Numerical Method
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Questions and Answers

What is the condition for the equation f(x) = 0 to have at least one real root between a and b?

  • f(a) and f(b) have the same sign
  • f(a) and f(b) are both positive
  • f(a) and f(b) have opposite signs (correct)
  • f(a) and f(b) are both negative
  • What is the graphical method used to find the real roots of an equation?

  • Plotting the graphs of y = f(x) and y = x
  • Plotting the graphs of y = f1(x) and y = f2(x)
  • Plotting the graphs of y = f(x) and y = 0
  • Plotting the graphs of y = f1(x) and y = f2(x) and finding the points of intersection (correct)
  • What is the condition for all roots of the equation x^3 - 3x + k = 0 to be real?

  • k > 2
  • k belongs to the range (-2, 2) (correct)
  • k belongs to the range [2, 3]
  • k < 2
  • What is the purpose of the Location Theorem in finding the roots of an equation?

    <p>To determine the intervals in which the root lies</p> Signup and view all the answers

    What is the method of finding the real roots of an equation by plotting the graphs of y = f1(x) and y = f2(x) and finding the abscissae of the points of intersection?

    <p>Graphical Method</p> Signup and view all the answers

    When a number is rounded off to n decimal places, what is the maximum bound on the magnitude of relative error?

    <p>0.5 × 10^(−n+1)</p> Signup and view all the answers

    What type of equation is ax^2 + bx + c = 0?

    <p>Algebraic equation</p> Signup and view all the answers

    When a number is truncated to n decimal places, what is the maximum bound on the absolute error?

    <p>10^(−n+1)</p> Signup and view all the answers

    What is the number of significant digits in the number 0.0001?

    <p>1</p> Signup and view all the answers

    What is the absolute error when the number 2.089 is rounded off to three significant digits?

    <p>-0.001</p> Signup and view all the answers

    What is the maximum number of real roots of the equation x²ⁿ - 1 = 0?

    <p>2</p> Signup and view all the answers

    What is the iterative method used for in solving algebraic and transcendental equations?

    <p>To approximate the root of the equation to any desired degree of accuracy</p> Signup and view all the answers

    What is the condition for all roots of the given equation to be real?

    <p>k + 2 &gt; 0 and k - 2 &lt; 0</p> Signup and view all the answers

    In the iterative method, what is the value of x₁ if x₀ is the initial approximation to the root α?

    <p>g(x₀)</p> Signup and view all the answers

    What is the interval in which the smallest positive root of the transcendental equation x - e^x = 0 lies?

    <p>(0, 1)</p> Signup and view all the answers

    What is the purpose of the iterative method in solving algebraic and transcendental equations?

    <p>To successively approximate the root of an equation to any desired degree of accuracy</p> Signup and view all the answers

    What is the nature of the function g(x) in the iterative method?

    <p>Non-constant</p> Signup and view all the answers

    What is the relationship between the successive approximations x₀, x₁, and x₂ in the iterative method?

    <p>x₂ = g(x₁) = g(g(x₀))</p> Signup and view all the answers

    What is the nature of the roots of the equation x²ⁿ - 1 = 0?

    <p>Exactly two real</p> Signup and view all the answers

    What is the purpose of locating the root of an equation in the successive bisection method?

    <p>To determine the interval in which the root of an equation lies</p> Signup and view all the answers

    What determines the number of significant digits in a number?

    <p>The position of the non-zero digits</p> Signup and view all the answers

    When rounding off a number to n significant digits, what happens to the (n+1)th digit if it is greater than 5?

    <p>It is discarded</p> Signup and view all the answers

    What is the purpose of rounding off numbers in numerical methods?

    <p>To decrease the complexity of the calculation</p> Signup and view all the answers

    How many significant digits are in the number 35.9200?

    <p>6</p> Signup and view all the answers

    What is the rule for rounding off a number to n significant digits if the (n+1)th digit is 5 and is followed by zero or zeros?

    <p>Increase the nth digit by 1 if it is odd and remain unchanged if it is even</p> Signup and view all the answers

    What is the characteristic of an algebraic equation?

    <p>It has a polynomial function</p> Signup and view all the answers

    What type of equation is ae^x + b sin x = 0?

    <p>Transcendental equation</p> Signup and view all the answers

    What is the purpose of the Location Theorem in finding the roots of an equation?

    <p>To find the interval where the root lies</p> Signup and view all the answers

    What is the iterative method used for in solving algebraic and transcendental equations?

    <p>To find the approximate root of the equation</p> Signup and view all the answers

    What is the condition for an equation to have at least one real root between a and b?

    <p>f(a) and f(b) have opposite signs</p> Signup and view all the answers

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