Category: Class 12

Tamilnadu State Board New Syllabus Samacheer Kalvi 12th Maths Guide Pdf Chapter 2 Complex Numbers Ex 2.1 Textbook Questions and Answers, Notes.

Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 2 Complex Numbers Ex 2.1

Question 1.
i¹⁹⁴⁷ + i¹⁹⁵⁰
Solution:
i¹⁹⁴⁷ + i¹⁹⁵⁰ = (i⁴)⁴⁸⁶ i³ + (i⁴)⁴⁸⁷ i²
[∵ i² = -1; i³ = i; i⁴ = 1]
= (1)⁴⁸⁶ (-i) + (1)⁴⁸⁷ (-1)
= – i – 1 = -(1 + i)

Question 2.
i¹⁹⁴⁸ – i^-1869
Solution:
= (i⁴)⁴⁸⁷ – [(i⁴)⁴⁶⁷ i¹]^-1
= (1)⁴⁸⁷ – (i)^-1 = 1 – \(\frac{1}{i}\) = 1 + i

Question 3.
\(\sum_{n=1}^{12} i^{n}\)
Solution:
= i¹ + i² + i³ + i⁴ + …. + i¹²
= (i – 1 – i + 1) + (i – 1 – i + 1) + (i – 1 – i + 1)
= 0

Question 4.
i⁵⁹ + \(\frac{1}{i^{59}}\)
Solution:
(-i) + \(\frac{1}{(-i)}\)
[∵ i⁵⁹ = i⁵⁶ i³ = (i⁴)¹⁴ (-i) = 1(-i) = -i]
= -i + i = 0

Question 5.
i i² i³ …… i²⁰⁰⁰
Solution:
= i^1+2+3+2000
[∵ 1 + 2 + 3 + …….. n = \(\frac{n(n+1)}{2}\)]
= i^{(\(\frac{2000×2001}{2}\))}
= i^(1000×2001)
= [i¹⁰⁰⁰]²⁰⁰¹
= i^2001000
= 1

Question 6.
\(\sum_{n=1}^{10} i^{n+50}\)
solution:
= i⁵¹ + i⁵² + i⁵³ + ……. + i⁶⁰
= i³ + i⁴ +i + i² + ……. + i⁴
= -i + 1 + i – 1 – i + 1 + i – 1 – i + 1
= 1 – i

Tamilnadu State Board New Syllabus Samacheer Kalvi 12th Maths Guide Pdf Chapter 2 Complex Numbers Ex 2.2 Textbook Questions and Answers, Notes.

Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 2 Complex Numbers Ex 2.2

Question 1.
Evaluate the following if z = 5 – 2i and w = -1 + 3i
(i) z + w
Solution:
z = 5 – 2i, w = -1 + 3i
z + w = (5 – 2i) + (-1 + 3i)
= (5 – 1) + (-2i + 3i)
= 4 + i

(ii) z – iw
Solution:
z – iw = (5 – 2i) – i (-1 + 3i)
= 5 – 2i + i + 3
= (5 + 3) + (-2i + i)
= 8 – i

(iii) 2z + 3w
Solution:
2z + 3w = 2(5 – 2i) + 3 (-1 +3i)
= 10 – 4i – 3 + 9i
= 7 + 5 i

(iv) zw
Solution:
zw = (5 – 2i) (-1 + 3i)
= -5 + 15i + 2i – 6i²
= -5 + 17i + 6
= 1 + 17i

(v) z² + 2zw + w²
Solution:
= (z + w)² [from (i)]
= (4 + i)²
= 16 – 1 + 8i
= 15 + 8i

(vi) (z + w)²
Solution:
(vi) (z + w)² = 15 + 8z [from (v)]

Question 2.
Given the complex number z = 2 + 3i, represent the complex numbers in the Argand diagram.
(i) z = 2 + 3i
iz = i(2 + 3i)
= (2i – 3)
= -3 + 2i
z + iz = (2 + 3i) + (-3 + 2i)
= -1 + 5i

(ii) z, – iz and z – iz
z = 2 + 3i
-iz = -i (2 + 3i)
= -2i – 3i² = -2i + 3
= 3 – 2i
z- iz= 2 + 3i -3 + 2i
= 5 + i

Question 3.
Find the values of the real numbers x and y. if the complex numbers
(3 – i)x – (2 – i) y + 2i + 5 and 2x + (-1 + 2i)y + 3 + 2i are equal.
Solution:
(3 – i) x – (2 – i) y + 2i + 5 = 2x + (-1 + 2i) y + 3 + 2i
⇒ 3x – ix – 2y + iy + 2i + 5 = 2x – y + 2yi + 3 + 2i
⇒ (3x – 2y + 5) + 1 (-x + y + 2) = (2x – y + 3) + i (2y + 2)
Equate real parts on both sides
3x – 2y + 5 = 2x – y + 3
x – y = -2 ……. (1)
Equate imaginary parts on both sides
-x + y + 2 = 2y + 2
-x – y = 0
x + y = 0 ……. (2)
(1) + (2) ⇒ 2x = -2
x = -1
Substituting x = -1 in (2)
-1 + y = 0
⇒ y = 1
∴ x = -1, y = 1

Tamilnadu State Board New Syllabus Samacheer Kalvi 12th Computer Science Guide Pdf Chapter 10 Python Classes and Objects Text Book Back Questions and Answers, Notes.

Tamilnadu Samacheer Kalvi 12th Computer Science Solutions Chapter 10 Python Classes and Objects

12th Computer Science Guide Python Classes and Objects Text Book Questions and Answers

I. Choose the best answer (1 Marks)

Question 1.
Which of the following are the key features of an Object Oriented Programming language?
a) Constructor and Classes
b) Constructor and Object
c) Classes and Objects
d) Constructor and Destructor
Answer:
c) Classes and Objects

Question 2.
Functions defined inside a class:
a) Functions
b) Module
c) Methods
d) section
Answer:
c) Methods

Question 3.
Class members are accessed through which operator?
a) &
b) .
c) #
d) %
Answer:
b) .

Question 4.
Which of the following method is automatically executed when an object is created?
a) _object _()
b) _del_()
c) _func_ ()
d) _init_ ()
Answer:
d) _init_ ()

Question 5.
A private class variable is prefixed with
a) _
b) &&
c) ##
d) **
Answer:
a) _

Question 6.
Which of the following method is used as destructor?
a) _init_ ()
b) _dest_ ()
c) _rem_ ()
d) _del_ ()
Answer:
d) del ()

Question 7.
Which of the following class declaration is correct?
a) class class_name
b) class class_name<>
c) class class_name:
d) class class_name[ ]
Answer:
c) class class_name:

Question 8.
Which of the following is the output of the following program?
class Student:
def_init_(self, name):
self.name=name
print(self.name)
S=Student(“Tamil”)
a) Error
b) Tamil
c) name
d) self
Answer:
b) Tamil

Question 9.
Which of the following is the private class variable?
a) num
b) ##num
c) $$num
d) &&num
Answer:
a) num

Question 10.
The process of creating an object is called as:
a) Constructor
b) Destructor
c) Initialize
d) Instantiation
Answer:
d) Instantiation

II. Answer the following questions (2 Marks)

Question 1.
What is the class?
Answer:
Classes and Objects are the key features of Object-Oriented Programming. Class is the main building block in Python. The object is a collection of data and functions that act on those data. Class is a template for the object. According to the concept of Object-Oriented Programming, objects are also called instances of a class or class variable.

Question 2.
What is instantiation?
Answer:
The process of creating an object is called “Class Instantiation”.
Syntax:
Object_name = class_name()

Question 3.
What is the output of the following program?
Answer:
class Sample:
_num=10
def disp(self):
print(self._num)
S=Sample( )
S.disp( )
print(S._num)
Output:
Error: Sample has no attribute S._num
10

RESTART:
C:/ Users/ COMPUTER / AppData / Local/ Programs / Python / Py thon37-32/ tst.py
Traceback (most recent call last):
File “C:/Users/COMPUTER/ AppData/ Local / Programs / Python/ Python37-32/ tst.py”, line
6, in <module>
S.disp()
AttributeError: ‘Sample’ object has no attribute ‘disp’
>>>

Question 4.
How will you create a constructor in Python?
Answer:

• In Python, there is a special function called “init” which acts as a Constructor.
• It must begin and end with a double underscore.
• This function will act as an ordinary function; but the only difference is, it is executed automatically when the object is created.
• This constructor function can be defined with or without arguments.
• This method is used to initialize the class variables.

General format of _init_method (Constructor function):
def _init_(self, [args………… ]):
< statements >

Question 5.
What is the purpose of Destructor?
Answer:
Destructor is also a special method gets executed automatically when an object exit from the scope. It is just the opposite to the constructor. In Python, _del_( ) method is used as the destructor.

III. Answer the following questions (3 Marks)

Question 1.
What are class members? How do you define it?
Answer:
In Python, a class is defined by using the keyword class. Every class has a unique name followed by a colon ( : ).
Syntax:
class class_name:
statement_1
statement_2
…………………
…………………
statement_n
Where, a statement in a class definition may be a variable declaration, decision control, loop or even a function definition. Variables defined inside a class are called as “Class Variable” and functions are called as “Methods”. Class variables and methods are together known as members of the class. The class members should be accessed through objects or instance of class. A class can be defined anywhere in a Python program.
Example:
Program to define a class
class Sample:
x, y = 10, 20 # class variables
In the above code, name of the class is Sample and it has two variables x and y having the initial value 10 and 20 respectively. To access the values defined inside the class, you need an object or instance of the class.

Question 2.
Write a class with two private class variables and print the sum using a method.
Answer:
Class with two private class variables and print the sum using a method:
class add:
def_init_(self,m,n):
self. _m=m
self. _n=n # m,n, – private variables
def display (self):
sum=self. m+self. n
print(‘ Enter first number=’,self. m,)
print(‘Enter second number=’,self. n)
print(“The sum is”,sum)
x=add(15,2)
x. display ()
Output:
>>>
RESTART: C:/Users/Computer/
AppData /Local/ Programs / Python /
Py thon3 7 / TW OLOC AL. py
Enter first number= 15
Enter second number= 2
The sum is 17
>>>

Question 3.
Find the error in the following program to get the given output?
class Fruits:
def_init_(self, f1, f2):
self.f1=f1
self.f2=f2
def display (self):
print(“Fruit 1 = %s, Fruit 2 = %s” %(self.fl, self.f2))
F = Fruits (‘Apple’, ‘Mango’)
del F.display
F.display( )
Output
Fruit 1 = Apple, Fruit 2 = Mango
Answer:
In line No. 8, del F.display will not come

Question 4.
What is the output of the following program?
Answer:
class Greeting:
def _init_ (self, name):
self. _name = name
def display(self):
print(“Good Morning “, self. _name)
obj=Greeting (‘BinduMadhavan’)
obj. display ()
Output:
Good Morning BinduMadhavan

Question 5.
How do define constructor and destructor in Python?
Answer:
Constructor:

• The constructor is the special function that is automatically executed when an object of a class is created.
• In Python, there is a special function called “init” which act as a Constructor.
• It must begin and end with double underscore.
• This function will act as an ordinary function; but the only difference is, it is executed automatically when the object is created.
• This constructor function can be defined with or without arguments. This method is used to initialize the class variables.

Syntax:
_init_ method (Constructor function)
def _init_(self, [args …. ]):
< statements >

Example:
class Sample:
def _init_(self, num):
print(” Constructor of class Sample…”)
self.num=num
print(“The value is :”, num)
S=Sample(10)

Destructor:

• Destructor is also a special method gets executed automatically when an object exit
• from the scope.
• It is just opposite to the constructor.
• In Python, _del_ () method is used as the destructor.

Example:
class Example:
def _init_ (self):
print “Object created”
# destructor
def _del_ (self):
print “Object destroyed”
# creating an object
myObj = Example ()
Output:
Object created
Object destroyed

IV. Answer the following questions (5 Marks)

Question 1.
Write a menu-driven program to add or delete stationery items. You should use I dictionary to store items and the brand.
Answer:
stationary = { }
while((ch == 1) or (ch == 2))
print(” 1. Add Item \n 2. Delete Item”)
ch = int(input(“Enter your choice “))
if(ch==1):
n = int(input(“Enter the number of items to be added in the stationary shop”))
for i in range(n):
item = input(“Enter an item “)
brand = input(“Enter the brand Name”)
stationary[item] = brand
print(stationary)
elif(ch == 2):
remitem = input(“Enter the item to be deleted from the shop”)
dict.pop(remitem)
print( stationary)
else:
print(“Invalid options. Type 1 to add items and 2 to remove items “)
ch = int(input(“Enter your choice :”)
Output:

1. Add item
2. Delete Item Enter your choice : 1

Enter the number of items to be added in the stationary shop : 2
Enter an item : Pen
Enter the brand Name : Trimax
Enter an item: Eraser
Enter the brand Name: Camlin
Pen: Trimax
Eraser: Camlin
Enter your choice: 2
Enter the item to be deleted from the shop: Eraser
Pen: Trimax
Enter your choice : 3
Invalid options. Type 1 to add items and 2 to remove items.

