Commutative Matrices:
(a)
Given,
3 0
A
0 2
4 k
B
h 1
We have to find the value of
h
and
k
such that
AB BA
Since,
AB BA
Therefore,
3 0 4
0 2 h
12
2h
k 4 k 3 0
1 h 1 0 2
3k 12 2k
2 3h 2
Compare the elements of the matrix of both sides as:
3k 2k
3k 2
Change of Variable:
Given,
D
be the region such that
1 x y 2
and
1 x y 3
(a)
We have to find the value of the integral
x
D
2
y 2 dA
First draw the region
D
as:
By the above figure,
Region
Now,
D
can be break into three sub regions
,
and .
D1 D2
D3
x
2
Hermite Polynomials are Mutually Orthogonal:
Given,
Hermite Polynomial:
Hk x e
x2
2
We have to that
Hk x e
are mutually orthogonal that we have to show that
x2
2
H k x H l x e x dx 0
2
Since,
H k x 1 e x
k
2
d k x2
e
dx k
Multiply the above expression by
Adjustable Rate Mortgage or ARM:
Given,
Mortgage Amount $52, 468
Number of Payments 25 12
300 Months
Margin rate 2%
Index Rate 6.6%
Periodic Rate Cap 2% per adjustment
Balance of Principle at the end of the first year $49, 602.02
(a)
We have to find the
X intercept of a function:
Given,
f x ax 2 2a 1 x 2
We have to find the point where the
x intercept
First factorize the given function as:
f x ax 2 2ax x 2
ax x 2 1 x 2
x 2 ax 1
For
x
intercept
f x 0
Therefore,
x 2 ax 1 0
x 2,
1
a
Therefore, there are
Complex Analysis (MA 412)
Instructor: Prof.Sourav Pal
January, 2017April, 2017
Problem set 2
1. Let : [0, 2] C be the curve (t) = 2eit . Prove that
Z
2
dz
z 3 + 2 2 1 .
2. Let be the circle (t) = 10eit , 0 t 2.Prove that
Z
dz
20
z 4 + 1 9999 .
1
1
3. Le
CBSE - Class X
Solved Paper 2014
Math
Section A
Q. 1. In a right angle triangle ABC , right angle at B , BC 12 cm and AB 5 cm . The radius of
the circle inscribed in the triangle (in cm) is
A.
B.
C.
D.
3
13
2
1
Solution: (C)
Given ABC is a right angle tri
CBSE - Class X
Solved Paper 2014
Math
Section A
Q. 1. In a right angle triangle
ABC
, right angle at ,
and
. The radius of
B BC 12 cm
AB 5 cm
the circle inscribed in the triangle (in cm) is
A.
B.
C.
D.
3
13
2
1
Solution: (C)
Given
ABC
is a right angle tri
Exercise
Samacheer-11
Physics
Wave Motion
General Instructions:
(i)
(ii)
(iii)
(iv)
Questions 1-15 are very short answer questions.
Questions 16-35 are short answer questions. Answers to them should
normally not exceed 70 words each.
Questions 36-45 are l
CBSE - Class X
Solved Paper 2012
Math
Section A
Q. 1. The roots of the quadratic equation
A.
B.
C.
D.
8 3 x
2
2
are
8 3 y
2
2
7 4 y
2
2
y 74
y3
8 3 x
2
2
x 83
x5
7 3 y
2
2
Solution: (A)
7 4 x
2
2
Incorrect options:
(B)
Since quadratic equation can have a
CBSE - Class X
Solved Paper 2012
Math
Section A
Q. 1. The roots of the quadratic equation
A.
2,
B.
C.
2x2 x 6 0
are
3
2
3 3
2, ,
2 2
1 2
,
2 3
D.
2,
3
2
Solution: (A)
2 x2 x 6 0
2 x 2 3x 4 x 6 0
x 2 x 3 2 2 x 3 0
x 2 2 x 3 0
x 2, x
3
2
Incorrect optio
CBSE - Class X
Solved Paper 2012
Math
Section A
Q. 1. The roots of the quadratic equation
A.
2,
B.
C.
2x2 x 6 0
are
3
2
3 3
2, ,
2 2
1 2
,
2 3
D.
2,
3
2
Solution: (A)
2 x2 x 6 0
2 x 2 3x 4 x 6 0
x 2 x 3 2 2 x 3 0
x 2 2 x 3 0
x 2, x
3
2
Incorrect optio
; MathType INI v1: Equation Preferences
[Styles]
Text=Times New Roman
Function=Times New Roman
Variable=Times New Roman,I
LCGreek=Symbol,I
UCGreek=Symbol
Symbol=Symbol
Vector=Times New Roman,B
Number=Times New Roman
User1=Courier New
User2=Times New Roman
CBSE - Class X
Solved Paper 2012
Math
Section A
Q. 1. The roots of the quadratic equation
A.
B.
C.
D.
8 3 x
2
2
are
8 3 y
2
2
7 4 y
2
2
y 74
y3
8 3 x
2
2
x 83
x5
7 3 y
2
2
Solution: (A)
7 4 x
2
2
Incorrect options:
(B)
Since quadratic equation can have a
CBSE - Class X
Solved Paper 2012
Math
Section A
Q. 1. The roots of the quadratic equation
A.
B.
C.
D.
8 3 x
2
2
are
8 3 y
2
2
7 4 y
2
2
y 74
y3
8 3 x
2
2
x 83
x5
7 3 y
2
2
Solution: (A)
7 4 x
2
2
Incorrect options:
(B)
Since quadratic equation can have a
Class IX
Math
Coordinate Geometry
Exercise
General instructions
(i)
(ii)
(iii)
(iv)
Question no. 1-15 are objective questions.
Questions no. 16-35 are short answer questions. Answers to them should normally not
exceed 70 words each.
Questions no. 36-45 ar
ICSE-Class X
Math
Exercise
Polynomials
General Instructions:
(i)
(ii)
(iii)
(iv)
Question no. 1-15 are objective questions.
Questions no. 16-35 are short answer questions. Answers to them should
normally not exceed 70 words each.
Questions no. 36-45 are l
Very short answer type questions
Question 1) What is the product of
Question 2) What is the product of
x2 x2 y
x 3 xy
Question 3) What is the product of
x4
Question 4) What is the product of
.
y
x
x2
y 2
y
2
Question 5) Find the product of
Question 6) Fi
CBSE - Class X
Solved Paper 2014
Math
Section A
Q. 1. If ,
and
are three consecutive terms of an A.P., then the value of is
k 2k 1
2k 1
k
A.
B.
C.
D.
2
3
3
5
Solution: (B)
In case of A.P. ,
,
,
.
a a d a 2d a 3d
Difference d is found by
,
a d a a 2d ( a
CBSE - Class X
Solved Paper 2014
Math
Section A
Q. 1. In a right angle triangle
ABC
, right angle at ,
and
. The radius of
B BC 12 cm
AB 5 cm
the circle inscribed in the triangle (in cm) is
A.
B.
C.
D.
3
13
2
1
Solution: (C)
Given
ABC
is a right angle tri
Mensuration MCQ
1. If all the triangles are equilateral triangle of length 7 cm. What is the perimeter of the
following figure below?
a.
b.
c.
d.
e.
84
112
56
3
None of these
The correct option is (a).
Solution:
Perimeter of the above figure is the sum of