Mahesh

Students can practice TS 10th Class Maths Solutions Chapter 2 Sets InText Questions to get the best methods of solving problems.

TS 10th Class Maths Solutions Chapter 2 Sets InText Questions

Do This

Question 1.
List the teeth under each of the following type.
i) Incisors
Solution:
Central incisors = 4
Lateral incisors = 4

Total incisors = 8

ii) Canines
Solution:
Total Canines = 4

iii) Pre-molars
Solution:
First premolars = 4
Second premolars = 4

Total premolars = 8

iv) Molars
Solution:
First molars = 4
Second molars = 4
Third molars = 4

Total molars = 12

Question 2.
Identify and write the “common property” of the following collections. (Page No. 26)

1) 2, 4, 6, 8 ….
2) 3, 5, 7, 11, ….
3) 1, 4, 9, 16, ………
4) January, February, March, April,………..
5) Thumb, index finger, middle finger, ring finger, pinky.
Solution:
1) For given integers 2, 4, 6, 8 …….. 2n, n = 1, 2, 3, 4,…. is any positive integer.
2) 2, 3, 5, 7, 11, …. prime numbers.
3) For given integers 1, 4, 9, 16, …. n², where ‘n’ is any positive integer and square of the numbers.
4) January, February, March, April,…. months of the every year.
5) Thumb, index finger, middle finger, ring finger, pinky fingers of the human hand.

Question 3.
Write the following sets. (Page No. 27)
1) Set of the first five positive integers.
2) Set of multiples of 5 which are more than 100 and less than 125.
3) Set of first five cubic numbers.
4) Set of digits in the Ramanujan number.
Solution:
1) { + 1, +2, +3, +4, +5}
2) {105, 110, 115, 120}
3) {1, 8, 27, 64,125} = {1³, 2³, 3³, 4³, 5³}
4) Ramanujan Number = 1729. So the set = {1, 2, 7, 9}

Question 4.
Some numbers are given below. Decide the numbers to which number sets they belong to and does not belong to and express with correct symbols. (Page No. 27)
i) 1
ii) 0
iii) -4
iv) \(\frac{5}{6}\)
v) \(\text { 1. } \overline{3}\)
vi) \(\sqrt{2}\)
vii) log 2
viii) 0.03
ix) π
x) \(\sqrt{-4}\)
Solution:
Set of natural numbers = N
set of integers = Z
Set of rational numbers = Q
Set of real numbers = R
i) 1 ∈ (N, Z, Q, R}
ii) 0 ∉ N, 0 ∈ {Z, Q, R}
iii) -4 ∉ N, -4 ∈ (Z, Q, R}
iv) \(\frac{5}{6}\) ∉ (N, Z} But \(\frac{5}{6}\) ∈ {Q, R}
v) \(1 . \overline{3}\) ∉ {N, Z} But \(1 . \overline{3}\) ∈ {Q, R}
vi) \(\sqrt{2}\) ∉ {N, Z} But \(\sqrt{2}\) ∈ {Q, R}
vii) log2 ∉ N,Z But log 2 ∈ {Q, R}
viii) 0.03 ∉ {N, Z}But 0.03 ∈ {Q, R}
ix) π ∉ {N, Z} But π ∈ (Q, R}
x) \(\sqrt{-4}\) ∉ (N, Z, Q} But \(\sqrt{-4}\) ∈ R.

Question 5.
List the elements of the following sets.
i) G = {all the factors of 20}
ii) F = {the multiples of 4 between 17 and 61 which are divisible by 7}
iii) S = {x : x is a letter in the word MADAM’}
iv) P = {x : x is a whole number between 3.5 and 6.7} (Page No. 29)
Answer:
i) G = {1, 2, 4, 5, 10, 20}
ii) Multiples of 4 between 17 and 61.
x = {20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60}
F = {28, 56}
iii) S = {M, D, A}
iv) P = {4, 5, 6}

Question 6.
Write the following sets in the roaster form. (Page No. 29)
i) B is the set of all months in a year having 30 days.
ii) P is the set of all prime numbers less than 10.
iii) X is the set of colours of the rainbow.
Answer:
i) B = {April, June, September, November}
ii) P = {2, 3, 5, 7}
iii) X = {Violet, Indigo, Blue, Green, Yellow, Orange, Red}

Question 7.
A is the set of factors of 12. Which one of the following is not a member of A ? (Page No. 29)
A) 1
B) 4
C) 5
D) 12
Answer:
[C]

Think – Discuss

Question 1.
Observe the following collections and prepare as many as generalised statements you can describing their more properties. (Page No. 26)
i) 2, 4, 6, 8, ….
ii) 1, 4, 9, 16
Answer:
i) 3, 6, 9, 12, ….
Property : multiples of 3.

ii) 1, 8, 27, 64,….
Property : cube of the numbers, i.e., 1³, 2³, 3³, 4³, ….

Question 2.
Can you write the set of rational numbers listing elements in it ? (Page No. 28)
Solution:
No

Try This

Question 1.
Write some sets of your choice, involving algebraic and geometrical ideas. (Page No. 29)
Answer:
1) The natural number greater than three and less than twelve.
2) A two digit number such that the sum of its digits is 8.
3) The set of quadrilaterals.
4) The set of all triangles in a plane.

Question 2.
Match roaster forms with the set builder form.
i) {P, R, I, N, C, A, L}
ii) {0}
iii) {1, 2, 3, 6, 9, 18}
iv) {3, -3} (Page No. 29)
a) {x : x is a positive integer and is a divisor of 18}
b) {x : x is an integer and x² – 9 = 0}
c) {x : x is an integer and x + 1 = 1}
d) {x: x is a letter of the word PRINCIPAL}
Answer:
i) d
ii) c
iii) a
iv) b

Do This

Question 1.
A = {1, 2, 3, 4}, B = {2, 4}, C = {1, 2, 3, 4, 7}, F = { } (Page No. 33)
Fill in the blanks with ⊂ or ⊄.
i) A………B
ii) C……..A
iii) B……..A
iv) A……..C
v) B……..C
vi) F……B
Answer:
i) A⊄B
ii) C⊄A
iii) B⊂A
iv) A⊂C
v) B⊂C
vi) F⊂B

Question 2.
State which of the following statements are true. (Page No. 33)
i) { } = ϕ
ii) ϕ = 0
iii) 0 = {ϕ}
Answer:
i) True (T)
ii) False (F)
iii) False (F)

Question 3.
Let A = {1, 3, 7, 8} and B = {2, 4, 7, 9}. Find A∩B.(Page No. 37)
Solution:
Given sets
A = {1, 3, 7, 8} and B = {2, 4, 7, 9}
A ∩ B = {1, 3, 7, 8} ∩ {2, 4, 7, 9}
= {7}

Question 4.
If A = {6, 9, 11}; ϕ = { }, find A ∪ ϕ. (Page No. 37)
Solution:
Given sets A = {6, 9, 11} and B = {2, 4, 7, 9}
A ∪ ϕ = {6, 9, 11} ∪ {ϕ}
= {6, 9, 11} = A
∴ A ∪ ϕ = A

Question 5.
A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10};
B = {2, 3, 5, 7}. Find A ∩ B. (Page No. 37) (June ’15(AP))
Solution:
Given sets A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and B = {2, 3, 5, 7}
A ∩ B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} ∩ {2, 3, 5, 7}
= {2, 3, 5, 7} = B
∴ A ∩ B = B

