TS 6th Class Maths Solutions Chapter 8 Data Handling Ex 8.1

Students can practice Telangana 6th Class Maths Textbook Solutions Chapter 8 Data Handling Ex 8.1 to get the best methods of solving problems.

TS 6th Class Maths Solutions Chapter 8 Data Handling Exercise 8.1

Question 1.
A child’s kiddy bank is opened and the coins collected are in the following denomination.
TS 6th Class Maths Solutions Chapter 8 Data Handling Ex 8.1 1
Represent the data in a frequency distribution table using tally marks.
Answer:
TS 6th Class Maths Solutions Chapter 8 Data Handling Ex 8.1 2

Question 2.
The favourite colours of 25 students in a class are given below :
Blue, Red, Green, White, Blue, Green, White, Red, Orange, Green, Blue, White, Blue, Orange, Blue, Blue, White, Red, White, White, Red, Green, Blue, Blue, White.
Write a frequency distribution table using tally marks for the data. Which is the least favourite colour for the students ?
Answer:
TS 6th Class Maths Solutions Chapter 8 Data Handling Ex 8.1 3
The least favourite colour for the students is Orange.

TS 6th Class Maths Solutions Chapter 8 Data Handling Ex 8.1

Question 3.
A TV channel invited a SMS poll on ‘Ban of Liquor’ giving options.
A – Complete ban B – Partial ban C – Continue sales
They received the following SMS; in the first hour.
TS 6th Class Maths Solutions Chapter 8 Data Handling Ex 8.1 4
Represent the data in a frequency distribution table using tally marks.
Answer:
SMS poll on ‘Ban of Liquor’
TS 6th Class Maths Solutions Chapter 8 Data Handling Ex 8.1 5

Question 4.
Vehicles that crossed a checkpost between 10AM and 11AM are as follows :
Car, lorry, bus, lorry, auto, lorry, lorry, bus, auto, bike, bus, lorry, lorry, zeep, lorry, bus, zeep, car, bike, bus, car, lorry, bus, lorry, bus, bike, car, zeep, bus, lorry, lorry, bus, car, car, bike, auto.
Represent the data in a frequency distribution table using tally marks.
Answer:
Vehicles that crossed a checkpost.
TS 6th Class Maths Solutions Chapter 8 Data Handling Ex 8.1 6

TS 6th Class Maths Solutions Chapter 6 Integers InText Questions

Students can practice Telangana 6th Class Maths Textbook Solutions Chapter 6 Integers InText Questions to get the best methods of solving problems.

TS 6th Class Maths Solutions Chapter 6 Integers InText Questions

Do This

Question 1.
Manasa has borrowed ₹ 50 and Swetha has borrowed ₹ 20 from their mother. How will you represent this on the number line ₹ Suppose their father gave them ₹ 100 each as pocket money, who will have more money after clearing the debit ?
Answer:
TS 6th Class Maths Solutions Chapter 6 Integers InText Questions 1
If each of them get ₹ 100 from their father
then amount at Manasa = 100 – 50 = ₹ 50
then amount at Swetha = 100 – 20 = ₹ 80
∴ Amount of Swetha is greater than that of Manasa.
TS 6th Class Maths Solutions Chapter 6 Integers InText Questions 2

Try These

Question 1.
Collect information about temperatures recorded in various places in India in the month of January and write them using integers.
Answer:
The temperatures of various places in the month of January
1. Hyderabad → 10°C
2. Siachain (J & K) → 20°C
3. Lambasingi → 5°C

Do this

Question 1.
Draw a vertical line and represent the following integers on the number line. – 5, 4, – 7, – 8, – 2, 9, 5, – 6, 2
Answer:
Vertical line :
TS 6th Class Maths Solutions Chapter 6 Integers InText Questions 3
On a number line :
TS 6th Class Maths Solutions Chapter 6 Integers InText Questions 4

TS 6th Class Maths Solutions Chapter 6 Integers InText Questions

Do This

Question 1.
Fill in the boxes using < or > signs.
(i) 0 ………. -1
Answer:
0 > -1

(ii) -3 ……. – 2
Answer:
-3 < -2

(iii) 5 ……… 6
Answer:
5 < 6

(iv) -4 ……… 0
Answer:
-4 < 0

Do This

Question 1.
Rajesh has a shop CM thp ground floor of a building. There are stairs going up to the terrace and stairs going down to the godown, where goods are stored.
Everyday his daughter Hasini, after coming back from school goes up to the terrace to play. She helps father in arranging things in the godown at night.
Observe the picture and try to answer the questions using integers marked on the steps.
TS 6th Class Maths Solutions Chapter 6 Integers InText Questions 5
(i) Go 7 steps up from the shop.
(ii) Go 3 steps down from the ground floor.
(iii) Go 5 steps up from the ground floor and then go 3 steps further up from there.
(iv) Go 4 steps down from the ground floor and then further 3 steps from there.
(v) Go down 5 steps down from the ground floor and 10 steps up fmin there.
(vi) Go 8 steps up from the ground floor and come down 9 steps down from there.
Answer:
(i) +7
(ii) -3
(iii) 5 + 3 = + 8
(iv) (-4) + (-3) = -7
(v) (-5) + 10 = +5
(vi) (8) + (-9) = (—1)

Do This

Question 1.
Find the values of the following.
(i) -7 + 8
(ii) -3 + 5
(iii) – 3 – 2
(iv) + 7 – 10
Answer:
(i) (- 7) + (8)
= (- 7) + (+ 7) + (+1)
= [(-7) + (+7)] + (+1)
= 0 + 1 = 1

(ii) (- 3) + (+ 5)
= [(-3) + (+3)] + (+2)
= 0 + 2 = 2

(iii) (-3) +(-2)
= [-(3 + 2)] = -5

(iv) (+7) + (-10)
= (+7) + [(-7) + (-3)]
= [(+7) + (-7)] + (-3)
= 0 – 3
= – 3

TS 6th Class Maths Solutions Chapter 6 Integers InText Questions

Try These

Question 1.
Find the value of the following using a number line.
(i) (- 3) + 5
(ii) (- 5) + 3
Make your own two new questions and solve them using the number line.
Answer:
(i) (- 3) + 5 = (- 3) + (3 + 2) = [- 3 + 3] + 2 = 0 + 2 = 2
TS 6th Class Maths Solutions Chapter 6 Integers InText Questions 6

(ii) (- 5) + 3 = [(- 3) + (- 2)] + 3 = [(- 3) + 3] + (- 2) = 0 + (- 2) = – 2
TS 6th Class Maths Solutions Chapter 6 Integers InText Questions 7

Ex: (i) (-4) + 1 = -3
TS 6th Class Maths Solutions Chapter 6 Integers InText Questions 8

(ii) 6 + (-4) = 2
TS 6th Class Maths Solutions Chapter 6 Integers InText Questions 9

Question 2.
Find the solution of the following:
(i) (+5) + (-5)
(ii) (+6) + (-7)
(iii) (-8) + (+2)
Ask your Mend to give five such questions and solve them.
Answer:
(i) (+ 5) + (- 5) = 5 – 5 = 0
(ii) (+ 6) + (- 7) = (+6) + [(- 6) + (- 1)] = [(+ 6) + (-6)] + (- 1) = 0 + (- 1) = – 1
(iii) – 8 + (+ 2) = [(- 6) + (- 2)] + (+ 2) = (- 6) + [(- 2) + (+ 2)] = (- 6) + 0 = – 6
Five related problems.
(i) (-6) + (-3)
(ii) (+8) + (-5)
(iii) (-16) + 15
(iv) 10 + (-6)
(v) (+11) + (-12)
Answer:
(i) (- 6) + (- 3)
= (- 6) + (- 3)
= – 9

(ii) (+ 8) + (- 5) = [(+ 3) + (+ 5)] + (-5)
= (+ 3) + [(+ 5) + (- 5)]
= (+ 3) + 0 = 3

(iii) (- 16) + 15
= [(- 15) + (- 1)] + 15
= (- 15) + (+ 15) + (- 1)
= – 1

(iv) 10 + (- 6)
= [(+ 4) + (+ 6)] + (- 6)
= (+4) + [(+ 6) + (- 6)]
= (+ 4) + 0 = + 4

(v) (+ 11) + (- 12)
= (+ 11) + [(- 11) + (- 1)]
= [(+ 11) + (- 11)] + (- 1)
= 0 + (- 1)
= – 1

Do This

Question 1.
Find the solution of the following.
(a) -5- (-3)
(b) – 7 – (+2)
(c) – 7 – (-5)
(d) 3 – (-4)
(e) 5 – (+7)
(f) 4 – (- 2)
Sol.
(a) – 5 – (- 3) = – 5 + 3 = – 2
(b) – 7 – (+2) = – 7 – 2 = – 9
(c) – 7 – (- 5) = – 7 + 5 = – 2
(d) 3 – (- 4) = 3 + 4 = + 7
(e) 5 – (+ 7) = 5 – 7 = – 2
(f) 4 – (- 2) = 4 + 2 = + 6

TS 6th Class Maths Solutions Chapter 6 Integers InText Questions

Think, Discuss And Write

Question 1.
Observe that as the number we subtract from 3 is decreasing, the result obtained is increasing. Is it true for all integer’s?
3 – 3 = 0
3 – 2 = 1
3 – 1 = 2
3 – 0 = 3
3 – (-1) = 4
3 – (- 2) = 5
3 – (- 3) = 6
Answer:
No, it is not true for all the integers.
Since 3 – 4 = – 1
3 – 5 = – 2
3 – 6 = – 3 it is decreasing.

TS Inter 1st Year Maths 1A Matrices Important Questions

Students must practice these TS Inter 1st Year Maths 1A Important Questions Chapter 3 Matrices to help strengthen their preparations for exams.

TS Inter 1st Year Maths 1A Matrices Important Questions

Very Short Answer Questions

Question 1.
Find the trace of A if A = \(\left[\begin{array}{rrr}
1 & 2 & -\frac{1}{2} \\
0 & -1 & 2 \\
-\frac{1}{2} & 2 & 1
\end{array}\right]\)
Solution:
The elements in the principal diagonal of A are 1, – 1, 1.
∴ Trace of A = 1 -1 + 1 = 1.

