TN SSLC 10th Maths All Chapters Formula Cheat Sheet & Top 3 Theorems (5 Marks)
Welcome to the ultimate Class 10 TN Board Maths Master Guide! Scoring a centum in Tamil Nadu State Board (TN SSLC) Class 10 Mathematics requires absolute command over formulas and flawless step-wise theorem proofs. To streamline your revision for the SSLC Public Examination, we have organized the essential formulas for all 8 chapters in interactive dropdown modules below. Expand each chapter to quickly revise formulas for Algebra, Coordinate Geometry, Trigonometry, and Mensuration.
Following the formula bank, you will find comprehensive, step-wise proof presentations for the Top 3 Most Expected 5-Mark Geometry Theorems (Basic Proportionality Theorem, Angle Bisector Theorem, and Pythagoras Theorem) complete with official board marking statements and diagrams.
Complete Class 10 Maths Formula Cheat Sheet (All 8 Chapters)
Exam Tip: Click on any chapter name below to expand and view the complete textbook formulas!
Chapter 1: Relations and Functions
- Cartesian Product: \(A \times B = \{(a,b) \mid a \in A, b \in B\}\)
- Number of Elements: If \(n(A) = p\) and \(n(B) = q\), then \(n(A \times B) = pq\)
- Relations: A relation \(R\) from \(A\) to \(B\) is a subset of \(A \times B\), i.e., \(R \subseteq A \times B\)
- Composition of Functions: \((f \circ g)(x) = f(g(x))\)
- Commutative Property: Generally, \(f \circ g \neq g \circ f\)
- Associative Property: Composition of three functions is always associative: \((f \circ g) \circ h = f \circ (g \circ h)\)
Chapter 2: Numbers and Sequences
- Euclid's Division Lemma: \(a = bq + r\), where \(0 \le r < b\)
- Arithmetic Progression (A.P): General form is \(a, a+d, a+2d, \dots\)
- \(n^{\text{th}}\) term of an A.P: \(t_n = a + (n-1)d\)
- Number of terms in an A.P: \(n = \frac{l-a}{d} + 1\)
- Sum of \(n\) terms of an A.P: \(S_n = \frac{n}{2}[2a + (n-1)d]\) (or) \(S_n = \frac{n}{2}(a + l)\)
- Geometric Progression (G.P): General form is \(a, ar, ar^2, \dots\)
- \(n^{\text{th}}\) term of a G.P: \(t_n = ar^{n-1}\)
- Sum of \(n\) terms of a G.P: \(S_n = \frac{a(r^n - 1)}{r - 1}\) (for \(r \neq 1\))
- Sum of Infinite terms of a G.P: \(S_\infty = \frac{a}{1 - r}\)
- Sum of first \(n\) Natural Numbers: \(\sum n = \frac{n(n+1)}{2}\)
- Sum of Squares of first \(n\) Natural Numbers: \(\sum n^2 = \frac{n(n+1)(2n+1)}{6}\)
- Sum of Cubes of first \(n\) Natural Numbers: \(\sum n^3 = \left[ \frac{n(n+1)}{2} \right]^2\)
- Sum of first \(n\) Odd Natural Numbers: \(\sum (2n-1) = n^2\)
Chapter 3: Algebra
- Relationship between LCM and GCD: \(f(x) \times g(x) = \text{LCM} \times \text{GCD}\)
- Quadratic Equation General Form: \(ax^2 + bx + c = 0\)
- Quadratic Formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
- Nature of Roots (Discriminant \(\Delta = b^2 - 4ac\)):
- If \(\Delta > 0\): Roots are real and unequal
- If \(\Delta = 0\): Roots are real and equal
- If \(\Delta < 0\): No real roots
- Sum of Roots: \(\alpha + \beta = -\frac{b}{a}\)
- Product of Roots: \(\alpha\beta = \frac{c}{a}\)
- Formation of Quadratic Equation: \(x^2 - (\alpha + \beta)x + \alpha\beta = 0\)
- Key Identity 1: \(\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta\)
- Key Identity 2: \((\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta\)
- Key Identity 3: \(\alpha^3 + \beta^3 = (\alpha + \beta)^3 - 3\alpha\beta(\alpha + \beta)\)
Chapter 4: Geometry
- Basic Proportionality Theorem (BPT): \(\frac{AD}{DB} = \frac{AE}{EC}\)
- Angle Bisector Theorem (ABT): \(\frac{AB}{AC} = \frac{BD}{CD}\)
- Pythagoras Theorem: \(AB^2 + AC^2 = BC^2\) (where \(\angle A = 90^\circ\))
- Ceva's Theorem: \(\frac{BD}{DC} \times \frac{CE}{EA} \times \frac{AF}{FB} = 1\)
- Menelaus' Theorem: \(\frac{BD}{DC} \times \frac{CE}{EA} \times \frac{AF}{FB} = -1\)
Chapter 5: Coordinate Geometry
