Copyright 2014 CBSE-SPOT
Time : 3 hours
F.M : 100
General Instructions :
All questions are compulsory.
Q. 1 to Q. 10 of Section A are of 1 mark each.
Q. 11 to Q. 22 of Section B are of 4 marks each.
Reliance Industries Limited
Equity and Liabilities
Tata MotoTata MotInfosys Infosys
Rs in CrsRs in CrsRs in CrsRs in CrsRs in Crs Rs in CrsRs in CrsRs in Crs
Supervised vs. unsupervised learning
From a theoretical point of view, supervised and unsupervised learning differ only in
the causal structure of the model. In supervised learning, the model defines the effect
one set of observations, called inputs, has
This is to certify that the PHYSICS
project titled ELECTROMAGNETIC
INDUCTION has been successfully
completed by NISHANT PANDEY of
Class XII in partial fulfillment of
curriculum of CENT
Sensitivity analysis determines how much variations in input values for a given variable will impact the
results for a mathematical model i.e. how sensitive the optimal solution is to changes in data values.
The changes may be either in:
1. An Objective F
CHEMISTRY, PHYSICS & MATHS
AIEEE-2010 QUESTION PAPER
PART-A : CHEMISTRY
The standard enthalpy of formation of NH3 is
46.0kJmol1. If the enthalpy of formation of
H2 from its atoms is 436 kJmol1 and that of
N2 is 712 kJmol1, the avera
International Securities Association For Institutional Trade Communication
TBA Market Practice
Presented by the North America Settlements/Cash/Treasury Working
This market practice document has been developed by the International S
Insurance Industry Analytics
Dimensions: The various aspects on which data can be evaluated
Metrics / KPIs the key output from the data (Key performance index) like mean median gross margin ,PE
Company who specialize providing data or
Description of the dataset:
A real estate agent is trying to understand the nature of housing stock and
home prices in and around a medium sized town in upstate New York. She
has collected data from a random sample of 1047 homes s
Question 2(d) Use Cayley-Hamilton theorem
to find the inverse of the following matrix
Solution. Characteristic polynomial is given by |xI A| = 0, where I is the 3 3 unit
1 x 2 3 = 0
3 1 x 1
x[x2 3x + 2 3] + 1[x + 1 9] 2[
Question 1(c) Find the least perimeter of an isoceles triangle in which a circle of radius r
can be inscribed.
Let be the semi-vertical angle. D is the
midpoint of BC. E is the point of contact
of AC and the circle or radius r inscribed
Range of T = T(V), kernel of T = cfw_v | T(v) = 0. If w1 , w2 T(V), then w1 =
T(v1 ), w2 = T(v2 ) for some v1 , v2 V, w1 + w2 = T(v1 ) + T(v2 ) = T(v1 + v2 ).
But v1 + v2 V w1 + w2 T(V), thus T(V) is a subspace of W. Note that
T(V) 6= 0 T(V) so T(V) is a
Question 5(b) Define a positive definite form. State and prove a necessary and sufficient
condition for a quadratic form to be positive definite.
Solution. See 1992 question 2(c).
Question 5(c) Show that the mapping T : R3 R3 defined by T (x, y, z) = (x y
1 0 0
Question 2(a) If A = 1 0 1 show that for all integers n 3, An = An2 + A2 I.
0 1 0
Hence determine A .
Solution. Characteristic equation of A is
1 0 0
1 = 0
or (1)(2 1) = 3 2 +1 = 0. From the Cayley-Hamilton theorem, A3 A2 A+I =
Question 2(b) Find the eigenvalues and eigenvectors of the matrix A =
= 0 (9
Solution. The characteristic equation of A =
2 ) 16 = 0 2 25 = 0 = 5, 5.
If (x1 , x2 ) is an eigenvector for = 5, then
= 0 2x1 4x2 =
Using (1), (2), (3) we get
2 2 H
+ 4x y
4x2 y 2 2 + 8x3 y
4x4 2 2(x2 + y 2 )
+ 4(y 4 x4 ) 2
8xy(x2 + y 2 )
Question 1(a) I. Show that the function
musingi, I 3321]
is continuous at .
2. tana is not continuous at :r = 3
1. Given 6 2% (1', let :5 = E1 then Im| E J =:- |mgsingi| E |33| 6: E, because IsinziI ii 1.
Thus |:.I:IJI s: :5: J
Question 1(a) Show that the vectors [1,1]. 1,cfw_l, 31,2 and cfw_Lil lm a basis of the
vector space Hamil].
Solution. Since EllIncfw_R3 = 3. it is enough to prove that these are linearly independent.
If possible: let
ecfw_1,.1+ em, 3. 2 + c[1, 2.1] = G
Cluster based on Constituencies