Math Formulas (Class 8-10)

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1. Number Systems & Algebra (Class 8-10) 1.1 Real Numbers Euclid's Division Lemma: $a = bq + r$, where $0 \le r Fundamental Theorem of Arithmetic: Every composite number can be expressed as a product of primes uniquely. HCF & LCM: For two positive integers $a, b$, $HCF(a,b) \times LCM(a,b) = a \times b$ Rational Numbers: Numbers of the form $p/q$, where $q \ne 0$. Irrational Numbers: Numbers that cannot be expressed as $p/q$ (e.g., $\sqrt{2}, \pi$). Terminating Decimals: Denominator $q$ has prime factors $2^m 5^n$. 1.2 Polynomials Degree of Polynomial: Highest power of the variable. Linear: $ax+b$ (degree 1) Quadratic: $ax^2+bx+c$ (degree 2) Cubic: $ax^3+bx^2+cx+d$ (degree 3) Zeros of a Polynomial: Values of $x$ for which $P(x)=0$. Relationship between Zeros and Coefficients (Quadratic $ax^2+bx+c$): Sum of zeros ($\alpha + \beta$): $-b/a$ Product of zeros ($\alpha \beta$): $c/a$ Relationship between Zeros and Coefficients (Cubic $ax^3+bx^2+cx+d$): Sum of zeros ($\alpha + \beta + \gamma$): $-b/a$ Sum of products of zeros taken two at a time ($\alpha\beta + \beta\gamma + \gamma\alpha$): $c/a$ Product of zeros ($\alpha\beta\gamma$): $-d/a$ 1.3 Algebraic Identities (Class 8-10) $(a+b)^2 = a^2+2ab+b^2$ $(a-b)^2 = a^2-2ab+b^2$ $a^2-b^2 = (a-b)(a+b)$ $(x+a)(x+b) = x^2+(a+b)x+ab$ $(a+b+c)^2 = a^2+b^2+c^2+2ab+2bc+2ca$ $(a+b)^3 = a^3+b^3+3ab(a+b) = a^3+b^3+3a^2b+3ab^2$ $(a-b)^3 = a^3-b^3-3ab(a-b) = a^3-b^3-3a^2b+3ab^2$ $a^3+b^3 = (a+b)(a^2-ab+b^2)$ $a^3-b^3 = (a-b)(a^2+ab+b^2)$ $a^3+b^3+c^3-3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ca)$ If $a+b+c=0$, then $a^3+b^3+c^3=3abc$ 1.4 Linear Equations in Two Variables General form: $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ Conditions for Solutions: Unique Solution: $\frac{a_1}{a_2} \ne \frac{b_1}{b_2}$ (Intersecting lines) No Solution: $\frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2}$ (Parallel lines) Infinitely Many Solutions: $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$ (Coincident lines) 1.5 Quadratic Equations General form: $ax^2+bx+c=0$, where $a \ne 0$. Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ Discriminant ($D$): $D = b^2-4ac$ If $D>0$, two distinct real roots. If $D=0$, two equal real roots. If $D 2. Coordinate Geometry (Class 9-10) Distance Formula: Between $(x_1, y_1)$ and $(x_2, y_2)$ is $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$. Section Formula: Point $(x,y)$ dividing line segment joining $(x_1, y_1)$ and $(x_2, y_2)$ in ratio $m:n$. Internal Division: $x = \frac{mx_2+nx_1}{m+n}$, $y = \frac{my_2+ny_1}{m+n}$ Midpoint: $x = \frac{x_1+x_2}{2}$, $y = \frac{y_1+y_2}{2}$ (for $m=n=1$) Area of Triangle: With vertices $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$. $$ \frac{1}{2} |x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)| $$ 3. Trigonometry (Class 10) 3.1 Trigonometric Ratios $\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$ $\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$ $\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{\sin \theta}{\cos \theta}$ $\csc \theta = \frac{1}{\sin \theta}$ $\sec \theta = \frac{1}{\cos \theta}$ $\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}$ 3.2 Trigonometric Identities $\sin^2 \theta + \cos^2 \theta = 1$ $1 + \tan^2 \theta = \sec^2 \theta$ $1 + \cot^2 \theta = \csc^2 \theta$ 3.3 Complementary Angles $\sin(90^\circ - \theta) = \cos \theta$ $\cos(90^\circ - \theta) = \sin \theta$ $\tan(90^\circ - \theta) = \cot \theta$ $\cot(90^\circ - \theta) = \tan \theta$ $\sec(90^\circ - \theta) = \csc \theta$ $\csc(90^\circ - \theta) = \sec \theta$ 3.4 Specific Angle Values $\theta$ $0^\circ$ $30^\circ$ $45^\circ$ $60^\circ$ $90^\circ$ $\sin \theta$ $0$ $1/2$ $1/\sqrt{2}$ $\sqrt{3}/2$ $1$ $\cos \theta$ $1$ $\sqrt{3}/2$ $1/\sqrt{2}$ $1/2$ $0$ $\tan \theta$ $0$ $1/\sqrt{3}$ $1$ $\sqrt{3}$ Undefined 4. Geometry (Class 8-10) 4.1 Triangles Area of Triangle: $\frac{1}{2} \times \text{base} \times \text{height}$ Pythagoras Theorem: In a right-angled triangle, $h^2 = p^2 + b^2$. Similar Triangles: AA, SSS, SAS criteria. Ratio of areas: If $\triangle ABC \sim \triangle PQR$, then $\frac{Area(ABC)}{Area(PQR)} = (\frac{AB}{PQ})^2 = (\frac{BC}{QR})^2 = (\frac{CA}{RP})^2$. Thales Theorem (Basic Proportionality Theorem): If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio. 4.2 Circles 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 ($90^\circ$). Opposite angles of a cyclic quadrilateral are supplementary ($180^\circ$). Tangent at any point of a circle is perpendicular to the radius through the point of contact. Length of tangents drawn from an external point to a circle are equal. 5. Mensuration (Class 8-10) 5.1 Areas & Perimeters of 2D Shapes Shape Perimeter/Circumference Area Square (side $a$) $4a$ $a^2$ Rectangle (length $l$, breadth $b$) $2(l+b)$ $l \times b$ Triangle (base $b$, height $h$) Sum of sides $\frac{1}{2} \times b \times h$ Circle (radius $r$) $2\pi r$ $\pi r^2$ Semicircle (radius $r$) $\pi r + 2r$ $\frac{1}{2}\pi r^2$ Sector of circle (radius $r$, angle $\theta$ in degrees) $2r + \frac{\theta}{360^\circ} \times 2\pi r$ $\frac{\theta}{360^\circ} \times \pi r^2$ Parallelogram (base $b$, height $h$) $2(\text{side}_1 + \text{side}_2)$ $b \times h$ Rhombus (diagonals $d_1, d_2$) $4 \times \text{side}$ $\frac{1}{2} d_1 d_2$ Trapezium (parallel sides $a, b$, height $h$) Sum of sides $\frac{1}{2}(a+b)h$ 5.2 Surface Areas & Volumes of 3D Shapes Shape Lateral/Curved Surface Area Total Surface Area Volume Cuboid ($l, b, h$) $2h(l+b)$ $2(lb+bh+hl)$ $l \times b \times h$ Cube (side $a$) $4a^2$ $6a^2$ $a^3$ Cylinder (radius $r$, height $h$) $2\pi rh$ $2\pi r(h+r)$ $\pi r^2 h$ Cone (radius $r$, height $h$, slant height $l$) ($l = \sqrt{r^2+h^2}$) $\pi rl$ $\pi r(l+r)$ $\frac{1}{3}\pi r^2 h$ Sphere (radius $r$) $4\pi r^2$ $4\pi r^2$ $\frac{4}{3}\pi r^3$ Hemisphere (radius $r$) $2\pi r^2$ $3\pi r^2$ $\frac{2}{3}\pi r^3$ Frustum of a Cone (radii $R, r$, height $h$, slant height $L$) ($L = \sqrt{h^2+(R-r)^2}$) $\pi (R+r)L$ $\pi L(R+r) + \pi R^2 + \pi r^2$ $\frac{1}{3}\pi h(R^2+Rr+r^2)$ 6. Statistics (Class 9-10) Mean (Ungrouped Data): $\bar{x} = \frac{\sum x_i}{n}$ Mean (Grouped Data): Direct Method: $\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$ Assumed Mean Method: $\bar{x} = A + \frac{\sum f_i d_i}{\sum f_i}$, where $d_i = x_i - A$ Step Deviation Method: $\bar{x} = A + \left(\frac{\sum f_i u_i}{\sum f_i}\right)h$, where $u_i = \frac{x_i - A}{h}$ Median (Grouped Data): $Median = L + \left(\frac{\frac{n}{2} - cf}{f}\right)h$ $L$: Lower limit of median class $n$: Total frequency $cf$: Cumulative frequency of class preceding the median class $f$: Frequency of median class $h$: Class size Mode (Grouped Data): $Mode = L + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right)h$ $L$: Lower limit of modal class $f_1$: Frequency of modal class $f_0$: Frequency of class preceding modal class $f_2$: Frequency of class succeeding modal class $h$: Class size Empirical Relationship: $3 \times Median = Mode + 2 \times Mean$ 7. Probability (Class 9-10) Probability of an Event E: $P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$ $0 \le P(E) \le 1$ Sum of probabilities of all elementary events is 1. $P(\text{event not E}) = 1 - P(E)$ or $P(\bar{E}) = 1 - P(E)$ Impossible event: $P(\text{impossible event}) = 0$ Sure event: $P(\text{sure event}) = 1$