12th Computer Science Guide Python Classes and Objects Additional Important Questions and Answers

I. Choose the best answer (1 Mark)

Question 1.
………… is not an object-oriented language.
a) C
b) C++
c) Java
d) Python
Answer:
a) C

Question 2.
All integer variables used in the python program is an object of class ……………………….
Answer:
int

Question 3.
………….. are called as Functions of the class.
a) Methods
b) Members
c) Variables
d) Loop
Answer:
a) Methods

Question 4.
In Python, every class name followed by ……………..delimiter.
a) ;
b) :
c) .
d) .
Answer:
b) :

Question 5.
A statement in a class definition may be a ………………………..
(a) variable declaration
(b) decision control
(c) loop
(d) all of these
Answer:
(d) all of these

Question 6.
……………… is a valid syntax for crating objects.
a) objectname = classname ()
b) objectname: classname ()
c) objectname = classname
d) classname = Objectname ()
Answer:
a) objectname = classname ()

Question 7.
…………….. is a valid syntax of accessing class members
a) objectname = classmember ()
b) objectname. classmember ()
c) objectname. Classmember
d) objectname.classmember
Answer:
b) objectname. classmember ()

Question 8.
………….. position of the argument named self in python class method.
a) First
b) Second
c) Third
d) Last
Answer:
a) First

Question 9.
The init function should begin and end with
(a) underscore
(b) double underscore
(c) #
(d) S
Answer:
(b) double underscore

Question 10.
……………. number of arguments can be taken by Python method even when a method is defined with one argument?
a) 1
b) 3
c) 2
d) 4
Answer:
c) 2

Question 11.
In Python. ……………… function will act as a constructor.
a) int
b) inti
c) class name
d) init
Answer:
d) init

Question 12.
…………………….. is a special function to gets executed automatically when an object exit from the scope.
(a) constructor
(b) init
(c) destructor
(d) object
Answer:
(c) destructor

Question 13.
…………… is used to initialize the class variables.
a) Destructor
b) Object
c) Constructor
d) Class member
Answer:
c) Constructor

Question 14.
The ……………..of the class should be accessed through the instance of a class.
a) Objects
b) Members
c) Functions
d) Tuples
Answer:
b) Members

Question 15.
Which variables can be accessed only within the class?
(a) private
(b) public
(c) protected
(d) local
Answer:
(a) private

Question 16.
In Python, the class method must name the first argument named as………….
a) this
b) new
c) self
d) var
Answer:
c) self

Question 17.
…………… and…………… are the key features of object-oriented programming.
a) List and tuples
b) Set and dictionary
c) Classes and objects
d) Variables and methods
Answer:
c) Classes and objects

Question 18.
By default, the class variables are ……………..
a) Private
b) Public
c) Protected
d) Method
Answer:
b) Public

II. Answer the following questions (2 and 3 Marks)

Question 1.
Write a note on public and private data members of the python class.
Answer:

• The variables which are defined inside the class is public by default.
• These variables can be accessed anywhere in the program using dot operator.
• A variable prefixed with double underscore becomes private in nature.
• These variables can be accessed only within the class.

class Sample:
def _init_ (self, n1, n2):
self.n1 =n1
self. _n2=n2
def display (self):
print(“Class variable 1 = “, self.n1)
print(“Class variable 2 = “, self._n2)
S=Sample(12,14)
S.display()
print(“Value 1 = “, S.n1)
print(“Value 2 = “, S._n2)

• In the above program, there are two class variables n1 and n2 are declared.
• The variable n1 is a public variable and n2 is a private variable.
• The display( ) member method is defined to show the values passed to these two variables.
• The print statements defined within the class will successfully display the values of n1 and n2, even though the class variable n2 is private.
• Because, in this case, n2 is called by a method defined inside the class.
• But, when we try to access the value of n2 from outside the class Python throws an error.
• Because private variables cannot be accessed from outside the class.

Question 1.
Write a note on self?
Answer:
The class method must have the first argument named as self. No need to pass a value for this argument when we call the method. Python provides its value automatically. Even if – a method takes no arguments, it should be defined with the first argument called self. If a method is defined to accept only one argument it will take it as two arguments i.e. self and the defined argument.

Question 3.
How Python class function differs from ordinary function.
Answer:

• Python class function or Method is very similar to ordinary function with, a small difference that, the class method must have the first argument named as self.
• No need to pass a value for this argument when we call the method. Python provides its value automatically.
• Even if a method takes no arguments, it should be defined with the first argument called self.
• If a method is defined to accept only one argument it will take it as two arguments ie. self and the defined argument.

Question 4.
Write the output of the following (March 2020)
Answer:
class Hosting:
def _init_ (self.name)
self. _name=name
def display (self):
print(“Welcome to”,self._name)
obj=Hosting(” Python Programming”)
obj.display()
Output:
Welcome to Python Programming

Question 5.
Write a program to calculate area and circumference of a circle?
Answer:
class Circle:
pi=3.14
def_init_(self,radius):
self.radius=radius
def area(self):
return Circle.pi*(self.radius**2)
def circumference(self):
return 2*Circle.pi*self.radius
r = int(input(“Enter Radius:”))
C=Circle(r)
print(“The Area =”,C.area( ))
print(“The Circumference =”, C.circumference( ))
Output:
Enter Radius: 5
The Area = 78.5
The Circumference = 31.400000000000002

HANDS ON PRACTICE

Question 1.
Rewrite the following Python program to get the given output:
OUTPUT:
Enter Radius: 5
The area = 78.5
The circumference = 34.10
CODE:
Class circle ()
pi=3.14
def _init_(self, radius):
self=radius
DEF area(SELF):
Return
Circle.pi + (self.radius * 2)
Def circumference(self):
Return 2*circle.pi * self.radius
r = input(“Enter radius=”)
c = circle(r)
print “The Area:”, c.area()
printf (“The circumference=”, c)

Correct Program:
class Circle:
pi=3.14
def init (self,radius):
self.radius=radius
def area (self):
return Circle.pi*(self.radius**2)
def circumference (self):
return 2*Circle.pi*self.radius
r=int(input(“Enter Radius: “))
C=Circle(r)
print(“The Area =”,C.area())
print(“The Circumference =”, C.circumference())

Question 2.
Write a menu driven program to read, display, add and subtract two distances.
Coding:
class Distances:
def init (self):
self.distl =”
self.dist2 = ”
self .result = ”
def inputdata(self):
self.dist1=float(input(/Enter the first point=’))
self.dist2=float(input(“Enter the second point= “))
def adddist(self):
self ,result=self. dist1+self. dist2
def subdist(self):
if self.distl >self.dist2:
self.result=self.distl-self.dist2
else:
self.result=self.dist2-self.distl
def DisplayDist(self):
return self, result
Dt=Distances() .
ch=,y’
while(ch==’y,):
print(“\nl. Add two points\n2.Subtract two points “)
choice=int(input(“\nEnter your choice:”))
if(choice==l):
Dt.inputdataQ
Dt.adddist()
print(“Sum of two points is:”,round
(Dt.DisplayDist(),3))
elif(choice==2):
Dt.inputdata()
Dt.subdist()
print(“Difference in between two points is:”,round(Dt.DisplayDist(),3))
else:
print(“Invalid input”)
ch=input(“do you want to continue y/n: “)
Output:
>>>

RESTART:
C:/Users / COMPUTER/ AppData/ Local / Programs / Python / Py thon37-32/ menudist-05.02.2020. py
1. Add two points
2. Subtract two points Enter your choice: 1
Enter the first point = 58.6
Enter the second point = 12.8
The Sum of two points is: 71.4
1. Add two points
2. Subtract two points
Enter your choice: 2 ,
Enter the first point= 47.5
Enter the second point= 23.6
The difference between the two points is: 23.9
1. Add two points
2.Subtract two points
Enter your choice: 4
Invalid input
do you want to continue y/n: n
>>>

Question 3.
What will be the output of the following Python code
class String:
def _init_(self):
self.uppercase=0
self.lowercase=0
self.vowels=0
self.consonants=0
self.spaces=0
self.string=””
def getstr (self):
self. string=” Welcome Puducherry”
def count_upper(self):
for ch in self.string:
if (ch.isupper())
self.uppercase+=1
def count_lo wer (self):
for ch in self.string:
if (ch.islower())
self.lowercase+=1
def count_vowers(self)
for ch in self.string:
if (ch in (‘A’, ‘a’, V, ‘E!, T, T, ‘o’, ‘O’, ‘u’,’U’)):
self.vowers+=1
def count_consonants(self):
for ch in self.string:
if (ch not in ((‘A’, ‘a’, V, ‘E’, Y, T, ‘o’, ‘O’, ‘u’, ‘U’,”)):
self.consonants+=1
def count_space (self):
for ch in self.string:
if (ch==” “):
self spaces+=1
def execute (self):
self.count_upper()
self.count_lower() ¦
self.count_vowels()
self.count_consonants()
self.count_space()
def display (self):
print (” The given string contains…”)
print(“%d Uppercase letters”% self .uppercase)
pring(“&d Lowercase letters” % self .lowrcase)
print(” % d V owels” % self .vowels)
print(” % d Consonants” % self, consonants)
pring(“%d Spaces” % self.spaces)
S = String()
S.getstr()
S.execute()
S.display()
Output:
The given string contains…
2 Uppercase letters
17 Lowercase letters
7 Vowels
12 Consonants
2 Spaces

Tamilnadu State Board New Syllabus Samacheer Kalvi 12th Maths Guide Pdf Chapter 1 Applications of Matrices and Determinants Ex 1.8 Textbook Questions and Answers, Notes.

Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 1 Applications of Matrices and Determinants Ex 1.8

Choose the Most Suitable Answer.

Question 1.
If |adj(adj A) |=| A|⁹, then the order of the square matrix A is:
(a) 3
(b) 4
(c) 2
(d) 5
Solution:
(b) 4
Hint:
Since |adj(adj A)| = |A|^{(n – 1)2}
(n – 1)² = 9 = 3²
n – 1 = 3
n = 4

Question 2.
If A is a 3 × 3 non-singular matrix such that AA^T = A^T A and B = A^-1A^T, then BB^T =
(a) A
(b) B
(C) I₃
(d) B^T
Solution:
(C) I
Hint:
AA^T = A^TA and B = A^-1A^T
B^T = (A^-1)^T(A^T)^T = (A^-1)^T A
BB^-1 = (A^-1 A^T) (A^-1)^T A = A^-1 (AA^-1)^T A
= A^-1 (I)^T A = A^-1 A = I₃

Question 3.
If A = \(\begin{bmatrix} 3 & 5 \\ 1 & 2 \end{bmatrix}\) B = adj A and C = 3A, then \(\frac{|adj B|}{|C|}\) =
(a) \(\frac{1}{3}\)
(b) \(\frac{1}{9}\)
(C) \(\frac{1}{4}\)
(d) 1
Solution:
(a) \(\frac{1}{3}\)
Hint:

Question 4.
If A \(\begin{bmatrix} 1 & -2 \\ 1 & 4 \end{bmatrix}\) = \(\begin{bmatrix} 6 & 0 \\ 0 & 6 \end{bmatrix}\), then A:

Solution:
(c) \(\begin{bmatrix} 4 & 2 \\ -1 & 1 \end{bmatrix}\)
Hint:

Solve the equations we get
a = 4, b = 2, c = -1, d = 1
A = \(\begin{bmatrix} 4 & 2 \\ -1 & 1 \end{bmatrix}\)

Question 5.
If A = \(\begin{bmatrix} 7 & 3 \\ 4 & 2 \end{bmatrix}\) then 9I₂ – A =
(a) A^-1
(b) \(\frac{A^{-1}}{2}\)
(c) 3A^-1
(d) 2A^-1
Solution:
(d) 2A^-1
Hint:
|A| = 14 – 12 = 2

Question 6.
If A = \(\begin{bmatrix} 2 & 0 \\ 1 & 5 \end{bmatrix}\) and B = \(\begin{bmatrix} 1 & 4 \\ 2 & 0 \end{bmatrix}\) then |adj (AB)| =
(a) -40
(b) -80
(c) -60
(d) -20
Solution:
(b) -80
Hint:

∴ |adj (AB)| = 8 – 88 = -80

Question 7.
If P = \(\left[ \begin{matrix} 1 & x & 0 \\ 1 & 3 & 0 \\ 2 & 4 & -2 \end{matrix} \right] \) is the adjoint of 3 × 3 matrix A and |A| = 4, then x is
(a) 15
(b) 12
(c) 14
(d) 11
Solution:
(d) 11
Hint:
Since |adj A| = |A|^{n – 1} (n = 3)
1(-6 + 0) – x(-2) = 4²
-6 + 2x = 16
2x = 16 + 6 = 22
x = 11

Question 8.

