Class 10 Mathematics: Complete Theory & Formulas

Chapter 1: Real Numbers

• Fundamental Theorem of Arithmetic: Every composite number can be written as a product of prime numbers in a unique way, ignoring the order of the factors.
• HCF and LCM Relationships:
  • The HCF of two co-prime numbers is 1.
  • The LCM of two co-prime numbers is equal to their product.
  • Formula for two numbers $a$ and $b$: $H.C.F.(a, b) \times L.C.M.(a, b) = a \times b$.
  • $L.C.M.(a, b) = \frac{a \times b}{H.C.F.(a, b)}$.
• Theorem 1.2: Let $p$ be a prime number; for a positive integer $a$, if $a^2$ is divisible by $p$, then $a$ is also divisible by $p$.

Chapter 2: Polynomials

• Standard Forms of Polynomials:
  • Linear Polynomial: $p(x) = ax + b$.
  • Quadratic Polynomial: $p(x) = ax^2 + bx + c$, where $a \neq 0$.
  • Cubic Polynomial: $P(x) = ax^3 + bx^2 + cx + d$.
• Zeroes of a Quadratic Polynomial:
  • Sum of zeroes: $\alpha + \beta = \frac{-b}{a}$.
  • Product of zeroes: $\alpha\beta = \frac{c}{a}$.
  • Forming a quadratic polynomial: $k[x^2 - (\alpha + \beta)x + \alpha\beta]$.
• Zeroes of a Cubic Polynomial:
  • $\alpha + \beta + \gamma = \frac{-b}{a}$.
  • $\alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a}$.
  • $\alpha\beta\gamma = \frac{-d}{a}$.
• Derived Algebraic Identities for Roots:
  • $\frac{1}{\alpha} + \frac{1}{\beta} = \frac{-b}{c}$.
  • $\alpha^2 + \beta^2 = \frac{b^2 - 2ac}{a^2}$.

Chapter 3: Pair of Linear Equations in Two Variables

• General Formula: $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$.
• Conditions for Solutions:
  • $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$: The lines are intersecting, yield a unique solution, and are consistent.
  • $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$: The lines are coincident, yield infinitely many solutions, and are consistent.
  • $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$: The lines are parallel, yield no solution, and are inconsistent.
• Two-Digit Number Problems:
  • If the tens digit is $x$ and the units digit is $y$, the original number is $10x + y$.
  • The new number formed by reversing the digits is $10y + x$.

Chapter 4: Quadratic Equations

• Standard Form: $ax^2 + bx + c = 0$, where $a \neq 0$.
• Discriminant: $D = b^2 - 4ac$.
• Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
• Nature of Roots:
  • If $b^2 - 4ac > 0$, the equation has two distinct real roots.
  • If $b^2 - 4ac = 0$, the equation has equal and real roots given by $x = \frac{-b}{2a}$.
  • If $b^2 - 4ac < 0$, the equation has no real roots.
• Special Conditions:
  • If $a = c$, the roots are reciprocals of each other.
  • If $b = 0$, the roots are opposites of each other.

Chapter 5: Arithmetic Progressions

• $n^{th}$ Term: $a_n = a + (n - 1)d$.
• Common Difference: $d = a_n - a_{n-1}$ or $d = \frac{a_m - a_n}{m - n}$.
• Sum of $n$ Terms ($S_n$):
  • $S_n = \frac{n}{2}[2a + (n - 1)d]$.
  • $S_n = \frac{n}{2}[a + l]$, where $l = a_n$.
• Relation between $a_n$ and $S_n$: $a_n = S_n - S_{n-1}$.
• Special Sum Formulas:
  • Sum of the first $n$ natural numbers: $S_n = \frac{n(n + 1)}{2}$.
  • Sum of the first $n$ even natural numbers: $S_n = n(n + 1)$.
  • Sum of the first $n$ odd natural numbers: $S_n = n^2$.

Chapter 6: Triangles

• Theorem 6.1 (Basic Proportionality Theorem / Thales Theorem): If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio ($\frac{AD}{BD} = \frac{AE}{EC}$).
• Theorem 6.2 (Inverse Thales Theorem): If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.
• Similarity Criteria:
  • Theorem 6.3 (AAA Similarity): If corresponding angles are equal, the triangles are similar.
  • Theorem 6.4 (SSS Similarity): If sides are proportional, the corresponding angles are equal, making the triangles similar.
  • Theorem 6.5 (SAS Similarity): If one pair of proportional sides includes equal corresponding angles, the triangles are similar.

Chapter 7: Coordinate Geometry

• Distance Formula: $AB = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$.
• Distance from Origin $O(0,0)$: $OA = \sqrt{x^2 + y^2}$.
• Section Formula: To divide a line segment in the ratio $m:n$, the coordinates are $x = \frac{mx_2 + nx_1}{m + n}$ and $y = \frac{my_2 + ny_1}{m + n}$.
• Mid-point Formula: $M(x,y) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$.
• Collinearity Rule: If points A, B, and C are collinear, then $AB + BC = AC$.

