Class 9 Maths Formulae

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1. Number Systems Rational Numbers: Numbers that can be expressed in the form $p/q$, where $p, q$ are integers and $q \neq 0$. Irrational Numbers: Numbers that cannot be expressed in the form $p/q$. E.g., $\sqrt{2}, \pi$. Real Numbers: Collection of rational and irrational numbers. Laws of Exponents: $a^m \cdot a^n = a^{m+n}$ $a^m / a^n = a^{m-n}$ $(a^m)^n = a^{mn}$ $(ab)^n = a^n b^n$ $(a/b)^n = a^n / b^n$ $a^0 = 1$ $a^{-n} = 1/a^n$ $a^{m/n} = (\sqrt[n]{a})^m = \sqrt[n]{a^m}$ 2. Polynomials Standard Form: $P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0$, where $a_n \neq 0$. Degree of a Polynomial: The highest power of the variable in a polynomial. Types of Polynomials: Linear: Degree 1 (e.g., $ax+b$) Quadratic: Degree 2 (e.g., $ax^2+bx+c$) Cubic: Degree 3 (e.g., $ax^3+bx^2+cx+d$) Remainder Theorem: If $P(x)$ is divided by $(x-a)$, the remainder is $P(a)$. Factor Theorem: $(x-a)$ is a factor of $P(x)$ if $P(a) = 0$. Algebraic Identities: $(x+y)^2 = x^2 + 2xy + y^2$ $(x-y)^2 = x^2 - 2xy + y^2$ $x^2 - y^2 = (x+y)(x-y)$ $(x+a)(x+b) = x^2 + (a+b)x + ab$ $(x+y+z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2zx$ $(x+y)^3 = x^3 + y^3 + 3xy(x+y) = x^3 + y^3 + 3x^2y + 3xy^2$ $(x-y)^3 = x^3 - y^3 - 3xy(x-y) = x^3 - y^3 - 3x^2y + 3xy^2$ $x^3 + y^3 = (x+y)(x^2 - xy + y^2)$ $x^3 - y^3 = (x-y)(x^2 + xy + y^2)$ $x^3 + y^3 + z^3 - 3xyz = (x+y+z)(x^2+y^2+z^2-xy-yz-zx)$ If $x+y+z=0$, then $x^3+y^3+z^3 = 3xyz$ 3. Coordinate Geometry Cartesian Plane: Formed by two perpendicular lines, X-axis (horizontal) and Y-axis (vertical). Coordinates of a point: $(x, y)$, where $x$ is the abscissa and $y$ is the ordinate. Origin: Point of intersection of X and Y axes, coordinates $(0,0)$. Quadrants: The plane is divided into four quadrants. 4. Linear Equations in Two Variables Standard Form: $ax + by + c = 0$, where $a, b, c$ are real numbers, and $a, b \neq 0$. Every point on the graph of a linear equation is a solution of the equation. 5. Euclid's Geometry Axioms: Self-evident truths that are not proved. Postulates: Specific axioms relating to geometry. Theorems: Statements that are proved using definitions, axioms, previously proved statements, and deductive reasoning. 6. Lines and Angles Types of Angles: Acute Angle: $0^\circ Right Angle: $\theta = 90^\circ$ Obtuse Angle: $90^\circ Straight Angle: $\theta = 180^\circ$ Reflex Angle: $180^\circ Complementary Angles: Sum is $90^\circ$. Supplementary Angles: Sum is $180^\circ$. Linear Pair of Angles: Two adjacent angles whose non-common arms are opposite rays (sum is $180^\circ$). Vertically Opposite Angles: When two lines intersect, vertically opposite angles are equal. Parallel Lines and a Transversal: Corresponding Angles are equal. Alternate Interior Angles are equal. Alternate Exterior Angles are equal. Interior Angles on the same side of the transversal are supplementary. Sum of angles in a triangle is $180^\circ$. An exterior angle of a triangle is equal to the sum of the two interior opposite angles. 7. Triangles Congruence Rules: SAS: Side-Angle-Side ASA: Angle-Side-Angle AAS: Angle-Angle-Side SSS: Side-Side-Side RHS: Right angle-Hypotenuse-Side (for right-angled triangles) Angles opposite to equal sides of an isosceles triangle are equal. Sides opposite to equal angles of a triangle are equal. In a triangle, the angle opposite to the longer side is larger. In a triangle, the side opposite to the larger angle is longer. The sum of any two sides of a triangle is greater than the third side. The difference of any two sides of a triangle is less than the third side. 8. Quadrilaterals Sum of angles in a quadrilateral is $360^\circ$. A diagonal divides a parallelogram into two congruent triangles. In