1
Lecture 9 : Sucient Conditions for Local Maximum, Point of Inection
In Lecture 6, we have seen a necessary condition for local maximum and local minimum. In this
lecture we will see some sucient conditions.
Sucient Conditions for Local Maximum and Local
Relations & Function
Name:
Maximum Marks: 200
Test Duration: 120 minutes
There are 50 questions in total. All the questions are objective type. Each question carries
equal marks and will be awarded 4 marks for correct response and 1 mark will be
deducted
BINOMIAL THOEREM / 1
USEFUL EXPANSIONS AND KEYPOINTS
Let
n
C0 ,
n
C1 ,
n
n
C2 , ,
Cn be the binomial coefficients.
1. x + y n = n C0 x n + n C1 x n1 y+ n C2 x n2 y 2 + + n Cr x nr y r + + n Cn y n
2. x + y n + x y n = 2[ n C0 x n + n C2 x n2 y 2 + n C4 x
1. If , 2 + 2 and 3 + 3 are first three terms of a G.P., then the fourth term is
(A) 27
(B) 27
(C) 13.5
3
5
(D) .
7
2. The sum to 50 terms of the series 12 + 12 +22 + 12 +22 +32 + is
(A)
(B)
150
(C)
17
200
50
(D) 17
51
3. Let 1 , 2 , 3 , 4 and 5 be such
1. If the mean of numbers 10, 20, 30, 40, 50 is 30, then the mean of the numbers 10+, 20+, 30+,
40+, 50+ is
(A) 30
(B) +
(C) 30 + 5
(D) 30 5
2. Mean of the first terms of the AP , + , + 2, is
(A) + 2
(B) +
(C) + 1
(D) +
3. The mean of the 50 observ
Set Theory Test
Name:
Maximum Marks: 80
Test Duration: 60 minutes
There are 20 questions in total. All the questions are objective type with single correct
choice. Each question carries equal marks and will be awarded 4 marks for correct
response and 1 m
Inequality Test
Name:
Maximum Marks: 20
Test Duration: 20 minutes
There are 10 questions in total. All the questions are objective type with single
correct choice. Each question carries equal marks and will be awarded 2 marks for
correct response. There
1. The value of
(A)
+
1
=1 log
2
(B)
1
2
log 2
(C) + 1 log 2
(D) 1 log 2
2. Total number of 9 digit numbers which have all the digits different is equal to
(A) !
(B) 9!
(C) 10!
(D) None of these
3. The total number of ways in which 5 balls of differe
EE210: HW-1
Q.1 For the equivalent circuit of the
amplifier shown in the figure, determine
the mid-band voltage gain, lower and
upper cutoff frequencies for a source
(RS) and load resistance (RL) of 1k.
Q.2 An amplifiers input-output relationship is descr
M6 : $o\u_SLCev~
Name : . Section:
Q. Analyze the circuit shown to determine expression for V01 / Vs.
You can use the relation gmro >1 to simplify your result
uuuuuu A c 322
\I x);
(9-;
mo ; 25 X \:x A f/w:
Hal/(4 9
9
a 57>
1Aq(w\&\gw13\ 2 5L5
3
Department of Physics, IIT Kanpur
Semester-1, 2016-17
PHY103 Problem Set # 9 Date: September 28, 2016 [RCB/Krishnacharya]
1. Two long cylinders (radii a and b) are separated by material of conductivity (s) =k/s
(where k is a constant), if they are maintai
Department of Mathematics, Indian Institute of Technology, Kanpur
MTH101A: End Semester Exam- 19-11-2014
Maximum Marks- 100
9:00a.m.-12:00 noon
i Please write your Name, Roll Number and Section Number correctly on the answer
booklet.
ii Attempt each quest
ADDITIONAL PROBLEMS FOR MTH-101
Problem 0.1. Show that
n=1 [0, 1/n] = cfw_0. What is n=1 [n, ) ?
