TS SSC 10th Class Mathematics
English Medium | Important Questions & Answers
Chapters 1 to 4 | Exam Focused
CHAPTER 1: REAL NUMBERS
Important Concepts
Euclid Division Lemma:
a = bq + r, where 0 ≤ r < b
HCF × LCM = Product of Two Positive Integers
Fundamental Theorem of Arithmetic:
Every composite number can be expressed as a product of prime numbers uniquely, apart from the order of factors.
A rational number p/q has a terminating decimal expansion if the denominator q is of the form:
2m × 5n
🔴 VERY IMPORTANT QUESTION
Q1. Use Euclid's Division Algorithm to find the HCF of 135 and 225.
Answer:
225 = 135 × 1 + 90
135 = 90 × 1 + 45
90 = 45 × 2 + 0
Therefore HCF = 45
🔴 VERY IMPORTANT QUESTION
Q2. Find the HCF and LCM of 72 and 120.
72 = 2³ × 3²
120 = 2³ × 3 × 5
HCF = 2³ × 3
HCF = 24
LCM = 2³ × 3² × 5
LCM = 360
🔴 VERY IMPORTANT QUESTION
Q3. Prove that √2 is irrational.
Assume √2 is rational.
Let √2 = p/q, where p and q are co-prime integers.
Squaring both sides:
2 = p²/q²
p² = 2q²
Therefore p is even.
Let p = 2k.
Then q is also even.
This means p and q have a common factor.
This contradicts our assumption.
Therefore √2 is irrational.
Q4. Does 13/250 have a terminating decimal expansion?
250 = 2 × 5³
The denominator contains only factors 2 and 5.
Therefore 13/250 has a terminating decimal expansion.
Q5. Evaluate log₃81.
3⁴ = 81
Therefore log₃81 = 4
CHAPTER 2: SETS
Important Concepts
Union: A ∪ B
Elements in A or B or both.
Intersection: A ∩ B
Common elements in A and B.
Difference: A − B
Elements in A but not in B.
Complement: A'
Elements in Universal Set but not in A.
Formula:
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
Q1. Let A = {1,2,3,4} and B = {3,4,5,6}. Find A ∪ B and A ∩ B.
A ∪ B = {1,2,3,4,5,6}
A ∩ B = {3,4}
Q2. Find A − B and B − A.
A = {1,2,3,4}
B = {3,4,5,6}
A − B = {1,2}
B − A = {5,6}
Q3. Find the complement of A.
U = {1,2,3,4,5,6,7,8,9,10}
A = {2,4,6,8,10}
A' = {1,3,5,7,9}
🔴 VERY IMPORTANT QUESTION
Q4. If n(A)=25, n(B)=18 and n(A∩B)=7, find n(A∪B).
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
= 25 + 18 − 7
= 36
Q5. How many subsets does a set with 4 elements have?
Number of subsets = 2ⁿ
= 2⁴
= 16
🔴 VERY IMPORTANT QUESTION
Q6. In a class of 50 students, 28 like Mathematics, 22 like Science and 10 like both. Find students who like at least one subject.
n(M ∪ S) = 28 + 22 − 10
= 40 Students
CHAPTER 3: POLYNOMIALS
Important Formulas
For ax² + bx + c:
α + β = −b/a
αβ = c/a
Remainder Theorem:
Remainder when p(x) is divided by (x−a) is p(a).
Factor Theorem:
(x−a) is a factor of p(x) if p(a)=0.
🔴 VERY IMPORTANT QUESTION
Q1. Find the zeroes of x² − 5x + 6.
x² − 5x + 6
= (x−2)(x−3)
Zeroes are 2 and 3.
Sum = 2 + 3 = 5
Product = 2 × 3 = 6
Q2. Find a quadratic polynomial whose zeroes are 4 and −2.
Sum = 4 + (−2) = 2
Product = 4 × (−2) = −8
Polynomial:
x² − (Sum)x + Product
x² − 2x − 8
🔴 VERY IMPORTANT QUESTION
Q3. Find the remainder when p(x)=2x³−3x+5 is divided by (x−2).