a parallelogram: Opposite sides are equal. Opposite angles are equal. Diagonals bisect each other. A quadrilateral is a parallelogram if: Opposite sides are equal. Opposite angles are equal. Diagonals bisect each other. A pair of opposite sides is equal and parallel. Mid-point Theorem: The line segment joining the mid-points of two sides of a triangle is parallel to the third side and half of it. Converse of Mid-point Theorem: A line drawn through the mid-point of one side of a triangle parallel to another side bisects the third side. 9. Areas of Parallelograms and Triangles Parallelograms on the same base and between the same parallels are equal in area. Area of a parallelogram = base $\times$ height. Area of a triangle = $\frac{1}{2} \times$ base $\times$ height. Triangles on the same base and between the same parallels are equal in area. A median of a triangle divides it into two triangles of equal areas. 10. Circles Equal chords of a circle subtend equal angles at the centre. If the angles subtended by the chords at the centre are equal, then the chords are equal. The perpendicular from the centre to a chord bisects the chord. The line drawn through the centre to bisect a chord is perpendicular to the chord. Equal chords are equidistant from the centre. Chords equidistant from the centre are equal. The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle. Angles in the same segment of a circle are equal. Angle in a semicircle is a right angle ($90^\circ$). The sum of either pair of opposite angles of a cyclic quadrilateral is $180^\circ$. If the sum of a pair of opposite angles of a quadrilateral is $180^\circ$, the quadrilateral is cyclic. 11. Constructions Construction of angle bisector. Construction of perpendicular bisector. Construction of angles $15^\circ, 30^\circ, 45^\circ, 60^\circ, 75^\circ, 90^\circ, 105^\circ, 120^\circ, 135^\circ, 150^\circ$. Construction of a triangle given its base, a base angle and sum/difference of other two sides. Construction of a triangle given its perimeter and two base angles. 12. Heron's Formula For a triangle with sides $a, b, c$: Semi-perimeter $s = (a+b+c)/2$ Area of triangle $= \sqrt{s(s-a)(s-b)(s-c)}$ Area of a quadrilateral can be found by dividing it into two triangles and applying Heron's formula. 13. Surface Areas and Volumes Cuboid: Length $l$, Breadth $b$, Height $h$ Lateral Surface Area (LSA) or Area of 4 walls $= 2h(l+b)$ Total Surface Area (TSA) $= 2(lb + bh + hl)$ Volume $= l \times b \times h$ Cube: Side $a$ Lateral Surface Area (LSA) $= 4a^2$ Total Surface Area (TSA) $= 6a^2$ Volume $= a^3$ Right Circular Cylinder: Radius $r$, Height $h$ Curved Surface Area (CSA) $= 2\pi rh$ Total Surface Area (TSA) $= 2\pi r(r+h)$ Volume $= \pi r^2 h$ Right Circular Cone: Radius $r$, Height $h$, Slant Height $l = \sqrt{r^2+h^2}$ Curved Surface Area (CSA) $= \pi rl$ Total Surface Area (TSA) $= \pi r(l+r)$ Volume $= \frac{1}{3} \pi r^2 h$ Sphere: Radius $r$ Surface Area $= 4\pi r^2$ Volume $= \frac{4}{3} \pi r^3$ Hemisphere: Radius $r$ Curved Surface Area (CSA) $= 2\pi r^2$ Total Surface Area (TSA) $= 3\pi r^2$ Volume $= \frac{2}{3} \pi r^3$ 14. Statistics Mean (Ungrouped Data): $\bar{x} = \frac{\sum x_i}{n}$ Mean (Grouped Data - Direct Method): $\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$ Median: The middle-most observation. If $n$ is odd, Median = $((n+1)/2)$-th observation. If $n$ is even, Median = Average of $(n/2)$-th and $((n/2)+1)$-th observations. Mode: The observation that occurs most frequently. Range: Highest value - Lowest value. 15. Probability Probability of an Event E: $P(E) = \frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}}$ $0 \le P(E) \le 1$ $P(\text{certain event}) = 1$ $P(\text{impossible event}) = 0$ $P(E) + P(\text{not E}) = 1$