Question 6.
If A = {4, 5, 6}; B = {7, 8}, then show that A ∪ B = B ∪ A. (Page No. 37)
Solution:
Given sets are
A = (4, 5, 6} and B = (7, 8}
A ∪ B = {4, 5, 6} ∪ {7, 8}
= {4, 5, 6, 7, 8}
B ∪ A = {7, 8} ∪ {4, 5, 6}
= {4, 5, 6, 7, 8}
∴ A ∪ B = B ∪ A

Question 7.
If A = {1, 2, 3, 4, 5}; B = {4, 5, 6, 7} then find A-B and B-A. Are they equal ? (Page No. 38)
Solution:
Given sets are A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7}
A – B = {1, 2, 3, 4, 5} – {4, 5, 6, 7} = {1, 2, 3}
B – A = {4, 5, 6, 7} – {1, 2, 3, 4, 5} = {6, 7}
No, A – B ≠ B – A

Question 8.
If V = {a, e, i, o, u} and B = {a, i, k, u}, find V – B and B – V. (Page No. 38)
Solution:
Given sets are
V = {a, e, i, o, u} and B = {a, i, k, u}
V – B = {a, e, i, o, u} – {a, i, k, u}
= {e, o}
B – V = {a, i, k, u} – {a, e, i, o, u}
= {k}

Try This

Question 1.
A = {set of quadrilaterals}, B = {square, rectangle, trapezium, rhombus}
State whether
A ⊂ B or B ⊂ A. Justify your answer. (Page No. 33)
Answer:
A ⊄ B
B ⊂ A every element of B is also an element of A.

Question 2.
If A = {a, b, c, d}. How many subsets does the set A have? (Remember null set and equal sets). (Page No. 33)
A) 5
B) 6
C) 16
D) 65
Solution:
A = {a, b, c, d}
Subsets of A = {a}, {b}, {c}, {d};
n(A) = 4
Number of subsets for a set, which is having ‘n’ elements is 2ⁿ.
So n(A) = 4
Number of subsets for A is 2⁴ =16.
Answer:
(C)

Question 3.
P is the set of factors 5, Q is the set of factors of 25 and R is the set of factors of 125.
Which one of the following is false?
A) P ⊂ Q
B) Q ⊂ R
C) R ⊂ P
D) P ⊂ R (Page No. 33)
Solution:
P = {1, 5}
Q = {1, 5, 25}
R = {1, 5, 25, 125}
Answer:
(C)

Question 4.
A is the set of prime numbers less than 10, B is the set of odd numbers < 10 and C is the set of even numbers < 10. How many of the following statements are true ? (Page No. 33)
i) A ⊂ B
ii) B ⊂ A
iii) A ⊂ C
iv) C ⊂ A
v) B ⊂ C
vi) X ⊂ A
Solution:
A = {2, 3, 5, 7}
B = {1, 3, 5, 7, 9}
C = {2, 4, 6, 8}
The given all statements are false.

Question 5.
List out some sets A and B and choose their elements such that A and B are disjoint. (Page No. 37)
Solution:
A and B are disjoint sets.
i) A = {2, 3, 5} B = {4, 6, 8}
ii) A = {1, 2, 3} B = {4, 5, 6}
iii) A = {1, 3, 5, 7} B = {2, 4, 6, 8}

Question 6
If A = {2, 3, 5}, find A∪ϕ and ϕ∪A and compare. (Page No. 37)
Solution:
A = {2, 3, 5}; ϕ = { }
A∪ϕ = {2, 3, 5} ∪ { } = {2, 3, 5}
ϕ∪A = { } ∪ {2, 3, 5} = {2, 3, 5}
∴ A∪ϕ = ϕ∪A

Question 7.
If A = {1, 2, 3, 4}; B = {1, 2, 3, 4, 5, 6, 7, 8} then find A ∪ B, A ∩ B. What do you notice about the result ? (Pg. No. 37)
Solution:
A = {1, 2, 3, 4};
B = {1, 2, 3, 4, 5, 6, 7, 8}
A∪B = {1, 2, 3, 4} ∪ {1, 2, 3, 4, 5, 6, 7, 8} = {1, 2, 3, 4, 5, 6, 7, 8}
The common elements in both A and B sets are 1, 2, 3, 4.
A∩B = {1, 2, 3, 4}
Result :
i) A∪B = B
ii) A∩B = A
iii) A ∩ B ⊂ A ∪ B

Question 8.
A = {1, 2, 3, 4, 5, 6}; B = {2, 4, 6, 8, 10}. Find the intersection of A and B. (Page No. 37)
Solution:
A = {1, 2, 3, 4, 5, 6}
B = {2, 4, 6, 8, 10}
The common elements in both A and B are 2, 4, 6.
∴ A ∩ B = {2, 4, 6}

Think : Discuss

Question 1.
Is empty set subset to every set? (Page No. 34)
Solution:
Yes.

Question 2.
Is any set subset to itself? (Page No. 34)
Solution:
Yes.

Question 3.
You are given two sets such that a set is not a subset of the other. If you have to prove this, how do you prove ? Justify your answers. (Page No. 34)
Solution:
The above statement is true only in the two sets does not have common elements.
Ex : A = {1, 2, 3}; B = {4, 5, 6}
So, A⊊B.

Question 4.
The intersection of any two disjoint sets is a null set. Justify your answer. (Page No. 37)
Answer:
Yes, this statement is true because A ∩ B = ϕ when A and B are disjoint sets.

Question 5.
The sets A – B, B – A and A ∩ B are mutually disjoint sets. Use examples to observe if this is true. (Page No. 38)
Solution:
Let the sets are
A = {1, 2, 3, 4} and B = {5, 6, 7, 8}
A – B = {1, 2, 3, 4} – {5, 6, 7, 8} = {1, 2, 3, 4}
B – A = {5, 6, 7, 8} – {1, 2, 3, 4} = {5, 6, 7, 8}
A ∩ B = {1, 2, 3, 4} ∩ {5, 6, 7, 8} = { } = ϕ
∴ A – B, B – A and A ∩ B are disjoint sets.

Do This

Question 1.
Which of the following are empty sets? Justify your answer. (Page No. 44)
i) Set of integers which lie between 2 and 3.
ii) Set of natural numbers that are less than 1.
iii)Set of odd numbers that have remainder zero, when divided by 2.
Answer:
i) This is null set. We know that there is no integer that lie between 2 and 3.
ii) This is also a null set. We know that there is natural number less than ‘1’.
iii) This is a null set. We know that odd numbers do not leave remainder zero when divided by 2.

Question 2.
State which of the following sets are finite and which are infinite. Give reasons for your answers.
i) A = {x: x ∈ N and x < 100}
ii) B = {x : x ∈ N and x ≤ 5}
iii)C = {1², 2², 3², ………… )
iv)D = {1, 2, 3, 4)
v) {x : x is a day of the week) (Page No. 44)
Answer:
i) A = {1, 2, 3, ………., 98, 99}
This set is finite, because there are 99 numbers possible to count.

ii) B = {1, 2, 3. 4, 5}
This set is finite, because there are 5 numbers possible to count.

iii) C = {1², 2², 3²,………)
This set is infinite, because there are infinite numbers.

iv) D = {1, 2, 3, 4}
This set is finite because there are 4 numbers that are possible to count.

v) E = {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday)
This set is finite, because there are 7 days in a week possible to count.