(a) 0
(b) -2
(c) -3
(d) -1
Solution:
(d) -1
Hint:

Question 9.
If A, B and C are invertible matrices of some order, then which one of the following is not true?
(a) adj A = |A|A^-1
(b) adj (AB) = (adj A)(adj B)
(c) det A^-1 = (det A)^-1
(d) (ABC)^-1 = C^-1B^-1A^-1
Solution:
(b) adj (AB) = (adj A)(adj B)
Hint:
adj (AB) ≠ (adj A)(adj B)

Question 10.

Solution:
(a) \(\begin{bmatrix} 2 & -5 \\ -3 & 8 \end{bmatrix}\)
Hint:
(AB)^-1 = B^-1 A^-1
(AB)^-1 A = B^-1 [A^-1]A
(AB)^-1 A = B^-1
(A^-1)^-1 = A
|A^-1| = 3 – 2 = 1

Question 11.
If A^TA^-1 is symmetric, then A² =
(a) A^-1
(b) (A^T)²
(c) A^T
(d) (A^-1)²
Solution:
(b) (A^T)²
Hint:
(A^T A^-1)^T = A^T A^-1
(A^-1)^T (A^T)^T = A^T A^-1
(A^-1)^T A = A^T A^-1
A^T (A^-1)^T AA = (A^T) A^T A^-1 A
(pre and post multiply by A^T and A)
A² = (A^T)²

Question 12.
If A is a non-singular matrix such that A^-1 = \(\begin{bmatrix} 5 & 3 \\ -2 & -1 \end{bmatrix}\), then (AT)^-1 =

Solution:
(d) \(\begin{bmatrix} 5 & -2 \\ 3 & -1 \end{bmatrix}\)
Hint:
A^-1 = \(\begin{bmatrix} 5 & 3 \\ -2 & -1 \end{bmatrix}\)
Since (A^T)^-1 = (A^-1)^T = \(\begin{bmatrix} 5 & -2 \\ 3 & -1 \end{bmatrix}\)

Question 13.

(a) \(\frac{-4}{5}\)
(b) \(\frac{-3}{5}\)
(c) \(\frac{3}{5}\)
(d) \(\frac{4}{5}\)
Solution:
(a) \(\frac{-4}{5}\)
Hint:
Since A^T = A^-1
AA^T =  I

Question 14.

(a) (cos² \(\frac{θ}{2}\))A
(b) (cos² \(\frac{θ}{2}\))A^T
(c) (cos²θ)I
(d) (sin² \(\frac{θ}{2}\))A
Solution:
(b) (cos² \(\frac{θ}{2}\))A^T
Hint:

Question 15.

(a) 0
(b) sin θ
(c) cos θ
(d) 1
Solution:
(d) 1
Hint:

cos² θ + sin² θ = k
Hence k = 1

Question 16.
If A = \(\begin{bmatrix} 2 & 3 \\ 5 & -2 \end{bmatrix}\) be such that λA^-1 = A, then λ is:
(a) 17
(b) 14
(c) 19
(d) 21
Solution:
(c) 19
Hint:
λA^-1 = A

⇒ λ = 19

Question 17.
If adj A = \(\begin{bmatrix} 2 & 3 \\ 4 & 1 \end{bmatrix}\) and adj B = \(\begin{bmatrix} 1 & -2 \\ -3 & 1 \end{bmatrix}\) then adj (AB) is:

Solution:
(b) \(\begin{bmatrix} -6 & 5 \\ -2 & -10 \end{bmatrix}\)
Hint:
adj (AB) = (adj B) (adj A)

Question 18.

(a) 1
(b) 2
(c) 4
(d) 4
Solution:
(a) 1
Hint:

∴ Number of non zero row = 1
ρ(A) = 1

Question 19.

(a) \(e^{(Δ_{2}/Δ_{1})}\), \(e^{(Δ_{3}/Δ_{1})}\)
(b) log(Δ₁/Δ₃), log(Δ₂/Δ₃)
(c) log(Δ₂/Δ₁), log(Δ₃/Δ₁)
(d) \(e^{(Δ_{1}/Δ_{3})}\), \(e^{(Δ_{2}/Δ_{3})}\)
Solution:
(d) \(e^{(Δ_{1}/Δ_{3})}\), \(e^{(Δ_{2}/Δ_{3})}\)
Hint:
x^ay^b = e^m
a log x + b log y = m
x^cy^d = eⁿ
c log x + d log y = n

Question 20.
Which of the following is/are correct?
(i) Adjoint of a symmetric matrix is also a symmetric matrix.
(ii) Adjoint of a diagonal matrix is also a diagonal matrix.
(iii) If A is a square matrix of order n and λ is a scalar, then adj(λA) = λⁿ adj (A).
(iv) A(adj A) = (adj A)A = |A|I
(a) Only (i)
(b) (ii) and (iii)
(c) (iii) and (iv)
(d) (i), (ii) and (iv)
Solution:
(d) (i), (ii) and (iv)

Question 21.
If ρ(A) ρ([A | B]), then the system AX = B of linear equations is:
(a) consistent and has a unique solution
(b) consistent
(c) consistent and has infinitely many solutions
(d) inconsistent
Solution:
(b) consistent
Hint:
If ρ(A) = ρ[A|B] = no. of unknowns
We say consistent an unique solutions
Hence ρ(A) = ρ[A|B] ← given

Question 22.
If 0 ≤ θ ≤ π and the system of equations x + (sin θ)y – (cos θ)z = 0, (cos θ) x – y + z = 0, (sin θ) x + y + z = 0 has a non-trivial solution then θ is:
(a) \(\frac{2π}{3}\)
(b) \(\frac{3π}{4}\)
(c) \(\frac{5π}{6}\)
(d) \(\frac{π}{4}\)
Solution:
(d) \(\frac{π}{4}\)
Hint:
If AX = 0 has non  trivial solution if |A| = 0

⇒ 1(1 – 1) – sinθ (-cosθ – sinθ) – cosθ (cosθ + sinθ) = 0
⇒ + sinθ cosθ + sin²θ – cos²θ – sinθ cosθ = 0
⇒ sin²θ – cos²θ = 0
⇒ sinθ = cosθ
θ =\(\frac{π}{4}\)

Question 23.

(a) λ = 7, µ ≠ -5
(b) λ = -7, µ = 5
(c) λ ≠ 7, µ ≠ -5
(d) λ = 7, µ = -5
Solution:
(d) λ = 7, µ = -5
Hint:
ρ(A) = ρ[A | B] = 2
λ – 7 = 0 and µ + 5 = 0
λ = 7, µ = -5

Question 24.

(a) 2
(b) 4
(c) 3
(d) 1
Solution:
(b) 4
Hint:
B = A^-1

Question 25.

Solution:
(a) \(\left[\begin{array}{rrr}
3 & -3 & 4 \\
2 & -3 & 4 \\
0 & -1 & 1
\end{array}\right]\)
Hint:
adj (adj A) = |A|^n-2 A = |A|² A
|A|= 3(-3 + 4) + 3(2) + 4 (-2)
= 3 + 6 – 8
= 1

Tamilnadu State Board New Syllabus Samacheer Kalvi 12th Maths Guide Pdf Chapter 1 Applications of Matrices and Determinants Ex 1.7 Textbook Questions and Answers, Notes.

Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 1 Applications of Matrices and Determinants Ex 1.7

Question 1.
Solve the following system of homogeneous equations.
(i) 3x + 2y + 7z = 0; 4x – 3y – 2z = 0; 5x + 9y + 23z = 0
Solution:

ρ(A) = 2 ρ[A | B] = 2
ρ(A) ρ[A | B] = 2 < n
The system is consistent. It has non trivial solution.
Writing the equivalent equations from echelon form
3x + 2y + 7z = 0 ………… (1)
-17y – 34z = 0 ……….. (2)
Put z = t
(2) ⇒ -17y = 34t
y = \(\frac {34t}{-17}\) = -2t
(1) ⇒ 3x + 2(-2t) + 7t = 0
3x – 4t + 7t = 0
3x + 3t = 0
3x = -3t
x = -t
(x, y, z) (-t, -2t, t) ∀ t ∈ R

(ii) 2x + 3y – z = 0, x – y – 2z = 0, 3x + y + 3z = 0
Solution:

ρ(A) = 3 ρ[A | B] = 3
ρ(A) = ρ[A | B] = 3
The system is consistent. It has trivial solution.
x = 0, y = 0, z = 0

Question 2.
Determine the values of λ for which the following system of equations.
x + y + 3z = 0; 4x + 3y + λz = 0, 2x + y + 2z = 0 has
(i) a unique solution
(ii) a non-trivial solution.
Solution:

Case (i):
if λ ≠ 8
ρ(A) = 3 ρ(A | B) = 3
ρ(A) = ρ(A | B) = 3 = n
The system is consistent. It has unique (trivial) solution.
∴ Solution x = 0, y = 0, z = 0

Case (ii):
if λ = 8
ρ(A)= ρ(A | B) = 2
ρ(A) = ρ(A | B) = 2 < n
The system is consistent. It has non trivial solution.

Question 3.
By using Gaussian elimination method, balance the chemical -reaction equation:
C₂H₆ + O₂ → H₂O + CO₂.
Solution:
We are searching for positive integers x₁, x₂, x₃ and x₄
x₁ C₂H₆ + x₂ O₂ → x₃ H₂O + x₄ CO₂ ……….(1)
The number of carbon atoms on the LHS of (1) should be equal to the number of carbon atoms on the RHS of (1) so we get a linear
homogeneous equation.
2x₁ x₄ = 2x₁ – x₄ = 0 ……..(2)
6x₁ = 2x₃ = 6x₁ – 2x₃ = 0
÷ 2 ⇒ 3x₁ – x₃ = 0 ………(3)
2x₂ = x₃ + 2x₄ ⇒ 2x₂ – x₃ – 2x₄ = 0 ……… (4)
Equation (2), (3) and (4) constitute a homogeneous system of linear equations in four unknowns.
Augmented matrix

The system is consistent and has an infinite number of solutions.
Writing the equations using the echelon form we get
-2x₃ + 3x₄ = 0 ……….. (1)
2x₂ – x₃ – 2x₄ = 0 ………… (2)
2x₁ – x₄ = 0 …………. (3)
Put x₄ = t
(3) ⇒ 2x₁ – t = 0
x₁ = \(\frac {t}{2}\)
(1) ⇒ -2x₃ + 3x₄ = 0
-2x₃ = -3t
x₃ = \(\frac {3}{2}\) t
(2) ⇒ 2x₂ – x₃ – 2x₄ = 0
2x₂ – \(\frac {3}{2}\) t – 2t = 0
2x₂ = \(\frac {3}{2}\) t + 2t = \(\frac {7t}{2}\)
x₂ = \(\frac {7t}{4}\)
(x₁, x₂, x₃, x₄) = (\(\frac {t}{2}\), \(\frac {7t}{4}\), \(\frac {3}{2}\)t, t) ∀ t ∈ R
since x₁, x₂, x₃ and x₄ are positive integers.
Let us choose t = 4

So the balanced equation is
2C₂H₆ + 7O₂ → 6H₂O + 4CO₂.

Tamilnadu State Board New Syllabus Samacheer Kalvi 12th Maths Guide Pdf Chapter 1 Applications of Matrices and Determinants Ex 1.6 Textbook Questions and Answers, Notes.

Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 1 Applications of Matrices and Determinants Ex 1.6

Question 1.
Test for consistency and if possible, solve the following systems of equations by rank method.
(i) x – y + 2z = 2, 2x + y + 4z = 7, 4x – y + z = 4
Solution:
Matrix form

The system is consistent.
ρ(A) ρ[A|B] = 3 = n
it has unique solution.
Writing the equivalent equations from echelon form
x – y + 2z = 2 ………… (1)
3y = 3 ⇒ y = 1
-7z = -7
z = 1
(1)⇒ x – y + 2z = 2
x – 1 + 2 = 2
x = 1
∴ Solution is x = 1, y = 1, z = 1

(ii) 3x + y + z = 2, x – 3y + 2z = 1, 7x – y + 4z = 5
Solution:

ρ(A) = 2 ρ[A | B] = 2
ρ(A) = ρ[A | B] = 2 < n
The system is consistent. It has infinitely many solution.
Writing the equivalent equations from echelon form.
x – 3y + 2z = 1 ………. (1)
10y – 5z = -1 ………. (2)
Put z = t.
(2) ⇒ 10y – 5z = -1
10y = -1 + 5z = 5t – 1

(iii) 2x + 2y + z = 5, x – y + z = 1, 3x + y + 2z = 4
Solution:

ρ(A) = 2 ρ[A | B] = 3
ρ(A) ≠ ρ[A | B] = 2 < n
∴ The system is inconsistent. It has no solution.