Chapter 8: Introduction to Trigonometry

• Trigonometric Ratios:
  • $\sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$, $\cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$, $\tan\theta = \frac{\text{Opposite}}{\text{Adjacent}}$.
  • $\csc\theta = \frac{\text{Hypotenuse}}{\text{Opposite}}$, $\sec\theta = \frac{\text{Hypotenuse}}{\text{Adjacent}}$, $\cot\theta = \frac{\text{Adjacent}}{\text{Opposite}}$.
• Relationship Formulas:
  • $\sin\theta \cdot \csc\theta = 1$, $\cos\theta \cdot \sec\theta = 1$, $\tan\theta \cdot \cot\theta = 1$.
  • $\tan\theta = \frac{\sin\theta}{\cos\theta}$, $\cot\theta = \frac{\cos\theta}{\sin\theta}$.
• Key Identities:
  • $\sin^2\theta + \cos^2\theta = 1$.
  • $\sec^2\theta - \tan^2\theta = 1$.
  • $\csc^2\theta - \cot^2\theta = 1$.

Chapter 9: Some Applications of Trigonometry

• Angle of Elevation: The angle formed between the line of sight and the horizontal line when the object observed is above the observer's level.
• Angle of Depression: The angle formed between the line of sight and the horizontal line when the object observed is below the observer's level.

Chapter 10: Circles

• Theorem 10.1: A tangent to a circle is perpendicular to the radius drawn at the point of contact ($OP \perp XY$).
• Theorem 10.2: The lengths of tangents drawn from an external point to a circle are equal ($PQ = PR$).
• Quadrilateral Rule: If a circle touches all four sides of a quadrilateral $ABCD$, then $AB + CD = AD + BC$.

Chapter 11: Areas Related to Circles

• Basic Formulas: Area of a circle $= \pi r^2$, Circumference $= 2\pi r$.
• Length of Minor Arc: $l = \frac{\pi r \theta}{180^\circ}$.
• Area of Minor Sector: $\frac{\pi r^2 \theta}{360^\circ}$ or $\frac{1}{2} l r$.
• Area of Major Sector: $\text{Area of circle} - \text{Area of minor sector}$.
• Area of Minor Segment: $\text{Area of Minor Sector} - \text{Area of Triangle}$.
• Area of Triangle (with central angle $\theta$): $\sin\left(\frac{\theta}{2}\right) \times \cos\left(\frac{\theta}{2}\right) \times r^2$.

Chapter 12: Surface Areas and Volumes

• Cube: Total Surface Area $= 6l^2$, Lateral Surface Area $= 4l^2$, Volume $= l^3$, Diagonal $= \sqrt{3}l$.
• Cuboid: Total Surface Area $= 2(lb + bh + hl)$, Lateral Surface Area $= 2h(l + b)$, Volume $= lbh$, Diagonal $= \sqrt{l^2 + b^2 + h^2}$.
• Cylinder: Curved Surface Area $= 2\pi rh$, Total Surface Area $= 2\pi r(h + r)$, Volume $= \pi r^2 h$.
• Cone: Curved Surface Area $= \pi rl$, Total Surface Area $= \pi r(l + r)$ (where $l^2 = h^2 + r^2$), Volume $= \frac{1}{3}\pi r^2 h$.
• Sphere: Surface Area $= 4\pi r^2$, Volume $= \frac{4}{3}\pi r^3$.
• Hemisphere: Curved Surface Area $= 2\pi r^2$, Total Surface Area $= 3\pi r^2$, Volume $= \frac{2}{3}\pi r^3$.
• Unit Conversions: $1 \text{ m}^3 = 1000 \text{ L}$ and $1000 \text{ cm}^3 = 1 \text{ L}$.

Chapter 13: Statistics

• Mean ($\bar{x}$):
  • Direct Method: $\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$.
  • Assumed Mean Method: $\bar{x} = a + \frac{\sum f_i d_i}{\sum f_i}$.
  • Step Deviation Method: $\bar{x} = a + \frac{\sum f_i u_i}{\sum f_i} \times h$.
• Mode ($Z$): $Z = l + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h$.
• Median ($M$):
  • Ungrouped Data (even observations): $M = \frac{\left(\frac{n}{2}\right)^{th} \text{ obs.} + \left(\frac{n}{2} + 1\right)^{th} \text{ obs.}}{2}$.
  • Ungrouped Data (odd observations): $M = \left(\frac{n + 1}{2}\right)^{th} \text{ observation}$.
  • Grouped Data: $M = l + \left(\frac{\frac{n}{2} - cf}{f}\right) \times h$.
• Empirical Relationship: $Z = 3M - 2\bar{x}$.

Chapter 14: Probability

• Probability Formula: $P(A) = \frac{\text{No. of favourable outcomes}}{\text{Total No. of outcomes}}$.
• Complementary Events: $P(A) + P(\overline{A}) = 1$.
• Range of Probability: $0 \le P(A) \le 1$.
• Certain and Impossible Events: The probability of a certain event is 1 and the probability of an impossible event is 0.