By Remainder Theorem:
Remainder = p(2)
= 2(8) − 3(2) + 5
= 16 − 6 + 5
= 15
Q4. Check whether (x−1) is a factor of x³−4x²+5x−2.
p(1)
= 1 − 4 + 5 − 2
= 0
Therefore (x−1) is a factor.
Q5. If α and β are zeroes of 2x²−7x+3, find α+β and αβ.
a = 2
b = −7
c = 3
α + β = −b/a = 7/2
αβ = c/a = 3/2
Answer:
α + β = 7/2
αβ = 3/2
CHAPTER 4: PAIR OF LINEAR EQUATIONS IN TWO VARIABLES
Important Concepts
Unique Solution:
a₁/a₂ ≠ b₁/b₂
No Solution:
a₁/a₂ = b₁/b₂ ≠ c₁/c₂
Infinite Solutions:
a₁/a₂ = b₁/b₂ = c₁/c₂
🔴 VERY IMPORTANT QUESTION
Q1. Solve: 2x+y=11 and x−y=1.
2x + y = 11
x − y = 1
Add both equations:
3x = 12
x = 4
Substitute:
4 − y = 1
y = 3
Answer: x=4, y=3
🔴 VERY IMPORTANT QUESTION
Q2. Solve: x+y=7 and x−y=3.
Add both equations:
2x = 10
x = 5
Substitute:
5 + y = 7
y = 2
Answer: x=5, y=2
Q3. Determine the number of solutions:
2x+4y=8
x+2y=4
Divide the first equation by 2:
x + 2y = 4
Both equations are the same.
Therefore: Infinitely Many Solutions
🔴 VERY IMPORTANT WORD PROBLEM
Q4. The sum of two numbers is 35 and their difference is 9. Find the numbers.
Let numbers be x and y.
x + y = 35
x − y = 9
Add:
2x = 44
x = 22
y = 35 − 22
y = 13
Answer: 22 and 13
🔴 VERY IMPORTANT WORD PROBLEM
Q5. 2 notebooks and 3 pens cost ₹54. 3 notebooks and 2 pens cost ₹61. Find the price of each.
Let notebook = x
Let pen = y
2x + 3y = 54
3x + 2y = 61
Multiply first equation by 3:
6x + 9y = 162
Multiply second equation by 2:
6x + 4y = 122
Subtract:
5y = 40
y = 8
Substitute:
3x + 16 = 61
3x = 45
x = 15
One Notebook = ₹15
One Pen = ₹8
FINAL REVISION TEST
Chapter 1
- Find HCF of 252 and 105 using Euclid's Algorithm.
- Find HCF and LCM of 96 and 144.
- Is 17/160 a terminating decimal?
- Evaluate log₅125.
- Prove that √3 is irrational.
Chapter 2
- Find A ∪ B and A ∩ B.
- Find complement of a set.
- Solve problems using n(A ∪ B).
- Find number of subsets.
- Draw Venn diagrams.
Chapter 3
- Find zeroes of quadratic polynomials.
- Verify relationship between zeroes and coefficients.
- Find polynomial from given zeroes.
- Use Remainder Theorem.
- Use Factor Theorem.
Chapter 4
- Solve using substitution method.
- Solve using elimination method.
- Identify unique, no and infinite solutions.
- Solve word problems.
- Practice graphical representation.
⭐ MOST IMPORTANT TOPICS FOR EXAM
🔴 Chapter 1: Real Numbers
- Euclid Division Algorithm
- HCF and LCM
- Irrational Numbers
- Decimal Expansion
- Logarithms
🔴 Chapter 2: Sets
- Union and Intersection
- Complement
- Venn Diagrams
- n(A ∪ B) Formula
- Word Problems
🔴 Chapter 3: Polynomials
- Zeroes of Polynomials
- Relation Between Zeroes and Coefficients
- Remainder Theorem
- Factor Theorem
🔴 Chapter 4: Pair of Linear Equations
- Elimination Method
- Substitution Method
- Consistency of Equations
- Word Problems