Question 3.
Tick the set which is infinite
A) The set of whole numbers < 10
B) The set of prime numbers < 10
C) The set of integers < 10
D) The set of factors of 10 (Page No. 44)
Answer:
[C]
The set of integers < 10
{……., -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

Try This

Question 1.
Which of the following sets are empty sets? Justify your answer.
i) A = {x : x² = 4 and 3x = 9}
ii) The set of all triangles in a plane having the sum of their three angles less than 180. (Page No. 44)
Answer:
i) Empty set.
To satisfy these equations same x value is not possible.
ii) Empty set.
The sum of the three angles of a triangle is equal to 180°.

Question 2.
B = {x : x + 5 = 5} is not an emptyset. Why? (Page No. 44)
Solution:
x + 5 = 5
x = 5 – 5
x = 0
For x = 0 it is true
Only one element is there.
So it is not an empty set.

Think — Discuss

Question 1.
An empty set is a finite set. Is this statement true or false? Why? (Page No. 44)
Answer:
Yes, it is a finite set because there is finite number i.e., ‘0’ elements it consists.

Question 2.
What is the relation between n(A), n(B), n(A ∩ B) and n (A ∪ B)? (Page No. 45)
Solution:
n(A) = elements in set A, n(B) = elements in set B
n(A ∩ B) = set of all elements which are common to both A and B.
n(A ∪ B) = elements in set A and set B (or) union of sets A and B.
n(A ∪ B) = n(A) + n(B) – n(A ∩ B)

Question 3.
If A and B are disjoint sets, then how can you find n(A ∪ B)? (Page No. 45)
Solution:
n(A) = elements in set A.
n(B) = elements in set B.
n(A ∩ B) = elements in set A and set B
Here it is ‘0’ (∴ A and B are disjoint sets)
n(A ∪ B) = elements in set A or set B.
∴ n(A ∪ B) = n(A) ÷ n(B) – n(A ∩ B)
= n(A) + n(B) – 0
= n(A) + n(B).

Students can practice TS 10th Class Maths Solutions Chapter 2 Sets Ex 2.3 to get the best methods of solving problems.

TS 10th Class Maths Solutions Chapter 2 Sets Exercise 2.3

Question 1.
Which of the following sets are equal ?
a) A = {x : x is a letter in the word FOLLOW’}
b) B = {x : x is a letter in the word ‘FLOW’}
c) C = {x : x is a letter in the word ‘WOLF’}
Answer:
a) Writing the given set in the roaster form, we have A = {F, O, L, W}
b) Writing the given set in the roaster form, we have B = {F, L, O, W}
c) Writing the given set in the roaster form, we have C = {W, O, L, F}
Therefore, A, B, C are equal sets.
[∴ The sets A, B, C have exactly the same elements]

Question 2.
Consider the following sets and fill up the blank in the statement given below with = or ≠ so as to make the statement true.
A = {1, 2, 3};
B = {The first three natural numbers}
C = {a, b, c, d};
D = {d, c, a, b}
E = {a, e, i, o, u};
F = {set of vowels in English Alphabet}
i) A ……… B
ii) A …….. E
iii) C ……. D
iv) D …… F
v) F ……. A
vi) D …… E
vii) F ……. B
Answer:
i) A = B
ii) A ≠ E
iii) C=D
iv) D ≠ F
v) F ≠ A
vii) D ≠ E
viii) F ≠ B

Question 3.
In each of the following, state whether
A = B or not.
i) A = {a, b, c, d}; B = {d, c, a, b}
ii) A = {4, 8, 12, 16}; B = {8, 4, 16, 18}
iii) A = (2, 4, 6, 8, 10)
B = {x: x is a positive even integer and x ≤ 10}
iv) A = {x: x ¡s a multiple of 10};
B = {10, 15, 20, 25, 30,……. }
Answer:
i) A = B because A and B have exactly the same elements i.e., a, b, c, d.
ii) A ≠ B because A and B have not exactly the same elements.
iii) A = B because A and B have exactly the same elements.
Writing B in roaster form, we have
B = {2, 4, 6, 8, 10}
iv) A = {10, 20, 30, 40,……..}
B = {10, 15, 20, 25,……..}
A ≠ B because A and B have not exactly the same elements.

Question 4.
State the reasons for the following:
i) {1,2, 3,…, 10} ≠ {x : x ∈ N and 1 < x < 10}
ii) {2, 4, 6, 8, 10} ≠ {x : x = 2n + 1 and x ∈ N}
iii) {5, 15, 30, 45} ≠ {x : x is a multiple of 15
iv) {2, 3, 5, 7, 9} ≠ {x : x is a prime number}
Solution:
The first set is {1, 2, 3, ……, 10}
Writing the second set in roaster form, we have {2, 3, 4, ……, 9}
The first set and the second set have not exactly the same elements.
∴ {1, 2, 3,……10} ≠ {x : x ∈ N and 1 < x < 10}

ii) The first set is {2, 4, 6, 8, 10}
Writing the second set in roaster form, we have {3, 5, 7, 9, ….}
∴ {2, 4, 6, 8, 10} ≠ {3, 5, 7, 9, ….}
x = 2n + 1 means x is odd.

iii) The first set is {5, 15, 30, 45}
Writing the second set in roaster form, we have {15, 30, 45, 60, …}
∴ {5, 15, 30, 45} ≠ {15, 30, 45, 60,…}
5 does not exist, since x is multiple of 15.

iv) The first set is {2, 3, 5, 7, 9}
Writing the second set in roaster form, we have {2, 3, 5, 7, 11, 13,…}
∴ {2, 3, 5, 7, 9} ≠ {2, 3, 5, 7, 11, 13 }
9 is not a prime number.

Question 5.
List all the subsets of the following sets.
i) B = {p, q}
ii) C = {x, y, z}
iii) D = {a, b, c, d}
iv) E = {1, 4, 9, 16}
v) F = {10, 100, 1000}
Solution:
i) {p}, {q}, {p, q}, { ϕ }
ii) {x}, {y}, {z}, {x, y}, {y, z}, {x, z}, {x, y, z}, {ϕ}
iii) {a}, {b}, {c}, {d}, {a, b}, {b, c}, {c, d}, {a, c}, [a, d}, {b, d}, {a, b, c}, {b, c, d}, {a, b, d}, {a, c, d}, {a, b, c, d}, {ϕ}
iv) {1}, {4}, {9}, {16}, {1, 4}, {4, 9}, {9, 16}, {1, 9}, {1, 16}, {4, 16}, {1, 4, 9}, {4, 9, 16}, {1, 4, 16}, {1, 9, 16}, {1, 4, 9, 16}, {ϕ}
v) {10}, {100}, {1000}, {10, 100}, {100, 1000}, {10, 1000}, {10, 100, 1000}, {ϕ}

Students can practice TS 10th Class Maths Solutions Chapter 2 Sets Ex 2.2 to get the best methods of solving problems.