TS Inter 1st Year Maths 1A Matrices Important Questions

Question 2.
If \(A=\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right], B=\left[\begin{array}{ll}
3 & 8 \\
7 & 2
\end{array}\right]\) and 2X+A=B then find X.
Solution:
2X + A= B ⇒ 2X = B – A
TS Inter 1st Year Maths 1A Matrices Important Questions 1

Question 3.
A certain bookshop has 10 dozen Chemistry books, 8 dozen Physics books, 10 dozen Economics books. Their selling prices are Rs. 80, Rs. 60 and Rs. 40 each respectively. Using matrix algebra, find the total value of the books in the shop.
Solution:
Number of books
TS Inter 1st Year Maths 1A Matrices Important Questions 2
TS Inter 1st Year Maths 1A Matrices Important Questions 3
= 120×80 +96×60+ 120×40
= 9600 + 5760 + 4800
= Rs. 20,160

Question 4.
If \(A=\left[\begin{array}{rrr}
2 & 3 & -1 \\
7 & 8 & 5
\end{array}\right] \text { and } B=\left[\begin{array}{rrr}
1 & 0 & 1 \\
2 & -4 & -1
\end{array}\right]\) then find A + B.
Solution :
TS Inter 1st Year Maths 1A Matrices Important Questions 4

Question 5.
If \(\left|\begin{array}{ccc}
x-1 & 2 & y-5 \\
z & 0 & 2 \\
1 & -1 & 1+a
\end{array}\right|=\left|\begin{array}{ccc}
1-x & 2 & -y \\
2 & 0 & 2 \\
1 & -1 & 1
\end{array}\right|\) then find the values of x, y, z and ‘a’.
Solution:
From the equality of matrices
x – 1=1 – x = 2x = 2 = x = 1
y – 5 = – y = 2y = 5 = y = \(\frac{5}{2}\) and z = 2,
Also 1 + a = 1 a = 0

TS Inter 1st Year Maths 1A Matrices Important Questions

Question 6.
If \(A=\left[\begin{array}{rr}
4 & -5 \\
-2 & 3
\end{array}\right]\) find -5A
Solution:
\(-5 A=-5\left[\begin{array}{rr}
4 & -5 \\
-2 & 3
\end{array}\right]=\left[\begin{array}{rr}
-20 & 25 \\
10 & -15
\end{array}\right]\)

Question 7.
Find the additive Inverse of A where
\(A=\left[\begin{array}{ccc}
i & 0 & 1 \\
0 & -i & 2 \\
-1 & 1 & 5
\end{array}\right]\)
Solution:
The additive inverse of A is – A
\(\text { i.e., }\left[\begin{array}{rrr}
-\mathrm{i} & 0 & -1 \\
0 & \mathrm{i} & -2 \\
1 & -1 & -5
\end{array}\right]\)

Question 8.
If \(A=\left[\begin{array}{rrr}
2 & 3 & 1 \\
6 & -1 & 5
\end{array}\right] \text { and } B=\left[\begin{array}{rrr}
1 & 2 & -1 \\
0 & -1 & 3
\end{array}\right]\) then find the matrix X such that A + B – X = 0. What is the order of the matrix?
Solution :
A + B – X = 0 = X = A + B
TS Inter 1st Year Maths 1A Matrices Important Questions 6

Question 9.
If \( A=\left[\begin{array}{lll}
0 & 1 & 2 \\
2 & 3 & 4 \\
4 & 5 & 6
\end{array}\right] \text { and } B=\left[\begin{array}{rrr}
1 & -2 & 0 \\
0 & 1 & -1 \\
-1 & 0 & 3
\end{array}\right]\)  then find A – B and 4B – 3A.
Solution :
TS Inter 1st Year Maths 1A Matrices Important Questions 7

TS Inter 1st Year Maths 1A Matrices Important Questions

Question 10.
Two factories I and II produce three varieties of pens namely Gel, Ball, and Ink pens. The sale In rupees of these varieties of pens by both the factories in the month of September and October in a year are given by the following matrices A and B. September sales (In Rupees)
\(A=\left\{\begin{array}{ccc}
\text { Gel } & \text { Ball } & \text { Ink } \\
1000 & 2000 & 3000 \\
5000 & 3000 & 1000
\end{array}\right\} \text { Factory I }\)
October sales (in Rupees)
\(B=\left\{\begin{array}{ccc}
\text { Gel } & \text { Ball } & \text { Ink } \\
500 & 1000 & 600 \\
2000 & 1000 & 1000
\end{array}\right\} \text { Factory I }\)
Solution:
i) Find the combined sales in September and October for each factory in each variety.
ii) Find the decrease in sales from September to October.
Solution:
i) Combined sales in September and October for each factory in each variety is
\(A+B=\left\{\begin{array}{ccc}
\text { Gel } & \text { Ball } & \text { Ink } \\
1500 & 3000 & 3600 \\
7000 & 4000 & 2000
\end{array}\right\} \text { Factory I }\)

ii) Decrease in sales from September to October is .
\(\mathrm{A}-\mathrm{B}=\left\{\begin{array}{ccc}
\text { Gel } & \text { Ball } & \text { Ink } \\
500 & 1000 & 2400 \\
3000 & 2000 & 0
\end{array}\right\} \begin{aligned}
& \text { Factory I } \\
& \text { Factory II }
\end{aligned}\)

TS Inter 1st Year Maths 1A Matrices Important Questions

Question 11.
Construct a 3 x 2 matrix whose elements are defined by \(a_{i j}=\frac{1}{2}|i-3 j|\).
Solution:
The matrix to be constructed is
TS Inter 1st Year Maths 1A Matrices Important Questions 8

Question 12.
If \(A=\left[\begin{array}{lll}
0 & 1 & 2 \\
1 & 2 & 3 \\
2 & 3 & 4
\end{array}\right] \text { and } B=\left[\begin{array}{rr}
1 & -2 \\
-1 & 0 \\
2 & -1
\end{array}\right]\) then find AB and BA.
Solution:
Since the number of columns of A is equal to number of rows of B, AB is defined and
TS Inter 1st Year Maths 1A Matrices Important Questions 9
Now the number of columns of B is not equal to the number of rows A, BA is not defined.

Question 13.
If \(A=\left[\begin{array}{rrr}
1 & -2 & 4 \\
2 & 3 & -1 \\
-3 & 1 & 2
\end{array}\right] \text { and } B=\left[\begin{array}{rrr}
1 & 0 & 2 \\
0 & 1 & 2 \\
1 & 2 & 0
\end{array}\right]\) then examine whether A and B commute with respect to multiplication of matrices.
Solution:
The two matrices A and B are square matrices of order 3. Hence AB and BA are both defined.
\(A B=\left[\begin{array}{rrr}
1 & -2 & 3 \\
2 & 3 & -1 \\
-3 & 1 & 2
\end{array}\right]\left[\begin{array}{rrr}
1 & 0 & 2 \\
0 & 1 & 2 \\
1 & 2 & 0
\end{array}\right]\)
TS Inter 1st Year Maths 1A Matrices Important Questions 10
∴ AB ≠ BA and hence A, B do not commute with respect to multiplication.

Question 14.
If \(A=\left[\begin{array}{rrr}
-2 & 1 & 0 \\
3 & 4 & -5
\end{array}\right] \text { and } B=\left[\begin{array}{rr}
1 & 2 \\
4 & 3 \\
-1 & 5
\end{array}\right]\) then find A +B’
Solution:
TS Inter 1st Year Maths 1A Matrices Important Questions 11

Question 15.
If \(A=\left[\begin{array}{rrr}
0 & 4 & -2 \\
-4 & 0 & 8 \\
2 & -8 & x
\end{array}\right]\) is a skew symmetric
Solution:
TS Inter 1st Year Maths 1A Matrices Important Questions 12
Since A is a skew symmetric matrix, A’ = – A
⇒ 2x = 0 ⇒ x = 0

Question 16.
If ω is a complex cube root of unity show that \(\left|\begin{array}{ccc}
1 & \omega & \omega^2 \\
\omega & \omega^2 & 1 \\
\omega^2 & 1 & \omega
\end{array}\right|=0\)
Solution:
TS Inter 1st Year Maths 1A Matrices Important Questions 13

TS Inter 1st Year Maths 1A Matrices Important Questions

Question 17.
Find the adjoint and inverse of the matrix
\(A=\left[\begin{array}{cc}
1 & 2 \\
3 & -5
\end{array}\right]\)
Solution:
TS Inter 1st Year Maths 1A Matrices Important Questions 14

Question 18.
Find whether the following 8ystem of linear homogeneous equations has a non-trivial solution.
X – y + z = 0
x+2y – z = 0
Zx+y+3z = 0
Solution:
The coefficient matrix is \(A=\left[\begin{array}{ccc}
1 & -1 & 1 \\
1 & 2 & -1 \\
2 & 1 & 3
\end{array}\right]\)
det A= 1(6+ 1)+ 1(3+2) + 1(1 – 4)
=7 +5 – 3 = 9 ≠ 0
Hence the system has the trivial solution x = y = z = 0 only.

TS Inter 1st Year Maths 1A Matrices Important Questions

Question 19.
If A is an invertible matrix then A’ is also invertible and \(\left(A^{\prime}\right)^{-1}=\left(A^{-1}\right)^{\prime}\)
Solution:
TS Inter 1st Year Maths 1A Matrices Important Questions 15

Question 20.
If A and B are two invertible matrices of same type then AB is also invertible and \((A B)^{-1}=B^{-1} A^{-1}\)
Solution:
TS Inter 1st Year Maths 1A Matrices Important Questions 16

Short Answer Questions

Question 1.
If \(A=\left[\begin{array}{cc}
\cos \theta & \sin \theta \\
-\sin \theta & \cos \theta
\end{array}\right]\) then show that for all the positive integers n \(A^{\mathbf{n}}=\left[\begin{array}{cc}
\cos n \theta & \sin n \theta \\
-\sin n \theta & \cos n \theta
\end{array}\right]\)
Solution:
We use the process of mathematical induction for proving this result statement
TS Inter 1st Year Maths 1A Matrices Important Questions 17
∴The statement P(n) is true for n = k + 1.
Hence by mathematical induction P(n) is true for all positive integral values of n.

TS Inter 1st Year Maths 1A Matrices Important Questions

Question 2.
If \(A=\left[\begin{array}{lll}
1 & 2 & 2 \\
2 & 1 & 2 \\
2 & 2 & 1
\end{array}\right]\) then show that A2 – 4A – 5I = 0
Solution:
TS Inter 1st Year Maths 1A Matrices Important Questions 18

Question 3.
For any n x n matrix A; Prove that A can be uniquely expressed as a sum of a symmetric matrix and a skew-symmetric matrix.
Solution:
For a square matrix of order n.
A + A’ is symmetric and A – A’ is a skew-symmetric matrix and
\(\mathrm{A}=\frac{1}{2}\left(\mathrm{~A}+\mathrm{A}^{\prime}\right)+\frac{1}{2}\left(\mathrm{~A}-\mathrm{A}^{\prime}\right)\)
Let B be a symmetric matrix and C be a skew symmetric matrix such that
A = B + C
TS Inter 1st Year Maths 1A Matrices Important Questions 19

TS Inter 1st Year Maths 1A Matrices Important Questions

Question 4.
Show that \(\left|\begin{array}{ccc}
1 & a & a^2 \\
1 & b & b^2 \\
1 & c & c^2
\end{array}\right|\) = (a – b) (b – c) (c -a)
Solution:
TS Inter 1st Year Maths 1A Matrices Important Questions 20

Question 5.
Show that \(\left|\begin{array}{ccc}
\mathbf{a}-\mathbf{b}-\mathbf{c} & 2 \mathbf{a} & 2 \mathbf{a} \\
2 \mathbf{b} & \mathbf{b}-\mathbf{c}-\mathbf{a} & 2 \mathbf{b} \\
2 \mathbf{c} & 2 \mathbf{c} & \mathbf{c}-\mathbf{a}-\mathbf{b}
\end{array}\right|\) = (a+ b+c)2
Solution:
TS Inter 1st Year Maths 1A Matrices Important Questions 21
TS Inter 1st Year Maths 1A Matrices Important Questions 22

Question 6.
Find the rank of \(A=\left[\begin{array}{lll}
0 & 1 & 2 \\
1 & 2 & 3 \\
3 & 2 & 1
\end{array}\right]\) using elementary transformations.
Solution:
TS Inter 1st Year Maths 1A Matrices Important Questions 23
Since there are two non-zero rows in above matrix the rank of the given matrix ρ(A) = 2.

