BOARD OF SECONDARY EDUCATION, TELANGANA
TS SSC CLASS 10 MATHEMATICS OFFICIAL FORMULA SHEET
Complete Chapter-Wise Formulas, Identities, Theorems & Exam Blueprint | Muqeem Tutorial
📊 CHAPTER-WISE MARKS WEIGHTAGE & BLUEPRINT
Exams Blueprint
| Chapter No. & Name | Unit / Branch | Expected Marks Weightage |
|---|---|---|
| Ch 1: Real Numbers | Number System | 6 - 8 Marks |
| Ch 2: Sets | Sets & Logic | 6 - 8 Marks |
| Ch 3: Polynomials | Algebra | 8 - 10 Marks |
| Ch 4: Pair of Linear Equations | Algebra | 8 - 10 Marks |
| Ch 5: Quadratic Equations | Algebra | 6 - 8 Marks |
| Ch 6: Progressions (AP & GP) | Algebra | 8 - 10 Marks |
| Ch 7: Coordinate Geometry | Coordinate Geometry | 10 - 12 Marks |
| Ch 8: Similar Triangles | Geometry | 8 - 10 Marks |
| Ch 9: Tangents & Secants to a Circle | Geometry | 6 - 8 Marks |
| Ch 10: Mensuration | Mensuration (3D) | 8 - 10 Marks |
| Ch 11: Trigonometry | Trigonometry | 8 - 10 Marks |
| Ch 12: Applications of Trigonometry | Trigonometry | 6 - 8 Marks |
| Ch 13: Probability | Statistics & Probability | 6 - 8 Marks |
| Ch 14: Statistics | Statistics & Probability | 10 - 12 Marks |
Chapter 1: Real Numbers
6-8 Marks
• Euclid's Division Lemma: Given positive integers a and b, there exist unique integers q and r satisfying:
a = bq + r (0 ≤ r < b)
• Fundamental Theorem of Arithmetic: Every composite number can be uniquely expressed as a product of prime numbers.
• Logarithmic Laws:
1. loga(xy) = loga x + loga y (Product Rule)
2. loga(x/y) = loga x - loga y (Quotient Rule)
3. loga(xn) = n · loga x (Power Rule)
4. loga a = 1 and loga 1 = 0
5. Change of Base: logb x = (loga x) / (loga b)
• Decimal Expansions: A rational fraction p/q has a terminating decimal if q = 2n · 5m (for non-negative integers n, m). Otherwise, it is non-terminating repeating.
a = bq + r (0 ≤ r < b)
• Fundamental Theorem of Arithmetic: Every composite number can be uniquely expressed as a product of prime numbers.
• Logarithmic Laws:
1. loga(xy) = loga x + loga y (Product Rule)
2. loga(x/y) = loga x - loga y (Quotient Rule)
3. loga(xn) = n · loga x (Power Rule)
4. loga a = 1 and loga 1 = 0
5. Change of Base: logb x = (loga x) / (loga b)
• Decimal Expansions: A rational fraction p/q has a terminating decimal if q = 2n · 5m (for non-negative integers n, m). Otherwise, it is non-terminating repeating.
Chapter 2: Sets
6-8 Marks
• Set Operations:
1. Union (A ∪ B): {x : x ∈ A or x ∈ B}
2. Intersection (A ∩ B): {x : x ∈ A and x ∈ B}
3. Difference (A \ B or A - B): {x : x ∈ A and x ∉ B}
• Cardinality Formula:
n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
• For disjoint sets (A ∩ B = ∅): n(A ∪ B) = n(A) + n(B)
• Subsets: Total subsets of a set with n elements = 2n (Proper subsets = 2n - 1).
1. Union (A ∪ B): {x : x ∈ A or x ∈ B}
2. Intersection (A ∩ B): {x : x ∈ A and x ∈ B}
3. Difference (A \ B or A - B): {x : x ∈ A and x ∉ B}
• Cardinality Formula:
n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
• For disjoint sets (A ∩ B = ∅): n(A ∪ B) = n(A) + n(B)
• Subsets: Total subsets of a set with n elements = 2n (Proper subsets = 2n - 1).
Chapter 3: Polynomials
8-10 Marks
• Quadratic Polynomial: p(x) = ax2 + bx + c (a ≠ 0)
1. Sum of zeroes (α + β) = -b / a
2. Product of zeroes (α · β) = c / a
3. Quadratic polynomial formula: k[x2 - (α + β)x + αβ]
• Cubic Polynomial: p(x) = ax3 + bx2 + cx + d
1. α + β + γ = -b / a
2. αβ + βγ + γα = c / a
3. αβγ = -d / a
• Division Algorithm: p(x) = g(x) · q(x) + r(x), where r(x) = 0 or deg(r) < deg(g).