TS 10th Class Maths Solutions Chapter 2 Sets Exercise 2.2

Question 1.
If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, then find A ∩ B and B ∩ C. Are they same ?
Solution:
A = {1, 2, 3, 4}; B = {1, 2, 3, 5, 6}
∴ A ∩ B = {1, 2, 3, 4} ∩ {1, 2, 3, 5, 6}
= {1, 2, 3}
B ∩ A = {1, 2, 3, 5, 6} ∩ {1, 2, 3, 4)
= {1, 2, 3}
Yes, A ∩ B and B ∩ A are same.

Question 2.
A = {0, 2, 4}, find A ∩ ϕ and A ∩ A. Comments. (June 15(A.P.))
Solution:
A = {0, 2, 4}
A ∩ ϕ = {0, 2, 4} ∩ ϕ
= ϕ
A ∩ A = {0, 2, 4} ∩ {0, 2, 4}
= {0, 2, 4} = A

Question 3.
If A = {2, 4, 6, 8, 10} and B = {3, 6, 9, 12, 15}, find A – B and B – A.
Solution:
A = {2, 4, 6, 8, 10);
B = {3, 6, 9, 12, 15)
A – B = {2, 4, 8, 10}
B – A = {3, 9, 12, 15}

Question 4.
If A and B are two sets such that A ⊂ B then what is A ∪ B?
Solution:
Let us consider A ⊂ B.
Set A = {1, 2, 3}
Set B = {1, 2, 3, 4, 5}
A ∪ B = {1, 2, 3} ∪ {1, 2, 3, 4, 5)
= {1, 2, 3, 4, 5}
= B
A ∪ B = B

Question 5.
If A = {x : x is a natural number}
B = {x : x is an even natural number}
C = {x : x is an odd natural number}
D = {x : x is a prime number
Find A∩B, A∩C, A∩D, B∩C, B∩D, C∩D.
Solution:
A = {1, 2, 3, 4, …….. }
B = {2, 4, 6, 8, ……….}
C = {1, 3, 5,7, ……….}
D = {2, 3, 5, 7, ……….}
A ∩ B = {1, 2, 3, 4,………. } ∩ {2, 4, 6, 8,………. }
= {2, 4, 6,…… }
= {even natural numbers)
A ∩ C = {1, 2, 3, 4,……} ∩ {1, 3, 5,……}
= {1, 3, 5,………}
= {odd natural numbers}
A ∩ D = {1, 2, 3, 4,……} ∩ {2, 3, 5, 7, ……..}
= {2, 3, ………}
= {prime natural numbers}
B ∩ C = {2, 4, 6, 8,…..} ∩ {1, 3, 5, 7, …….}
= ϕ
= Null set (Or) empty set
B ∩ D = {2, 4, 6, 8,…….. } ∩ {2, 3, 5, 7,…….. }
= {2} = {even natural number}
C ∩ D = {1, 3, 5, 7,……. } ∩ {2, 3, 5, 7,…….}
= {3, 5, 7,…….. }
= {All odd prime numbers)

Question 6.
If A = {3, 6, 9, 12, 15, 18, 21);
B = {4, 8, 12, 16, 20};
C = {2, 4, 6, 8, 10, 12, 14, 16};
D = {5, 10, 15, 20}, find
i) A – B
ii) A – C
iii) A – D
iv) B – A
v) C – A
vi) D – A
vii) B – C
viii) B – D
ix) C – B
x) D – B
Solution:
i) A – B = {3, 6, 9, 12, 15, 18, 21} – {4, 8, 12, 16, 20}
= {3, 6, 9, 15, 18, 21}

ii) A – C = {3, 6, 9, 12, 15, 18, 21} – {2, 4, 6, 8, 10, 12, 14, 16)
= {3, 9, 15, 18, 21}

iii) A – D = {3, 6, 9, 12, 15, 18, 21} – {5, 10, 15, 20}
= {3, 6, 9, 12, 18, 21}

iv) B – A = {4, 8, 12, 16, 20} – {3, 6, 9, 12, 15, 18, 21}
= {4, 8, 16, 20}

v) C – A = {2, 4, 6, 8, 10, 12, 14, 16} – {3, 6, 9, 12, 15, 18, 21}
= {2, 4, 8, 10, 14, 16}

vi) D – A = {5, 10, 15, 20} — {3, 6, 9, 12, 15, 18, 21)
= (5, 10, 20)

vii) B – C = {4, 8, 12, 16, 20) – {2, 4, 6, 8, 10, 12, 14, 16}
= {20}

viii) B — D = {4, 8, 12, 16, 20} – {5, 10, 15, 20}
= {4, 8, 12, 16)

ix) C – B = {2, 4, 6, 8, 10, 12, 14, 16} – {4, 8, 12, 16, 20}
= {2, 6, 10, 14)

x) D – B = {5, 10, 15, 20} – {4, 8, 12, 16, 20}
= {5, 10, 15}

Question 7.
State whether each of the following statement is true or false. Justify your answers.
i) {2, 3, 4, 5} and {3, 6) are disjoint sets.
ii) {a, e, i, o, u) and {a, b, c, d) are disjoint sets.
iii){2, 6, 10, 14) and {3, 7, 11, 15) are disjoint sets.
iv) (2, 6, 10) and {3, 7, 11) are disjoint sets.
Answer:
Two sets are said to be disjoint sets when they have no elements in common.
i) In the given two sets. ‘3’ is common. So, they are not disjoint sets. Hence, the given statement is false.
ii) In the given two sets, ‘a’ is common. So, they are not disjoint sets. Hence, the given statement is false.
iii) There are no elements common in the given two sets. So they are disjoint sets. Hence, the given statement is true.
iv) There are no elements common in the given two sets, So they are disjoint sets. Hence. the given statement is true.

Students can practice Telangana 6th Class Maths Textbook Solutions Chapter 7 Fractions and Decimals Ex 7.3 to get the best methods of solving problems.

TS 6th Class Maths Solutions Chapter 7 Fractions and Decimals Exercise 7.3

Question 1.
Write shaded portion as fraction. Arrange them in ascending or descending order using sign ‘<‘, ‘=’, ‘>’ between the fractions.
(i)

Answer:
\(\frac{3}{8}, \frac{6}{8}, \frac{4}{8}, \frac{1}{8}\); Arranging them in ascending order, we get \(\frac{1}{8}<\frac{3}{8}<\frac{4}{8}<\frac{6}{8}\)
Descending order: \(\frac{6}{8}>\frac{4}{8}>\frac{3}{8}>\frac{1}{8}\)

(ii)

Answer:
\(\frac{8}{9}, \frac{4}{9}, \frac{3}{9}, \frac{6}{9}\); Arranging them in ascending order, we get \(\frac{3}{9}<\frac{4}{9}<\frac{6}{9}<\frac{8}{9}\)
Descending order: \(\frac{8}{9}>\frac{6}{9}>\frac{4}{9}>\frac{3}{9}\)

Question 2.
Show \(\frac{2}{6}, \frac{4}{6}, \frac{8}{6}, \frac{5}{6}\) and \(\frac{6}{6}\) on the number line. Also arrange them in ascending order.
Answer:

Question 3.
Look at the figures and write ‘<‘ or’>’, ‘=‘ between the given pairs of fractions:

