Class 9 Maths Formulas

Cheatsheet Content

### Number Systems - **Natural Numbers (N):** $\{1, 2, 3, ...\}$ - **Whole Numbers (W):** $\{0, 1, 2, 3, ...\}$ - **Integers (Z):** $\{..., -2, -1, 0, 1, 2, ...\}$ - **Rational Numbers (Q):** Numbers that can be written as $p/q$, where $p, q \in Z$ and $q \ne 0$. - **Irrational Numbers:** Numbers that cannot be written as $p/q$. E.g., $\sqrt{2}, \pi$. - **Real Numbers (R):** Collection of all rational and irrational numbers. - **Laws of Exponents:** - $a^m \cdot a^n = a^{m+n}$ - $(a^m)^n = a^{mn}$ - $a^m / a^n = a^{m-n}$ - $a^0 = 1$ - $a^{-n} = 1/a^n$ - $(ab)^n = a^n b^n$ - $(a/b)^n = a^n / b^n$ - $\sqrt[n]{a} = a^{1/n}$ - $\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}$ - $\sqrt[n]{a/b} = \sqrt[n]{a} / \sqrt[n]{b}$ ### Polynomials - **General Form:** $P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0$ - **Degree:** Highest power of the variable. - **Types:** - **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:** - $(a+b)^2 = a^2 + 2ab + b^2$ - $(a-b)^2 = a^2 - 2ab + b^2$ - $a^2 - b^2 = (a-b)(a+b)$ - $(x+a)(x+b) = x^2 + (a+b)x + ab$ - $(a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$ - $(a+b)^3 = a^3 + b^3 + 3ab(a+b) = a^3 + 3a^2b + 3ab^2 + b^3$ - $(a-b)^3 = a^3 - b^3 - 3ab(a-b) = a^3 - 3a^2b + 3ab^2 - b^3$ - $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$ - $a^3 - b^3 = (a-b)(a^2 + ab + b^2)$ - $a^3 + b^3 + c^3 - 3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ca)$ - If $a+b+c=0$, then $a^3+b^3+c^3=3abc$ ### Linear Equations in Two Variables - **General Form:** $ax + by + c = 0$, where $a, b, c$ are real numbers and $a, b \ne 0$. - **Solution:** A pair of values $(x,y)$ that satisfies the equation. - **Graph:** A straight line. Every point on the line is a solution. ### Coordinate Geometry - **Coordinates of a point:** $(x, y)$ - **Origin:** $(0, 0)$ - **X-axis:** $y=0$ - **Y-axis:** $x=0$ - **Quadrants:** - I: $(+, +)$ - II: $(-, +)$ - III: $(-, -)$ - IV: $(+, -)$ ### Euclid's Geometry - **Axioms:** Basic facts that are taken for granted without proof. - **Postulates:** Specific axioms for geometry. - **Theorems:** Statements proved using definitions, axioms, postulates, and previously proved statements. ### Lines and Angles - **Intersecting Lines:** Have one common point. - **Parallel Lines:** Never intersect. - **Transversal:** A line intersecting two or more parallel lines. - **Angle Pairs:** - **Vertically opposite angles:** Equal. - **Corresponding angles:** Equal (if lines are parallel). - **Alternate interior angles:** Equal (if lines are parallel). - **Interior angles on the same side of transversal:** Sum is $180^\circ$ (if lines are parallel). - **Linear Pair:** Sum of two adjacent angles on a straight line is $180^\circ$. - **Complementary Angles:** Sum is $90^\circ$. - **Supplementary Angles:** Sum is $180^\circ$. - **Triangle Angle Sum Property:** Sum of angles in a triangle is $180^\circ$. - **Exterior Angle Property:** An exterior angle of a triangle equals the sum of the two interior opposite angles. ### Triangles - **Congruence Rules:** - **SSS:** Side-Side-Side - **SAS:** Side-Angle-Side - **ASA:** Angle-Side-Angle - **AAS:** Angle-Angle-Side - **RHS:** Right-Hypotenuse-Side (for right-angled triangles) - **Isosceles Triangle Property:** Angles opposite to equal sides are equal, and vice-versa. - **Inequalities in a Triangle:** - Side opposite to the larger angle is