1. Sum of zeroes (α + β) = -b / a
2. Product of zeroes (α · β) = c / a
3. Quadratic polynomial formula: k[x2 - (α + β)x + αβ]
• Cubic Polynomial: p(x) = ax3 + bx2 + cx + d
1. α + β + γ = -b / a
2. αβ + βγ + γα = c / a
3. αβγ = -d / a
• Division Algorithm: p(x) = g(x) · q(x) + r(x), where r(x) = 0 or deg(r) < deg(g).
Chapter 4: Pair of Linear Equations in Two Variables
8-10 Marks
For equations a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0:
| Ratio Comparison | Graphical Representation | Algebraic Interpretation | Consistency |
|---|---|---|---|
| a1/a2 ≠ b1/b2 | Intersecting Lines | Unique Solution | Consistent |
| a1/a2 = b1/b2 = c1/c2 | Coincident Lines | Infinitely Many Solutions | Dependent & Consistent |
| a1/a2 = b1/b2 ≠ c1/c2 | Parallel Lines | No Solution | Inconsistent |
Chapter 5: Quadratic Equations
6-8 Marks
• Standard Form: ax2 + bx + c = 0 (a ≠ 0)
• Quadratic Formula:
x = [-b ± √(b2 - 4ac)] / (2a)
• Discriminant (D = b2 - 4ac) & Nature of Roots:
1. D > 0: Two distinct real roots
2. D = 0: Two equal real roots (x = -b / 2a)
3. D < 0: No real roots
• Quadratic Formula:
x = [-b ± √(b2 - 4ac)] / (2a)
• Discriminant (D = b2 - 4ac) & Nature of Roots:
1. D > 0: Two distinct real roots
2. D = 0: Two equal real roots (x = -b / 2a)
3. D < 0: No real roots
Chapter 6: Progressions
8-10 Marks
• Arithmetic Progression (AP):
1. n-th term: an = a + (n - 1)d
2. Sum of first n terms: Sn = (n/2)[2a + (n - 1)d] or Sn = (n/2)[a + l] (where l is last term)
• Geometric Progression (GP):
1. n-th term: an = a · rn-1
2. Common Ratio: r = a2 / a1 = a3 / a2
1. n-th term: an = a + (n - 1)d
2. Sum of first n terms: Sn = (n/2)[2a + (n - 1)d] or Sn = (n/2)[a + l] (where l is last term)
• Geometric Progression (GP):
1. n-th term: an = a · rn-1
2. Common Ratio: r = a2 / a1 = a3 / a2
Chapter 7: Coordinate Geometry
10-12 Marks
• Distance Formula: d = √[(x2 - x1)2 + (y2 - y1)2]
• Distance from origin (0, 0): √(x2 + y2)
• Section Formula (Ratio m1 : m2):
P(x, y) = [ (m1x2 + m2x1)/(m1 + m2) , (m1y2 + m2y1)/(m1 + m2) ]
• Midpoint Formula: M = [ (x1 + x2)/2 , (y1 + y2)/2 ]
• Centroid of Triangle: G = [ (x1 + x2 + x3)/3 , (y1 + y2 + y3)/3 ]
• Area of Triangle:
Area = (1/2) | x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2) |
• Heron's Formula: Area = √[s(s - a)(s - b)(s - c)] where s = (a + b + c) / 2
• Slope of Line: m = (y2 - y1) / (x2 - x1) = tan θ
• Distance from origin (0, 0): √(x2 + y2)
• Section Formula (Ratio m1 : m2):
P(x, y) = [ (m1x2 + m2x1)/(m1 + m2) , (m1y2 + m2y1)/(m1 + m2) ]
• Midpoint Formula: M = [ (x1 + x2)/2 , (y1 + y2)/2 ]
• Centroid of Triangle: G = [ (x1 + x2 + x3)/3 , (y1 + y2 + y3)/3 ]
• Area of Triangle:
Area = (1/2) | x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2) |
• Heron's Formula: Area = √[s(s - a)(s - b)(s - c)] where s = (a + b + c) / 2
• Slope of Line: m = (y2 - y1) / (x2 - x1) = tan θ
Chapter 8: Similar Triangles
8-10 Marks
• Basic Proportionality Theorem (BPT / Thales Theorem): If line DE || BC in ΔABC, then:
AD / DB = AE / EC
• Ratio of Areas of Similar Triangles: If ΔABC ~ ΔPQR, then:
Area(ΔABC) / Area(ΔPQR) = (AB/PQ)2 = (BC/QR)2 = (AC/PR)2
• Pythagoras Theorem: In a right-angled triangle:
Hypotenuse2 = Base2 + Height2
AD / DB = AE / EC
• Ratio of Areas of Similar Triangles: If ΔABC ~ ΔPQR, then:
Area(ΔABC) / Area(ΔPQR) = (AB/PQ)2 = (BC/QR)2 = (AC/PR)2
• Pythagoras Theorem: In a right-angled triangle:
Hypotenuse2 = Base2 + Height2
Chapter 9: Tangents and Secants to a Circle
6-8 Marks
• Radius ⊥ Tangent: Tangent at any point is perpendicular to the radius through point of contact.
