Numerical Method

Maths, HSSC-Punjab Second Year. This chapter is split into 55 topics and has 1,293 practice questions. The topics covered are listed below, followed by 10 free sample questions from the chapter.

Topics in this chapter

1. Applications: heartbeat signal and bisection 23 questions
2. Newton-Raphson: second iterate for polynomial equations 24 questions
3. Trapezium rule: overestimate, underestimate and refining n 21 questions
4. Newton-Raphson: first iterate for polynomial equations 24 questions
5. Bisection: first midpoint and its function value 26 questions
6. Applications: diode voltage and cryptography with Regula-Falsi 22 questions
7. Applications: equilibrium, blood flow, circuits and decoding problems 25 questions
8. Newton-Raphson: later iterates for transcendental equations 23 questions
9. Bisection: stopping iteration and accuracy of the root 24 questions
10. Regula-Falsi: new interval after the first step 24 questions
11. Trapezium rule: estimates for polynomial integrals 24 questions
12. Trapezium rule: step size and division points 25 questions
13. Roots and zeros of a function 22 questions
14. Applications: chemical equilibrium and Newton-Raphson 20 questions
15. Regula-Falsi: second iteration for polynomial equations 24 questions
16. Newton-Raphson: stopping criterion and converged roots 24 questions
17. Regula-Falsi: third iteration and fixed end point 23 questions
18. Newton-Raphson: square roots and cube roots 22 questions
19. Bisection: trigonometric, exponential and logarithmic equations (later steps) 26 questions
20. Newton-Raphson: first iterate for transcendental equations 24 questions
21. Trapezium rule: doubling the number of subintervals 22 questions
22. Location of Root Theorem 22 questions
23. Accuracy, tolerance and calculator setup 23 questions
24. Choosing an interval of sign change 26 questions
25. Regula-Falsi: trigonometric, exponential and logarithmic equations 26 questions
26. Trapezium rule: estimates for radical, rational and logarithmic integrals 23 questions
27. Regula-Falsi: secant line slope and x-intercept 22 questions
28. Regula-Falsi: first approximation for polynomial equations 24 questions
29. Regula-Falsi compared with bisection 26 questions
30. Trapezium rule: area bounded by a curve and the x-axis 21 questions
31. Newton-Raphson: when the method fails and speed of convergence 22 questions
32. Bisection: interval after the first iteration 26 questions
33. Newton-Raphson: derivative values and tangent x-intercepts 24 questions
34. Trapezium rule: area of a single strip 24 questions
35. Comparing bisection, Regula-Falsi and Newton-Raphson 23 questions
36. Bisection: trigonometric, exponential and logarithmic equations (first steps) 26 questions
37. Trapezium rule: end ordinates and coarser subdivisions 21 questions
38. Bisection: second iteration of polynomial equations 24 questions
39. Bisection: third iteration of polynomial equations 24 questions
40. Trapezium rule: idea and formula 22 questions
41. Bisection method: principle and steps 23 questions
42. Regula-Falsi: later iterations for transcendental equations 26 questions
43. Newton-Raphson method: idea, formula and conditions 23 questions
44. Regula-Falsi method: idea and formula 22 questions
45. Nonlinear equations and the need for numerical methods 23 questions
46. Graphical method for approximating real roots 23 questions
47. Trapezium rule: error compared with the exact value 24 questions
48. Evaluating a function to find sign changes 21 questions
49. Regula-Falsi: worked equations x cubed minus 3x minus 10, sine and quintic 21 questions
50. Trapezium rule: step width and ordinate values 26 questions
51. Trapezium rule: worked examples (Gaussian and rational integrands) 22 questions
52. Trapezium rule: estimates for exponential and trigonometric integrals 24 questions
53. Graphical method: roots of exercise functions 26 questions
54. Bisection: worked equation x cubed minus 3x minus 10 22 questions
55. Bisection: interval length and number of iterations 26 questions

Sample questions: Numerical Method

10 sample questions with the correct answer and an explanation. Try to answer each one before reading the explanation.

Q1. For f(x) = x⁴ − 200x − 50, the two intervals [−1, 0] and [5, 6] of sign change show that f has:

• A. at least two real roots(correct answer)
• B. no real root
• C. exactly one real root
• D. exactly four real roots

Answer: A.Each interval of sign change contains at least one root by the Location of Root Theorem; the two intervals are disjoint, so there are at least two roots.