- Distance between two points: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
- Midpoint Formula: \(M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\)
- Centroid of a Triangle: \(G = \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right)\)
- Section Formula (Internal Division): \(P = \left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)\)
- Area of Triangle: \(\frac{1}{2} \left| (x_1y_2 + x_2y_3 + x_3y_1) - (x_2y_1 + x_3y_2 + x_1y_3) \right|\)
- Slope (\(m\)): \(m = \tan \theta\) (or) \(m = \frac{y_2 - y_1}{x_2 - x_1}\)
- Parallel Lines Condition: \(m_1 = m_2\)
- Perpendicular Lines Condition: \(m_1 \times m_2 = -1\)
- Slope-Intercept Form: \(y = mx + c\)
- Point-Slope Form: \(y - y_1 = m(x - x_1)\)
- Two-Point Form: \(\frac{y - y_1}{y_2 - y_1} = \frac{x - x_1}{x_2 - x_1}\)
- Intercept Form: \(\frac{x}{a} + \frac{y}{b} = 1\)
Chapter 6: Trigonometry
- \(\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}\), \(\csc \theta = \frac{1}{\sin \theta}\)
- \(\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}\), \(\sec \theta = \frac{1}{\cos \theta}\)
- \(\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}\), \(\cot \theta = \frac{1}{\tan \theta}\)
- Fundamental Identities:
- \(\sin^2 \theta + \cos^2 \theta = 1\)
- \(1 + \tan^2 \theta = \sec^2 \theta\)
- \(1 + \cot^2 \theta = \csc^2 \theta\)
- Complementary Angles: \(\sin(90^\circ - \theta) = \cos \theta\), \(\tan(90^\circ - \theta) = \cot \theta\), \(\sec(90^\circ - \theta) = \csc \theta\)
| \(\theta\) | \(0^\circ\) | \(30^\circ\) | \(45^\circ\) | \(60^\circ\) | \(90^\circ\) |
|---|---|---|---|---|---|
| \(\sin \theta\) | \(0\) | \(\frac{1}{2}\) | \(\frac{1}{\sqrt{2}}\) | \(\frac{\sqrt{3}}{2}\) | \(1\) |
| \(\cos \theta\) | \(1\) | \(\frac{\sqrt{3}}{2}\) | \(\frac{1}{\sqrt{2}}\) | \(\frac{1}{2}\) | \(0\) |
| \(\tan \theta\) | \(0\) | \(\frac{1}{\sqrt{3}}\) | \(1\) | \(\sqrt{3}\) | \(\infty\) |
Chapter 7: Mensuration
| Solid Shape | Curved / Lateral Surface Area (CSA/LSA) | Total Surface Area (TSA) | Volume |
|---|---|---|---|
| Cylinder | \(2\pi rh\) | \(2\pi r(h+r)\) | \(\pi r^2h\) |
| Hollow Cylinder | \(2\pi h(R+r)\) | \(2\pi(R+r)(h+R-r)\) | \(\pi h(R^2-r^2)\) |
| Cone | \(\pi rl\) | \(\pi r(l+r)\) | \(\frac{1}{3}\pi r^2h\) |
| Frustum | \(\pi (R+r)l\) | \(\pi (R+r)l + \pi R^2 + \pi r^2\) | \(\frac{1}{3}\pi h(R^2+r^2+Rr)\) |
| Sphere | \(4\pi r^2\) | \(4\pi r^2\) | \(\frac{4}{3}\pi r^3\) |
| Hemisphere | \(2\pi r^2\) | \(3\pi r^2\) | \(\frac{2}{3}\pi r^3\) |
| Cube | \(4a^2\) | \(6a^2\) | \(a^3\) |
| Cuboid | \(2h(l+b)\) | \(2(lb+bh+hl)\) | \(lbh\) |
Chapter 8: Statistics and Probability
- Range (\(R\)): \(L - S\) (Largest value − Smallest value)
- Coefficient of Range: \(\frac{L-S}{L+S}\)
- Standard Deviation (Direct Method): \(\sigma = \sqrt{\frac{\sum x^2}{n} - \left(\frac{\sum x}{n}\right)^2}\)
- Standard Deviation of first \(n\) natural numbers: \(\sigma = \sqrt{\frac{n^2 - 1}{12}}\)
- Coefficient of Variation (C.V): \(\text{C.V} = \frac{\sigma}{\bar{x}} \times 100\%\)
- Probability of Event A: \(P(A) = \frac{n(A)}{n(S)}\)
- Complement of Event: \(P(\bar{A}) = 1 - P(A)\)
- Addition Theorem of Probability (2 Events): \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)
- Addition Theorem of Probability (3 Events): \(P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(B \cap C) - P(C \cap A) + P(A \cap B \cap C)\)
Top 3 Most Expected Theorems (5-Mark Questions)
In Part III of the TN SSLC Mathematics question paper, one geometry theorem is almost always tested. Board examiners award marks according to a strict 5-part structure: Statement, Given, To Prove, Construction, and Proof Table.
1. Thales Theorem or Basic Proportionality Theorem (BPT)
Statement: A straight line drawn parallel to a side of a triangle intersecting the other two sides, divides the sides in the same ratio.
