Master Formula Sheet and Theorems Guide for CBSE Class 10 Mathematics

When preparing for the CBSE Class 10 Mathematics board examination, high-scoring students do not rely on frantic last-minute textbook skimming. They rely on a condensed, precision-crafted Formula Cheat Sheet that summarizes all essential algebraic identities, geometric theorems, trigonometric values, mensuration formulas, and statistical algorithms into a single structured reference.

This comprehensive revision guide brings together every single formula, theorem statement, and mathematical relationship across all 14 chapters of the CBSE Class 10 syllabus.

Unit 1: Number Systems & Algebra

1. Real Numbers (Chapter 1)

• Fundamental Theorem of Arithmetic: Every composite number can be uniquely expressed as a product of primes, apart from the order of factors.
• HCF & LCM Relationship (for two numbers $a$ and $b$): $\mathbf{\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b}$
• Irrationality Lemma: If prime $p$ divides $a^2$, then $p$ divides $a$.

2. Polynomials (Chapter 2)

For a quadratic polynomial $p(x) = ax^2 + bx + c$ with zeroes $\alpha, \beta$:

• Sum of zeroes: $\alpha + \beta = -\frac{b}{a} = -\frac{\text{coeff of } x}{\text{coeff of } x^2}$
• Product of zeroes: $\alpha \beta = \frac{c}{a} = \frac{\text{constant term}}{\text{coeff of } x^2}$
• Forming a Quadratic Polynomial: $k[x^2 - (\alpha + \beta)x + \alpha \beta]$

3. Pair of Linear Equations in Two Variables (Chapter 3)

For $a_1 x + b_1 y + c_1 = 0$ and $a_2 x + b_2 y + c_2 = 0$:

• Intersecting Lines (Unique Solution / Consistent): $\frac{a_1}{a_2} \ne \frac{b_1}{b_2}$
• Coincident Lines (Infinitely Many Solutions / Dependent): $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$
• Parallel Lines (No Solution / Inconsistent): $\frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2}$

4. Quadratic Equations (Chapter 4)

For standard form $ax^2 + bx + c = 0$ ($a \ne 0$):

• Discriminant ($D$): $\mathbf{D = b^2 - 4ac}$
• Quadratic Formula: $x = \frac{-b \pm \sqrt{D}}{2a}$
• Nature of Roots:
  • $D > 0 \implies$ Two distinct real roots
  • $D = 0 \implies$ Two equal real roots ($x = -b/2a$)
  • $D < 0 \implies$ No real roots

5. Arithmetic Progressions (Chapter 5)

For an AP with first term $a$ and common difference $d$:

• $n^{\text{th}}$ Term: $\mathbf{a_n = a + (n - 1)d}$
• $n^{\text{th}}$ Term from the End: $a_n' = l - (n - 1)d$
• Sum of First $n$ Terms ($S_n$): $\mathbf{S_n = \frac{n}{2}[2a + (n - 1)d] = \frac{n}{2}[a + l]}$
• Relationship: $a_n = S_n - S_{n-1}$

Unit 2: Coordinate Geometry & Triangles

6. Coordinate Geometry (Chapter 7)

• Distance Formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ (Distance from origin $= \sqrt{x^2 + y^2}$)
• Section Formula (Internal division in ratio $m_1 : m_2$): $\mathbf{(x, y) = \left( \frac{m_1 x_2 + m_2 x_1}{m_1 + m_2}, \; \frac{m_1 y_2 + m_2 y_1}{m_1 + m_2} \right)}$
• Mid-Point Formula: $(x, y) = \left( \frac{x_1 + x_2}{2}, \; \frac{y_1 + y_2}{2} \right)$
• Centroid of a Triangle: $G = \left( \frac{x_1 + x_2 + x_3}{3}, \; \frac{y_1 + y_2 + y_3}{3} \right)$

7. Triangles & Circles (Chapters 6 & 10)

• Basic Proportionality Theorem (Thales' Theorem): If $DE \parallel BC$, then $\frac{AD}{DB} = \frac{AE}{EC}$.
• Tangent Perpendicularity (Theorem 10.1): The tangent at any point of a circle is perpendicular to the radius through the point of contact ($OP \perp XY$).
• Equal Tangent Lengths (Theorem 10.2): Tangents drawn from an external point to a circle are equal in length ($PQ = PR$).

Unit 3: Trigonometry

8. Introduction to Trigonometry (Chapter 8)

• $\sin \theta = \frac{P}{H}, \quad \cos \theta = \frac{B}{H}, \quad \tan \theta = \frac{P}{B} = \frac{\sin \theta}{\cos \theta}$
• $\csc \theta = \frac{1}{\sin \theta}, \quad \sec \theta = \frac{1}{\cos \theta}, \quad \cot \theta = \frac{1}{\tan \theta}$

Fundamental Trigonometric Identities (MUST MEMORIZE!):

1. $\mathbf{\sin^2 \theta + \cos^2 \theta = 1} \implies \sin^2 \theta = 1 - \cos^2 \theta$
2. $\mathbf{1 + \tan^2 \theta = \sec^2 \theta} \implies \sec^2 \theta - \tan^2 \theta = 1$
3. $\mathbf{1 + \cot^2 \theta = \csc^2 \theta} \implies \csc^2 \theta - \cot^2 \theta = 1$

Specific Angle Values:

Unit 4: Mensuration

9. Areas Related to Circles (Chapter 11)

• Circumference: $C = 2\pi r$; Area: $A = \pi r^2$
• Perimeter of a Semicircle: $\mathbf{r(\pi + 2)}$
• Area of Circular Ring: $\pi(R^2 - r^2) = \pi(R - r)(R + r)$
• Arc Length of Sector: $l = \frac{\theta}{360^\circ} \times 2\pi r$
• Area of Sector: $\text{Area} = \frac{\theta}{360^\circ} \times \pi r^2 = \mathbf{\frac{1}{2} l r}$
• Area of Segment: $\text{Area of Sector} - \text{Area of } \Delta OAB$
  • For $\theta = 60^\circ$ or $120^\circ$: $\text{Area}(\Delta) = \frac{\sqrt{3}}{4} r^2$
  • For $\theta = 90^\circ$: $\text{Area}(\Delta) = \frac{1}{2} r^2$

10. Surface Areas and Volumes (Chapter 12)

Unit 5: Statistics & Probability

11. Statistics (Chapter 13)

• Mean (Direct): $\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$
• Mean (Assumed Mean): $\bar{x} = a + \frac{\sum f_i d_i}{\sum f_i}$
• Mean (Step-Deviation): $\bar{x} = a + \left( \frac{\sum f_i u_i}{\sum f_i} \right) \times h$
• Mode Formula: $\mathbf{\text{Mode} = l + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h}$
• Median Formula: $\mathbf{\text{Median} = l + \left( \frac{\frac{N}{2} - cf}{f} \right) \times h}$
• Karl Pearson's Empirical Formula: $\mathbf{3\,\text{Median} = \text{Mode} + 2\,\text{Mean}}$

12. Probability (Chapter 14)

• Classical Probability: $P(E) = \frac{n(E)}{n(S)}$
• Complementary Event: $P(\bar{E}) = 1 - P(E)$
• Universal Range: $\mathbf{0 \le P(E) \le 1}$

Important: <u>Keep this master sheet accessible during your revision sessions. Reviewing these formulas daily for 10 minutes guarantees that you will never blank out on an identity during board exams!</u>