Q2. The Graphical Method for roots is mainly suitable for obtaining a root correct up to:

• A. exactly the true value
• B. ten decimal places
• C. one or two decimal places(correct answer)
• D. five significant digits always

Answer: C.The book states that the graphical method gives roots to one or two decimal places and serves as an initial estimate for other numerical methods.

Q3. Besides the function-value criterion, another way of judging a root's accuracy mentioned in the chapter is:

• A. finding the area under f
• B. measuring the slope at the origin
• C. counting the degree of f
• D. reducing the interval length(correct answer)

Answer: D.The text notes that accuracy can be checked by reducing the interval length or by checking the function value; the chapter follows the function-value criterion.

Q4. Regula-Falsi on f(x) = x³ − 3x − 10 starting from [2, 3] is stopped when |f(c)| < 0.01. At which iteration?

• A. 2 (c = 2.5949, |f| = 0.3113)
• B. 4 (c = 2.6125, |f| = 0.0075)(correct answer)
• C. 5 (c = 2.6128, |f| = 0.0012)
• D. 3 (c = 2.6101, |f| = 0.0485)

Answer: B.|f| = 0.3113, 0.0485, 0.0075 for iterations 2, 3, 4 (the book prints 0.0068 for the last); the first value below 0.01 is at iteration 4, giving x ≈ 2.61.

Q5. Regula-Falsi on f(x) = x⁵ − 10x + 5 starting from [1, 2]: the approximation in iteration 1 (4 decimals) is:

• A. 1.2207
• B. 1.1905(correct answer)
• C. 1.1404
• D. 1.5000

Answer: B.Because f is convex here, only a moves: 1.1905, 1.3603, 1.4808, 1.5517 (the book's table, rounded at each step, differs slightly from the full-precision values in the fourth decimal).

Q6. Regula-Falsi on f(x) = x⁵ − 10x + 5 starting from [1, 2]: the approximation in iteration 2 (4 decimals) is:

• A. 1.5952
• B. 1.3102
• C. 1.3905
• D. 1.3603(correct answer)

Answer: D.Because f is convex here, only a moves: 1.1905, 1.3603, 1.4808, 1.5517 (the book's table, rounded at each step, differs slightly from the full-precision values in the fourth decimal).

Q7. Regula-Falsi on f(x) = x⁵ − 10x + 5 starting from [1, 2]: the approximation in iteration 3 (4 decimals) is:

• A. 1.4307
• B. 1.4808(correct answer)
• C. 1.5110
• D. 1.6802

Answer: B.Because f is convex here, only a moves: 1.1905, 1.3603, 1.4808, 1.5517 (the book's table, rounded at each step, differs slightly from the full-precision values in the fourth decimal).

Q8. Regula-Falsi on f(x) = x⁵ − 10x + 5 starting from [1, 2]: the approximation in iteration 4 (4 decimals) is:

• A. 1.5016
• B. 1.5819
• C. 1.7404
• D. 1.5517(correct answer)

Answer: D.Because f is convex here, only a moves: 1.1905, 1.3603, 1.4808, 1.5517 (the book's table, rounded at each step, differs slightly from the full-precision values in the fourth decimal).

Q9. Use the Trapezium Rule with n = 8 equal subintervals (h = 0.5) to estimate ∫₀⁴ exp(x²) dx (4 decimals):

• A. 4660726.2341
• B. 2275945.4986
• C. 108835.4869
• D. 2330363.1170(correct answer)

Answer: D.h/2 [f(x₀) + 2(f(x₁) + ... + f(x₇)) + f(x₈)] with interior sum 217670.4738, f(x₀) = 1.0000, f(x₈) = 8886110.5205 gives 2330363.1170 (exact value 1149400.6346).

Q10. The Trapezium Rule with n = 8 for the area under y = √(8 − 2x − x²) on [−4, 2] gives (1 decimal):

• A. 15.0
• B. 14.1
• C. 12.0
• D. 13.5(correct answer)

Answer: D.The ordinates are 0, 1.9843, 2.5981, 2.9047, 3, 2.9047, 2.5981, 1.9843, 0; the sum of the interior ordinates is 17.9742, so T = 0.375 × 35.9484 = 13.48 ≈ 13.5.