• Equal Tangents: Tangents drawn from an external point are equal in length (PA = PB).
• Area of Sector: A = (θ / 360°) × πr2
• Length of Arc (l): l = (θ / 360°) × 2πr
• Area of Segment: Area of Sector - Area of Corresponding Triangle
• Equal Tangents: Tangents drawn from an external point are equal in length (PA = PB).
• Area of Sector: A = (θ / 360°) × πr2
• Length of Arc (l): l = (θ / 360°) × 2πr
• Area of Segment: Area of Sector - Area of Corresponding Triangle
Chapter 10: Mensuration (3D Geometry)
8-10 Marks
| Solid Shape | Curved / Lateral Surface Area (CSA/LSA) | Total Surface Area (TSA) | Volume (V) |
|---|---|---|---|
| Cube | 4a2 | 6a2 | a3 |
| Cuboid | 2h(l + b) | 2(lb + bh + hl) | l · b · h |
| Cylinder | 2πrh | 2πr(r + h) | πr2h |
| Cone | πrl (l = √(r2 + h2)) | πr(r + l) | (1/3)πr2h |
| Sphere | 4πr2 | 4πr2 | (4/3)πr3 |
| Hemisphere | 2πr2 | 3πr2 | (2/3)πr3 |
Chapter 11: Trigonometry
8-10 Marks
• Trigonometric Table Values:
• Complementary Angles:
sin(90° - θ) = cos θ | tan(90° - θ) = cot θ | sec(90° - θ) = cosec θ
• Trigonometric Identities:
1. sin2 θ + cos2 θ = 1
2. 1 + tan2 θ = sec2 θ ⇒ sec2 θ - tan2 θ = 1
3. 1 + cot2 θ = cosec2 θ ⇒ cosec2 θ - cot2 θ = 1
| Ratio \ Angle | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin θ | 0 | 1/2 | 1/√2 | √3/2 | 1 |
| cos θ | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| tan θ | 0 | 1/√3 | 1 | √3 | Undefined |
sin(90° - θ) = cos θ | tan(90° - θ) = cot θ | sec(90° - θ) = cosec θ
• Trigonometric Identities:
1. sin2 θ + cos2 θ = 1
2. 1 + tan2 θ = sec2 θ ⇒ sec2 θ - tan2 θ = 1
3. 1 + cot2 θ = cosec2 θ ⇒ cosec2 θ - cot2 θ = 1
Chapter 12: Applications of Trigonometry
6-8 Marks
• Line of Sight: Line drawn from observer's eye to the viewed object.
• Angle of Elevation: Formed when looking upward above horizontal level.
• Angle of Depression: Formed when looking downward below horizontal level.
• Primary Formula: tan θ = Height (Opposite) / Distance (Adjacent)
• Angle of Elevation: Formed when looking upward above horizontal level.
• Angle of Depression: Formed when looking downward below horizontal level.
• Primary Formula: tan θ = Height (Opposite) / Distance (Adjacent)
Chapter 13: Probability
6-8 Marks
• Probability Formula:
P(E) = n(E) / n(S) = (Favorable Outcomes) / (Total Outcomes)
• Key Rules:
1. Range: 0 ≤ P(E) ≤ 1
2. Sure Event: P(E) = 1
3. Impossible Event: P(E) = 0
4. Complementary Event: P(E) + P(NOT E) = 1 ⇒ P(NOT E) = 1 - P(E)
P(E) = n(E) / n(S) = (Favorable Outcomes) / (Total Outcomes)
• Key Rules:
1. Range: 0 ≤ P(E) ≤ 1
2. Sure Event: P(E) = 1
3. Impossible Event: P(E) = 0
4. Complementary Event: P(E) + P(NOT E) = 1 ⇒ P(NOT E) = 1 - P(E)
Chapter 14: Statistics
10-12 Marks
• Mean (x̄):
1. Direct Method: x̄ = Σfixi / Σfi
2. Assumed Mean Method: x̄ = a + (Σfidi / Σfi) (where di = xi - a)
3. Step-Deviation Method: x̄ = a + (Σfiui / Σfi) × h (where ui = (xi - a)/h)
• Mode:
Mode = L + [ (f1 - f0) / (2f1 - f0 - f2) ] × h
• Median:
Median = L + [ (N/2 - cf) / f ] × h
• Empirical Relationship:
Mode = 3 Median - 2 Mean
1. Direct Method: x̄ = Σfixi / Σfi
2. Assumed Mean Method: x̄ = a + (Σfidi / Σfi) (where di = xi - a)
3. Step-Deviation Method: x̄ = a + (Σfiui / Σfi) × h (where ui = (xi - a)/h)
• Mode:
Mode = L + [ (f1 - f0) / (2f1 - f0 - f2) ] × h
• Median:
Median = L + [ (N/2 - cf) / f ] × h
• Empirical Relationship:
Mode = 3 Median - 2 Mean
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