TS Inter 1st Year Maths 1A Matrices Important Questions

Question 7.
Find the rank \( A=\left[\begin{array}{rrrr}
1 & 2 & 0 & -1 \\
3 & 4 & 1 & 2 \\
-2 & 3 & 2 & 5
\end{array}\right]\) using elementary transformations.
Solution:
TS Inter 1st Year Maths 1A Matrices Important Questions 24
Since these are 3 non-zero rows in above reduced form, the rank of the given matrix ρ(A) = 3.

Question 8.
Show that matrix multiplication is associative.
Solution:
TS Inter 1st Year Maths 1A Matrices Important Questions 25
∴ (AB)C = A(BC); Hence matrix multiplication is associative.

Long Answer Questions

Question 1.
Without expanding the determinant show that
TS Inter 1st Year Maths 1A Matrices Important Questions 26
Solution:
LHS = Use R1 + (R2 + R3)
TS Inter 1st Year Maths 1A Matrices Important Questions 27

Question 2.
Show that \(\left|\begin{array}{ccc}
1 & a^2 & a^3 \\
1 & b^2 & b^3 \\
1 & c^2 & c^3
\end{array}\right|\) = (a  –  b) (b – c) (c – a) (ab bc + ca).
Solution:
Use R2 – R1, R3 – R1 on LHS
TS Inter 1st Year Maths 1A Matrices Important Questions 28

TS Inter 1st Year Maths 1A Matrices Important Questions

Question 3.
Find the value of x if
TS Inter 1st Year Maths 1A Matrices Important Questions 29
Solution:
TS Inter 1st Year Maths 1A Matrices Important Questions 30

Question 4.
Find the adjoint and the inverse of the matrix A = \(\left[\begin{array}{lll}
1 & 3 & 3 \\
1 & 4 & 3 \\
1 & 3 & 4
\end{array}\right]\)
Solution :
TS Inter 1st Year Maths 1A Matrices Important Questions 31

Question 5.
Show that \( A=\left[\begin{array}{lll}
1 & 2 & 1 \\
3 & 2 & 3 \\
1 & 1 & 2
\end{array}\right]\) is non-singular and find A-1
Solution:
TS Inter 1st Year Maths 1A Matrices Important Questions 32
TS Inter 1st Year Maths 1A Matrices Important Questions 33

Question 6.
Apply the test of rank to examine whether the following equations are consistent 2x-y+3z = 8, -x+2y+z=4, 3x+y-4Z = 0 and if consistent find the complete solution.
Solution:
TS Inter 1st Year Maths 1A Matrices Important Questions 34
Now rank (A) = rank (AB) = 3
∴ The system has unique solution From above we have so
– x + 2y + z = 4 ……………………… (1)
3y + 5z = 16 ……………………… (2)
z = 2 ……………………… (3)
3y+10 = 16 ⇒ 3y = 6 ⇒ y = 2
∴ From (1), – x + 4 + 2 = 4
⇒ x = 2
∴ x = 2, y= 2, z = 2 is the solution.

TS Inter 1st Year Maths 1A Matrices Important Questions

Question 7.
Show that the following system of equations is consistent and solve it completely
x + y + z = 3, 2x +2y – z = 3, x + y – z = 1.
Solution :
TS Inter 1st Year Maths 1A Matrices Important Questions 35
Here ρ(A) = 2 and ρ(AB) = 2.
∴ The system is consistent and has infinitely many solutions.
From the above-reduced form
x+y+z=3
z= 1
∴ x + y = 2
Hence x = k, y = 2 – k, z = 1, k ∈ R is a solution set.

TS Inter 1st Year Maths 1A Matrices Important Questions

Question 8.
Solve the following slmultaneou8 linear equations by using Cramer’s rule.
3x+4y+5z =18
2x- y+8z = 13
5x – 2y+ 7z = 20
Solution :
TS Inter 1st Year Maths 1A Matrices Important Questions 36
TS Inter 1st Year Maths 1A Matrices Important Questions 37
∴ The solution of the given system of equations is x = 3, y = 1, z = 1.

Question 9.
Solve 3x+3y+5z=18
2x – y + 8z = 13 .
and 5x – 2y+7z=20 by using matrix inversion method.
Solution:
TS Inter 1st Year Maths 1A Matrices Important Questions 38

TS Inter 1st Year Maths 1A Matrices Important Questions

Question 10.
Solve the following equations by Gauss Jordan method.
3x+4y+5z=18
2x – y+8z= 13
5x – 2y+7z= 20
Solution:
The augmented matrix is
TS Inter 1st Year Maths 1A Matrices Important Questions 39
TS Inter 1st Year Maths 1A Matrices Important Questions 40
ρ(A) = ρ(AB) = 3; and unique soLution exists given by x = 3, y = 1 and z = 1.

Question 11.
Solve the following system of equations by Gauss Jordan method
x+y+z=3,
2x+2y – z=3,
x+y – z=1
Solution:
Augmented matrix of the system is
TS Inter 1st Year Maths 1A Matrices Important Questions 41
Here ρ(A) 2, ρ(AB) = 2.
So the system is consistent and has infinite number of solutions.
∴ x + y + z = 3, – 3z = – 3 ⇒ z = 1
∴ x + y = z
Suppose x = k, then y = 2 – k and solution set
= (x = ky = 2 – k, z= 1 where k ∈ R)

Question 12.
By using Gauss Jordan method show that the following system has no solution
2x+4y – z= o,
x + 2y + 2z = 5,
3x+ 6y – 7z = 2.
Solution:
Augmented matrix of the system is
TS Inter 1st Year Maths 1A Matrices Important Questions 42
2x+4y – z=0 ………………. (1)
5z = 10 = z = 2 ……………….. (2)
and 0.x + 0.y + 0.z = 130 ………………. (3)
Clearly no value of x, y, z satisfy (3).
∴ Given system of equations has no solution.

TS Inter 1st Year Maths 1A Matrices Important Questions

Question 13.
Find the non-trivial solutions, if any, for the following system of equations.
2x + 5y + 6z = 0
x – 3y – 8z = 0
3x+y – 4z = 0
Solution:
Coefficient matrix is
TS Inter 1st Year Maths 1A Matrices Important Questions 43
Since two rows are identical, ρ(A) = 2.
Hence the system has a non-trivial solution
x – 3y + 8z = 0 …………….. (1)
y + 2z = 0 …………. (2)
Suppose z = k, then y – 2k and
x = 3y – 8z = – 6k – 8k = – 14k for k ≠ 0
so we obtain non-trivial solutions.

Question 14.
If A is a non-singular matrix then prove that \(A^{-1}=\frac{1}{\operatorname{det} A}(\operatorname{Adj} A)\)
Solution:
TS Inter 1st Year Maths 1A Matrices Important Questions 44
TS Inter 1st Year Maths 1A Matrices Important Questions 45

TS 6th Class Maths Solutions Chapter 4 Basic Geometrical Ideas InText Questions

Students can practice TS 6th Class Maths Solutions Chapter 4 Basic Geometrical Ideas InText Questions to get the best methods of solving problems.

TS 6th Class Maths Solutions Chapter 4 Basic Geometrical Ideas InText Questions

Do This

Question 1.
Four points are marked in the given rectangle. Name them.
TS 6th Class Maths Solutions Chapter 4 Basic Geometrical Ideas InText Questions 1
Answer:
Their names taken as P, Q, R, S.
TS 6th Class Maths Solutions Chapter 4 Basic Geometrical Ideas InText Questions 2

Question 2.
Take a geo-board. Select any two nails and tie tightly a thread from one end to the other. The thread you have fixed is a line which can extend in both directions and only in these two directions.
TS 6th Class Maths Solutions Chapter 4 Basic Geometrical Ideas InText Questions 3
Answer:
Student activity

Think: Discuss And Write

Question 1.
Here is a ray \(\overrightarrow{\mathrm{O A}}\). It starts at O and passes through the points A and B.
Can you name ray \(\overrightarrow{\mathrm{O A}}\) as \(\overrightarrow{\mathrm{O B}}\)? Why?
TS 6th Class Maths Solutions Chapter 4 Basic Geometrical Ideas InText Questions 4
Can you write the ray \(\overrightarrow{\mathrm{O A}}\) as \(\overrightarrow{\mathrm{A O}}\) ? Why ? Give reasons.
Answer:
TS 6th Class Maths Solutions Chapter 4 Basic Geometrical Ideas InText Questions 5
The above ray starts from the point ‘O’ towards B so, it is named as \(\overrightarrow{\mathrm{O B}}\).
\(\overrightarrow{\mathrm{OA}} \neq \overrightarrow{\mathrm{AO}}\) since the ray starts from 0 i.e., it should be represented by only \(\overrightarrow{\mathrm{O A}}\).

TS 6th Class Maths Solutions Chapter 4 Basic Geometrical Ideas InText Questions

Think. Discuss And Write

Question 1.
Move your pencil along the following English letters and state which are open and which are closed?
TS 6th Class Maths Solutions Chapter 4 Basic Geometrical Ideas InText Questions 6
Answer:
D and O are closed letters
G, L, M are open letters.

Question 2.
Tell which letter is an example of simple curve.
Answer:
O is an example of simple curve.

Try These

Question 1.
Identify which are simple curves and which are not?
TS 6th Class Maths Solutions Chapter 4 Basic Geometrical Ideas InText Questions 7
Answer:
(i) and(ii) are simple curves.
(iii) and (iv) are not simple curves.

Do This

Question 1.
Take some match sticks and try to make simple figures. Identify closed figures in them.
Answer:
TS 6th Class Maths Solutions Chapter 4 Basic Geometrical Ideas InText Questions 8

Question 2.
TS 6th Class Maths Solutions Chapter 4 Basic Geometrical Ideas InText Questions 9
What is the least number of sticks needed to form a closed figure ? Obviously three. Can you explain why two match sticks can not make a closed figure.
Answer:
Minimum number of sticks that are needed to form a closed figure are 3. If we take less than 3 sticks it will become a open figure.