(iv) 2x – y + z = 2, 6x – 3y + 3z = 6, 4x – 2y + 2z = 4
Solution:

ρ(A) = 1 ρ[A | B] = 1
ρ(A) = ρ[A | B] = 1 < n.
∴ The system reduces into a single equation.
∴ It is consistent and has infinitely many solutions.
Writing the equivalent equations from echelon form
2x – y + z = 2
Put y = s, z = t
2x – s + t = 2
2x = 2 + s – t
x = \(\frac {2+s-t}{2}\)
(x, y, z) = (\(\frac {2+s-t}{2}\), s, t) ∀ s, t ∈ R

Question 2.
Find the value of k for which the equations kx – 2y + z = 1, x – 2ky + z = -2, x – 2y + kz = 1 have
(i) no solution
(ii) unique solution
(iii) infinitely many solution.
Solution:

[∵ 2 – k – k² = -(k² + k – 2)
= -(k + 2)(k – 1)
= (k + 2)(1 – k)]
case (i)
If k = 1
ρ(A) = 2, ρ(A | B) = 3.
ρ(A) ≠ ρ(A | B)
The system is inconsistent and it has no solution.

Case (ii)
If k ≠ 1, k ≠ -2
ρ(A) = 3, ρ(A | B) = 3 = n
The system is consistent and it has unique solution.

Case (iii)
If k = -2
ρ(A) = 2, ρ(A | B) = 2
The system is consistent and it has infinitely many solution.

Question 3.
Investigate the values of λ and µ the system of linear equations 2x + 3y + 5z = 9, 7x + 3y – 5z = 8, 2x + 3y + λz = µ, have
(i) no solution
(ii) a unique solution
(iii) an infinite number of solutions.
Solution:

Case (i)
If λ = 5, µ ≠ 9
ρ(A) = 2, ρ(A | B) = 3
ρ(A) ≠ ρ(A | B)
The system is inconsistent. It has no solution.

Case (ii)
If λ = 5, µ ≠ 9
ρ(A) = 3, ρ(A | B) = 3
ρ(A) = ρ(A | B) = 3 = n
The system is consistent. It has unique solution.

Case (iii)
If λ = 5, µ = 9
ρ(A) = 2, ρ(A | B) =2
ρ(A) = ρ(A | B) = 2 < n
The system is consistent. It has infinitely many solution.

Students can Download Tamil Nadu 12th English Model Question Paper 4 Pdf, Tamil Nadu 12th English Model Question Papers helps you to revise the complete Tamilnadu State Board New Syllabus, helps students complete homework assignments and to score high marks in board exams.

TN State Board 12th English Model Question Paper 4

Time: 2 1/2 Hours
Maximum Marks: 90

General Instructions:

• The question paper comprises of four parts.
• You are to attempt all the sections in each part. An internal choice of questions is provided wherever applicable.
• All questions of Part I, II, III, and IV are to be attempted separately.
• Question numbers 1 to 20 in Part I are Multiple Choice Questions of one mark each. These are to be answered by writing the correct answer along with the corresponding option code.
• Part II has got two sections. The questions are of two marks each. Question numbers 21 to 26 in Section I and Question numbers 27 to 30 in Section II are to be answered in about one or two sentences each.
• Question numbers 31 to 40 in Part III are of three marks each and have been divided in three sections. These are to be answered as directed.
• Question numbers 41 and 47 in Part IV are of five marks each. These are to be answered as directed.

Part -1

I. Answer all the questions. [20 x 1= 20]

Choose the correct synonyms for the underlined words from the options given:

Question 1.
Of the rest, many will end up as mental or physical cripples.
(a) rare gifts (b) able-bodied (c) disabilities (d) perfect health
Answer:
(c) disabilities

Question 2.
Little dangling baskets under the spout to catch the stray leaves.
(a) loose (b) gathered (c) harmful (d) fresh
Answer:
(a) loose

Question 3.
They were doing brisk business in the public square.
(a) dull (b) indolent (c) lethargic (d) active
Answer:
(d) active

Choose the correct antonyms for the underlined words from the options given:

Question 4.
The trolley was commandeered by the two boys.
(a) snatched (b) abandoned (c) usurped (d) hijacked
Answer:
(b) abandoned

Question 5.
The advent of brain-machine interfaces is certain to blur the boundary between humans and machines.
(a) departure (b) danger (c) drawback (d) dispute
Answer:
(d) dispute

Question 6.
Makalu was unexplored.
(a) expelled (b) explained (c) explored (d) expedited
Answer:
(c) explored

Question 7.
Choose the correct combination for the compound word light show.
(a) Noun + Verb (b) Verb + Noun (c) Noun- + Gerund (d) Preposition + Noun
Answer:
(a) Noun + Verb

Question 8.
Choose the correct expansion of NSC.
(a) National Savings Certificate (b) National Service Certificate (c) National Savings Career (d) National Service Certificate
Answer:
(a) National Savings Certificate

Question 9.
Choose the meaning of the foreign word in the sentence: My cousin Charlotte is a real klutz.
(a) stranger (b) thief (c) clumsy fool (d) coward
Answer:
(c) clumsy fool

Question 10.
Choose the right combination for the blended word Bionic.
(a) Bio + Electric (b) Biology + Electronic (c) Bio + Nic (d) Biology + Nic
Answer:
(b) Biology + Electronic

Question 11.
Choose the clipped word for dormitory.
(a) dormi (b) doritory (c) dory (d) dorm
Answer:
(d) dorm

Question 12.
A craze for establishing banks is known as …………………… .
(a) monotonist (b) islomania (c) bancomania (d) barbarian
Answer:
(c) bancomania

Question 13.
Form a derivative by adding the right prefix to the word ‘open’.
(a) re- (b) en- (c) un (d) dis-
Answer:
(a) re-

Question 14.
Fill in the blanks with a suitable relative pronoun.
I have a friend …………………….. cat is very cute.
(a) which (b) whom (c) whose (d) when
Answer:
(c) whose

Question 15.
Fill in the blanks with a suitable preposition.
I heard that news …………………….. the radio.
(a) in (b) on (c) at (d) through
Answer:
(b) on

Question 16.
Choose the correct question tag for the following statement.
He reads a lot of comics, ……………………..?
(a) should he (b) won’t he (c) will he (d) doesn’t he
Answer:
(d) doesn’t he

Question 17.
Choose the suitable meaning or idiom found in the following sentence.
Sumathi was sick of bending over backwards to entertain her nieces.
(a) doing gymnastics (b) dancing (c) trying very hard (d) doing yoga
Answer:
(c) trying very hard

Question 18.
Substitute the underlined word with the appropriate polite alternative.
Till the next trial, the suspects will be kept in the prison camp.
(a) relocation center (b) jail (c) dungeon (d) dormitory
Answer:
(a) relocation center

Question 19.
Choose the correct sentence pattern for the following sentence.
We aren’t complaining.
(a) SVIODO (b) SVC (c) SVO (d) SVCA
Answer:
(b) SVC

Question 20.
Fill in the blank with a suitable phrasal verb.
The following afternoon we …………………….. the tiny village set high upon the hillside.
(a) set aside (b) called on (c) got away (d) drove to
Answer:
(d) drove to

Part II
Section -1

Read the following sets of poetic lines and answer any four from it. [4 x 2 = 8]

Question 21.
“For what, we thought, had we to fear
With our arms and provender, load on load,”
(a) What was the mood of the soldiers?
(b) What made the soldiers confident?
Answer:
(a) The soldiers were confident of winning the war.
(b) They had enough grains to survive a siege and plenty of arms to fight the war. These things made them confident.

Question 22.
“…………….. Free imaginations
Bringing changes into a world resenting change.”
(a) How does free imagination help the world?
(b) Identify the figure of speech.
Answer:
(a) Free imagination brings changes in the world.
(b) Personification

Question 23.
“Death closes all: but something ere the end,
Some work of noble note, may yet be done,
Not unbecoming men that strove with Gods.”
(a) The above lines convey the undying spirit of Ulysses. Explain.
(b) Pick out the words in alliteration in the above lines.
Answer:
(a) Ulysses is aware of ageing and substantial decrease in his physical strength. He knows that will close in on him sooner or later. But before that happens, he wants to sail beyond the sunset/horizon and if possible meet warriors like Achilles. He wants to achieve something worthy of those who challenged and fought with Gods. Thus these lines show the undying spirit of Ulysses.
(b) ere, end noble, note are the words that alliterate.

Question 24.
“And then the lover,
Sighing like furnace, with a woeful ballad
Made to his mistress’ eyebrow.”
(a) What does the lover do for his mistress?
(b) Explain, ‘sighing like furnace’.
Answer:
(a) The lover is always sighing and longing for his beloved. He writes a sad ballad describing the eyebrow of his mistress.
(b) It means moaning, breathing deeply and sadly like a fire place.

Question 25.
“But not because of its magnificence Dear is the Casuarina to my soul:
Beneath it we have played; though years may roll,”
(a) What is not the cause for Toru Dutt’s love for the Casuarina tree?
(b) What makes the tree dear to the poet?
Answer:
(a) As children, the poet and her friends had played under the tree. This experience has made the tree dear to the poet.
(b) The poet is unable to forget the wonderful time she had under the tree with her friends.

Question 26.
“Just as perhaps he mused, ‘My plans’
That soar, to earth may fall,”
(а) What may hamper the soaring plans of Napoleon?
(b) What is the figure of speech employed in these lines?
Answer:
(a) The negative outcome of battle at Ratisbon may hamper his soaring plans.
(b) ‘plans that soar’ – Metaphor.

Section – 2

Answer any three of the following questions. [3 x 2 = 6]

Question 27.
Report the following dialogue:
Answer:
Lady to the Judge : This man is responsible for the accident.
Man : No, my Lord. This lady does not know how to drive a car.

The lady told the Judge that, that man was responsible for the accident. The man denied the lady’s statement respectfully and said that that lady did not know how to drive a car.

Question 28.
Obey the traffic rules. Otherwise, you will be prosecuted, (combine using if)
Answer:
If you do not obey the traffic rules, you will be prosecuted.

Question 29.
Rewrite the sentence making an inversion in the conditional clause.
If you don’t wish to sign the contract, you must let them know by the end of this month.
Answer:
Should you not wish to sign the contract, you must let them know by the end of this month.

Question 30.
When he walked through the wood, he saw a fox that was following him.(Change the following into a simple sentence)
Answer:
Walking through the wood, he saw a fox following him.

Part-III
Section -1

Explain any two of the following with reference to the context. [2 x 3 = 6]

Question 31.
Unto thy honor, Tree, beloved of those Who now in blessed sleep for aye repose,
Answer:
Reference: These lines are from the poem, ‘Our Casuarina Tree’ written by Torn Dutt.

Context: The poet says this while consecrating the memories of the tree to the dear departed.

Explanation: The poet wants to freeze the love for the Casuarina tree in her poem. She remembers with pain three younger Dutts who had succumbed to Tuberculosis. She remembers with poignance the numerous days they had spent under the Casuarina tree.

Question 32.
We are not now that strength which in old days Moved earth and heaven;
Answer:
Reference: These lines are from the poem, ‘Ulysses’ written by Alfred Tennyson.

Context: The poet says these words through Ulysses when he wants to justify the reasons for resuming the daring voyage.

Explanation: Ulysses admits the decline in the compatriots’ physical strength with which they were able to move heaven and earth in their youth. He asks his compatriots to ignore the infinity of age and draw on their inner spiritual strength to resume their voyage beyond sunset.

Question 33.
Then off there flung in smiling joy,
And held himself erect
Answer:
Reference: These words are from the poem, ‘Incident of the French Camp” written by Robert Browning.

Context: The narrator says these words while describing the arrival of a boy soldier at the mound where Napoleon was anxiously awaiting news about the battle.

Explanation: Even though the boy-soldier was split into two, he sped fast in a horse amidst the smoke of cannon fire. On seeing Napoleon he jumped off the horse with a beaming face. Being at the verge of embracing death, he declared the good news that they had won the battle.

Section – 2

Answer any two of the following questions in about 30 words. [2 x 3 = 6]

Question 34.
What were the boys doing at night in a deserted square?
Answer:
The boys were resting on the stone pavement in the windy and deserted square beneath the street light. They were waiting for the last bus from Padua to sell the unsold newspapers.

Question 35.
What are the subsidiary uses of tea leaves?
Answer:
Telling fortunes, predicting the arrival of visitors, feeding rabbits, healing bums and sweeping the carpet are some of the subsidiary’s uses of tea leaves.

