30 Questions
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 opposite signs
What is the graphical method used to find the real roots of an equation?
Plotting the graphs of y = f1(x) and y = f2(x) and finding the points of intersection
What is the condition for all roots of the equation x^3 - 3x + k = 0 to be real?
k belongs to the range (-2, 2)
What is the purpose of the Location Theorem in finding the roots of an equation?
To determine the intervals in which the root lies
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?
Graphical Method
When a number is rounded off to n decimal places, what is the maximum bound on the magnitude of relative error?
0.5 × 10^(−n+1)
What type of equation is ax^2 + bx + c = 0?
Algebraic equation
When a number is truncated to n decimal places, what is the maximum bound on the absolute error?
10^(−n+1)
What is the number of significant digits in the number 0.0001?
1
What is the absolute error when the number 2.089 is rounded off to three significant digits?
-0.001
What is the maximum number of real roots of the equation x²ⁿ - 1 = 0?
2
What is the iterative method used for in solving algebraic and transcendental equations?
To approximate the root of the equation to any desired degree of accuracy
What is the condition for all roots of the given equation to be real?
k + 2 > 0 and k - 2 < 0
In the iterative method, what is the value of x₁ if x₀ is the initial approximation to the root α?
g(x₀)
What is the interval in which the smallest positive root of the transcendental equation x - e^x = 0 lies?
(0, 1)
What is the purpose of the iterative method in solving algebraic and transcendental equations?
To successively approximate the root of an equation to any desired degree of accuracy
What is the nature of the function g(x) in the iterative method?
Non-constant
What is the relationship between the successive approximations x₀, x₁, and x₂ in the iterative method?
x₂ = g(x₁) = g(g(x₀))
What is the nature of the roots of the equation x²ⁿ - 1 = 0?
Exactly two real
What is the purpose of locating the root of an equation in the successive bisection method?
To determine the interval in which the root of an equation lies
What determines the number of significant digits in a number?
The position of the non-zero digits
When rounding off a number to n significant digits, what happens to the (n+1)th digit if it is greater than 5?
It is discarded
What is the purpose of rounding off numbers in numerical methods?
To decrease the complexity of the calculation
How many significant digits are in the number 35.9200?
6
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?
Increase the nth digit by 1 if it is odd and remain unchanged if it is even
What is the characteristic of an algebraic equation?
It has a polynomial function
What type of equation is ae^x + b sin x = 0?
Transcendental equation
What is the purpose of the Location Theorem in finding the roots of an equation?
To find the interval where the root lies
What is the iterative method used for in solving algebraic and transcendental equations?
To find the approximate root of the equation
What is the condition for an equation to have at least one real root between a and b?
f(a) and f(b) have opposite signs
Solve problems on functions, equations and calculus. This quiz includes problems on finding roots of equations, range of values of k, and solving transcendental equations.
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