Download CUSTOMER_CODE SMUDE DIVISION_CODE SMUDE

CUSTOMER_CODE
SMUDE
DIVISION_CODE
SMUDE
EVENT_CODE
Jan2017
ASSESSMENT_CODE BT0063_Jan2017
QUESTION_TYPE
DESCRIPTIVE_QUESTION
QUESTION_ID
4986
QUESTION_TEXT
Define a group and give one example.
Definition of Group:
A non-empty set G is said to be a group with respect to the binary
operation * if the following axioms are satisfied:
a.Closure law: For every a, b in G, a*b belongs to G
b.Associative law: For every a, b, c in G,
a*(b*c)=(a*b)*c
c.Existence of identity element: There exists an element e in G such
that a*e=e*a=a for every a in G.
d.Existence of inverse: For every a in G, there exists an element b in G
such that,
a*b=b*a=e. here b is called the inverse of a.
SCHEME OF
EVALUATION
a group G with respect to binary operation * is denoted by(G, *)
Example:
The set Z of integers is a group with respect to the usual addition as the
binary operation.
a.Closure law: We know that the sum of two integers is also an integer.
Hence for every m, n in Z, m+n belongs to Z.
b.Associative law: It is well known that the addition of integers is
associative
c.Existence of identity element: There exists 0 in Z such that,
m+0=0+m=m for every m in Z. Hence 0 is the additive identity.
d.Existence of inverse: For every m in Z, there exists –m in Z such that
m+(–m)=(–m)+m=0. Here –m is called the additive inverse of m or
simply the negative of m.
Therefore (Z, +) is a group.
QUESTION_TYPE
DESCRIPTIVE_QUESTION
QUESTION_ID
72348
QUESTION_TEXT
Differentiate between mean, median and mode. Write two merits of
median and mode.
Mean: Arithmetic mean of a set of values is obtained by dividing the
sum of the values by the number of values in the set.
Median: Median of a set of values is the middle most value when they
are arranged in the ascending order of magnitude. It is a value the is
greater than half of the values lesser than the remaining half.
Mode: Mode is the value, which has the highest frequency. It is the
most frequently occurring value.
Two merits of median:
SCHEME OF
EVALUATION
i. Even when some of the extreme values are missing, it can be
calculated.
ii.
It can be graphically found out.
Two merits of mode:
1.
It can be used for the study of qualitative data also.
2.
It can be graphically found out.
QUESTION_TYPE
DESCRIPTIVE_QUESTION
QUESTION_ID
72351
QUESTION_TEXT
Define the following terms:
i. Null set
ii. Finite and infinite sets
iii. Equal and equivalent sets
iv. Subsets
v. Universal set
SCHEME OF
EVALUATION
i. Null set: A set which does not contain any element is called the
empty set or null set.
ii. Finite and infinite sets: A set which is empty or consists of a
definite number of elements is called finite otherwise the set is called
infinite.
iii. Equal and equivalent sets: Given two sets of A and B . If every
element of A is also an element of B and if every element of B is also an
element of A, the sets A and B are said to be equal. Clearly, the two sets
have exactly the same elements. Two finite sets A and B are said to be
equivalent if they have the same number of elements.
iv. Subsets: If every element of a set A is also an element of a set B,
then A is called subset of B or A is contained in B.
v. Universal set: If in any particular context of sets, we find a set U
which contains all the sets under consideration as subsets of U, then set
U is called the universal set.
QUESTION_TYPE
DESCRIPTIVE_QUESTION
QUESTION_ID
72353
QUESTION_TEXT
Briefly define the following :
i. Standard deviation
ii. Uses of standard deviation
iii. Variance
iv. Two merits of standard deviation
v. Two demerits of standard deviation
SCHEME OF
EVALUATION
(i) Standard deviation: Standard deviation is the root mean square
deviation of the value from their arithmetic mean. Standard deviation
is the positive square root of variance.
Uses of standard deviation : Standard deviation is the best absolute
measure of dispersion. It is a part of many statistical concepts such as
skewness, kurtosis, correlation, regression, estimation sampling, test of
significance and statistical quality control.
Variance: Variance is the mean square deviation of the values from
their arithmetic mean. Standard deviation is the positive square of
variance.
Two merits of standard deviation:
i. It is calculated on the basis of the magnitudes of all the items.
ii. The combined standard deviation can be calculated further.
Two demerits of standard deviation:
i. Compared with other absolute measures of dispersion, it is difficult
to understand.
ii. It gives more weightage to the items away from the mean that
those near the mean as the deviations are squared.
QUESTION_TYPE
DESCRIPTIVE_QUESTION
QUESTION_ID
72355
QUESTION_TEXT
a.
b.
SCHEME OF
EVALUATION
a.
Let e and e1 be the two identity elements of a group G. then for
every a in G, ae = ea = a and
Ae1 = e1a = a
Substitute a = e1 in (i) and a = e in (ii).
Then, e1e = ee1 = e1
Hence e1 = ee1 = e
Therefore identity element in a group G is unique.
b.
Let b and c be the two inverse of an element a in G.
Then, ab = ba = e
Ac = ca = e
Now consider, b = be
= b (ac)
= (ba) c
= ec
=c
Therefore inverse of every element in a group G is unique.
Prove that the identity element in a group is unique.
Prove that in a group G the inverse of an element is unique.
QUESTION_TYPE
DESCRIPTIVE_QUESTION
QUESTION_ID
118223
Prove that
i.
sin 3A = 3 sin A – 4 sin3 A
ii.
cos 3A = 4 cos3 A – 3 cos A
QUESTION_TEXT
i)
SCHEME OF EVALUATION ii)
sin 3A = sin(2A + A)
= sin 2 A cos A + cos 2A sinA
= (2 sin A cos A) cos A + (1 – 2 sin2 A) sin A
sin 2A = 2 sub A cos A, cos 2A = 1 – 2 sin2 A)
= 2 sin A cos2 A + sin A – 2 sin3 A
= 2 sin A(1 – sin3 A) + sin A – 2 sin3 A
= 2 sin A – 2 sin3 a + sin A – 2 sin3 A
= 3 sin A – 4 sin3 A
sin 3A = 3 sin A – 4 sin3 A
cos 3A =cos(2A + A)
= cos 2 A cos A – sin 2A sin A
= (2 cos2 A – 1) cos A – 2 sin A cos A sin A
cos 2A = 2 cos2 A – 1, sin 2A = 2 sin A cos A)
= 2 cos3 A – cos A – 2 cos A sin2 A
= 2 cos3 A – cos A – 2 cos A(1 – cos2 A)
= 2 cos3 A – cos A – 2 cos A + 2 cos3 A
= 4 cos3 A – 3 cos A
cos 3A =4 cos3 A – 3 cos A