Question 36.
When did Hillary feel a sense of freedom and well being?
Answer:
Firsdtly, their partly-filled bottle of oxygen got exhausted. They had only one oxygen bottle to carry. With reduced load of 20 litre bottle, Hillary cut steps down off the South Summit. So, he felt a sense of freedom and well-being.

Section – 3

Answer any three of the following questions in about 30 words. [3 x 3 = 9]

Question 37.
Study the pie chart given and answer the questions that follow.

Popular Brands of Bluetooth Speakers
(a) Which is the most favourite brand of bluetooth speaker?
(b) Name the brand preferred by the least number of customers?
(c) Which two brands have equal number of customers?
Answer:
(a) UE Boom 3 is the most favourite brand preferred by the customers.
(b) Marshall Kilburn is the least preferred brand.
(c) UE Wonderbbom and Bose SoundLink Mini II.

Question 38.
Write a dialogue of minimum 3 exchanges between two friends discussing about a Book Fair.
Answer:
Rajini : Did you visit the Book Fair that started last week?
Ravi : Yes, I did. I was in fact waiting for the Book Fair to start.
Rajini : I know that you are a Voracious reader. How many did you buy?
Ravi : I bought a couple of mystery and detective novels by Tamizhvanan, Agatha Christie and Sujatha. What are you planning to buy?
Rajini : I need to buy a good set of Encyclopedias
Ravi : Do you want me to accompany you? I too want to buy some books published by Scholastic India.

Question 39.
Describe the process of making ice cream.
Answer:
Boil one litre of milk till it becomes thick and creamy and is reduced to one fourth its quantity. Add sugar to taste.
Keep aside five tablespoons of cold milk and mix two heaped spoons of ice cream powder. Pour the ice cream powder mixed milk to the boiling milk and stir continuously till it thickens. Add two drops of vanilla, strawberry or chocolate essence to taste and stir well.
After it cools, freeze for four hours.

Question 40.
Complete the proverbs using the word given below.
(a) He who hesitates is (gone, lost, dead)
(b) If God had meant us to fly, he would have given us (wings, cars, tails)
Answer:
(c) Little pitchers have big (ears, mouths, bottom)
(a) lost (b) wings (c) bottom

Part – IV

Answer the following questions: [7 x 5 = 35]
Answer in a paragraph in about 150 words.

Question 41.
What are the aspects that contribute to humour in the essay?
Answer:
The author claims the best manner of making tea is the subject matter of violent disputes. He compliments china tea for being economical but also says one does not feel rejuvenated after drinking it. One doesn’t feel braver, wiser, more optimistic or stimulated after drinking the Chinese tea. He says army tea tastes of grease and white wash. In Britain there are two schools of thought (i.e.) milk first school and tea first school (i.e.) one school claiming that milk must be poured first and tea added later and another school claiming that tea must be poured first and milk to be added next.

He says those who take tea with sugar for its sweetness alone as misguided people. He says that they could very well add salt and pepper to tea and drink it. He claims that there are some mysterious social etiquette surrounding teapot. It is vulgar to drink tea out of one’s saucer. There are some subsidiary uses of tea leaves such as telling fortunes, predicting the arrival of visitors, feeding rabbits, healing burns and sweeping the carpet.

[OR]

“My right to swing my fist ends, where your nose begins.” Elucidate with reference to, ‘On the Rule of the Road’.
Answer:
Rights are not completely individual affairs. In order to enjoy one’s rights one has to respect the rights of others too. The lady in Petrograd had the right to walk on the pavement. The right to move stops when the other person’s right to drive starts. A person may have a walking stick and roll it too. But his right just stops where the other person’s nose begins. No one has the right to violate the rights of others. The right, one exercises, must not affect or erode the rights of others.

One should not think of one’s own rights but also the rights of others. A.G.Gardiner beautifully illustrates this idea by emphasising the metaphor of traffic rules. Rules of road are in fact rules of politeness and unselfishness. One may have absolute freedom in the choice of food, religion, fashionable dress, up keep of hair, funny hairstyle, etc. But one must be conscious of the rights of others. So, the statement “my right to swing my fist ends, where your nose begins” fits well with the central theme of the essay ‘ On the Rule of the Road’.

Question 42.
Human greed led to the mighty fall of the citadel. Explain.
Answer:
The loyal soldiers and their brave captain expected enemies from outside the castle. Their arms and army was ready to fight them. But they could not identify the enemy within. The soldiers were proud that no might would tear their castle down. But they were unaware of the invisible soul-dead enemy within. The ingredients of personal downfall went unnoticed by them.

If a person never looks within, the faults that can be their doom go overlooked. Their reality could crumble while they gaze outward and pride themselves on their sureness. This is what happened precisely with the soldiers of the castle. They only focused on the strength of their physical surroundings and what was beyond the castle. Human greed-propelled betrayal from within caused the castle’s downfall.

[OR]

Explain how the poet guides his son who is at the threshold of manhood, to face the challenges of life.
Answer:
The poet shares his wisdom with his son who is at the threshold of manhood. He persuades his son to be hard like steel or rock to withstand challenges and unforeseen betrayals in life. A person with soft heart will crumble before a breach of trust. Similarly he wants his son to be discerning enough to be soft when needed to grow like a frail flower plant splitting a rock.

Occasionally one has to go with the current because life is at times fertile with a lot of opportunities to grow even among the harshest circumstances. ‘Rich soft wanting’ can help a person to win against all odds. He reiterates this idea by explaining how gentleness can reform a hardened criminal when lashes would, in contrast, harden them further.

Question 43.
Write a paragraph of about 150 words by developing the following hints:
Pi convinced – water on board – divining rod – Pi’s mind – genuine regulation lifeboat – outfitted with supplies – captain – ensure safety and survival – ship chandler – extra money-saving lives – water on board.
Answer:
Pi had never before experienced physical hell than that putrid taste and pasty feeling in the mouth. It was an unbearable pressure at the back of his throat. The divining rod in Pi’s mind dipped sharply and spring gushed forth. He remembered that he was on a genuine regulation lifeboat and such a lifeboat was surely outfitted with supplies. A captain would never fail in so elementary way of preserving water to ensure safety and survival of his crew in the event of a disaster. Besides, it is natural that a ship chandler would think of making a little extra money under the noble pretext of saving lives. It was settled beyond doubt that there was water on board.

[OR]

Bond of friendship – Baldwin and Gresham – boyhood chums – school together – lifetime bondage – thirty five years – baptism – 60 dollars a week – Governor of the private bank – Gresham siphoning money – loyalty and honesty – part ways – upright character – close the bank – return the deposit – reorganization of the bank – arrest – court could not nail Gresham – one hundred thousand dollars – three words – slapped him – body language – not compromise with conscience.
Answer:
Baldwin and Gresham were boyhood chums. They went to school together. Their friendship flowered into a lifetime bondage as it was sustained for thirty five years. When Baldwin’s son was being baptised in the church, Gresham was present. Baldwin expressed his wish that his son John Gresham Baldwin grew up to Gresham’s standard in life. Baldwin was paid only 60 dollars a week for working as a Governor of the private bank founded by Gresham.

Accidentally, Baldwin found out how Gresham was siphoning money very cleverly from the deposits technically without really causing any loss to the depositors. But Baldwin was loyal to Gresham only as long as he was honest. Once he parted ways with honesty, Baldwin would also part ways with Gresham. By the force of his upright character, Baldwin advised him to close the bank and return the deposit to the customers.

Third National Bank was ready to help the reorganization of the bank. Meanwhile, Gresham was arrested. The court could not nail Gresham as there was not a shred of evidence to prove his guilt. If Baldwin testified, the court would indict him. Just before being arrested Gresham offered him one hundred thousand dollars to just say “I don’t remember”. The three words to let him off the hook. He claimed it was the difference between the salary he had been paid and what he ought to have received. Had it been someone else, Baldwin would have slapped him.

But it was Gresham who understood his body language and did not press the matter further. He was proud of the fact that Gresham understood how he could not compromise with his conscience. His family members Martha, Evie and John felt that it was a compliment that cost him one hundred thousand dollars. Baldwin said that Gresham’s compliment about his uprightness was worth a hundred thousand dollars.

Question 44.
Write a summary or Make notes of the following passage.
Answer:
Occasional self-medication has always been part of normal living. The making and selling of drugs has a long history and is closely linked, like medical practice itself, with belief in magic. Only during the last hundred years or so, as the development of scientific techniques made it possible, diagnosis has become possible. The doctor is now able to follow up the correct diagnosis of many illnesses – with specific treatment of their causes. In many other illnesses of which the causes remain unknown, he is still limited, like the unqualified prescriber, to the treatment of symptoms.

The doctor is trained to decide when to treat symptoms only and when to attack the cause. This is the essential difference between medical prescribing and self-medication. The advance of technology has brought about much progress in some fields of medicine, including the development of scientific drug therapy. Parallel with such beneficial trends are two which have an adverse effect. One is the use of high pressure advertising by the pharmaceutical industry which has tended to influence both patients and doctors and has led to the overuse of drugs generally.

The other is emergence of eating, insufficient sleep, excessive smoking and drinking. People with disorders arising from faulty habits such as these, as well as from unhappy human relationships, often resort to self-medication. Advertisers go to great lengths to catch this market. Clever advertising, aimed at chronic sufferers who will try anything because doctors have not been able to cure them, can induce such faith in a preparation, particularly if steeply priced, that it will produce -by suggestion- a very real effect in some people.

It is doubtful whether taking these things ever improves a person’s health, it may even make it worse. Worse, because the preparation may contain unsuitable ingredients; worse because the taker may become dependent on them; worse because they might be taken excess; worse because they may cause poisoning, and worst of all because symptoms of some serious underlying cause may be asked and therefore medical help may not be sought. Self-diagnosis is a greater danger than self-medication.

Summary

No. of words given in the original passage: 347
No. of words to be written in the summary: 347/3 = 115 ± 5
Rough Draft
Self-medication is part of normal living. Medicinal experts are required for diagnosis and treatment of diseases according to symptoms and causes. The development of drug therapy and improvement in public health organizations and nutritional standards have helped progress in medicinal science. Excessive advertising by pharmaceutical companies and emergence of the sedentary society are two counter trends. Self-medication is dangerous as the preparation may be toxic or contain unsuitable ingredients; the user becomes dependent and consumes medicine in excess. Self-diagnosis is worse than self-medication.

No. of words in the summary: 83
Notes:
Title: Self-Medication

1. Self-medication
(a) Part of normal living—last 100 yrs
(b) Advance in diagnostic technology
(c) Doctors required – diagnosis & treatment of disease
(d) Self-medication differs from medical prescription

2. Technological Advmnt in medicine
(a) Drug therapy
(b) Improvement in public health organizations
(c) Increase in nutritional standards

3. Clever advertising by pharmaceutical companies
(a) Take advantage of people’s need
(b) Chronic sufferers
(c) Faulty lifestyle
(i) Lack of exercise, overeating, insufficient sleep, etc.
(ii) Stress, unhappy rela’ps, etc.

4. Dangers of self-medication
(a) Preparation of some drugs contains unsuitable ingredients
(b) Taker becomes dependent
(c) Taker consumes medicine in excess
(d) Preparations may cause poisoning .
(e) Real cause of illness gets suppressed or untreated

Abbreviations used: yrs – years; Advmnt – Advertisement; rela’ps – relapses

Question 45.
Write a letter to the Port Trust, Chennai, asking permission for a group of 25 students of your school to visit a ship that has collection of books of all languages of the world. The ship would remain in Chennai only for a fortnight. Get permission mentioning the date, time and the number of visitors the teachers in – charge of the bona fide students of your school.
Answer:
14th January, 2020
From
The Principal
TM Hr. Sec. School
Chennai – 45

To
The Naval Officer
Port Trust of India
Chennai
Respected Officer,

Sub: Reg. Permission for visiting the Ship – Queen Mary

Given to understand that the Ship named Queen Mary with a collection of books of all languages of the world has arrived at the Chennai Port for public viewing, I hereby request permission for our school children to visit the ship on one of the working days.
We would be grateful if you can permit us to view the books in the ship on Friday, 6th February, 2020 at 10 a.m. There will be 15 students from Std IX accompanied by two teachers with one male office staff.

Please find enclosed the bona fide certificates and Photocopy of the staff identity card accompanying them.
Staff in-charge: Mrs. Shalini Varma and Mrs. Sharmila Das.

Students of Std IX:

• Anitha.M
• Asha.L
• Balaji.G
• Bharath.U
• Chandhini.G
• Christopher. Y
• rthy.T
• Firoz.P
• Janaki.R
• Kalai.A
• Lalitha.G
• Leena.B
• Mohan.C
• Niranjan.F
• Rohit.R

Looking forward to an enriching experience and safe visit. Kindly send us a reply acknowledging our request.

