CUSTOMER_CODE
SMUDE
DIVISION_CODE
SMUDE
EVENT_CODE
SMUAPR15
ASSESSMENT_CODE BCA3010_SMUAPR15
QUESTION_TYPE
DESCRIPTIVE_QUESTION
QUESTION_ID
10676
QUESTION_TEXT
Define algebraic and transcendental equation. List out any three
basic properties of an algebraic equation.
SCHEME OF
EVALUATION
An equation f(x)=0 is called an algebraic equation if it is purely a
polynomial in x (2 Marks)
An equation f(x)=0 is called an transcendental equation if f(x)
contains trigonometric, exponential or logarithmic functions. (2
Marks)
Properties:
1.Every algebraic equation of nth degree, has ne only n roots (2
Marks)
2.Complex roots occur in pairs (2 Marks)
3.(x-a) is a factor of f(x) (2 Marks)
Or
4.Descartes rule of signs (2 Marks)
QUESTION_TYPE
DESCRIPTIVE_QUESTION
QUESTION_ID
10681
QUESTION_TEXT
Explain Inherent errors and numerical errors with their
component
SCHEME OF
EVALUATION
Inherent errors (2 Marks)
Data error (2 Marks)
Conversion error (2 Marks)
Numerical errors (2 Marks)
Truncation numerical error (2 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
125707
QUESTION_TEXT
Explain the bisection method?
Step 1: Consider two trail points which by enclose the roots. Two
points a and b enclose a root if f(a) < 0 (negative) and f(b) > 0
(positive) are of opposite signs.
SCHEME OF
EVALUATION
Step 2: Bisect the interval (a, b) and denote the mid-point by x1, so
that
x1 =
equation.
. If f(x1) = 0, we conclude that x1 is a root of the
Otherwise
Step 3: The root lies either between x1 and b if f(x1) < 0, or the root
lies between x1 and a if f(x1) > 0.
If f(x1) > 0 (positive), then
Step 4: Replace b by x1 and search for the root in this new interval
which is half the previous interval. Then the second approximation
to the root is
x2 =
. If f(x2) = 0 then x2 is a root of f(x) = 0.
If f(x2) < 0 (negative), then
Step 5: The root lies between x1 and x2. Then the third
approximation to the root is x3
QUESTION_TYPE
DESCRIPTIVE_QUESTION
QUESTION_ID
125708
QUESTION_TEXT
Explain the power method.
=
.
In many engineering problems, it is required to compute the
numerically largest eigen value and the corresponding eigen vector.
In such cases, the following power method is quite convenient
which is also well suited for solid machine computation.
Method: (To find the largest eigen value and the corresponding eigen
vector).
SCHEME OF
EVALUATION
Suppose A is the given square matrix.
Step 1: Choose the initial vector such that the largest element is
unity.
(choose initially an eigen vector X(0) = (1, 0, 0)t or (0, 1, 0)t or (0, 0, 1)t
etc)
Step 2: This normalized (taking the largest component out as a
common factor) vector X(0) is pre-multiplied by the given matrix.
(evaluate the matrix product AX(0) which is written as (1) X(1) after
normalization).
Step 3: This gives the first approximation (1) to the eigen value and
X(1) to the eigen vector.
Step 4: Compute AX(1) and again put in the form AX(1) = (2) X(2) by
normalization which gives the second approximation. Similarly we
evaluate AX(2) and put it in the form AX(2) = (3) X(3).
Step 5: Repeat this process till the difference between two successive
iterations is negligible. The values so obtained are respectively the
largest eigen value and the corresponding eigen vector of the given
square matrix A
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 sub-interval. (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)