/5d(i)]

24. State the principle of duality in Boolean algebra and given the dual of the Boolean expressions $(X+Y)\cdot (\overline{X}\cdot \overline{Z})\cdot (Y+Z)$ and $X\cdot \overline{X}=0$ [2008/6m/5d(i)]

25. Represent $(\overline{A}+\overline{B}+\overline{C})(A+\overline{B}+C)(A+B+\overline{C})$ in NOR to NOR logic network. [2008/6m/5d(ii)]

6.7] Algorithms and flow charts

11 questions

1. Draw a flow chart and write a basic algorithm (in FORTRAN/C/C${}^{++}$) for evaluating $y={\int }_0^6\frac{dx}{1+x^2}$ using Trapezoidal rule. [2019/10m/5e]

2. Write down the basic algorithm for solving the equation : $x{e}^x-1=0$ by bisection method, correct to 4 decimal places. [2018/10m/5e]

3. Write an algorithm in the form of a flow chart for Newton-Raphson method. Describe the cases of failure of this method. [2017/15m/8b]

4. Draw a flowchart for Simpson’s one-third rule. [2014/15m/7b]

5. Develop an algorithm for Newton — Raphson method to solve $f\left(x\right)=0$ starting with initial iterate $x_0$, $n$ be the number of iterations allowed, eps be the prescribed relative error and delta be the prescribed lower bound for $f\prime \left(x\right)$. [2013/20m/7a]

6. Provide a computer algorithm to solve an ordinary differential equation $\frac{dy}{dx}=f(x,y)$ in the interval $[a,b]$ for $n$ number of discrete points, where the initial value is $y\left(a\right)=\alpha$, using Euler’s method. [2012/12m/5c]

7. In a certain examination, a candidate has to appear for one major and two minor subjects. The rules for declaration of results are : marks for major are denoted by $M_1$ and for minors by $M_2$ and $M_3$. If the candidate obtains 75% and above marks in each of the three subjects, the candidate is declared to have passed the examination in first class with distinction. If the candidate obtains 60% and above marks in each of the three subjects, the candidate is declared to have passed the examination in first class. If the candidate obtains 50% or above in major, 40% or above in each of the two minors and an average of 50% or above in all the three subjects put together, the candidate is declared to have passed the examination in second class. All those candidates, who have obtained 50% and above in major and 40% or above in minor, are declared to have passed the examination. If the candidate obtains less than 50% in major or less than 40% in any one of the two minors, the candidate is declared to have failed in the examinations. Draw a flow chart to declare the results for the above. [2012/20m/7c]

8. Draw a flow chart for Lagrange’s interpolation formula. [2011/20m/7c]

9. (i) Find the hexadecimal equivalent of the decimal number ${\left(587632\right)}_{10}$ (ii) For the given set of data points $(x_1,f\left(x_1\right)),(x_2,f\left(x_2\right)),\dots \dots ,(x_n,f\left(x_n\right))$ write an algorithm to find the value of $f\left(x\right)$ by using Lagrange’s interpolation formula. (iii) Using Boolean algebra, simplify the following expressions (i) $a+a\prime b+a\prime b\prime c+a\prime b\prime c\prime d+\dots \dots$ (ii) $x\prime y\prime z+yz+xz$ where $x\prime$ represents the complement of $x$. [2010/20m/7c]

10. Develop an algorithm for Regula-Falsi method to find a root of $f\left(x\right)=0$ starting with two initial iterates $x_0$ and $x_1$ to the root such that sign $\left(f\left(x_0\right)\right)\ne$ sign $\left(f\left(x_1\right)\right)$. Take $n$ as the maximum number of iterations allowed and eps be the prescribed error. [2009/30m/7a]

11. Draw a flow chart for solving equation $F\left(x\right)=0$ correct to five decimal places by Newton-Raphson method. [2008/30m/7b]

« Previous: Partial Differential Equations  |  Next: Mechanics and Fluid Dynamics »

Back to: Mathematics Optional PYQs – Topic-wise | Also see: Mathematics Optional year-wise question papers

Last updated: 30 October 2026