Yours sincerely,
Meena
(Principal)

Address on the envelope:
The Naval Officer
Port Trust of India
Chennai

[OR]

Write a paragraph of 150 words on “Punctuality”.
Answer:
It has been said, “Never put off for tomorrow what you can do today”. Yet there are many people who enjoy postponing things. Such people do not realize the dangers of delaying. Work does not disappear if we postpone it. The difficulty of the work does not get reduced by postponing it. The more we postpone, the more the work piles up. Finally, the load of work seems too much. We then have to work for many days at a stretch under great strain and tension. Finally we do it hurriedly and in a careless manner.

It is as bad as not doing the work at all. Again, when some work is delayed, the time which would have been profitably used is wasted. Time wasted is time lost forever. Hence procrastination, that is the habit of postponing things, is rightly called the thief of time. Precious time wasted means opportunities lost, and lost forever. By the time we realize how much time we have wasted and how many opportunities we have missed by delaying word, it may be too late.

And then nothing remains for us except to regret and repent. Sometimes a very heavy price has to be paid for delaying things. The person who delays insuring his factory will regret it when the factory is gutted by a sudden fire. Delay in the treatment of a disease may make it worse, and may even result in death. There are other proverbs conveying similar meaning. Thus we say: “Time and tide wait for no man” and “A stitch in time saves nine”. All these proverbs warn us against the dangers of delay in actions, and stress the importance of timely action and punctuality.

Question 46.
Spot the errors and rewrite the sentences correctly
(a) Many a man have been arrested at the meeting.
(b) Most of the work have been completed.
(c) Neither of the books are illustrated.
(d) One of the thieves, were arrested.
(e) Rani sat besides the new student.
Answer:
(a) Many a man had been arrested at the meeting.
(b) Most of the work has been completed.
(c) Neither of the books is illustrated.
(d) One of the thieves, was arrested.
(e) Rani sat beside the new student.

[OR]

Fill in the blanks correctly.
(а) Someone who decides to …………….. a car has committed a crime, but auto parts are made of aluminium and …………….. (steel/steal)
(b) They look just the same! I am sure that boy …………….. be his son. (Use a modal in the given blank.)
(c) All my clothes are dirty! I …………….. do some laundry, (use a semi-modal)
(d) He …………….. never …………….. (study) art before he came to college. (use a proper tense)
Answer:
(a) steal/steel (b) must (c) need to (d) had/studied

Question 47.
Identify each of the following sentences with the fields given below:
(a) Shakespeare’s plays are read by many people.
(b) The yield of wheat has increased.
(c) Fast food is a growing health hazard.
(d) My brother is planning to go to the US.
(e) Dhoni was declared the Man of the Series.
[Sports, Nutrition and Dietetics, Travel, Literature, Agriculture]
Answer:
(a) Literature
(b) Agriculture
(c) Nutrition and Dietetics
(d) Travel
(e) Sports

[OR]

Read the following passage and answer the questions in your own words.
Answer:
Housed in an 18th century style Heritage building with wooden paneling running through the thirty seat dining area, “Lean and Lovely” is the latest attraction in Siddhapuram Nagar. Says the owner Chef Virina “People come to us for the unusual fare that we serve. Fairly ordinary Indian recipes are ignited with a dash of sauce and spiced with colour. There is harmony and balance between taste, lightness and tradition. Wholly organically grown vegetables and flour are used. We use the freshest of ingredients that are cut and cooked so as to display their colour and individual texture.

Questions:
a. What is special about the vegetables and flour used by “Lean and Lovely”?
b. In which town is “Lean and Lovely” located?
c. Who is a ‘Chef’?
d. How do the people in “Lean and Lovely” make the ordinary Indian recipes more attractive?
e. How do they keep the colour and texture of the food items?
Answers:
(a) Wholly organically grown vegetables and flour is the speciality of “Lean and Lovely’.
(b) “Lean and Lovely” is located in Siddhapuram Nagar.
(c) A chef is a skilled cook, who prepares a variety of dishes in a restaurant and manages them.
(d) Fairly ordinary Indian recipes are ignited with a dash of sauce and spiced with colour in “Lean and Lovely”. There is also harmony and balance between taste, lightness and tradition attracting the people.
(e) Freshest of ingredients to retain the colour and texture of the food items.

Tamilnadu State Board New Syllabus Samacheer Kalvi 12th Maths Guide Pdf Chapter 1 Applications of Matrices and Determinants Ex 1.5 Textbook Questions and Answers, Notes.

Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 1 Applications of Matrices and Determinants Ex 1.5

Question 1.
Solve the following systems of linear equations by Gaussian elimination method:
(i) 2x – 2y + 3z = 2, x + 2y – z = 3, 3x – y + 2z = 1
Solution:
Augmented matrix

Writing the equivalent equations from echelon from.
x – y + 2z = 3 …………. (1)
5y – 6z = -4 ………….. (2)
-z = -4
z = 4
(2) ⇒ 5y – 6z = -4
5y – 24 = -4
5y = -4 + 24
5y = 20
y = 4
(1) ⇒ x – y + 2z = 3
x – 4 + 8 = 3
x = 3 + 4 – 8
x = -1
∴ x = -1, y = 4, z = 4

(ii) 2x + 4y + 6z = 22, 3x + 8y + 5z = 27, -x + y + 2z = 2.
Solution:
Augmented matrix

Writing the equivalent equations from echelon from.
x + 2y + 3z = 11 …………. (1)
y – 2z = -3 ………….. (2)
11z = 22
z = 2
(2) ⇒ y – 2z = -3
y – 4 = -3
y = -3 + 4
y = 1
(1) ⇒ x + 2y + 3z = 11
x + 2(1) + 3(2) = 11
x + 2 + 6 = 11
x = 11 – 8 = 3
∴ x = 3, y = 1, z = 2

Question 2.
If ax² + bx + c is divided by x + 3, x – 5, and x – 1, the remainders are 21, 61 and 9 respectively. Find a, b and c. (Use Gaussian elimination method.)
Solution:
Given: f(x) = ax² + bx + c
In Remainder Theorem
f(-3) = 21
a(-3)² + b(-3) + c = 21
9a – 3b + c = 21 ……….. (1)
f(5) = 61
25a + 5b + c = 61 …………. (2)
f(1) = 9
a + b + c = 9 ………… (3)
Augmented matrix

Writing the equivalent equations from echelon from.
a + b + c = 9 …………. (1)
b + 2c = 5 ………….. (2)
-4c = -8
c = 2
(2) ⇒ b + 2c = 5
b + 4 = 5
b = 5 – 4
b = 1
(1) ⇒ a + b + c = 9
a + 1 + 2 = 9
a = 9 – 3
a = 6
a = 6, b = 1, c = 2

Question 3.
An amount of Rs 65,000 is invested in three bonds at the rates of 6%, 8% and 10% per annum respectively. The total annual income is Rs 5,000. The income from the third bond is Rs 800 more than that from the second bond. Determine the price of each bond. (Use Gaussian elimination method.)
Solution:
Let the amounts of 3 bounds be x, y, z
x + y + z = 65,000

Writing the equivalent equations from echelon from.
x + y + z = 65000 …………. (1)
2y + 3z = 90000 ………….. (2)
21z = 42000
z = 20000
(2) ⇒ 2y = 90000 – 3 × 20000
2y = 30000
y = 15000
(1) ⇒ x + 15000 + 20000 = 65000
x = 30000
∴ x = 30000, y = 15000, z = 20000

Question 4.
A boy is walking along the path y = ax² + bx + c through the points (-6, 8),(-2, -12), and (3, 8). He wants to meet his friend at P(7, 60). Will he meet his friend? (Use Gaussian elimination method.)
Solution:
y = ax² + bx + c
At(-6, 8) ⇒ 8 = 36a – 6b + c ………… (1)
At(-2, -12) ⇒ -12 = 4a – 2b + c ………… (2)
At(3, 8) ⇒ 8 = 9a + 3b + c ………… (3)
Augmented matrix

Writing the equivalent equations from the echelon.
36a – 6b + c = 8 …………. (1)
3b – 2c = 29 ………….. (2)
5c = -50
c = -10
(2) ⇒ 3b – 2c = 29
3b – 20 = 29
3b = 9
b = 3
(1) ⇒ 36a – 18 – 10 = 8
36a = 8 + 18 + 10
36a = 36
a = 1
At P (7, 60), y = ax² + bx + c
60 = 1(7²) + 3(7) – 10
60 = 49 – 21 – 10
60 = 60
He will meet his friend at P (7, 60)

Students can Download Tamil Nadu 12th English Model Question Paper 5 Pdf, Tamil Nadu 12th English Model Question Papers helps you to revise the complete Tamilnadu State Board New Syllabus, helps students complete homework assignments and to score high marks in board exams.

TN State Board 12th English Model Question Paper 5

Time: 2 1/2 Hours
Maximum Marks: 90

General Instructions:

• The question paper comprises of four parts.
• You are to attempt all the sections in each part. An internal choice of questions is provided wherever applicable.
• All questions of Part I, II, III, and IV are to be attempted separately.
• Question numbers 1 to 20 in Part I are Multiple Choice Questions of one mark each. These are to be answered by writing the correct answer along with the corresponding option code.
• Part II has got two sections. The questions are of two marks each. Question numbers 21 to 26 in Section I and Question numbers 27 to 30 in Section II are to be answered in about one or two sentences each.
• Question numbers 31 to 40 in Part III are of three marks each and have been divided in three sections. These are to be answered as directed.
• Question numbers 41 and 47 in Part IV are of five marks each. These are to be answered as directed.

Part -1

I. Answer all the questions. [20 x 1= 20]

Choose the correct synonyms for the underlined words from the options given:

Question 1.
The policeman is not a symbol of tyranny.
(a) autocracy (b) plutocracy (c) aristocracy (d) democracy
Answer:
(d) democracy

Question 2.
I dragged our oxygen sets into the tent.
(a) pushed (b) pulled (c) lauded (d) loaded
Answer:
(b) pulled

Question 3.
His shoulder and arm were amputated.
(a) frustrated (b) attached (c) transplanted (d) removed
Answer:
(d) removed

Choose the correct antonyms for the underlined words from the options given:

Question 4.
She was eager to do so.
(a) anxious (b) apathetic (c) impatient (d) repulsive
Answer:
(b) apathetic

Question 5.
Not the flat, shallow type…
(a) wide (b) wide (c) direct (d) deep
Answer:
(d) deep

Question 6.
Why people should suffer.
(a) undergo pain (b) enjoy (c) heal (d) struggle
Answer:
(b) enjoy

Question 7.
Choose the correct combination for the compound word ‘milk white’.
(a) Noun+ Gerund (b) Noun + Adjective (c) Gerund + Noun (d) Preposition + Noun
Answer:
(b) Noun + Adjective

Question 8.
Choose the correct expansion of TPS.
(a) Thermal Power Supply (b) Thermal Private Sector (c) Thermal Power Station (d). Thermal Power Sector
Answer:
(c) Thermal Power Station

Question 9.
Choose the meaning of the foreign word in the sentence:
I planned to go downtown to meet with Betty, but I ran into a glitch.
(a) big problem (b) small problem (c) gutter (d) accident
Answer:
(b) small problem

Question 10.
Choose the correct combination for the blended word Globish.
(a) Globe+ Fish (b) Globe + frisk (c) Globe + ish (d) Global + English
Answer:
(d) Global + English

Question 11.
Choose the clipped form for graduate.
(a) gradud (b) gad (c) grad (d) duate
Answer:
(c) grad

Question 12.
A desire to read and enjoy eulogy is known as ……………………. .
(a) eulogomania (b) misogamist (c) oleograph (d) monogram
Answer:
(a) eulogomania

Question 13.
Form a derivative by adding the right prefix to the word ‘closed’.
(a) in- (b) re- (c) un- (d) en-
Answer:
(d) en-

Question 14.
Fill in the blanks with a suitable relative pronoun. This is the house ………………… my father built.
(a) where (b) by (c) that (d) who
Answer:
(c) that

Question 15.
Fill in the blanks with a suitable preposition. You should explain this ……………… them.
(a) for (b) with (c) from (d) to
Answer:
(d) to

Question 16.
Choose the correct question tag for the following statement.
Heera can read many languages ……………………….?
(a) won’t she (b) can’t she (c) isn’t she (d) wasn’t she
Answer:
(b) can’t she

Question 17.
Choose the suitable meaning or idiom found in the following sentence.
When the management wanted to cut down on the man power, they offered a golden hand shake to many of their newly recruited employees.
(a) extension of job
(b) renewal of contract
(c) large amount of money given to a person when they leave
(d) give part-time work
Answer:
(c) large amount of money given to a person when they leave

Question 18.
Substitute the underlined word with the appropriate polite alternative.
I think the garbage man didn’t come on his usual rounds.
(a) cleaner (b) sanitizer (c) garbage disposer (d) sanitation engineer
Answer:
(d) sanitation engineer

Question 19.
Choose the correct sentence pattern for the following sentence.
There was a grand finale of scattered plates and silverware.
(a) SVIODO (b) SVAA (c) SVCA (d) AVSC
Answer:
(c) SVCA

Question 20.
Fill in the blank with a suitable phrasal verb.
I ………………………… to the fountain to have my shoes shined.
(a) went across (b) went over (c) went into (d) put aside
Answer:
(b) went over

Part II
Section – 1

Read the following sets of poetic lines and answer any four from it. [4 x 2 = 8]

Question 21.
“And the enemy half a mile away They seemed no threat to us at all”
(a) What were the soldiers expecting?
(b) Where were the enemies?
Answer:
(a) The soldiers were expecting a war.
(b) The enemies were half a mile away.