Given: In \(\Delta ABC\), \(D\) is a point on \(AB\) and \(E\) is a point on \(AC\).
To Prove: \(\frac{AD}{DB} = \frac{AE}{EC}\)
Construction: Draw a line \(DE \parallel BC\).
| S.No | Statement | Reason |
|---|---|---|
| 1. | \(\angle ABC = \angle ADE = \angle 1\) | Corresponding angles are equal because \(DE \parallel BC\) |
| 2. | \(\angle ACB = \angle AED = \angle 2\) | Corresponding angles are equal because \(DE \parallel BC\) |
| 3. | \(\angle DAE = \angle BAC = \angle 3\) | Both triangles have a common angle |
| 4. | \(\Delta ABC \sim \Delta ADE\) | By AAA Similarity |
| 5. | \(\frac{AB}{AD} = \frac{AC}{AE}\) | Corresponding sides of similar triangles are proportional |
| 6. | \(\frac{AD + DB}{AD} = \frac{AE + EC}{AE}\) | Splitting \(AB\) and \(AC\) using intermediate points \(D\) and \(E\) |
| 7. | \(\frac{AD}{AD} + \frac{DB}{AD} = \frac{AE}{AE} + \frac{EC}{AE}\) | Separating fractions |
| 8. | \(1 + \frac{DB}{AD} = 1 + \frac{EC}{AE}\) | Simplifying fractions |
| 9. | \(\frac{DB}{AD} = \frac{EC}{AE}\) | Cancelling 1 from both sides |
| 10. | \(\frac{AD}{DB} = \frac{AE}{EC}\) | Taking reciprocals. Hence Proved |
2. Angle Bisector Theorem (ABT)
Statement: The internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the corresponding sides containing the angle.
Given: In \(\Delta ABC\), \(AD\) is the internal bisector of \(\angle A\).
To Prove: \(\frac{AB}{AC} = \frac{BD}{CD}\)
Construction: Draw a line through \(C\) parallel to \(AB\). Extend the bisector line \(AD\) to meet this parallel line at point \(E\).
| S.No | Statement | Reason |
|---|---|---|
| 1. | \(\angle AEC = \angle BAE = \angle 1\) | Alternate interior angles are equal (\(AB \parallel CE\) cut by transversal \(AE\)) |
| 2. | \(\Delta ACE\) is an isosceles triangle | In \(\Delta ACE\), \(\angle CAE = \angle CEA\) |
| 3. | \(AC = CE\) | Sides opposite to equal angles in a triangle are equal — (Equation 1) |
| 4. | \(\Delta ABD \sim \Delta ECD\) | By AA Similarity (alternate angles and vertically opposite angles are equal) |
| 5. | \(\frac{AB}{CE} = \frac{BD}{CD}\) | Corresponding sides of similar triangles are proportional |
| 6. | \(\frac{AB}{AC} = \frac{BD}{CD}\) | Substituting \(CE = AC\) from Equation 1. Hence Proved |
3. Pythagoras Theorem (Baudhayana Theorem)
Statement: In a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.
Given: In \(\Delta ABC\), \(\angle A = 90^\circ\).
To Prove: \(AB^2 + AC^2 = BC^2\)
Construction: Draw \(AD \perp BC\).
| S.No | Statement | Reason |
|---|---|---|
| 1. | Compare \(\Delta ABC\) and \(\Delta DBA\): \(\angle B\) is common \(\angle BAC = \angle BDA = 90^\circ\) |
Given \(\angle BAC = 90^\circ\) and by construction \(\angle BDA = 90^\circ\) |
| 2. | \(\Delta ABC \sim \Delta DBA\) | By AA Similarity |
| 3. | \(\frac{AB}{BD} = \frac{BC}{AB}\) | Corresponding sides of similar triangles are proportional |
| 4. | \(AB^2 = BC \times BD\) | Cross multiplication — (Equation 1) |
| 5. | Compare \(\Delta ABC\) and \(\Delta DAC\): \(\angle C\) is common \(\angle BAC = \angle ADC = 90^\circ\) |
Given \(\angle BAC = 90^\circ\) and by construction \(\angle ADC = 90^\circ\) |
| 6. | \(\Delta ABC \sim \Delta DAC\) | By AA Similarity |
| 7. | \(\frac{BC}{AC} = \frac{AC}{DC}\) | Corresponding sides of similar triangles are proportional |
| 8. | \(AC^2 = BC \times DC\) | Cross multiplication — (Equation 2) |
| 9. | \(AB^2 + AC^2 = (BC \times BD) + (BC \times DC)\) | Adding Equations (1) and (2) |
| 10. | \(AB^2 + AC^2 = BC(BD + DC)\) | Factoring out common term \(BC\) |
| 11. | \(AB^2 + AC^2 = BC \times BC\) | Since \(BD + DC = BC\) from the diagram |
| 12. | \(AB^2 + AC^2 = BC^2\) | Hence Proved |
Exam Tip: Whenever proving a theorem in Part III, always maintain the 5-step format: Statement → Given → To Prove → Construction → Proof Table. Leaving out the statement or construction will result in the loss of 1 to 2 marks!