longer. - Sum of any two sides is greater than the third side. - Difference of any two sides is less than the third side. ### Quadrilaterals - **Angle Sum Property:** Sum of angles in a quadrilateral is $360^\circ$. - **Parallelogram Properties:** - Opposite sides are equal. - Opposite angles are equal. - Diagonals bisect each other. - Adjacent angles are supplementary. - **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 its length. ### Areas of Parallelograms and Triangles - **Parallelograms on the same base and between the same parallels** have equal area. - **Triangles on the same base and between the same parallels** have equal area. - If a triangle and a parallelogram are on the same base and between the same parallels, the **area of the triangle is half the area of the parallelogram**. ### Circles - **Radius (r):** Distance from center to any point on the circle. - **Diameter (d):** $d = 2r$. - **Chord:** Line segment joining any two points on the circle. - **Arc:** A part of the circumference. - **Sector:** Region between two radii and an arc. - **Segment:** Region between a chord and an arc. - **Angle subtended by an arc at the center** is double the angle subtended by it at any point on the remaining part of the circle. - **Angles in the same segment** are equal. - **Angle in a semicircle** is a right angle ($90^\circ$). - **Cyclic Quadrilateral:** A quadrilateral whose all four vertices lie on a circle. - Sum of opposite angles is $180^\circ$. ### Constructions - Bisecting a line segment. - Bisecting an angle. - Constructing angles of $30^\circ, 45^\circ, 60^\circ, 90^\circ$. - Constructing a triangle given specific conditions (e.g., base, base angle, sum/difference of other two sides). ### Heron's Formula - For a triangle with sides $a, b, c$: - **Semi-perimeter (s):** $s = (a+b+c)/2$ - **Area of Triangle:** $\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$ - **Area of an equilateral triangle:** $(\sqrt{3}/4) a^2$ - **Area of a right-angled triangle:** $(1/2) \times \text{base} \times \text{height}$ ### Surface Areas and Volumes - **Cuboid:** - **Lateral Surface Area:** $2h(l+b)$ - **Total Surface Area:** $2(lb+bh+hl)$ - **Volume:** $l \times b \times h$ - **Cube:** - **Lateral Surface Area:** $4a^2$ - **Total Surface Area:** $6a^2$ - **Volume:** $a^3$ - **Right Circular Cylinder:** - **Curved Surface Area:** $2\pi rh$ - **Total Surface Area:** $2\pi r(r+h)$ - **Volume:** $\pi r^2 h$ - **Right Circular Cone:** - **Slant Height (l):** $l = \sqrt{r^2+h^2}$ - **Curved Surface Area:** $\pi rl$ - **Total Surface Area:** $\pi r(l+r)$ - **Volume:** $(1/3) \pi r^2 h$ - **Sphere:** - **Surface Area:** $4\pi r^2$ - **Volume:** $(4/3) \pi r^3$ - **Hemisphere:** - **Curved Surface Area:** $2\pi r^2$ - **Total Surface Area:** $3\pi r^2$ - **Volume:** $(2/3) \pi r^3$ ### Statistics - **Measures of Central Tendency:** - **Mean:** Sum of observations / Number of observations - **Median:** Middle value of ordered data. - 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 most frequently occurring observation. - **Bar Graphs, Histograms, Frequency Polygons:** Visual representations of data. ### Probability - **Probability of an Event E (P(E)):** - $P(E) = \text{Number of favorable outcomes} / \text{Total number of outcomes}$ - **$0 \le P(E) \le 1$** - **$P(\text{event}) + P(\text{not event}) = 1$** - **Impossible Event:** $P(E) = 0$ - **Sure Event:** $P(E) = 1$