Question 22.
“And then the justice,
In fair round belly with good capon lin’d,
With eyes severe and beard of formal cut,
Full of wise saws and modern instances
(a) Whom does justice refer to?
(b) Describe his appearance.
Answer:
(a) Justice refers to a man in his fifth stage when he becomes critical of everyone else’s opinion in life.
(b) He has a pot belly and is fond of eating delicacies.

Question 23.
“A creeper climbs, in whose embraces bound No other tree could live.”
(a) How does the tree survive the tight hold of the creeper?
(b) Why does Toru Dutt use the expression ‘a creeper climbs’?
Answer:
(a) The tree takes the tight hold of the creeper like the embraces of a lady love. So, it doesn’t hurt the tree. It grows stronger bearing the bite marks of love.
(b) A creeper cannot grow without the support of another tree or a pole. While climbing, it tries to sap the energy from the living tree. It twines its body around the tree and keeps climbing.

Question 24.
“That ever with a frolic welcome took
The thunder and the sunshine, and opposed”
(a) What do ‘thunder’ and ‘sunshine’ refer to?
(b) What do we infer about the attitude of the sailors?
Answer:
(a) Thunder and sunshine refers to misfortunes and happy days. Ulysses and his comrades had undergone both kinds of experiences.
(b) The sailors shared the undying quest for exploration, adventure and for seeking newer knowledge in the untravelled world. They even welcomed dangers in fighting with Gods. They enjoyed the thrill of action and never worried about the outcome of battles or quests. They have equal temper of heroic hearts.

Question 25.
“Just as perhaps he mused, ‘My plans That soar, to earth may fall,”
(а) Who does ‘he’ refer to?
(b) What may hamper the soaring plans of Napoleon?
Answer:
(a) ‘He’ refers to Napoleon Bonaparte.
(b) The negative outcome of battle at Ratisbon may hamper his soaring plans.

Question 26.
“Tell him to be a fool ever so often and to have no shame over having been a fool yet learning something out of every folly hoping to repeat none of the cheap follies”
(a) Is it a shame to be a fool at times?
(b) What does one learn from every folly?
Answer:
(a) No, everyone does commit funny mistakes in life. One might just laugh at them.
(b) Every folly teaches a person his limitations and vulnerabilities. By making conscious efforts to avoid them in future, one will become stronger and wiser.

Section – 2

Answer any three of the following questions. [3 x 2 = 6]

Question 27.
Report the following dialogue:
Answer:
James : Can you come to my apartment tomorrow?
Simon : Thank you, I will come tomorrow after lunch.
James asked Simon if he could come to his apartment the next day. Simon thanked him and said that he would come to his apartment the following day after lunch.

Question 28.
Sundar wants to pass the test. He has to work hard, (combine using ‘if’)
Answer:
If Sundar wants to pass the test, he has to work hard.

Question 29.
Rewrite the sentence making an inversion in the conditional clause.
If we were to have children, we would certainly need to move to a bigger house.
Answer:
Were we to have children, we would certainly need to move to a bigger house.

Question 30.
The house was destroyed in the fire, but the whole family was saved. (Change the following into a complex sentence)
Though the house was destroyed in the fire, the whole family was saved.

Part-III
Section -1

Explain any two of the following with reference to the context. [2 x 3 = 6]

Question 31.
I will maintain until my death
Answer:
Reference: This line is from Edwin Muir’s poem “The Castle”.

Context: The poet says this while recounting the shameful act of betrayal by the aged warder of the wicket gate.

Explanation: The narrator was overconfident of the invincibility of their castle, their stock piled arms and well-stocked granary. They had a brave captain and loyal soldiers. In an open war, they need not fear defeat as their friendly neighbours were also willing to join them during a war against any invader. But their enemy was within their fortified castle. They were sold for a bag of gold. Without a groan the citadel was captured. It was a shameful way to lose one’s side. So, the narrator prefers not to recount this shameful secret to anyone so long as he is alive.

Question 32.
Last scene of all,
That ends this strange eventful history,
Answer:
Reference: These lines are from the poem ‘All the world’s a stage’ written by William Shakespeare.

Context: The poet says these words while describing the preparedness of the old man in the last stage of life to exit from this lonely planet.

Explanation: The poet beautifully says the “eventful history” (i.e.) life which was spiced up with many interesting things is now coming to a dramatic close. The eternal jewel of life, ‘the soul’, is going to depart the body which had kept it imprisoned for long. The soul celebrates the joy of freedom in death.

Question 33.
He will be lonely enough to have time for the work
Answer:
Reference: These lines are from the poem ‘A Father to his Son’ written by Carl August Sandburg.

Context: The poet says these words to explain how creative thinkers and those who strive to bring about changes are left alone to fend for themselves.

Explanation: The poet asks his son to take advantage of the loneliness. He must enjoy the advantages of solitude. Solitude would help him to be creative. He would invariably learn that final decisions are always taken in silent rooms. In solitude he can pursue his creative imagination and succeed like Shakespeare, Wright brothers, Pasteur, Pavlov and Faraday.

Section – 2

Answer any two of the following questions in about 30 words. [2 x 3 = 6]

Question 34.
How was Dr. Barnard’s attitude to suffering different from that of his father’s?
Answer:
Dr. Barnard’s father accepted suffering as God’s will. He also believed that suffering ennobles humans. But Dr. Barnard found no meaning in the agony and suffering of patients and especially of the young children.

Question 35.
What was thrilling to watch 8000 feet below them?
Answer:
In a number of places, the overhanging ice cornices were very large. In order to escape them, Hillary cut a line of steps down to where the snow met the rocks on the west. It was a great thrill to look straight down that enormous rock face to see 8000 ft below them the tiny tents of Camp 4 in the Western Cwm.

Question 36.
‘Curtailment of private liberty is done to establish social order’ – Do you agree?
Answer:
Yes, curtailment of private liberty is done to establish social order. Liberty is not a personal affair only, it is a social contract. There are a lot of people in the world who need to accommodate their liberty to the liberties of others. A reasonable consideration of the rights or feelings of others in the foundation of social conduct. When one has submitted to the curtailment of private liberty, one may enjoy a social order which makes his liberty, a reality.

Section – 3

Answer any three of the following questions in about 30 words. [3 x 3 = 9]

Question 37.
Study the pie-chart given below and answer the questions that follow:

(a) How much percentage in this group have AB blood group?
(b) What is the percentage of people in this group who do not have the donor group?
(c) According to the given percentage, how many people have either A or B blood groups?
Answer:
(a) 19% of the people have AB Blood group.
(b) 60% of the people do not have the donor group that is Type O.
(c) Approximately Question Question 41.1 % of people do not have either A or B blood groups.

Question 38.
Write a dialogue of minimum 3 exchanges between a Mr. Karnan and a Policeman.
Answer:
Mr. Karnan : Thank you sir for responding to us at this time. There has been a theft in my house.
Policeman : It’s our duty, sir. I hope you haven’t touched anything inside the house.
Mr. Karnan : No sir, I made a call from my mobile as soon as we returned from the night show and found the front door open.
Policeman : Did you enter the house?
Mr. Karnan : Yes sir, just now. The burglars have stolen many of my household articles including money and jewellery.
Policeman: Don’t worry. We will find the culprits at the earliest.

Question 39.
Describe the process of making an omelette.
Answer:
Take an egg and beat the egg well with a fork or egg beater.
Finely chop a small onion, one green chilly, one pod of garlic, coriander leaves Mix the chopped ingredients to the beaten egg.
Add salt to taste and beat.
Heat a tava and add a spoon of oil.
Pour the mixture into the hot tava like a dosa.
Flip sides till it gets cooked on both sides.

Question 40.
Complete the proverbs using the word given below.
(а) Familiarity breeds (friendship, contempt, malice)
(b) Fools rush in where angels fear to (tread, exit, enter)
(c) Give the devil his (share, profit, due)
Answer:
(a) contempt (b) tread (c) due

Part – IV

Answer the following questions: [7 x 5 = 35]
Answer in a paragraph in about 150 words.

Question 41.
How did a casual incident in a hospital help Dr. Barnard perceive a new dimension of life?
Answer:
Initially Dr. Barnard was grumbling. He wondered why on earth he and his wife should have been subjected to agony and inconvenience. He couldn’t take his dad’s view that suffering ennobles human beings. But the little boys taught him a profound lesson of life. One should get on with the business of living irrespective of whatever misfortune strikes one.

You don’t become a better person because you suffered, your suffering does not ennoble you. But you become a better person because you have experienced suffering. It is not what you have lost is important. What is important is what you have left. We can appreciate light better once we have experienced darkness. Similarly, we can appreciate warmth only after experiencing cold.

[OR]

The ridge had Hillary and Tenzing two and half hours, but it seemed like lifetime. Why?
Answer:
Both Hillary and Tenzing reached a wide ledge. Hillary deeply felt the fierce determination that nothing could stop them from scaling the Himalayas. He took a stronghold on the ledge and signalled Tenzing to come up. With great difficulty Tenzing reached the way up the crack and collapsed like a giant fish hauled up from the ocean. There were giant cornices on the right and steep rock sloped on the left. The ridge curved away to the right. They had no idea where the peak was.

As Hillary cut around the back of one hump, another would swing anew to his view. Time seemed never-ending. Their original zest started diminishing. The climb was becoming a grim struggle. To their surprise, the ridge ahead now dropped sharply away. A few more whacks of the ice-axe in the firm snow, they stood on top. It was Question 11.30, the ridge had taken two and a half hours. But it seemed like a lifetime as they had to negotiate numerous ridges on the way.

Question 42.
Give the summary of the poem, “Our Casuarina Tree’.
Answer:
The poet Torn Dutt describes the perpetually young Casuarina tree in this poem. The tree is tall and mighty. A creeper winds around its huge and wide trunk like a python. Birds chirp happily. The poet is ecstatic seeing the tree from her casement / window. In fact, she greets the tree everyday soon after getting up from bed. She finds a baboon sitting like a statue on top of the tree. The cows are grazing in the lush green meadows around the tree. Water lilies add to the charm of the pond. She recalls that the tree is dear not because of its majestic looks but because of the nostalgic memories attached to it.

She remembers the happy times she had spent with her siblings, who are now no more, under the tree. They died of consumption. Their absence makes the tree much more endearing to the poet. Mysteriously she hears the grief- filled lament of the tree when she is on the shores of France and Italy. She consecrates the tree’s memory to her dear departed siblings. Taking inspiration from William Wordsworth’s poem ‘Yew-trees’, she also wishes to immortalize the Cauarina tree. So she sings about it in her poem.

[OR]

What is the role of the young soldier in the victory of the French at Ratisbon?
Answer:
The young soldier was one of the soldiers in the infantry division leading the battle. On storming Ratisbon, unmindful of the cannon fire, he climbed the flag post with French flag and hoisted it. He received the bullets in turn for his service to the emperor and French army. He did not succumb to the bullets immediately, he galloped on horse back to convey the news to emperor Napoleon Bonaparte himself.

He held on to life till he reached Napoleon and conveyed the happy news. He waited with abated breath to know the reaction of his great leader. When he expressed his sadness, his pride was hurt. He denied the emperor’s sympathy and said emphatically that he was killed. He fell down beside emperor Napoleon with a smiling face and died.

Question 43.
Write a paragraph of about 150 words by developing the following hints:
Ivan – tongue-tied – knife – moral shock – innocent victim – faith in justice – wife’s suspicion – loses hopes – belief in God – divine justice. Tsar’s judgement- mills of justice grind slow – Semyonich confesses – cruel joke
Answer:
Ivan is tongue-tied when the police finds knife in his bag. The moral shock he undergoes as an innocent victim of circumstances allows him to have faith injustice. But the moment his wife expresses her suspicion over his involvement in the murder, he loses all hopes. It is then that he starts believing in God and divine justice.