(i) \(\frac{1}{6}\) ________ \(\frac{1}{3}\)
(ii) \(\frac{3}{4}\) ________ \(\frac{2}{6}\)
(iii) \(\frac{2}{3}\) ________ \(\frac{2}{4}\)
(iv) \(\frac{6}{6}\) ________ \(\frac{3}{3}\)
(v) \(\frac{5}{6}\) ________ \(\frac{5}{5}\)
Make five more such problems and ask your friends to solve them.
Answer:
(i) \(\frac{1}{6}\) < \(\frac{1}{3}\)
(ii) \(\frac{3}{4}\) > \(\frac{2}{6}\)
(iii) \(\frac{2}{3}\)> \(\frac{2}{4}\)
(iv) \(\frac{6}{6}\) = \(\frac{3}{3}\)
(v) \(\frac{5}{6}\) < \(\frac{5}{5}\)

Question 4.
Fill with the appropriate sign. (‘<‘, ‘=’, ‘>’)
(i) \(\frac{1}{2}\) _______ \(\frac{1}{5}\)
Answer:
\(\frac{1}{2}\) > \(\frac{1}{5}\)

(ii) \(\frac{2}{4}\) _______ \(\frac{3}{6}\)
Answer:
\(\frac{2}{4}\) = \(\frac{3}{6}\)

(iii) \(\frac{3}{5}\) _______ \(\frac{2}{3}\)
Answer:
\(\frac{3}{5}\) < \(\frac{2}{3}\)

(iv) \(\frac{3}{4}\) _______ \(\frac{2}{8}\)
Answer:
\(\frac{3}{4}\) > \(\frac{2}{8}\)

(v) \(\frac{3}{5}\) _______ \(\frac{6}{5}\)
Answer:
\(\frac{3}{5}\) < \(\frac{6}{5}\)

(vi) \(\frac{7}{9}\) _______ \(\frac{3}{9}\)
Answer:
\(\frac{7}{9}\) > \(\frac{3}{9}\)

Question 5.
Answer the following. Also write how you solved them.
(i) Is \(\frac{5}{9}\) equal to \(\frac{4}{5}\)?
Answer:
We write the fractions \(\frac{5}{9}\) and \(\frac{4}{5}\) having the same denominators

(ii) Is \(\frac{9}{16}\) equal to \(\frac{5}{9}\)?
Answer:
We write the fractions \(\frac{9}{16}\) and \(\frac{5}{9}\) having the same denominators

(iii) Is \(\frac{4}{5}\) equal to \(\frac{16}{20}\)?
Answer:
We write the fractions \(\frac{4}{5}\) and \(\frac{16}{20}\) having the same denominators

(iv) Is \(\frac{1}{15}\) equal to \(\frac{4}{30}\)?
Answer:
We write the fractions \(\frac{1}{15}\) and \(\frac{4}{30}\) having the same denominators

Question 6.
Varshith read 25 pages of a story book containing 100 pages. Lalitha read \(\frac{2}{5}\) of the same story book. Who read less? Give reason.
Answer:
Total number of pages in the story book = 100
Number of pages that Varshith read = 25
Number of pages that Lalitha read
= \(\frac{2}{5}\) of 100
= \(\frac{2}{5}\) × 100 = 40
So, Varshith read less pages.

Question 7.
Fill the appropriate (+ or -) sign in the blank space.
(i)

Answer:
+

(ii)

Answer:
–

(iii)

Answer:
+

Question 8.
simplify:
(i) \(\frac{1}{18}+\frac{1}{18}\)
Answer:
\(\frac{1}{18}+\frac{1}{18}=\frac{1+1}{18}=\frac{2}{18}=\frac{1}{9}\)

(ii) \(\frac{8}{15}+\frac{3}{15}\)
Answer:
\(\frac{8}{15}+\frac{3}{15}=\frac{8+3}{15}=\frac{11}{15}\)

(iii) \(\frac{7}{7}-\frac{5}{7}\)
Answer:
\(\frac{7}{7}-\frac{5}{7}=\frac{7-5}{7}=\frac{2}{7}\)

(iv) \(\frac{1}{22}+\frac{21}{22}\)
Answer:
\(\frac{1}{22}+\frac{21}{22}=\frac{1+21}{22}=\frac{22}{22}\) = 1

(v) \(\frac{12}{15}-\frac{7}{15}\)
Answer:
\(\frac{12}{15}-\frac{7}{15}=\frac{12-7}{15}=\frac{5}{15}=\frac{1}{3}\)

(vi) \(\frac{5}{8}+\frac{3}{8}\)
Answer:
\(\frac{5}{8}+\frac{3}{8}=\frac{5+3}{8}=\frac{8}{8}\) = 1

(vii) 1 – \(\frac{2}{3}\)
Answer:
1 – \(\frac{2}{3}=\frac{3}{3}-\frac{2}{3}=\frac{3-2}{3}=\frac{1}{3}\) [∵ 1 = \(\frac{3}{3}\)]

(viii) \(\frac{1}{4}+\frac{0}{4}\)
Answer:
\(\frac{1}{4}+\frac{0}{4}=\frac{1+0}{4}=\frac{1}{4}\)

(ix) 3 – \(\frac{2}{15}\)
Answer:
3 – \(\frac{12}{5}=\frac{3 \times 5}{1 \times 5}-\frac{12}{5}=\frac{15}{5}-\frac{12}{5}=\frac{15-12}{5}=\frac{3}{5}\)

Question 9.
Fill in the missing fractions.
(i) \(\frac{7}{10}\) – ___________ = \(\frac{3}{10}\)
Answer:

(ii) ____________ – \(\frac{3}{21}\) = \(\frac{5}{21}\)
Answer:

(iii) ___________ – \(\frac{3}{3}\) = \(\frac{3}{6}\)
Answer:

(iv) ___________ + \(\frac{5}{27}\) = \(\frac{12}{27}\)
Answer:

Question 10.
Narendra painted \(\frac{2}{3}\) area of the wall in his room. His brother Ritesh helped and painted \(\frac{1}{3}\) area of the wall. How much did they paint together?
Answer:
Area of the wall painted by Narendra = \(\frac{2}{3}\)
Area of the wall painted by Ritesh = \(\frac{1}{3}\)
Area of the wall painted by both Narendra and Ritesh = \(\frac{2}{3}+\frac{1}{3}\)
= \(\frac{2+1}{3}\) = \(\frac{3}{3}\) = 1
∴ Narendra and his brother Ritesh painted the complete wall.

Question 11.
Neha was given \(\) of a basket of bananas. What fraction of bananas was left in the basket?
Answer:
The part of a basket of bananas given to Neha = \(\frac{5}{7}\)
The, part of bananas left in the basket
= 1 – \(\frac{5}{7}=\frac{1 \times 7}{1 \times 7}-\frac{5}{7}=\frac{7}{7}-\frac{5}{7}=\frac{7-5}{7}=\frac{2}{7}\)

Question 12.
A piece of rod \(\frac{7}{8}\) metre long is broken into two pieces. One piece was \(\frac{1}{4}\) metre long. How long is the other piece?
Answer:
Length of a piece of rod = \(\frac{7}{8}\) metre
Length of one broken piece of rod = \(\frac{1}{4}\) metre.
Length of the other piece = \(\frac{7}{8}-\frac{1}{4}\)
= \(\frac{7}{8}-\frac{1 \times 2}{4 \times 2}=\frac{7}{8}-\frac{2}{8}=\frac{7-2}{8}\) = \(\frac{5}{8}\)m
∴ \(\frac{5}{8}\)m long is the other piece.