Question 3.
Take some straw pieces of diffrent size. Pass thread into any 3 pieces and make different triangles. Draw figures for the tiangles in your notebook.
Answer:
TS 6th Class Maths Solutions Chapter 4 Basic Geometrical Ideas InText Questions 10

Think, Discuss And Write

Question 1.
Take four points A, B, C and D such that A, B, C lie on the same line and D is not on it. Can the four line segments \(\overline{\mathrm{A B}}, \overline{\mathrm{B C}}, \overline{\mathrm{C D}}\) and \(\overline{\mathrm{AD}}\) form a quadrilateral? Give reason.
TS 6th Class Maths Solutions Chapter 4 Basic Geometrical Ideas InText Questions 11
Answer:
TS 6th Class Maths Solutions Chapter 4 Basic Geometrical Ideas InText Questions 12
No, the given four line segments \(\overline{\mathrm{A B}}, \overline{\mathrm{B C}}, \overline{\mathrm{C D}}\) and \(\overline{\mathrm{AD}}\)
Can not form a quadrilateral.
To form a quadrilateral maximum two points should be collinear.

TS 6th Class Maths Solutions Chapter 4 Basic Geometrical Ideas InText Questions

Do This

Question 1.
Draw a circle on a paper and cut ¡t along its edge. Fold it Into half and again fold it to one fourth to make folding marks as shown.
TS 6th Class Maths Solutions Chapter 4 Basic Geometrical Ideas InText Questions 13
You will observe a point in the middle. Mark this O. This is the centre of the circle. You can also indicate its radius. How many radii can you draw in a circle ?
Answer:
TS 6th Class Maths Solutions Chapter 4 Basic Geometrical Ideas InText Questions 14
Infinite number of radii we can draw in a circle.
Because infinite number of points are there on the circumference of the circle.

Question 2.
Draw a circle and draw at least 5 chords in it. Make sure at least one of them passes through the centre. Name them and fill the table.

S.No.ChordLengthPasses through the centre (Yes/No)
1
2
3
4
5

What do you notice?
Answer:

S.NoChordLengthPasses through the centre (Yes/No)
1AB5Yes
2CD2No
3EB1.5No
4GH2.7No
5FT2No

TS 6th Class Maths Solutions Chapter 4 Basic Geometrical Ideas InText Questions 15
I notice that a chord which passes through the centre of the circle is the largest chord of all the chords.

TS 6th Class Maths Solutions Chapter 4 Basic Geometrical Ideas InText Questions

Think And Discuss

Question 1.
Is it possible to draw more than one diameter in a circle ? Are all the diameters equal in length ? Discuss with your friends and find the answer.
Answer:
We can draw infinite number of diameters in a circle.
TS 6th Class Maths Solutions Chapter 4 Basic Geometrical Ideas InText Questions 16
All the lengths of diameters are equal in a circle.
Since \(\overline{\mathrm{AF}}=\overline{\mathrm{BG}}=\overline{\mathrm{CH}}\) = 2.5 cm

TS 6th Class Maths Solutions Chapter 2 Whole Numbers Ex 2.3

Students can practice TS 6th Class Maths Solutions Chapter 2 Whole Numbers Ex 2.3 to get the best methods of solving problems.

TS 6th Class Maths Solutions Chapter 2 Whole Numbers Exercise 2.3

Question 1.
Study the pattern:
1 × 8 + 1 = 9
12 × 8 + 2 = 98
123 × 8 + 3 = 987
1234 × 8 + 4 = 9876
12345 × 8 + 5 = 98765
Write the next four steps. Can you find out how the pattern works?
Answer:
123456 × 8 + 6 = 987654 .
1234567 × 8 + 7 = 9876543
12345678 × 8 + 8 = 98765432
123456789 × 8 + 9 = 987654321

Working pattern is as follows:
1) Write the smallest of the natural numbers, multiply it by 8 and add 1 to product obtained. You get a single digit number (i.e.) 9. i.e., 1 × 8 + 1 = 9

2) Write the number formed by the first two natural numbers in the same order, multiply it by 8 and add 2 to the product obtained. You get a two digit number with 9 tens place and the next digit (succeeding digit) decreases by 1.
i.e., 12 × 8 + 2 = 98

3) Write the number formed by the first three natural numbers in the same order, multiply it by 8 and add 3 to the product obtained. You get a three digit number with 9 in hundreds place and the next two digits gradually decrease by 1,
i.e., 123 × 8 + 3 = 987 and so on.

TS 6th Class Maths Solutions Chapter 2 Whole Numbers Ex 2.3

Question 2.
Study the pattern:
91 × 11 × 1 = 1001
91 × 11 × 2 = 2002
91 × 11 × 3 = 3003
Write next seven steps. Check, whether the result is correct.
Try the pattern for 143 × 7 × 1, 143 × 7 × 2 ………..
Answer:
91 × 11 × 4 = 4004
91 × 11 × 5 = 5005
91 × 11 × 6 = 6006
91 × 11 × 7 = 7007
91 × 11 × 8 = 8008
91 × 11 × 9 = 9009
91 × 11 × 10= 10010

Verification :
91 × 11 × 4 = 91 × 44
= 91 × (40 + 4)
= (91 × 40) + (91 × 4)
= 3640 + 364
= 4004

143 × 7 × 1 = 1001
143 × 7 × 2 = 2002
143 × 7 × 3 = 3003
143 × 7 × 4 = 4004
143 × 7 × 5 = 5005

Question 3.
How would we multiply the numbers 13680347, 35702369 and 25692359 with 9 mentally? What is the pattern that emerges.
Answer:
(i) 13680347 × 9 = 13680347 × (10 – 1)
= 13680347 × 10 – 13680347 × 1
= 136803470 – 13680347
= 123123123

(ii) 35702369 × 9 = 35702369 × (10 – 1)
= 35702369 × 10 – 35702369 × 1
= 357023690 – 35702369
= 321321321

(iii) 25692359 × 9
= 25692359 × (10 – 1)
= 25692359 × 10 – 25692359 × 1
= 256923590 – 25692359
= 231231231
Working pattern : If any number is to be multiplied by 9, first multiply the given number by 10 (write ‘0’ at the end of the number) and subtract the given number from the product obtained. (∴ 9 = 10-1)

TS 6th Class Maths Solutions Chapter 2 Whole Numbers Ex 2.2

Students can practice TS 6th Class Maths Solutions Chapter 2 Whole Numbers Ex 2.2 to get the best methods of solving problems.

TS 6th Class Maths Solutions Chapter 2 Whole Numbers Exercise 2.2

Question 1.
Give the results without actually performing the operations using the given information.
(i) 28 × 19 = 532 then 19 × 28 =
(ii) 1 × 47 = 47 then 47 × 1 =
(iii) a × b = c then b × a =
(iv) 58 + 42 = 100 then 42 + 58 =
(v) 85 + 0 = 85 then 0 + 85 =
(vi) a + b = d then b + a =
Answer:
(i) 28 × 19 = 532 then 19 × 28 = 532
(ii) 1 × 47 = 47 then 47 × 1 = 47;
(iii) a × b = c then b × a = c
(iv) 58 + 42 = 100 then 42 + 58 = 100
(v) 85 + 0 = 85 then 0 + 85 = 85
(vi) a + b = d then b + a = d

Question 2.
Find the sum by suitable rearrangement:
(i) 238 + 695 + 162
Answer:
238 + 695 + 162 = (238 + 162) + 695
= 400 + 695 = 1095

(ii) 154 + 197 + 46 + 203
Answer:
154 + 197 + 46 + 203
= (154 + 46) + (197 + 203) = 200 + 400 = 600

TS 6th Class Maths Solutions Chapter 2 Whole Numbers Ex 2.2

Question 3.
Find the product by suitable rearrangement.
(i) 25 × 1963 × 4
Answer:
25 × 1963 × 4 = (25 × 4) × 1963
= 100 × 1963
= 196300

(ii) 20 × 255 × 50 × 6
Answer:
20 × 255 × 50 × 6 = (20 × 50) × (255 × 6)
= 1000 × 1530
= 1530000

Question 4.
Find the value of the following:
(i) 368 × 12 + 18 × 368
Answer:
(368 × 12) + (18 × 368)
= 368 × [12 + 18]
= 368 × 30
= 11040

(ii) 79 × 4319 + 4319 × 11
Answer:
(79 × 4319) + (4319 × 11)
= 4319 × [79 + 11]
= 4319 × 90
= 388710

Question 5.
Find the product using suitable properties:
(i) 205 × 1989
Answer:
1989 × 205 = 1989 × (200 + 5) = (1989 × 200) + (1989 × 5)
= 397800 + 9945 = 407745

(ii) 1991 × 1005
Answer:
1991 × 1005 = 1991 × [1000 + 5] = (1991 × 1000) + (1991 × 5)
= 1991000 + 9955 = 2000955

Question 6.
A milk vendor supplies 56 liters of milk in the morning and 44 liters of milk in the evening to a hostel. If the milk costs ₹ 30 per liter, how much money he gets per day?
Answer:
Quantity of milk supplied to the hostel in the morning = 56 litres
Quantity of milk supplied to the hostel in the evening = 44 litres
Quantity of milk supplied to the hostel per day = (56 + 44) litres = 100 litres
Cost of 1 litre of milk = ₹ 30
Cost of 100 litres of milk = ₹ 30 × 100 = ₹ 3000

TS 6th Class Maths Solutions Chapter 2 Whole Numbers Ex 2.2

Question 7.
Chandana and Venu purchased 12 note books and10 note books respectively. The cost of each note book is ₹ 15,then how much amount should they pay to the shop keeper?
Answer:
Cost of one note book = ₹ 15
Number of note books that Chandana bought =12
Amount that Chandana has to pay the shopkeeper = ₹ 15 × 12
Number of notebooks that Venu bought = 10
Amount that Venu has to pay to the shopkeeper = ₹ 15 × 10
Total amount that Chandana and Venu have to pay the shopkeeper =
₹ 15 × 12 + ₹ 15 × 10 = ₹ 15 [12 + 10] = ₹ 15 × 22 = ₹ 330

Question 8.
Match the following:

(i) 1991 + 7 = 7 + 1991(a) Additive identity
(ii) 68 × 50 = 50 × 68(b) Multiplicative identity
(iii) 1(c) Commutative under addition
(iv) 0(d) Distributive property of multiplication over addition
(v) 879 × (100 + 30) = 879 × 100 + 879 × 30(e) Commutative under multiplication

Answer:

(i) 1991 + 7 = 7 + 1991(c) Commutative under addition
(ii) 68 × 50 = 50 × 68(e) Commutative under multiplication
(iii) 1(b) Multiplicative identity
(iv) 0(a) Additive identity
(v) 879 × (100 + 30) = 879 × 100 + 879 × 30(d) Distributive property of multiplication over addition

TS 6th Class Maths Solutions Chapter 2 Whole Numbers InText Questions

Students can practice TS 6th Class Maths Solutions Chapter 2 Whole Numbers InText Questions to get the best methods of solving problems.

TS 6th Class Maths Solutions Chapter 2 Whole Numbers InText Questions

Do This

Question 1.
Which is the smallest whole number ?
Answer:
‘0’ is the smallest whole

Think, Discuss And Write

Question 1.
Are all natural numbers whole numbers ?
Answer:
Yes, all natural numbers are whole numbers.