Tsar can’t give him freedom. His judgement is based on evidences given by crooked people. God need not be given any evidences of innocence. He knows the truth but his mills of justice grind very slow. Ivan loses interest in escaping or leaving the prison. He had no one to go back to. At this juncture, Semyonich confesses his guilt of murdering the merchant. When the pardon arrives as a cruel joke, Ivan is dead. So, ‘God knows the truth but waits’ is the most appropriate title for this story.

[OR]

Sermon on diligence – Roger demands half-day leave – value of hard work – an impartial judge – switches off attention – take notes – self-indulgent absent-mindedness – escapes – rewinds memory – hoodwinks the judge – interview the award committee – judge was quite impressed – Mr. Clay’s office – orderly mind – excel in any learned profession
Answer:
In the beginning of the play Roger demands half-day leave and that was enough for the judge to start his sermon on the value of hard work and his standing in the society as an impartial judge. Roger conveniently switches off his attention and pretends to take notes of the instructions of the judge. The judge almost catches in his self-indulgent absent-mindedness. Roger escapes by the skin of his teeth by just rewinding from his memory the oft-repeated story of his success.

He hoodwinks the judge as to the purpose of his demand for half-a-day leave by saying that he wanted to personally interview the award committee as to why they had not forwarded the documents. The judge was quite impressed. The judge asked him to walk to Mr. Clay’s office in the afternoon. He reiterated that diligence, courage and attention to details are required to cultivate an orderly mind. Without an orderly mind no man can hope to excel in any learned profession.

Question 44.
Write a summary or Make notes of the following passage.
Answer:
Teaching is the noblest of vocations. A teacher has a sacred duty to perform. It is he on whom rests the responsibility of shaping the character of young children. Apart from developing their intellect, he can inculcate in them qualities of good citizenship, remaining neat and clean, talking decently and sitting properly. These virtues are not easy to be absorbed.

Only he who himself leads a life of simplicity, purity and rigid discipline can successfully cultivate these habits in his pupils. Besides a teacher always remains young. He may grow old in age, but not in spite. Perpetual contact with budding youth keeps him happy and cheerful. There are moments when domestic worries weigh heavily on his mind but the delightful company of innocent children makes him overcome his temporary moods of despair.

Summary

No. of words given in the original passage: 133
No. of words to be written in the summary: 133/3 = 41 ± 5

Rough Draft
Teaching is the noblest vocation. A teacher has a sacred duty to perform the responsibility of shaping the character of young children. Apart from developing their intellect, he can inculcate good citizenship. Besides a teacher always remains young perpetual contact with the youth keeps him overcome his temporary moods of despair.

Fair draft Noble Profession
Teaching is the noblest vocation. A teacher himself leading a simple, pure and disciplined life can shape the character of the young children and make them neat and good mannered citizens. Besides he remains every young forgetting his own domestic worries in the constant company of the young.

No. of words in the summary: 48

Or

Notes
Title: Noble Profession
Teachg – Noble Profession
sacred duty – mouldg charac
qlts A good citizenship
neat, clean,

Teachr – Forever Young
grow old in age but not in spite.
contact c budg youth .
domestic worries weigh heavily-delightful co. A innocent children overcomes

Abbreviations used: Mouldg – moulding; charac – character; A – of; qlts – qualities; Teachg – teaching; Teachr – teacher; c – with; budg-budding; co. – company

Question 45.
You are Roshini of 27-Railway Quarters, ICF, Chennai. Last week you bought a bluetooth speaker from ‘Universal Mobile’, 20L, Luz Corner, Chennai. The bluetooth speaker developed a problem within a few days of its purchase. Write a complaint letter to the dealer giving details of the nature of the problem and asking him/her to rectify the defect or replace the phone.
Answer:
26th March, 2019
From
Roshini
27-Railway Quarters
ICF
Chennai

To
M/s Universal Mobile
20L, Luz Corner
Chennai
Dear Sirs,

Sub: Defective Mobile Phone
I am a resident of ICF, Chennai. I purchased a JBL Clip 3 bluetooth speaker from Universal Mobile on 19th March, 20Question 20.1 am sorry to say that the speaker developed a problem within a few days of its purchase. The sound is quite unclear and irritating. I feel cheated to have such a defective mobile phone after spending more than twelve thousand mpees.

It is quite unfortunate that even after sending two reminders, you have shown no urgency to rectify the defects or replace the defective mobile set at the earliest. I hope you will do the needful within a week. I am sure you will not compel me to knock the doors of the Consumer Court for this unpleasant lapse of time on your part.

Yours sincerely
Roshini

Address on the envelope:
To
Universal Mobile, Luz Comer
Chennai

[OR]

Write a paragraph of 150 words on “My ambition in life.”.
Answer:

My ambition in life

Taking a photo or posing for a photo is something that I have always enjoyed even as a small little kid. There’s one thing that gets me to do and excites me. It is photography and that really stands out from all the rest. I always think of going all over the world and take shots of everything worth-seeing. I want to stick to photography all the time. I think I will never get tired of it. Even as a kindergartener, I used to admire pictures taken and collect pictures.

I always asked my parents to get me a camera. For my 14th birthday, I was gifted an SLR because of my passion for photography. I know there’s so much I have to leam about photography and I am trying to leam more and more about it. I joined a photography class after my tenth exams. I always used to take part in competitions and win prizes. I think there are many people in my community who are really good at photography and present outstanding photographs regarding nature’s beauty and others.

Many of them held photography contests to motivate and encourage new photographers. I now have a craze to do Candid Photography. I know for sure that I can pursue this as a hobby as well as my profession. If you love your job, you will never feel the pressure of working and hence I have decided to become a photographer for my life.

Question 46.
Spot the errors and rewrite the sentences correctly
(a) If I had known you were ill, I would visit you.
(b) He spoke the English perfectly.
(c) I am writing the essay when the bell rang.
(d) I and Sheela have booked tickets for the latest Harry Potter movie.
(e) It is an great honour.
Answer:
(a) If I had known you were ill, I would have visited you.
(b) He spoke English perfectly.
(c) I was writing the essay when the bell rang.
(d) Sheela and I have booked tickets for the latest Harry Potter movie.
(e) It is a great honour.

[OR]

Fill in the blanks correctly.
(а) My daughter who is just four …………………… (ears/years) old loves to play in the (sun/son)
(b) If it were not cold outside, I …………………… be much happier! (Use a modal in the given blank.)
(c) You …………………… finish reading this book by this weekend, (use a semi-modal)
(d) They …………………… (buy) sponges this week, (use a proper tense)
Answer:
(a) years/sun (b) would (c) need to (d) will be buying

Question 47.
Identify each of the following sentences with the fields given below:
(a) At the Sarangkheda horse fair in Maharashtra, local breeds attract buyers from as far as Saudi Arabia.
(b) My brother was awestruck at the western classical concert.
(c) Dhoni was declared the Man of the Series.
(d) French actress Catherine’s clothing was designed by her friend Yas for her next project.
(e) The investigations included megascopic and microscopic studies of rocks.
[Geology; Tourism; Music; Sports; Costume]
Answer:
(a) Tourism (b) Music (c) Sports (d) Costume (e) Geology

[OR]

Read the following passage and answer the questions in your own words.
Answer:
Black holes are one of the most mysterious and powerful forces in the universe. But what exactly are they? Are they simply holes that are black? Well, there’s more to it than that. A black hole is not a regular object with a surface area, like a planet. Instead, it’s an area in space where matter has literally collapsed onto itself. Black holes are formed when giant stars run out of energy. They end their life cycle and explode. This massive explosion is called a supernova. If the star has enough mass, it will collapse onto itself into a super small size.

Imagine an enormous mass squeezed into a tiny space. That space would become compact and dense. This causes the gravitational pull to be extremely strong. It will gobble up and absorb anything in its way, even gigantic stars. Nothing can move fast enough to escape its gravity. Not even light, the fastest thing in the universe! Black holes will gobble up anything and everything in their path, including light.

What happens to an object that gets sucked into a black hole? It will be literally stretched to its breaking point. Luckily, we don’t have to worry about black holes swallowing up our solar system. That’s because the nearest black hole is about 27,000 light-years away! Here’s a strange fact: black holes cannot actually be seen. That’s because they don’t deflect light. So how can we be certain they exist? Scientists discovered them by observing light and objects around them.

They noticed that black holes affected their surroundings like nearby dust, stars, and galaxies. Two different scientists from the 18th century named John Michell and Pierre-Simon Laplace first observed them. Then in 1967, a physicist named John Archibald Wheeler came up with the term “black hole”.

Black holes come in different sizes. Some are about the mass of one star. These are called “stellar” black holes. Others can grow and become huge. They may continue to absorb light, mass, and even stars around them. These are called “super-massive black holes.” They can be one million times more massive than our sun. Many scientists believe these super-massive black holes often exist right at the center of galaxies. In fact, we have one right in the middle of our very own Milky Way.

Questions:
a. Based on the information in the article, what is a black hole?
b. Who first discovered the existence of black holes?
c. Why are black holes dangerous?
d. Are all black holes the same size? Explain.
e. Why can’t black holes be seen?
Answers:
(a) A black hole is one of the most mysterious and powerful forces in the universe, an area in space where matter has collapsed in on itself.
(b) Two different scientists from the 18th century named John Michell and Pierre-Simon Laplace first discovered the existence of black holes.
(c) Black holes pull in anything that gets too close and stretch things to their breaking point and hence they are dangerous.
(d) No. Some black holes are the size of a single star. These are called stellar black holes. Others are the size of a million stars the size of our sun, or larger. These are called super- massive black holes.
(e) Black holes cannot actually be seen because they don’t deflect light. They can only be discovered by observing light and objects around them.

Tamilnadu State Board New Syllabus Samacheer Kalvi 12th Maths Guide Pdf Chapter 1 Applications of Matrices and Determinants Ex 1.1 Textbook Questions and Answers, Notes.

Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 1 Applications of Matrices and Determinants Ex 1.1

Question 1.
Find the adjoint of the following:

Solution:

Question 2.
Find the inverse (if it exists) of the following.

Solution:
\(\begin{bmatrix} -2 & 4 \\ 1 & -3 \end{bmatrix}\)
|A| = 6 – 4 = 2 ≠ 0
∴ A^-1 exists. A is non singular.


|A| = 2(8-7)-3(6-3)+1(21-12)
= 2 – 9 + 9 = 2 ≠ 0. A^-1 exists.

Question 3.

Solution:
\(\left[\begin{array}{ccc}
\cos \alpha & 0 & \sin \alpha \\
0 & 1 & 0 \\
-\sin \alpha & 0 & \cos \alpha
\end{array}\right]\)
|F(α)| = cos α(cos α – 0) – 0 + sin α(0 + sin α)
= cos²α + sin²α = 1
|f(α)| = 1 ≠ 0. [F(α)]^-1 exists.


[∵ cos (-θ) = cos θ ; sin(-θ) = -sin θ]
from (1) and (2) we have
[F(α)]^-1 = F(-α)

Question 4.
If A = \(\begin{bmatrix} 5 & 3 \\ -1 & -2 \end{bmatrix}\), show that A² – 3A – 7I₂ = O₂. hence find A^-1
Solution:

∴ A² -3A – 7I₂ = O₂
Post multiply this equation by A^-1
A²A^-1 – 3A A^-1 – 7I₂ A^-1 = 0
A – 3I – 7A^-1 = 0
A – 3I = 7 A^-1
A^-1 = \(\frac {1}{7}\) (A – 3I)

Question 5.

Solution:

Question 6.
If A = \(\begin{bmatrix} 8 & -4 \\ -5 & 3 \end{bmatrix}\) verify that A(adj A) = (adj A) A = \(\left| A \right|\)I₂.
Solution:

(1), (2) and (3) ⇒ A (adj A) = (adj A)A = |A| I₂.

Question 7.

Solution:

Question 8.

Solution:

|adj (A)| = 2 (24 – 0) + 4 (- 6 – 14) + 2(0 + 24)
= 48 – 80 + 48 = 16

Question 9.

Solution:

|adj A| = 0 + 2(36 – 18) + 0 = 2(18) = 36

Question 10.

Solution:

Question 11.

Solution:


Hence proved

Question 12.

Solution:

Question 13.

Solution:
Given A × B × C
⇒ A^-1 A × B B^-1 = A^-1 C B^-1
I × I = A^-1 C B^-1
⇒ X = A^-1 CB^-1
let us find A^-1 and B^-1

Question 14.

Solution:


Hence proved.

Question 15.
Decrypt the received encoded message [2 -3] [20 4] with the encryption matrix \(\begin{bmatrix} -1 & -1 \\ 2 & 1 \end{bmatrix}\) and the decryption matrix as its inverse, where the system of codes are described by the numbers 1 – 26 to the letters A – Z respectively, and the number 0 to a blank space.
Solution:

So the sequence of decoded row matrics is [8 5] [12 16]
The receiver reads the message as “HELP”.