Question 13.
Renu takes 2\(\frac{1}{5}\) minutes to walk around the school ground. Snigdha takes \(\frac{7}{4}\) minutes to do the same. Who takes less time and by what fraction?
Answer:
Time taken by Renu to walk around the school ground = 2\(\frac{1}{5}\) minutes
= \(\frac{11}{5}\)
Time taken by Snigdha to walk around the school ground = \(\frac{7}{4}\)minutes
To find the person who takes less time to do the same, we write the fractions \(\frac{11}{5}\) and \(\frac{7}{4}\) having the same denominators.
\(\frac{11}{5} \times \frac{4}{4}=\frac{44}{20}\); \(\frac{7}{4} \times \frac{5}{5}=\frac{35}{20}\)
We know that \(\frac{35}{20}\) < \(\frac{44}{20}\)
Therefore, Snigdha takes \(\frac{9}{20}\) minutes less time to walk around the school ground.
(∵ \(\frac{44}{20}-\frac{35}{20}=\frac{44-35}{20}=\frac{9}{20}\))

Students can practice Telangana 6th Class Maths Textbook Solutions Chapter 7 Fractions and Decimals Ex 7.2 to get the best methods of solving problems.

TS 6th Class Maths Solutions Chapter 7 Fractions and Decimals Exercise 7.2

Question 1.
Which group of fractions are like fractions among the following?
(i) \(\frac{2}{7}, \frac{3}{7}, \frac{4}{7}\)
Answer:
\(\frac{2}{7}, \frac{3}{7}, \frac{4}{7}\) are like fractions

(ii) \(\frac{1}{9}, \frac{2}{9}, \frac{4}{9}\)
Answer:
\(\frac{1}{9}, \frac{2}{9}, \frac{4}{9}\) are also like fractions

(iii) \(\frac{3}{7}, \frac{4}{9}, \frac{7}{11}\)
Answer:
\(\frac{3}{7}, \frac{4}{9}, \frac{7}{11}\) are unlike fractions

Question 2.
Write five groups of like fractions.
Answer:

Question 3.
From each of these, Identify like fractional numbers.
(i) \(\frac{2}{3}, \frac{5}{3}, \frac{1}{3}, \frac{4}{6}\)
Answer:
\(\frac{2}{3}, \frac{5}{3}, \frac{1}{3}, \frac{4}{6}\left(=\frac{2}{3}\right)\) are like fractional numbers

(ii) \(\frac{1}{7}, \frac{3}{5}, \frac{2}{5}, \frac{1}{9}\)
Answer:
\(\frac{3}{5}\) and \(\frac{2}{5}\) are like fractional numbers

(iii) \(\frac{7}{8}, \frac{8}{7}, \frac{2}{8}, \frac{7}{5}\)
Answer:
\(\frac{7}{8}\) and \(\frac{2}{8}\) are like fractional numbers

Students can practice Telangana 6th Class Maths Textbook Solutions Chapter 7 Fractions and Decimals Ex 7.1 to get the best methods of solving problems.

TS 6th Class Maths Solutions Chapter 7 Fractions and Decimals Exercise 7.1

Question 1.
Out of these which are proper fractional numbers?
(i) \(\frac{3}{2}\)
(ii) \(\frac{2}{5}\)
(iii) \(\frac{1}{7}\)
(iv) \(\frac{8}{3}\)
Answer:
\(\frac{2}{5}\) and \(\frac{1}{7}\) are proper fractional numbers.

Question 2.
Which of these are improper fractional numbers?
(i) \(\frac{2}{7}\)
(ii) \(\frac{7}{11}\)
(iii) \(\frac{9}{7}\)
(iv) \(\frac{13}{2}\)
(v) \(\frac{7}{3}\)
Write where each of the above improper fractional numbers would lie on the number line.
Answer:
\(\frac{7}{3}\) and \(\frac{13}{2}\) are improper fractional numbers.
\(\frac{7}{3}\) lies in between 2 and 3.

\(\frac{13}{2}\) lies in between 6 and 7.

Question 3.
Pick out the mixed fractions from these.
(i) \(\frac{3}{5}\)
(ii) 1\(\frac{2}{7}\)
(iii) \(\frac{7}{2}\)
(iv) 2\(\frac{3}{5}\)
Answer:
1\(\frac{2}{7}\) and 2\(\frac{3}{5}\) are mixed fractions.

Question 4.
Convert the following Improper fractions Into mixed fractions.
(i) \(\frac{7}{3}\)
(ii) \(\frac{11}{2}\)
(iii) \(\frac{9}{4}\)
(iv) \(\frac{27}{4}\)
Answer:

Question 5.
Convert the following mixed fractions into Improper fractions.
(i) 1\(\frac{2}{7}\)
(ii) 3\(\frac{2}{8}\)
(iii) 10\(\frac{2}{9}\)
(iv) 8\(\frac{7}{9}\)
Answer:

Students can practice TS 10th Class Maths Solutions Chapter 2 Sets Ex 2.1 to get the best methods of solving problems.

TS 10th Class Maths Solutions Chapter 2 Sets Exercise 2.1

Question 1.
Which of the following are sets? Justify your answer.
(i) The collection of all the months of a year begining with the letter “J”.
Answer:
The months of a year which begin with the letter “J” are January, June and July.
The required set is {January, June, July}.
∴ The given statement is a well defined collection of objects. So, it is a set.

(ii) The collection of ten most talented writers of India.
Answer:
The given statement is not a set because the criterion for determining as most talented writers of India may vary from person to person. Thus, it is not a well defined collection.

(iii) A team of eleven best cricket batsmen of the world.
Answer:
The given statement is not a set because the criterion for determining eleven best cricket batsmen of the world may vary from person to person.

(iv) The collection of all boys in your class.
Answer:
The given statement is a set because given a student we can divide whether he/she belongs to the set or not.

(v) The collection of all even integers.
Answer:
The given statement is a set because given a number we can divide whether the number belongs to the given set or not.

Question 2.
If A = {0, 2, 4, 6}, B = {3, 5, 7} and C = {p, q, r}, then fill the appropriate symbol, ∈ or ∉ in the blanks.
(i) 0 ….. A
Answer:
0 ∈ A

(ii) 3 …… C
Answer:
3 ∉ C

(iii) 4 ….. B
Answer:
4 ∉ B

(iv) 8 ….. A
Answer:
8 ∉ A

(v) p …… C
Answer:
p ∈ C

(vi) 7 …… B
Answer:
7 ∈ B

Question 3.
Express the following statements using symbols.
(i) The element ‘x’ does not belong to ‘A’.
Answer:
x ∉ A

(ii) ‘d’ is an element of the set ‘B’.
Answer:
d ∈ B

(iii) ‘1’ belongs to the set of Natural numbers.
Answer:
1 ∈ N

(iv) ‘8’ does not belong to the set of prime numbers P.
Answer:
8 ∉ P

Question 4.
State whether the following statements are true or false. Justify your answer
(i) 5 ∉ set of prime numbers
Answer:
Not true (5 is a prime number)

(ii) S ∈ {5, 6, 7} implies 8 ∈ S.
Answer:
Not true (8 is not a member of S)

(iii) -5 ∉ W where ‘W’ is the set of whole numbers
Answer:
True (-5 is not a member of W)

(iv) \(\frac{8}{11}\) ∈ Z Where ‘Z’ is the set of integers.
Answer:
Not true (\(\frac{8}{11}\) is a rational number)

Question 5.
Write the following sets in roster form.