Question 2.
Are all whole numbers natural numbers ?
Answer:
No, all whole numbers are not natural numbers. Since 0 ∉ N.

TS 6th Class Maths Solutions Chapter 2 Whole Numbers InText Questions

Do This

Question 1.
Show these on number line :
1) 5 + 3
Answer:
TS 6th Class Maths Solutions Chapter 2 Whole Numbers InText Questions 1

2) 5 – 3
Answer:
TS 6th Class Maths Solutions Chapter 2 Whole Numbers InText Questions 2

3) 5 + 3
Answer:
TS 6th Class Maths Solutions Chapter 2 Whole Numbers InText Questions 3

4) 10 + 1
Answer:
TS 6th Class Maths Solutions Chapter 2 Whole Numbers InText Questions 4

Try These
Find the following by using the number line.

Question 1.
What number should be deducted from 8 to get 5 ?
TS 6th Class Maths Solutions Chapter 2 Whole Numbers InText Questions 5
Answer:
3 should be deducted from 8 to get 5.

Question 2.
What number should be deducted from 6 to get 1 ?
TS 6th Class Maths Solutions Chapter 2 Whole Numbers InText Questions 6
5 should be deducted from 6 to get 1.

Question 3.
What number should be added to 6 to get 8 ?
TS 6th Class Maths Solutions Chapter 2 Whole Numbers InText Questions 7
Answer:
2 should be added to 6 to get 8.

Question 4.
How many 6 are needed to get 30 ?
TS 6th Class Maths Solutions Chapter 2 Whole Numbers InText Questions 8
Answer:
The first leap takes you to 6.
From there, the second leap takes you to 12.
From there, the third leap takes you to 18.
From there, the fourth leap takes you to 24.
From there, the fifth leap takes you to 30.
∴ Five 6 are needed to get 30.

TS 6th Class Maths Solutions Chapter 2 Whole Numbers InText Questions

Think, Discuss And Write

Question 1.
Are the whole numbers closed under subtraction ?
Answer:
7 – 5 = 2, is a whole number.
5 – 7 = – 2, is not a whole number.
10 – 4 = 6, is a whole number.
4 – 10 = – 6, is not a whole number.
∴ The whole numbers are not closed under subtraction.

Question 2.
Are the whole numbers closed under division ?
Answer:
6 + 3 = 2, is a whole number,
5 ÷ 2 = \(\frac{5}{2}\), is not a whole number
14 ÷ 2 = 7, is a whole number.
19 ÷ 2 = \(\frac{19}{2}\), is not a whole number.
∴ The whole numbers are not closed under division.

Do This

Question 1.
Find out 12 ÷ 3 and 42 ÷ 7.
Answer:
12 ÷ 3 = 4 and 42 ÷ 7 = 6

Question 2.
What would 6 ÷ 0 and 9 ÷ 0 be equal to?
Answer:
6 ÷ 0 is not defined
9 ÷ 0 is also not defined.

Try These

Take a few examples and check whether.

Question 1.
Subtraction is commutative for whole numbers or not ?
Answer:
Observe the following:
i) 4 – 3 = 1; 3 – 4 = – 1
ii) 5 – 2 = 3; 2 – 5 = -3
1 and – 1 are not equal. 3 and – 3 are also not equal.
∴ Subtraction is not commutative for whole numbers.

Question 2.
Division is commutative for whole numbers or not ?
Answer:
Observe the following:
i) 6 ÷ 2 = 3; 2 ÷ 6 = \(\frac{1}{3}\)
ii) 11 ÷ 3 = \(\frac{11}{3}\); 3 ÷ 11 = \(\frac{3}{11}\)
3 and \(\frac{1}{3}\) are not equal; \(\frac{11}{3}\) and \(\frac{3}{11}\) are not equal.
∴ Division is not commutative for whole numbers.

Do this

Question 1.
Verify the following.
(i) (5 × 6) × 2 = 5 × (6 × 2)
Answer:
(5 × 6) × 2 = 30 × 2 = 60 (∵ 5 × 6 = 30)
5 × (6 × 2) = 5 × 12 = 60 (∵ 6 × 2= 12)
(5 × 6) × 2 = 5 × (6 × 2)

(ii)(3 × 7) × 5 = 3 × (7 × 5)
Answer:
(3 × 7) × 5 = 21 × 5 = 105 (∵ 3 × 7 = 21)
3 × (7 × 5) = 3 × 35 = 105 (∵ 7 × 5 = 35)
(3 × 7) × 5 = 3 × (7 × 5)
We see that multiplication is associative over whole numbers.

Do This

Question 1.
Use the commutative and associative properties to simplify the following.
(i) 319 + 69 + 81
(ii) 431 + 37 + 69 + 63
(iii) 2 × (71 × 5)
(iv) 50 × 17 × 2
Answer:

CommutativeAssociative
(i) 319 + 69 + 81 →319 + (81 + 69) →(319 + 81) + 69 = 400 + 69 = 469
(ii) 431 + 37 + 69 + 63 →(431 + 69) + (37 + 63) = 500 + 100 = 600
(iii) 2 × (71 × 5) →2 (5 × 71) →(2 × 5) × 71 → 10 × 71 = 710
(iv) 50 × 17 × 2 →50 × (2 × 17) →(50 × 2) × 17 → 100 × 17 = 1700

TS 6th Class Maths Solutions Chapter 2 Whole Numbers InText Questions

Think, Discuss And Write

Question 1.
Is (16 ÷ 4) + 2 = 16 ÷ (4 ÷ 2) ?
Does the associative property for division hold for the set of whole numbers? Check if the property holds for subtraction of whole numbers too.Give 5 examples each for substantiate your answer.
Answer:
(16 ÷ 4) ÷ 2 = 16 ÷ (4 ÷ 2)
⇒ 4 ÷ 2 = 16 ÷ (4 ÷ 2)
⇒ 4 ÷ 2 = 16 ÷ 2
⇒ 2 = 8 (False)
For division associative property doesn’t applicable.
Examples for Associative property for division
1) (54 ÷ 9) ÷ 3 = 18 ÷ (9 ÷ 3)
6 ÷ 3 = 18 ÷ 3
2 = 6 (false)

2) (64 ÷ 8) ÷ 4 = 64 ÷ (8 ÷ 4)
8 ÷ 4 = 64 ÷ 2
2 = 32(false)

3) (24 ÷ 6) ÷ 2 = 24 ÷ (6 ÷ 2)
4 ÷ 2 = 24 ÷ 3
2 = 8 (false)

4) (32 ÷ 8) ÷ 4 = 32 ÷ (8 ÷ 4)
4 ÷ 4 = 32 ÷ 2
1 = 16 (false)

5) (49 ÷ 7) ÷ 7 = 49 ÷ (7 ÷ 7)
7 ÷ 7 = 49 ÷ 1
1 = 49 (false)

Similarly, 3 – (2 – 1) = (3 – 2) – 1 ⇒ 3 – 1 = 1 – 1 ⇒ 2 = 0 (False)
∴ For subtraction also associative property doesn’t applicable.
Examples for associative property for subtraction :

1) 4 – (3 – 1) = (4 – 3) – 1
4 – 2 = 1 – 1
2 = 0 (false)

2) 8 – (4 – 3) = (8 – 4) – 3
8 – 1 = 4 – 3
7 = 1 (false)

3) 12 – (4 – 1) = (12 – 4) – 1
12 – 3 = 8 – 1
9 = 7 (false)

4) 13 – (8 – 7) = (13 – 8) – 7
13 – 1 = 5 – 7
12 = -2(false)

5) 15 – (12 – 3) = (15 – 12) – 3
15 – 9 = 3 – 3
6 = 0 (false)

Do This

Question 1.
Find the values of 25 × 78; 17 × 26; 49 × 68 + 32 × 49 using distributive property.
Answer:
(i) 25 × 78; a × (b + c) = a × b + a × c =>.25x 78 = 25 × (80 – 2) = 25 × 80 – 25 × 2 = 2000 – 50 = 1950
(ii) 17 × 26 = 17 × (30 – 4) = 17 × 30 – 17 × 4 = 510 – 68 = 442
(iii) 49 × 68 + 32 × 49 = 49 [68 + 32] = 49 × 100 = 4900

(Try These)

Question 1.
Which numbers can be shown as a line only ?
Answer:
Every number can be shown as a line.
But out of these numbers some numbers can be shown as triangles, some as squares and some as rectangles.
∴ The numbers that can be shown only as lines are 2, 5, 7, 11, 13, ……………..

Question 2.
Which numbers can be shown as rectangles ?
Answer:
The numbers that can be expressed as the product of two numbers can be shown as rectangles.
6 = 2 × 3
8 = 2 × 4
10 = 2 × 5
12 = 2 × 6 (or) 3 × 4
…………….

Question 3.
Which numbers can be shown as squares ?
Answer:
The perfect square numbers such as 4, 9, 16, 25, 36, 49, 64, 81, ………. can be shown as squares.

Question 4.
Which numbers can be shown as triangles ? eg. 3, 6, ……..
Answer:
The numbers that can be shown as triangles are 3, 6, 10, 15, 21, etc.

TS Inter 1st Year Maths 1A Mathematical Induction Important Questions

Students must practice these TS Inter 1st Year Maths 1A Important Questions Chapter 2 Mathematical Induction to help strengthen their preparations for exams.

