CUSTOMER_CODE
SMUDE
DIVISION_CODE
SMUDE
EVENT_CODE
JULY15
ASSESSMENT_CODE IMC3010_JULY15
QUESTION_TYPE
DESCRIPTIVE_QUESTION
QUESTION_ID
10677
QUESTION_TEXT
Define curve fitting. What are the methods of curve fitting.
SCHEME OF
EVALUATION
The process of finding the equation of the curve of best fit which may
be suitable for predicting the unknown values in known is curve
fitting. (2 Marks)
Methods:
i.Graphic method (1 Mark)
ii.Method of group average (1 Mark)
iii.Method of moments (1 Mark)
iv.Method of least squares (1 Mark)
Graphical method explanation (4 Marks)
QUESTION_TYPE
DESCRIPTIVE_QUESTION
QUESTION_ID
10678
QUESTION_TEXT
Define interpolation and extrapolation. When the following formulate
are useful
i. Newton’s forward difference interpolation formula
ii. Newton’s backward difference interpolation formula
iii. Central difference formula
SCHEME OF
EVALUATION
For a given table of values (Xk, Yk), K=0, 1,2, 3…….n the process of
estimating value for Y, for any intermediate value of X is called
interpolation (2 Marks)
Computing Y for X, lying outside the table of values of X
extrapolation (2 Marks)
i.Interpolation near the beginning of set of tabular values (2 Marks)
ii.Near the end of the tabular values (2 Marks)
iii.Near middle of the table (2 Marks)
QUESTION_TYPE
DESCRIPTIVE_QUESTION
QUESTION_ID
73415
QUESTION_TEXT
Explain Blunders and Data uncertainty.
SCHEME OF
EVALUATION
Blunders: These errors can be either due to human imperfection or
computer malfunctioning. It can occur at any stage of the mathematical
modeling process and can contribute to all the other components of
error.
Blunders are usually not considered when we discuss about the
different types of error this is just because mistakes are to some extent
unavoidable no matter how efficiently we are going to work.
(5 marks)
Data uncertainty: This type of error is also known as noise. Uncertainty
error is due to uncertainty in physical data upon which a model is
based. This error shows both inaccuracy and imprecision. If the given
data has n significant digits of accuracy then the result obtained from it
will contain n significant digits of accuracy. For example if a = 2.467 and
b = 0.03241 both have 4 significant digits of accuracy then a-b = 2.43459
. Although a-b have 6 significant digits the correct answer will have four
significant digits only that is 2.434 so the answer will be 2.434. (5
marks)
QUESTION_TYPE
DESCRIPTIVE_QUESTION
QUESTION_ID
73416
QUESTION_TEXT
Explain six steps to apply Cramer’s Rule
SCHEME OF
EVALUATION
Step i. Write the given equations in order so
that
constant terms all on the right side (1.5
Marks)
Step ii. Take = the determinant formed by
the
coefficients of x, y, z (1.5 Marks)
Step iii. Replace the first column of  by constant
terms of
the equations and denote as x
(1.5 Marks)
Step iv. Replace the second column of by
constant
terms of the equations and denote
as y (1.5 Marks)
Step v. Replace the third column of by
constant
terms of the equations and denote
as z (1.5 Marks)
Step vi. Write the solution
(3 marks)
QUESTION_TYPE DESCRIPTIVE_QUESTION
QUESTION_ID
73417
Using the given figure explain Regula–Falsi method.
QUESTION_TEXT
Choose two points xo and x1 such that f(x1) and f(x2) are of opposite
signs. Since the graph of y=f(x) crosses the X–axis between these two
points.
This indicates that a root lies between these two points x1 and x2.
SCHEME OF
EVALUATION
Equation of the chord joining the points A(x1, f(x1)) and B(x2, f(x2)) is
y–f(x1) = f(x2)–f(x1) divided by x2–x1 Whole multiplied by (x–x1)-------(i) (3.5 marks)
Where f(x2)–f(x1) divided by x2–x1 is the slope of the line AB.
The method consists in replacing the curve AB by means of the Chord
AB and taking the point of intersection of the chord with the X–axis as
an approximation to the root. The point of intersection in the present
case is given by putting y=0 in (i).
Thus we obtain
0–f(x1)=f(x2)–f(x1) divided by x2–x1 whole multiplied by (x–x1). Solve
for x,
We get x=x1–f(x1)(x2–x1) divided by f(x2)–f(x1)-------(ii) (3.5 marks)
Hence the second approximation to the root of f(x)=0 is given by
x3=x=x1–f(x1)(x2–x1) divided by f(x2)–f(x1)------(iii)
If f(x3) and f(x1) are of opposite signs, then the root lies between x1
and x3, and we replace x2 by x3 in (iii), and obtain the next
approximation. Otherwise, f(x3) and f(x1) are of same sign; we replace
x1 by x3 and generate the next approximation. The procedure is
replaced till the root is obtained to the desired accuracy. (3 marks)
QUESTION_TYPE
DESCRIPTIVE_QUESTION
QUESTION_ID
164973
i.
Why Runge Kutta method is better than Taylor’s series
method and Euler’s method of solving differential equations.
QUESTION_TEXT
ii.
What are the merits and demerits of Taylor’s series method of
solving differential equation?
Runge-Kutta method The Taylor’s series method of solving differential
equations is restricted by the labour involved in the determination of
higher order derivatives. Euler’s method is less efficient in practical
problems since it requires h to be small for obtaining reasonable
accuracy. A class of method known as Runge-Kutta method does
not require the calculations of higher order derivatives and they are
designed to give greater accuracy with the advantage of requiring
only the function values at some selected points on the subinterval. (5 marks)
SCHEME OF
EVALUATION
Merits: i. The method of numerical solution by using Taylor series is of
the single-step untruncated type.
ii. The method is very powerful if we can calculate the successive
derivatives of y in an easy manner.
iii. If there is a simple expression for the higher derivatives in terms of
the previous derivatives of y, Taylor’s method will work very well.
Demerits: The differential equation dy/dx= f(x, y), the function f(x, y)
may have a complicated algebraic structure. Then the evaluation of
higher order derivatives may become tedious and so this method has
little application for computer programmes. (5 marks)