(i) B = {x : x is a natural number smaller than 6}
Answer:
B = {1, 2, 3, 4, 5}

(ii) C = {x : x is a two-digit natural number such that the sum of its digits is 8}.
Answer:
C = {17, 26, 35, 44, 53, 62, 71}

(iii) D = {x : x is a prime number which is a divisor of 60}.
Answer:
D = {3, 5}

(iv) E = {x : x is an alphabet in BETTER}.
Answer:
E = {B, E, T, R}

Question 6.
Write the following sets in the set-builder form.
(i) {3, 6, 9, 12}
Answer:
A = {x: x is multiple of 3 and less than 13}

(ii) {2, 4, 8, 16, 32}
Answer:
B = {x : x is in power of 2 and x is less than 6}

(iii) {5, 25, 125, 625}
Answer:
C = {x : x is in power of 5 and x is less than 5}

(iv) {1, 4, 9, 16, 25, 100}
Answer:
D = {x : x is in square of natural number and not greater than 10)

Question 7.
Write the following sets in roster form.
(i) A = {x : x is a natural number greater than 50 but smaller than 100}
Answer:
A = {51, 52, 53, 54….., 98, 99}

(ii) B = {x : x is an integer, x = 4}
Answer:
B = {+2, -2}

(iii) D = {x : x is a letter in the word “LOYAL”}
Answer:
D = {L, O, Y, A}

Question 8.
Match the roster form with set builder form.

Answer:
i) c
ii) a
iii) d
iv) b

Students can practice Telangana 6th Class Maths Textbook Solutions Chapter 5 Measures of Lines and Angles Ex 5.2 to get the best methods of solving problems.

TS 6th Class Maths Solutions Chapter 5 Measures of Lines and Angles Exercise 5.2

Question 1.
Write ‘True’ or ‘False’. Correct all those that are false.
(i) An angle smaller than right angle is acute angle. ___________
Answer:
True

(ii) A right angle measures 180°. ___________
Answer:
False
A right angle measures 90°. (correct)

(iii) A straight angle measures 90°. ___________
Answer:
False
A straight angle measures 180°. (correct)

(iv) The measure greater than 180° is a reflex angle. ___________
Answer:
True

(v) A complete angle measures 360°. ___________
Answer:
True

Question 2.
Which angles in the adjacent figure are acute and which are obtuse ? Check your estimation by measuring them. Write their measures too.

Answer:
In the adjacent figure The measure of ∠1 is 80°
The measure of ∠2 is 100°
The measure of ∠3 is 80°
The measure of ∠4 is 100°
∠l and ∠3 are acute angles because they are less than 90°.
∠2 and ∠4 are obtuse angles because they are greater than 90° and less than 180°.

Question 3.
What is the measure of these angles ? Which is the largest angle ? Draw an angle larger than the largest angle.

Answer:
∠ABC = 70°
∠FED = 120°
∠RQP = 90°
∠FED is the largest angle.

An angle larger than the largest ∠DEF is ∠STU = 150°

Question 4.
Write the type of angle formed between the long hand and short hand of a clock at the given timings. (Take the small hand as the base)
(i) At 9’0 clock in the morning
(ii) At 6’0 clock in the evening
(iii) At 12 noon
(iv) At 4’0 clock in the afternoon
(v) At 8’0 clock in the night.
Answer:
The angle formed between the long hand and short hand of a clock.
(i) At 9’O clock in the morning is right angle.
(ii) At 6’O clock in the evening is straight angle.
(iii) At 12 noon no angle is formed because the two hands coincide.
(iv) At 4’O clock in the afternoon is obtuse angle.
(v) At 8’O clock in the night is reflex angle.

Question 5.
Match the angles by measure. Draw figures for these as well.

Answer:

Students can practice TS 6th Class Maths Solutions Chapter 3 Playing with Numbers Ex 3.6 to get the best methods of solving problems.

TS 6th Class Maths Solutions Chapter 3 Playing with Numbers Exercise 3.6

Question 1.
Find the LCM and HCF of the following numbers.
(i) 15, 24
(ii) 8, 25
(iii) 12,48
check their relationship.
Answer:
(i) Factors of 15= 3 × 5
Factors of 24 = 3 × 2 × 2 × 2
LCM of 15 and 24 = 3 × 5 × 2 × 2 × 2= 120
HCF of 15 and 24 = 3
LCM × HCF = 120 × 3 = 360
Product of the two numbers = 15 × 24 = 360
∴ LCM × HCF = Product of the two numbers

(ii) Factors of 8 = 2 × 2 × 2
Factors of 25 = 5 × 5
LCM of 8 and 25 = 2 × 2 × 2 × 5 × 5 = 200
HCF of 8 and 25 = 1
∴ LCM × HCF = 200 × 1 = 200
Product of the two numbers = 8 × 25 = 200
∴ LCM × HCF = Product of the two numbers

(iii)

LCM of 12 and 48 = 2 × 2 × 3 × 2 × 2 = 48
HCF of 12 and 48 = 2 × 2 × 3 = 12
LCM × HCF = 12 × 48576
Product of the two numbers = 12 × 48 = 576
∴ LCM × HCF = Product of the two numbers

Question 2.
If the LCM of two numbers is 216 and their product is 7776, what will be its HCF?
Answer:
Product of the two numbers = 7776
LCM of two numbers = 216
We know, LCM × HCF
= Product of the two numbers
∴216 × HCF = 7776
∴HCF = \(\frac{7776}{216}\) = 36

Question 3.
The product of two numbers is 3276.
If their HCF is 6, find their LCM.
Answer:
Product of the two numbers = 3276
HCF of the two numbers = 6
We know, LCM × HCF = Product of the two numbers
LCM × 6 = 3276
∴ LCM = \(\frac{3276}{6}\) = 3276

Question 4.
The HCF of two numbers is 6 and their LCM is 36. If one of the numbers is 12, find the other.
Answer:
The HCF of two numbers = 6
The LCM of two numbers = 36
One of the numbers = 12
Let the other number be x.
HCF × LCM = 6 × 36
Product of the two numbers = 12 × x
We know, LCM × HCF = Product of the two numbers
6 × 36 = 12 × x
(i.e.) 12 × x = 6 × 36
∴ x = \(\frac{6 \times 36}{12}\) = 18
∴ The other number is 18.

Students can practice TS 10th Class Maths Solutions Chapter 1 Real Numbers Ex 1.5 to get the best methods of solving problems.

TS 10th Class Maths Solutions Chapter 1 Real Numbers Exercise 1.5

Question 1.
Determine the value of the following.