TS Inter 1st Year Maths 1A Mathematical Induction Important Questions

Question 1.
By using mathematical induction show that ∀ n ∈ N
\(\frac{1}{1 \cdot 4}+\frac{1}{4 \cdot 7}+\frac{1}{7 \cdot 10}+\) ………….. upto n terms \(=\frac{n}{3 n+1}\)
Solution:
1, 4, 7. are in A.P whose nth term is 1 (n – 1) 3 = 3n – 2
and 4, 7, lo are in A.P. whose nth term is 4 (n – 1) 3 = 3n + 1
∴ The nth term of the given series
TS Inter 1st Year Maths 1A Mathematical Induction Important Questions 1
TS Inter 1st Year Maths 1A Mathematical Induction Important Questions 2
Hence S(k + 1) is true ∀ n ∈ N = k + 1
∴ By the principle of MathematicaL
Induction, S(n) is true ∀ n ∈ N

TS Inter 1st Year Maths 1A Mathematical Induction Important Questions 3

TS Inter 1st Year Maths 1A Mathematical Induction Important Questions

Question 2.
Use mathematical induction to prove the statement 2 + 3 . 2 + 4 . 22 + upto n terms = n 2n∀n∈N
Solution :
TS Inter 1st Year Maths 1A Mathematical Induction Important Questions 4
∴ S(n) is true for n = k + 1 also
∴ By the principle of Mathematical Induction.
S(n) is true for all n ∈ N.
∴ 2+3 . 2 + 4 . 22+ ………………….. + (n+1) 2n-1
= n . 2n ∀ n ∈ N

Question 3.
If x and y are natural numbers and x ≠ y using mathematical induction show that xn-yn is divisible by x – y, ∀n∈N
Solution:
Let S(n) be the statement
”xn-yn is divisible by (x – y)”
For n = 1, x1 – y1 is divisible by x’ – y’
We have S(1) is true for n = 1
Assume S(n) is true for n = k
Then xk – yk is divisible by (x-y)
Then (xk – yk) = (x – y) p where p ∈ Z …………………. (1)
We have prove that the statement S(n) is true for n = k + 1 also
TS Inter 1st Year Maths 1A Mathematical Induction Important Questions 5

TS Inter 1st Year Maths 1A Mathematical Induction Important Questions

Question 4.
Show that 49n + 16n – 1 is divisible by 64 for all positive integers ‘n’.
Solution:
Let S(n) be the statement.
49n + 16n – 1 is divisible by 64.
Since 491 + 16 (1) – 1 = 64 is divisible by 64.
The statement S(n) is true for n = 1.
Suppose the statement is true for S = k then
49k + 16k – 1 = 64t for t ∈ N
We have to prove that S(n) is true for
n = k + 1 also
We have to prove that S(n) is true for
n = k + 1 also
Consider 49k+1. + 16 (k + 1) – 1
49k . 49 + 16k+ 16 – 1
=(64t – 16k+ 1)49 + 16k + 15
= (49) 64t + 16k (1 – 49) + 64
= 64 [49t – 12k + 1] and 49t – 12k + 1 is an integer.
∴ 49k+1 + 16 (k + 1) – 1 is divisible by 64
∴ The statement is true for n = k + 1.
∴ By the principle of mathematical induction S(n) is true ∀ n ∈ N
∴ 49n + 16n – 1 is divisible by 64, ∀ n∈ N

Question 5.
Use mathematical induction to prove the statement
\(1^3+2^3+3^3+\ldots \ldots+n^3=\frac{n^2(n+1)^2}{4}, \forall n \in N\)
Solution:
Let S(n) be the statement
\(1^3+2^3+3^3+\ldots +n^3=\frac{n^2(n+1)^2}{4}\)
TS Inter 1st Year Maths 1A Mathematical Induction Important Questions 6
∴ Statement is true for n = k + 1 also
∴ By the principle of Mathematical Induction, S(n) is true ∀ n ∈ N
TS Inter 1st Year Maths 1A Mathematical Induction Important Questions 7

TS Inter 1st Year Maths 1A Mathematical Induction Important Questions

Question 6.
Use mathematical induction to prove the statement.
\(\sum_{k=1}^n(2 k-1)^2=\frac{n(2 n-1)(2 n+1)}{3}, \forall n \in N\)
Solution:
Let S(n) be the statement
TS Inter 1st Year Maths 1A Mathematical Induction Important Questions 8
TS Inter 1st Year Maths 1A Mathematical Induction Important Questions 9
TS Inter 1st Year Maths 1A Mathematical Induction Important Questions 10

Question 7.
Use mathematical induction to prove that
2n – 3 ≤ 2n-2  ∀ n ≥ 5, n ∈ N
Solution:
Let S(n) be the statement
2n – 3 ≤ 2n-2  ∀ n ≥ 5, n ∈ N
∴ Since 2(5) – 3 ≤ 25-2
TS Inter 1st Year Maths 1A Mathematical Induction Important Questions 11
∴ The statement S(n) is true for
n k + 1, k ≥ 5
∴ By the principle of mathematical induction S(n) is true for all n ≥ 5, n ∈ N.

TS Inter 1st Year Maths 1A Mathematical Induction Important Questions

Question 8.
Use mathematical induction to prove that
(1 + x)n > 1 + nx for n ≥ 2, x > – 1, x ≠ 0.
Solution:
Let S(n) be the statement
(1+ x)n > 1 + nx for n ≥ 2, x > – 1, x ≠ 0.
The basis of induction is 2
and x ≠ 0, x > – 1 ⇒ x+1>0
For n = 2,
(1 + x)2 = 1 + 2x + x2 > 1 + 2x; the statement
S(n) is true for n = 2
Assume that the statement is true for n = k, k ≥ 2
TS Inter 1st Year Maths 1A Mathematical Induction Important Questions 12
∴ By the principle of mathematical induction the statement S(n) is true for all n≥2.
∴ (1 + x)n > 1 + nx, ∀ n ≥ 2, X > 1, X ≠ 0

Question 9.
Using mathematical induction show that xm + ym is divisible by x + y, if in is an odd natural number and x, y are natural numbers.
Solution:
Since m is an odd natural number
Let m = 2n – 1 where n is a non negative integer
Let S(n) be the statement
TS Inter 1st Year Maths 1A Mathematical Induction Important Questions 13
Now px2 – (x y) y2k+1 + is an integer
∴ x2k+3 + y2k+3 is divisible by (x + y)
∴ The statement is true for n = k + 1.
∴ By the principle of mathematical induction S(n) is true ∀n.
∴ x2n+1 + y2n -1 is divisible by (X y) for all non negative integers.
∴ xm + ym is divisible by (x + y) if n is an odd natural number.

TS Inter 1st Year Maths 1A Mathematical Induction Important Questions

Question 10.
Use mathematical induction to prove that 2.4(2n+1) + 3(3n+1) + is divisible by 11, ∀ n ∈ N.
Solution:
Let S(n) be the statement
2.4(2n+1) + 3(3n+1) is divisible by 11
For n = 1
S(I) is 2.43 + 34 = 128 + 81 = 209 is divisible by 11.
∴ S(n) is true is for n = I
Suppose S(n) is true for n = k then
2.4(2n+1) + 3(3n+1) is divisible by 11
TS Inter 1st Year Maths 1A Mathematical Induction Important Questions 14

TS Inter 2nd Year English Study Material Revision Test-II

Telangana TSBIE TS Inter 2nd Year English Study Material Revision Test-II Exercise Questions and Answers.

TS Inter 2nd Year English Study Material Revision Test-II

Time: 1 1/2 Hrs
Marks : 50

Section – A

Question 1.
Annotate ANY ONE of the following in about 100 words. [1 × 4 = 4]

a) Undoubtedly women in ancient India enjoyed a much higher status than their descendants in the eighteenth and nineteenth centuries.
Answer:
The given lines occur in the informative essay “The Awakening of Women”. This article was composed by a committed writer K.M. Phanikkar. The article deals with the status of women over various periods. Every statement is backed with supporting details.

The essay focuses mainly on the impact the Gandhian Movement had on the progress of women. Yet, the writer states how women’s status was in the past. Women ancient India had a respectable position. It is only in the eighteenth and nineteenth centuries that women’s condition touched a pathetic low. The given lines highlight the fact that writer is balanced but not biased.

TS Inter 2nd Year English Study Material Revision Test-II

b) It was a matter of surprise to the outside world that independence of India should have appointed women to the highest posts so freely, as members of the Cabinet,
Answer:
The given lines occur in the informative essay “The Awakening of Women”. This article was composed by a committed writer, K.M. Phanikkar. The article deals with the status of women’s over various periods. Every statement is backed with supporting details. The position of women started to improve with their active participation in the Gandhian Movement, showed constant progress in all fields.

In pre-independent India, legislation was made in favour of their rights. After India became independent, women were appointed in both key government and administrative posts. This surprised the world. People outside India thought that India was very conservative regarding women’s position. Thus the lines play an important role in clearing certain prejudices.

Question 2.
Annotate ANY ONE of the following in about 100 words. [1 × 4 = 4]

a) He rests at case beneath some pleasant weed.
Answer:
Introduction:
The above line is taken from the sonnet “On the Grasshopper and Cricket” written by John Keats. He denoted his life to the perfection of poverty.

Context and Meaning:
Here the poet expresses his feelings, regarding natures song. The Grasshopper and the Cricket are used as symbols. Seasons may come and go. But Nature never fails to inspire us with its songs. When birds, stop singing in extreme heat, during the summer. The earth is filled with songs of a grasshopper. We can hear the voice of the grasshopper who runs from hedge to hedge. He keeps singing tiredlessly and when he gets tired with fun, he goes under some pleasant weed to take rest.

Critical Comment:
The poet sends the message that nature is beautiful all the line, irrespective of the season.

TS Inter 2nd Year English Study Material Revision Test-II

b) And seems to one in drowsiness half lost, The Grasshopper’s among some grassy hills.
Answer:
Introduction:
These are the conducting lines of the poem “On the Grasshopper and Cricket?? written by John Keats, a Romantic poet. He devoted his life to the perfection of poetry.

Context and Meaning:
John Keats celebrates the music of the Earth. He finds beauty in hot summer as well as in the cold winter. Here, the grasshopper is symbol of hot summer and cricket is symbol of cold winter. During the winter season in the frosty evening, the birds stop singing songs. At that time the cricket begins to sing. He spreads the warmth ofjoy everywhere. The people who are half sleep feel that it is the grasshopper song which is coming from the grassy hills. Thus, he depicts the beauty of Nature.

Critical Comment:
The poet sends the message that nature is beautiful all the time, irrespective of the season. In a similar way, we should be joyful in our life and be happy in all situations.

Question 3.
Answer ANY ONE of the following questions in about 100 words. [1 × 4 = 4]

a) But when the movement was actually started, women were everywhere at the forefront. Elaborate.
Answer:
The essay “The Awakening of Women” traces the evolution women’s progress in India over ages. K.M. Panikkar, a multifaceted genius, discusses the theme at length. Facts have been presented in a systematic order. Supporting details have been provided, Gandhiji understood the power of women. He believed that women could be an inexhaustible source of power.

He gave a call to them to participate in his movement. But, he had certain doubts about their readiness. His doubts were proved to be baseless. Women were very active in every area. They picketed liquor shops. They boycotted foreign goods. They took part in civil disobedience. Nowhere were women inferior to men. It was in fact the other way round.

TS Inter 2nd Year English Study Material Revision Test-II

b) Name some legislative reforms mentioned in the essay The Awakening of Women that seek to establish the equality of women.
Answer:
“The Awakening of Women” traces the evolution women’s progress in India over ages. K.M. Panikkar, a multifaceted genius, discusses the theme at length. Facts have been presented in a systematic order. Supporting details have been provided. Women’s active part in the struggle for freedom initiated a positive change in their status.

Even before India attained independence, laws were enacted and enforced in their favour. And that process continued after independence. Rights to property, to freedom of marriage, to education and employment, raising the age of marriage and the prevention of the dedication of women to temple services were some major legislative reforms.

Question 4.
Answer ANYONE of the following questions in about 100 words [1 × 4 = 4]

a) What is the theme of the poem On the Grasshopper and Cricket?
Answer:
The poem “On the Grasshoppers and Cricket’ is written by John Keats, an English Romantic poet. He has devoted his life to the perfection of poetry. In this poem, John Keats depicts the beauty of Nature. He says that the poetry of earth as symbols to praise Nature is never ending beauty. Seasons may come and go but Nature never fails to inspire us with its songs.

