UNIT-I
Solution of Algebraic and Transcendental equations and
Numerical Integration
1. Using bisection method find the negative root of 𝑥 3 − 4𝑥 + 9 = 0
correct to three decimal places.
2. Find the root of the equation 𝑥 3 − 𝑥 2 + 𝑥 − 7 = 0 correct to three
decimal places using the bisection method.
3. Find a real root of the equation 𝑥 3 + 2𝑥 2 + 10𝑥 − 20 = 0 by using
Newton-Raphson method.
4. Solve s𝑖𝑛𝑥 = 1 + 𝑥 3 by using Newton-Raphson method.
(Hint: Get any one of the root of given equation)
5. Find a real root of the equation 𝑥 3 − 2𝑥 − 5 = 0 by the method of false
position correct to three decimal places.
6. Find a real root of the equation 𝑥 3 + 𝑥 2 − 1 = 0 by iteration method.
7. Find the least positive root of equation 𝑥 3 – 2 𝑥 + 0.5 = 0
by using Newton- Raphson method.
8. Find the real root of the equation
2𝑥−𝑙𝑜𝑔10 𝑥 = 7 which lies between 3.5
and 4 by using Regula-falsi method.
9. Find a real root for 𝑥 𝑡𝑎𝑛𝑥 + 1 = 0 by using Newton-Raphson method.
1
10. Find a seventh approximate root of the equation 𝑠𝑖𝑛𝑥 =
𝑥
by using bisection method.
11. Construct Newton’s-Raphson formula to find the cube root and reciprocal
of a number N, where 𝑁(≠ 0) is natural number and
3
hence find the approximate value of √7.
10 𝑑𝑥
12. Evaluate ∫0 𝑑𝑥 by using
1+𝑥 2
1 𝑟𝑑
(i) Trapezoidal rule (ii) Simpson’s ( ) rule (iii) Simpson’s 3/8th rule
3
and compare the results with the actual value.
13. A rocket is launched from the ground. Its acceleration measured every 5
seconds is tabulated below. Find the velocity and the position of rocket
at 𝑡 = 40 seconds. Use trapezoidal rule and simpson’s 1/3rd rule
𝑡 0 5 10 15 20 25 30 35 40
𝑎(𝑡) 40 45.25 48.5 51.25 54.35 59.48 61.5 64.3 68.7
𝜋
14. Evaluate ∫0 𝑡𝑠𝑖𝑛𝑡 𝑑𝑡 by using the trapezoidal rule.
2 2
15. Evaluate ∫0 𝑒 −𝑥 𝑑𝑥 by using Simpson’s rule taking ℎ = 0.25
UNIT- II
Interpolation
1. Find the Newton’s forward difference interpolating polynomial for the data:
𝑥 0 1 2 3
𝑓(𝑥) 1 3 7 13
2. From the following tables values of 𝑥 and 𝑦, interpolate value of 𝑦 when
𝑥 = 1.91. Use Gauss's forward interpolation formula.
𝑥 1.7 1.8 1.9 2 2.1 2.2
𝑦 5.4739 6.0496 6.6859 7.3891 8.1662 9.0250
3. The population of a town in the decimal census was given below.
Estimate the population for the years 1895 and 1925
Year 𝑥 1891 1901 1911 1921 1931
Population 𝑦 46 66 81 93 101
(in thousands)
4. The following data give the percentage of criminals for different age groups
Age (less than 𝑥) 25 30 40 50
% of criminals 52 67.3 84.1 94.4
Using Lagrange’s formula, find the percentage of criminals under the age of 35
5. Construct a forward difference table from the following data and hence
Evaluate ∆3 𝑦1
𝑥 0 1 2 3 4
𝑦 1 1.5 2.2 3.1 4.6
6. Find the interpolating polynomial for the following data
𝑥 0 1 2 5
𝑓(𝑥) 2 3 12 147
7. Find the parabola passing through the points (0,1) (1,3) (3,55)
using Lagrange’s formula.
8. By using Lagrange interpolation formula find 𝑓 (3.5)
𝑥 0 2 3 6
𝑓(𝑥) 659 705 729 804
9. Given that 𝑠𝑖𝑛450 = 0.7071, 𝑠𝑖𝑛500 = 0.7660, 𝑠𝑖𝑛550 = 0.8192, 𝑠𝑖𝑛600 = 0.8660
𝑡hen find 𝑠𝑖𝑛520 Using Newton’s interpolation formula. Ans: 0.788032
10. Find by Gauss’s backward formula the value of 𝑦 at 𝑥 = 1936,
using the following table
𝑥 1901 1911 1921 1931 1941 1951
𝑦 12 15 20 27 39 52
11. Find the number of students who got less than 45 marks from
the following frequency table
Marks 30-40 40-50 50-60 60-70 1941
Frequency 31 42 51 35 31
12. Find the missing terms of 𝑦 in the following table
𝑥 4 5 6 7 8 9
𝑦 72 --- 146 192 --- 302
UNIT – III
Numerical Solution of ordinary Differential Equations
1. Using Taylor’s series method, find an approximate value of 𝑦 at 𝑥 = 0.3 for
the differential equation 𝑦′ − 2𝑦 = 3𝑒 𝑥 , 𝑦(0) = 0 and compare
the numerical solution with an exact solution.
2. Given that 𝑦′ = 𝑥 2 − 𝑦, 𝑦(0) = 1, compute 𝑦(0.1), 𝑦(0.2), 𝑦(0.3) and
𝑦(0.4) (correct to 4 decimal places) by using Taylor's series method.
3. Find the value of 𝑦 at 𝑥 = 0.1 by Picard’s method, given that
𝑑𝑦 𝑦 − 𝑥
= , 𝑦(0) = 1
𝑑𝑥 𝑦 + 𝑥
4. Given that 𝑦 ′ = 𝑥 + 𝑦, 𝑦(0) = 1 and find 𝑦(0.3) taking step size ℎ = 0.1 by
using Euler's method and compare the results with the exact solution.
5. Using Runge-Kutta method of fourth order, find 𝑦 at 𝑥 = 1.1 and 𝑥 = 1.2
given that 2𝑦 ′ = 2𝑥 3 + 𝑦, 𝑦(1) = 2.
6. Solve 𝑦′ = 𝑥 + 𝑦, 𝑦(0) = 1 to find y at 𝑥 = 0.1, 0.2, 0.3 by using R-K method.
7. Apply the fourth order R-K method, to find an approximate value of 𝑦 when
𝑥 = 1.2 in steps of 0.1 given that: 𝑦′ = 𝑥 2 + 𝑦 2 , 𝑦(1) = 1.5
8. Use Milne’s predictor – corrector formula to find y (1.4) from
𝑑𝑦
− 𝑥 2 𝑦 = 𝑥 2 , 𝑦(1) = 1. Find the initial values 𝑦 (1.1), 𝑦 (1.2), 𝑦 (1.3)
𝑑𝑥
from the Taylor’s series method.
𝑑𝑦
9. By using Milne's method to the value 𝑦(0.4) from = 1 + 𝑦 2 , 𝑦(0) = 0.
𝑑𝑥
Find the initial values 𝑦 (0.1), 𝑦 (0.2), 𝑦 (0.3) by using Euler's method.