(i) log₂₅5
Solution:
Let log₂₅5 = x ∵ log_aN = x ⇒ a^x = N
25^x = 5; 5^2x = 5¹ ⇒ a^x = N
⇒ 2x = 1 ⇒ x = \(\frac{1}{2}\)
log₂₅5 = \(\frac{1}{2}\)

(ii) log₈₁3
Solution:
log₈₁3 = x ∵ log<sub.aN = x
81^x = 3 ⇒ (3⁴)^x = 3¹ ⇒ a^x = N
⇒ 4x = 1 ⇒ x = \(\frac{1}{4}\)
log₈₁3 = \(\frac{1}{4}\)

(iii) log₂ \(\left(\frac{1}{16}\right)\)
Solution:
log₂\(\frac{1}{16}\) = x ∵ log_aN = x ⇒ a^x = N
⇒ Then 2^x = \(\frac{1}{16}\) ⇒ 2^x = \(\frac{1}{2^4}\) = 2^-4
⇒ x = – 4
los₂(\(\frac{1}{16}\)) = -4

(iv) log₇1
Solution:
log₇1 = x ∵ log_aN = x ⇒ a^x = N
⇒ Then 7^x = 1 ⇒ 7^x – 7⁰ ⇒ x = 0
⇒ log₁a = 0

(v) log_x \(\sqrt{x}\)
Solution:
log_x \(\sqrt{x}\) ∵ log_aN = x ⇒ a^x = N
⇒ Then x^y = \(\sqrt{x}\) ⇒ x^y = x^1/2
⇒ y = \(\frac{1}{2}\)
log_x \(\sqrt{x}\) = \(\frac{1}{2}\)

(vi) log₂512
Solution:
log₂ 512 ∵ log_aN = x ⇒ a^x = N
⇒ Then 2^x = 512 ⇒ 2^x = 2⁹
⇒ x = 9
log₂512 = 9

(vii) log₁₀0.01
Solution:
log₁₀0.01 = x ∵ log_aN = x ⇒ a^x = N
Then 10^x = 0.01
⇒ 10^x = 10^-2
⇒ x = -2
log₁₀0.01 = -2

(viii) \(\log _{\frac{2}{3}}\left(\frac{8}{27}\right)\)
Solution:
\(\log _{\frac{2}{3}}\left(\frac{8}{27}\right)\) ∵ log_aN = x ⇒ a^x = N
\(\left(\frac{2}{3}\right)^x\) = \(\frac{8}{27}\)
\(\left(\frac{2}{3}\right)^2\) = \(\left(\frac{2}{3}\right)^3\) ⇒ x = 3
\(\log _{\frac{2}{3}}\left(\frac{8}{27}\right)\) = 3

(ix) \(2^{2+\log _2^3}\)
Solution:
\(2^{2+\log _2^3}\) ∵ \(a^{\log _a^m}\) = m
= 2² × \(2^{\log _2^3}\)
= 4 × 3 = 12

Question 2.
Write the following expressions as log N and find their values.

(i) log 2 + log 5
Solution:
log 2 + log 5
∵ log x + log y = log xy
log 2 + log 5 = log (2 × 5)
= log (2 × 5)
= log 10

(ii) log 16 – log 2
Solution:
log 16 – log 2
∵ log x – log y = log \(\left(\frac{x}{y}\right)\)
log 16 – log 2 = log \(\left(\frac{16}{2}\right)\) = log 8

(iii) 3 log 4
Solution:
3 log 4
∵ m log_ax = log_ax^m
3 log 4 = log 4³ = log 64

(iv) 2 log 3 – 3 log 2
Solution:
2 log 3 – 3 log 2 ∵ m log a = log a^m
log x – log y = log\(\frac{x}{y}\)

log 3² – log 2³
log \(\frac{3^2}{2^3}\) = log \(\frac{9}{8}\)

(v) log 243 + log 1
Solution:
log 243 + log 1 ∵ log x + log y = log xy
= log 243 × 1
= log 243

(vi) log 10 + 2 log 3 – log 2
Solution:
log 10 + 2 log 3 – log 2
= log 10 + log 3² – log 2
= log 10 + log 9 – log 2
= log 90 – log 2
= log \(\frac{90}{2}\) = log 45

Question 3.
Evaluate each of the following in terms of x and y, if it is given that x = log₂3 and y = log₂5
(i) log₂ 15
(ii) log₂ 7.5
(iii) log₂60
(iv) log₂6750
Solution:

Question 4.
Expand the following.

(i) log 1000
Solution:

= 3 log 2 + 3 log 5
= 3 [log 2 + log 5]

(ii) log₂\(\left(\frac{128}{625}\right)\)
Solution:
log \(\left(\frac{128}{625}\right)\)

(iii) log x²y³z⁴
Solution:
log x²y³z⁴
= log x² + log y³ + log z⁴
= 2 log x + 3 log y + 4 log z

(iv) log \(\frac{p^2 q^3}{r}\)
Solution:
log \(\frac{p^2 q^3}{r}\)
log (p²q³) – log r
= log p² + log q³ – log r
= 2 log p + 3 log q – log r

(v) \(\log \sqrt{\frac{x^3}{y^2}}\)
Solution:
log \(\sqrt{\frac{x^3}{y^2}}\) ∵ log x^m = m log x
= log \(\left(\frac{x^3}{y^2}\right)^{1 / 2}\) = \(\frac{1}{2} \log \left(\frac{x^3}{y^2}\right)\)
= \(\frac{1}{2}\)[log x³ – log y²]
= \(\frac{1}{2}\)[3 log x – 2 log y]

Question 5.
If x² + y² = 25xy, then prove that 2 log(x + y) = 3log3 + logx + logy.
Solution:
Given x² + y² = 25xy
Adding ‘2xy’ on both sides.
x² + y² + 2xy = 25xy + 2xy
(x + y)² = 27xy
Applying ‘log’ on both sides
log(x + y)² = log 27xy
2log(x + y) = log(3³ × x × y)
= log3³ + log x + log y
∴ 2log(x + y) = 31og3 + logx + logy

Question 6.
If log\(\left(\frac{x+y}{3}\right)\) = \(\frac{1}{2}\)log(x + y) = 3log3 + logx + logy.
Solution:

Squaring on both sides

Question 7.
If (2.3)^x = (0.23)^y = 1000, then find the value of \(\frac{1}{x}\) – \(\frac{1}{y}\).
Solution:
Given : (2.3)^x = (0.23)^y = 1000
(2.3)^x = 1000 = 10³
∴ 2.3 = \(10^{\frac{3}{x}}\)
Also (0.23)^y = 10³
∴ 0.23 = \(10^{\frac{3}{y}}\)
Now 0.23 = \(\frac{2.3}{10}\)

Question 8.
If 2^x+1 = 3^1-x then find the value of x.
Solution:
Given : 2^{x + 1} = 3^{1 – x}
Taking log on b.t.s
log 2^{x + 1} = log 3^{1 – x}
(x + 1) log 2 = (1 – x) log 3
x log 2 + log 2 = log 3 – x log 3
x log 2 + x log 3 = log 3 – log 2
x (log 3 + log 2) = log 3 – log 2
∴ x = \(\frac{\log 3-\log 2}{\log 3+\log 2}\) = \(\frac{\log \frac{3}{2}}{\log 6}\)

Question 9.
Is (i) log 2 rational or irrational? Justify your answer.
(ii) log 100 rational or irrational? Justify your answer.
Solution:
i) log2 is rational. Since the value of log₁₀2
= 0.3010

(ii) log 100 rational or irrational? Justify your answer.
Solution:
log 100 is rational
∴ log₁₀100 = log₁₀10²
= 2 log₁₀10 = 2