When birds stop singing in extreme heat, the earth is filled with the songs of a grasshopper. He sings endlessly, but when tired rests under some pleasant weed. During winter birds stop singing. There is a deathly silence. Frost spreads its blanket over Nature. Regardless, a shrill second cOmes from beneath stones and it is the cricket singing. Its song restores warmth. Thus, the small creatures prove to the world that the poetry of earth never ceases.

TS Inter 2nd Year English Study Material Revision Test-II

b) Discuss the common features between the grasshopper and cricket.
Answer:
The poem “ On the Grasshopper and Cricket” is written by John Keats. He is an English Romantic poet. He has developed his life to the perfections of poetry. In this poem, he depicts the beauty of nature. He says that the poetry of earth never ceases. He uses the Grasshopper and the Cricket as symbol to praise nature’s never ending beauty.

Seasons may come and go. But, nature never fails to inspire us with its songs. Therefore both the grasshopper and the cricket are the representative voices of nature’s music or poetry. Both offer a soothing effect to the extremities of climate. The grasshoppers song balances the extreme heat during the summer by providing music that is comforting and pleasing the cricket does the same during winter.

Question 5.
Answer ANYONE of the following questions in about 100 words. [1 × 4 = 4]

a) “Love, sacrifice and generosity are the essential elements for happy living”. Explain the statement with reference to the story A Gift for Christmas.
Answer:
A Gift for Christmas” is a well-known short story by O. Henry. The original name of the author is William Sydney Porter. This story was first published in 1905. A Gift for Christmas” is a Christmas story, and it functions as a parable about both the nature of love and the true meaning of generosity. Della’s earnest desire to buy a meaningful Christmas gift for Jim drives the plot of the story, and Jim’s reciprocity of that sentiment is shown when he presents Della with the tortoise-shell combs. Both Jim and Della give selflessly, without expectation of reciprocity. Their sole motivation is to make the other person happy. This, combined with the personal meaning imbued in each of the gifts, conveys the story’s moral that true generosity is both selfless and thoughtful.

TS Inter 2nd Year English Study Material Revision Test-II

b) Sketch the character of Jim.
Answer:
A Gift for Christmas” is a well-known short story by O. Henry. The original name of the author is William Sydney Porter. This story was first published in1905. Jim is a thin man of twenty two. He does not have enough income to support his wife. He bears the burden of fulfilling everyday demands of his wife. He is a very punctual person that why he constantly looks at his watch.

Section – B

Question 6.
Read the following passage and answer ANV FIVE questions given below: [5 × 1 = 5]

Della’s white fingers quickly opened the package. And then at first a scream of joy followed by a quick feminine change to tears. For, there lay The Combs — the set of combs, side and back, that Della had seen in a Broadway window and liked so much. They were beautiful combs, so expensive and they were hers now. But alas, the hair in which she was to wear them was sold and gone! She took them up lovingly, smiled through her tears and said, ‘My hair grows so fast, Jim!’

i) Who gave the package to Delia?
Answer:
to sharpen it

ii) What was Della’s reaction at first?
Answer:
We become better persons

iii) How did her joy change?
Answer:
No

iv) What did Della find in the package?
Answer:
the graphite inside

v) Where had she seen the combs earlier?
Answer:
because every action leaves a mark

TS Inter 2nd Year English Study Material Revision Test-II

vi) The combs were inexpensive. Write true or false.
Answer:
True

vii) Write the antonym of the word happy from the passage.
Answer:
exterior

viii) Write he synonym of the word shout from the passage.
an:
sorrows

Question 7.
Study the following advertisement and answer ANY FIVE questions that follow. [5 × 1 = 5]

TS Inter 2nd Year English Study Material Revision Test-II 1
i) Expand SHIP.
Answer:
Swach Hyderabad Internship Programme

ii) What are the eligibility criteria for participating in the programme?
Answer:
Social responsibility, passion, above 18years of age.

iii) Can very young boys and girls participate in this programme?
Answer:
no

iv) State any two themes of the internship programme.
Answer:
Lake cleanup, plantation

v) Registration is both online and offline. Write true or false.
Answer:
false

TS Inter 2nd Year English Study Material Revision Test-II

vi) When is the programme scheduled to begin?
Answer:
21st January 2022

vii) The number of hours of-the schedule is ___________ . (Fill in the blank.)
Answer:
20

viii) Write the word used in the ad to mean ‘a short term training period for practical experience’.
Answer:
Internship

Question 8.
Study the bar graph below and answer ANT FIVE questions given after it. [5 × 1 = 5]

TS Inter 2nd Year English Study Material Revision Test-II 2
i) What does the bar graph present?
Answer:
Sales of ice-creams of different flavours

ii) Ice cream of which flavour do people like the most in shop A?
Answer:
Ice-cream of mango flavours

iii) How many ice creams of almond flavour are sold in shop B?
Answer:
65

TS Inter 2nd Year English Study Material Revision Test-II

iv) Find the total number of ice creams of chocolate flavour sold in shop A and shop B.
Answer:
165

v) 30 ice creams of coconut flavour are sold in ___________. (Fill in the blank.)
Answer:
shop B

vi) Ice cream of which flavour do people like more in shop B, Chocolate or Vanilla?
Answer:
Chocolate

vii) How many ice creams of mango flavour are sold in shop A?
Answer:
100

viii) How many ice cream flavours are shown in the graph?
Answer:
5

Section – C

Question 9.
Rewrite the following passage using FIVE punctuation marks wherever necessary- [5 × 1 = 5]

The brahmo samaj led the movement for emancipation the ancient rules of purdah were broken and brahmo women moved freely in society: but this was. but a false dawn as it was far in advance of popular opinion.
Answer:
The Brahmo Samaj led the movement for emancipation. The ancient rules of purdah were broken and Brahmo women moved freely in society: but this was but a false dawn as it was far in advance of popular opinion.

TS Inter 2nd Year English Study Material Revision Test-II

Question 10.
Match the following words in Column ‘A’ with their definitions in Column ‘B’. [5 × 1 = 5]
Column A —– Column B
i) amphibious   ( ) a) lasting for a very short time
ii) antidote       ( ) b) designed to cause death
iii) ephemeral  ( ) c) living on land as well as in water
iv) lethal          ( ) d) someone who has a lot of experience in a field
v) veteran        ( ) e) a substance that can act against the effect of poison
Answer:
i) c
ii) e
iii) a
iv) b
v) d

TS Inter 2nd Year English Study Material Revision Test-I

Telangana TSBIE TS Inter 2nd Year English Study Material Revision Test-I Exercise Questions and Answers.

TS Inter 2nd Year English Study Material Revision Test-I

Time : 1 1/2 Hrs
Marks : 50

Section – A

Question 1.
Annotate ANY ONt of the following in about 100 words. [1 × 4 = 4]

a) If someone maintains that two and two are five or that Iceland is on the equator, you feel pity rather than anger.
Answer:
Introduction:
These beautiful lines are taken from the thought-provoking essay,” How to Avoid Foolish Opinions” written by Bertrand Russell. His clarity of thought and fluency of expression lend beauty to his style.

Context and Meanings:
Russell gives us tips on how to avoid foolish opinions. He says that there are many ways to avoid being dramatic. To avoid foolish opinions, no super human or genius is required. Many matters are less easily brought to the test of experience. If we hear views opposite to our opinions, it makes us angry. It is a sign that we actually have no good reason for our opinion. If someone has very stupid and wrong opinions, we feel pity rather than anger. So, we must carefully reconsider our ideas.

Critical Comment:
Though the article discusses many mistakes mankind is prove to make, it ends with a lively ray of hope.

TS Inter 2nd Year English Study Material Revision Test-I

b) Persecution is used in theology, not in arithmetic because in arithmetic, there is knowledge, but in theology, there is only opinion.
Answer:
Introduction:
These lines are taken from the thought provoking essay, “How to Avoid Foolish Opinion” written by Bertrand Russell. His clarity of thought and fluency of expression lend beauty to his style.

Context and Meanings:
Russell gives us tips on how to avoid foolish opinion. Here, he teas us about controversies. The worst controversies are about matters which have no good evidence either way. If you can not observe an issue, think about any biases you might have about it. This is because belief can go beyond facts. Persecution means annoying others deliberately all the time, the theology is required belief where as arithmetic is require facts and figures. Hence, persecution is not used in arithmetic but in theology – the study of God and religion.

Critical Comment:
The author says that belief can go beyond facts.

Question 2.
Annotate ANY ONE of the following in about 100 words. [1 × 4 = 4]

a) We are meeting today to wish her bon voyage.
Answer:
Reference:
These lines are taken from the satirical and humorous poem “Goodbye party for Miss Pushpa TS” by Nission Ezekid, a.versalite Indo-Anglian poet with a great sense of humour and wit.

Context and Explanation:
It is a farewell speech for miss pushpa, who is leaving the country the poem is a parody of English as used by some indians. The speaker adresses his colleagues as friends and miss pushpa as sister.

Critical Comment:
The lines highlight the speaker’s good nature and good intention. The style is simple and clear.

TS Inter 2nd Year English Study Material Revision Test-I

b) That is showing good spirit. I am always appreciating the good spirit
Answer:
Reference:
These lines are extracted from the satirical and humorous peom “Goodbye party for Miss Pushpa TS” written by Nission Ezhekiel, a versatile Indo-Anglion poet with a great sense of humour and wit. The poem is a parody of English as used by some Indians.

Context and Explanation:
It is a farewell speech for Miss pushpa, who is leaving the country. On the occasion, the speaker is praising the helpful nature of Miss. Pushpa. He praises her good nature. Whenever he asked her to do anything, she was saying that she would do it just now only. This reply from her is showing her good spirit and he is always appreciating the good spirit.

Critical Comment:
This shows her good spirit and her readiness to do any work she is a willing worker.

Question 3.
Answer ANY ONE of the following questions in about 100 words [1 × 4 = 4]

a) How can we prevent developing a dogmatic attitude as per Russell’s suggestion?
Answer:
The thought provoking essay,”How to Avoid Foolish Opinion’s” is written by Bertrand Russell. In this essay, he says that there are many ways to avoid being dogmatic. Making a keen observation where it can settle the bias is the first way. Next to know what other people think.

One has to be aware of what they think. This can be done by going on vacation and talking to people with different ideas. The third is arguing with an imaginary character. The fourth one is to deal with one’s sense of self worth. To overcome conceit, we must remember that we live for a short while on a small planet in vast cosmos.

TS Inter 2nd Year English Study Material Revision Test-I

b) What does Bertrand Russell say about a person getting angry about a difference of opinion?
Answer:
The thought provoking essay,”How to Avoid Foolish Opinions” is wiitten by Bertrand Russell. In this essay, he gives us tips on how to avoid foolish opinions and being dogmatic. He advices us to identify our weak points and reconsider our opinions. when we hear views opposite to our opinions. It makes us angry. It is a clear sign of something wrong with our beliefs, then we must carefully reconsider our ideas. We have to be aware of what other people think. Thus, we can avoid such problem.