Hint: If x
n=1 [0, 1/n] then 0 x 1/n. By the Archimedean
property, x = 0. Answer to second part is empty set.
Problem 0.2. Prove that the sequence (sin(n) i
Indian Institute Of Technology Kanpur
FIRST MIDSEM EXAM, MTH-101
Date: 17-09-2014
Duration: 2 Hour
1
2
FIRST MIDSEM EXAM, MTH-101
1 (a) Let x1 = 1 and xn+1 =
1
2+xn
for n N. Show that (xn ) satisfies Cauchy
criterion. Find the limit of (xn ).
[6]
Solution
MTH 101-2016
Assignment 1 : Real Numbers, Sequences
m
1. Find the supremum of the set cfw_ |m+n
: n N, m Z.
2. Let A be a non-empty subset of R and R. Show that = supA if and only if
not an upper bound of A but + n1 is an upper bound of A for every n N.
EE210: Microelectronics-I
Lecture-5 : PN Junction Diode-1
B. Mazhari
Dept. of EE, IIT Kanpur
B. Mazhari, IITK
1G-Number
Outline
Semiconductor: Basics
PN Junction Diode
Basic Operation
dc model
small signal model
B. Mazhari, IITK
2G-Number
Semiconductors:
EE210: Microelectronics-I
Lecture-4
Small Signal Device Model-2
B. Mazhari
Dept. of EE, IIT Kanpur
B. Mazhari, IITK
16
G-Number
Vx
)
I x k exp(
.026
I XQ ix k exp(
I XQ k exp(
B. Mazhari, IITK
VXQ
.026
VXQ v x
.026
)
)
17
G-Number
Vx
)
I x k exp(
.026
VXQ
ELECTRODYNAMICS
Ohms Law
To make the current flow, you have to push the charges
If J is the current density and f the force per unit charge
J=sf
What is the nature of this force?
(gravity, electrochemical potential, thermal emf, etc)
In electrodynamics th
Intensity of EM wave ? = the average power per unit area = I
What is the pressure exerted by the radiation on area A in time Dt?
Pressure = force/area
= momentum transferred/ A.Dt
A
v x B force
E
k
B
Electron velocity v
Pressure
on
Reflection
2P
Electroma
PHY103 IInd Half
Instructor R. C. Budhani
Room SL113
Email: [email protected]
Book Introduction to Electrodynamics
(D J Griffiths, IIIrd Edition)
Chapters 6, 7, 8, 9 and 12
Some dos and donts
Magnetostatics an overview
1.
What does it mean?
There are no magn
FLUX RULE
Let us define EMF in a different manner.
We consider the total magnetic field passing through an area
(-ve sign because x is decreasing with time) RHS is nothing but the emf e
A tricky situation
Dics radius a
Magnetic field (constant) as shown
S
Department of Physics, IIT Kanpur
Semester-1, 2016-17
PHY103 Problem Set # 11 Date: October 18, 2016 [RCB/Krishnacharya]
1. In a laboratory experiment a muon is observed to travel 800m before disintegrating. A
researcher looks up the lifetime of a muon (2
1
Lecture 23: Review of vectors, equations of lines and planes; Sequences in R3
In the next two lectures we will deal with the functions from R to R3 . Such functions are called
vector valued functions. After two lectures we will deal with the functions o
1
Lecture 28 : Directional Derivatives, Gradient, Tangent Plane
The partial derivative with respect to x at a point in R3 measures the rate of change of the
function along the X-axis or say along the direction (1, 0, 0). We will now see that this notion c
1
Lecture 24 : Calculus of vector valued functions
In the previous lectures we had been dealing with functions from a subset of R to R. In this
lecture we will deal with the functions whose domain is a subset of R and whose range is in R3 (or
Rn ). Such f
1
Lecture 33 : Change of Variable in a Double Integral; Triple Integral
We used Fubinis theorem for calculating the double integrals. We have also noticed that
Fubinis theorem can be applied if the domain is in a particular form. In this lecture, we will