Question 4.
Answer ANY ONE of the following questions in about 100 words [1 × 4 = 4]

a) How does the speaker describe Miss Pushpa in the poem?
Answer:
The poem “Goodbye Party for Miss Pushpa T.S.” is written by Nission Ezekiel. He is a versatile poet with a great sense of humor and wit. His present poem is a parody of English as used by some Indians. It is a farewell speach for Miss Push pa, who is going abroad. On this occasion, they have gathered there to hid farewell to her. A person comes and gives a speech. the poet creates humour through the speakers description of Miss Pushpa.

The speaker describes not only her internal but external sweetness. He says that she always ‘smiles’ without reason. He describes her good and amicasle nature. He also refers to her helpful qualities. He appreciates her concern for friends. He says that she always says ‘yes’ to any of their request. Thus, he is trying to exaggesate to show his love and respect for her. But, he doesn’t realize that he is describing her humorously and very lovsely. He doesn’t mind that his English is wrong linguestically. Thus, he describes her humorously with his Babu English.

TS Inter 2nd Year English Study Material Revision Test-I

b) Does the poem bring out the sweetness of Miss Pushpa? Justify your answer.
Answer:
The poem “Goodbye party for Miss Pushpa TS” is written by Nission Ezekiel. He is a versatile poet with a great sense of humour and wit. The poem is an extract from his volume of pems. ‘Hymns in Darkness’. It is an parody of English as used by Some Indians. It is farewell party for Miss. Pushpa, who is going abroad for better prospects. The Speaker announces in the very beginning that they have gathered there to bid farewell to her.

They want to wish her a happy journey. On this occasion, the speaker brings out the sweetness Miss Pushpa. He starts praising her sweetness which is both internal and external. She is beautiful not only because of her charms, but her honesty also. She is a sweet lady with all smile on her face. She smiles without a reason. She belongs to a reputed family. She is very popular among people. She is always ready to do anything for everyone. Thus the poem brings out the sweetness of Miss Pushpa. It appreciates her concern for friends. These are sweet qualities of Miss Pushpa.

Question 5.
Answer ANY ONE of the following questions in about 100 words; [1 × 4 = 4]

a) Describe the character of Arun, the boarding school boy?
Answer:
Ruskin Bond is a well-known contemporary Indian writer of British descent. He wrote many books inspirational children’s books and was honoured with the Sahitya Akademi Award for his literary work. The present short story represent from “The women on plat form No. 8” the main idea of is a story about love and affection that overcomes all sense of belonging barriers.

Arun was a boarding school student. He was returning to school. His parents thought he was old enough to travel alone. So he took a bus from his hometown to Ambala, arriving early in the evening. The train he needed to catch left at 12 a.m. He was waiting for the northbound train on platform 8 at Ambala station. It had been a long time for him. He walked up and down the platform, browsing the book stall and feeding Street dogs biscuits. He stood there watching the trains come and go. Whenever a train arrived, the platform became a center for activity, and it would be quiet after the train had left. He sat down on his suitcase, tired of pacing around the platform, and stare at the railway tracks.

A soft voice asked from behind if he was alone. Arun saw a middle aged woman in white sari, with dark kind eyes leaning over him. There was some kind of dignity about her which made Arun stand up respectfully and answer. He told her that he was alone and that he was going to school. She asked him if his parents had not come to see him off. Arun said that he did not live there and he could travel alone. The lady agreed with him. Arun liked her for her simplicity, her deep soft voice and the serenity of her face.

TS Inter 2nd Year English Study Material Revision Test-I

b) What made Arun call the strange woman ‘mother’ at the end?
Answer:
Ruskin Bond is a well-known contemporary English Indian writer. He wrote a number of inspirational children’s books and was awarded the Sahitya Akademi Award for his literary work. The following is a short story from “The Women on Plat Form No. 8.” This short story’s core concept is about love and affection overcoming all obstacles to belonging.

Arun called the stranger woman ‘mother’ at the end, because she had treated him tenderly and offered him tea and sweets. She listened to him and showed trust in him. He liked her kindness and graceful behaviour. She introduced herself as his mother. She supported Arun against satish’s mother, Arun wanted to repay her kindness by acknowledging her as mother.

Section – B

Question 6.
Read the following passage and answer ANY SIX questions given below: [6 × 1 = 6]

I was going to refuse out of shyness and suspicion, but she took me by the hand. Then I felt it would be silly to pull my hand away. She told a porter to look after my suitcase and then she led me down the platform. Her hand was gentle. She held my hand neither too firmly nor too lightly. I looked up at her again. She was not young, but she was not old.

i) Name the short story from which this passage is taken.
Answer:
The Woman on Platform No. 8

ii) What was Arun going to do?
Answer:
To Refuse out of shyness and suspicion

iii) How did she hold his hand?
Answer:
neighter to firmly nor too lightly

iv) How did Arun feel of taking back his hand away?
Answer:
He felt silly

v) What did she tell a porter?
Answer:
To look after Arun’s suitcase

TS Inter 2nd Year English Study Material Revision Test-I

vi) Pick out the word from the passage which means tender.
Answer:
Gentle

vii) ‘She held my hand neither too firmly nor too lightly.’ Say true or false.
Answer:
True

viii) Rewrite the sentence “She was not young, but she was not old.” using “neither. nor……”.
Answer:
She was neither young nor old.

Question 7.
Read the following passage and answer ANY SIX questions given below: [6 × 1 = 6]

The Warrior Who Broke All Barriers The very word COVID spreads dread worldwide. But, Dr Annam Srinivasa Rao of Khammam stands out as an exception. Putting his life at risk, he served a large number of COVID patients in various ways. His organization served food and extended medical facilities to them while alive. When died, the Foundation arranged for transportation and last rites of hundreds of bodies when their own families abandoned the bodies!

When he was tested positive, he was worried, not of his health, but of the people who needed his service! By looking after hundreds of disabled (both physically and mentally) orphans from far and near for over a decade, his organization – Annam Seva Foundation, Danvaigudem, Khammam – has redefined philanthropy. He was inspired into this service when he himself grew up in an orphanage. His struggles to secure some employment helped him master the art of taking everything in his stride. Thus, he continues his services undeterred by police cases, neighbours’ wrath, financial constraints etc. An admirable spirit indeed!

i) What makes one describe Dr Annam Srinivasa Rao as the warrior who broke all barriers?
Answer:
Has he put his life at risk and served COVID patients also

ii) How does the world respond to the word COVID, according to the passage?
Answer:
Reads the word COVID

iii) Why was Dr Annam Srinivasa Rao worried when he tested positive?
Answer:
Of the people who needed his service and missed it

TS Inter 2nd Year English Study Material Revision Test-I

iv) Name the organization that he founded and its location.
Answer:
Annam Seva Foundation: Danavayagudem, Khammam

v) Why did Dr Annam Srinivasa Rao choose this path?
Answer:
Has he him self growup in a orphanage

vi) Pick out from the passage the one-word substitute that means selfless service out of love for humanity.
Answer:
Phinlanthropy

vii) Find out from the passage the phrasal verb that means taking full care and responsibility of.
Answer:
Looking after

viii) Write the idiom used in the passage that means accepting and dealing with difficulties without letting them worry one too much
Answer:
Taking something in one’s stried

Question 8.
Study the pie-chart below and answer any Six questions given after it. [6 × 1 = 6]

TS Inter 2nd Year English Study Material Revision Test-I 1
i) What does the pie chart show?
Answer:
favourite beverages of people

ii) How many beverages are taken into account?
Answer:
5

iii) What is the most preferred beverage?
Answer:
Coconut Water (38%)

TS Inter 2nd Year English Study Material Revision Test-I

iv) How many people preferred coffee?
Answer:
15%

v) What is the least preferred beverage?
Answer:
Sugarcane Juice

vi) People who preferred tea are _____________.
Answer:
25%

vii) People who preferred cola are __________.
Answer:
12%

viii) People who preferred Coconut water are 27%. Write true or false.
Answer:
False

Section – C

Question 9.
Rewrite’th6 following passage using SIX of the punctuation marks wherever necessary. [6 × 1 = 6]

It is more difficult to deal with the self-esteem of man as a man because we cannot argue out the matter with some non-human mind, the only way I know of dealing with this general human conceit is to remind ourselves that man is a brief episode in the life of a small planet in a little comer of the universe and that for aught we know other parts of the cosmos may contain beings as superior to ourselves as we are to jellyfish.
Answer:
It is more difficult to deal with the self-esteem of man as a man because we cannot argue out the matter with some non- human mind. The only way I know of dealing with this general human conceit is to remind ourselves that man is a brief episode in the life of a small planet in a little corner of the universe and that, for aught we know, other parts of the cosmos may contain beings as superior to ourselves as we are to jellyfish.

TS Inter 2nd Year English Study Material Revision Test-I

Question 10.
Match the following words in Column ‘A’ with their definitions in Column ‘B’. [6 × 1 = 6]
Column A —- Column B
i) ambidextrous   ( ) a) one who eats excessively
ii) contemporary ( ) b) something which is out of date
iii) glutton           ( ) c) a person walking on a street
iv) invincible       ( ) d) able to use both hands equally well
v) obsolete         ( ) e) living or occurring at the same time
vi) pedestrian     ( ) f) too strong to be defeated
Answer:
i) d
ii) e
iii) a
iv) f
v) b
vi) c

TS 6th Class Maths Solutions Chapter 4 Basic Geometrical Ideas Ex 4.2

Students can practice TS 6th Class Maths Solutions Chapter 4 Basic Geometrical Ideas Ex 4.2 to get the best methods of solving problems.

TS 6th Class Maths Solutions Chapter 4 Basic Geometrical Ideas Exercise 4.2

Question 1.
Tick the figures which are simple curves.
TS 6th Class Maths Solutions Chapter 4 Basic Geometrical Ideas Ex 4.2 1
Answer:
TS 6th Class Maths Solutions Chapter 4 Basic Geometrical Ideas Ex 4.2 2

Question 2.
State which curves are open and which are closed.
TS 6th Class Maths Solutions Chapter 4 Basic Geometrical Ideas Ex 4.2 3
Answer:
Figure (i) and (v) are open
Figure (ii), (iii) & (iv) are closed

Question 3.
Name the points that lie in the interior, on the boundary and in the exterior of the figure.
TS 6th Class Maths Solutions Chapter 4 Basic Geometrical Ideas Ex 4.2 4
Answer:
The points that lie in the interior of the figure are A, B, E, G and I.
The points that lie on the boundary of the figure are C, F and K.
The points that lie in the exterior of the figure are D and J.

TS 6th Class Maths Solutions Chapter 4 Basic Geometrical Ideas Ex 4.2

Question 4.
Draw three simple closed figures :
i) by straight lines only
ii) by straight lines and curved lines both
Answer:
i) Simple closed figures drawn by straight lines only
TS 6th Class Maths Solutions Chapter 4 Basic Geometrical Ideas Ex 4.2 5

ii) Simple closed figures drawn by straight lines and curved lines both
TS 6th Class Maths Solutions Chapter 4 Basic Geometrical Ideas Ex 4.2 6