• Euclid's Division Lemma: \(a = bq + r\), where \(0 \leq r < b\)
• HCF × LCM: \(\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b\)
• Fundamental Theorem of Arithmetic: Every composite number can be expressed as a product of primes

For polynomial \(ax^2 + bx + c\):

• Sum of zeros: \(\alpha + \beta = -\frac{b}{a}\)
• Product of zeros: \(\alpha \beta = \frac{c}{a}\)
• Quadratic polynomial from zeros: \(x^2 - (\alpha + \beta)x + \alpha\beta\)

For polynomial \(ax^3 + bx^2 + cx + d\):

• \(\alpha + \beta + \gamma = -\frac{b}{a}\)
• \(\alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a}\)
• \(\alpha\beta\gamma = -\frac{d}{a}\)

where degree of \(r(x) < \) degree of \(g(x)\)

For equations \(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\):

• Unique solution (Intersecting lines): \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\)
• No solution (Parallel lines): \(\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}\)
• Infinite solutions (Coincident lines): \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\)

• If \(D > 0\): Two distinct real roots
• If \(D = 0\): Two equal real roots (\(x = -\frac{b}{2a}\))
• If \(D < 0\): No real roots

• nth term: \(a_n = a + (n-1)d\)
• Sum of n terms: \(S_n = \frac{n}{2}[2a + (n-1)d]\)
• Alternative sum formula: \(S_n = \frac{n}{2}[a + l]\), where \(l\) is last term
• Common difference: \(d = a_n - a_{n-1}\)
• Middle term (odd n): \(a_{\frac{n+1}{2}}\)

• AAA Similarity: If corresponding angles are equal
• SSS Similarity: \(\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}\)
• SAS Similarity: Two sides proportional and included angle equal

For similar triangles:

• Distance Formula: \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\)
• Section Formula (Internal): \(\left(\frac{m_1x_2 + m_2x_1}{m_1+m_2}, \frac{m_1y_2 + m_2y_1}{m_1+m_2}\right)\)
• Mid-point Formula: \(\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\)
• Area of Triangle: \(\frac{1}{2}|x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)|\)

• \(\sin\theta = \frac{\text{Perpendicular}}{\text{Hypotenuse}} = \frac{P}{H}\)
• \(\cos\theta = \frac{\text{Base}}{\text{Hypotenuse}} = \frac{B}{H}\)
• \(\tan\theta = \frac{\text{Perpendicular}}{\text{Base}} = \frac{P}{B}\)
• \(\csc\theta = \frac{H}{P}\)
• \(\sec\theta = \frac{H}{B}\)
• \(\cot\theta = \frac{B}{P}\)

• \(\sin\theta \cdot \csc\theta = 1\)
• \(\cos\theta \cdot \sec\theta = 1\)
• \(\tan\theta \cdot \cot\theta = 1\)
• \(\tan\theta = \frac{\sin\theta}{\cos\theta}\)
• \(\cot\theta = \frac{\cos\theta}{\sin\theta}\)

• \(\sin^2\theta + \cos^2\theta = 1\)
• \(1 + \tan^2\theta = \sec^2\theta\)
• \(1 + \cot^2\theta = \csc^2\theta\)

• \(\sin(90° - \theta) = \cos\theta\)
• \(\cos(90° - \theta) = \sin\theta\)
• \(\tan(90° - \theta) = \cot\theta\)
• \(\cot(90° - \theta) = \tan\theta\)
• \(\sec(90° - \theta) = \csc\theta\)
• \(\csc(90° - \theta) = \sec\theta\)

• Tangent perpendicular to radius: The tangent at any point is perpendicular to the radius through point of contact
• Length of tangent from external point: If two tangents are drawn from an external point, they are equal in length
• Tangent formula: \(PA = \sqrt{OP^2 - r^2}\) where \(PA\) is tangent, \(OP\) is distance from center to external point, \(r\) is radius

• Area of circle: \(A = \pi r^2\)
• Circumference: \(C = 2\pi r\)
• Area of sector: \(A = \frac{\theta}{360°} \times \pi r^2\)
• Length of arc: \(l = \frac{\theta}{360°} \times 2\pi r\)
• Area of segment: \(A = \text{Area of sector} - \text{Area of triangle}\)

• Volume: \(V = a^3\)
• Total Surface Area: \(TSA = 6a^2\)
• Lateral Surface Area: \(LSA = 4a^2\)

• Volume: \(V = l \times b \times h\)
• Total Surface Area: \(TSA = 2(lb + bh + hl)\)
• Lateral Surface Area: \(LSA = 2h(l + b)\)

• Volume: \(V = \pi r^2 h\)
• Curved Surface Area: \(CSA = 2\pi rh\)
• Total Surface Area: \(TSA = 2\pi r(r + h)\)

• Volume: \(V = \frac{1}{3}\pi r^2 h\)
• Curved Surface Area: \(CSA = \pi rl\) where \(l = \sqrt{r^2 + h^2}\)
• Total Surface Area: \(TSA = \pi r(r + l)\)

• Volume: \(V = \frac{4}{3}\pi r^3\)
• Surface Area: \(SA = 4\pi r^2\)

• Volume: \(V = \frac{2}{3}\pi r^3\)
• Curved Surface Area: \(CSA = 2\pi r^2\)
• Total Surface Area: \(TSA = 3\pi r^2\)

• Volume: \(V = \frac{1}{3}\pi h(r_1^2 + r_2^2 + r_1r_2)\)
• Curved Surface Area: \(CSA = \pi l(r_1 + r_2)\)
• Total Surface Area: \(TSA = \pi[l(r_1 + r_2) + r_1^2 + r_2^2]\)

• Direct Method: \(\bar{x} = \frac{\sum f_ix_i}{\sum f_i}\)
• Assumed Mean Method: \(\bar{x} = a + \frac{\sum f_id_i}{\sum f_i}\) where \(d_i = x_i - a\)
• Step Deviation Method: \(\bar{x} = a + \frac{\sum f_iu_i}{\sum f_i} \times h\) where \(u_i = \frac{x_i - a}{h}\)

where \(l\) = lower limit of median class, \(n\) = total frequency, \(cf\) = cumulative frequency before median class, \(f\) = frequency of median class, \(h\) = class width

where \(l\) = lower limit of modal class, \(f_1\) = frequency of modal class, \(f_0\) = frequency of class before modal class, \(f_2\) = frequency of class after modal class, \(h\) = class width

• Probability: \(P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\)
• \(0 \leq P(E) \leq 1\)
• \(P(\text{not } E) = 1 - P(E)\)
• Sum of all probabilities: \(\sum P(E_i) = 1\)
• Impossible event: \(P(E) = 0\)
• Sure event: \(P(E) = 1\)

• \((a + b)^2 = a^2 + 2ab + b^2\)
• \((a - b)^2 = a^2 - 2ab + b^2\)
• \((a + b)(a - b) = a^2 - b^2\)
• \((a + b)^3 = a^3 + b^3 + 3ab(a + b)\)
• \((a - b)^3 = a^3 - b^3 - 3ab(a - b)\)
• \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)
• \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)
• \((x + a)(x + b) = x^2 + (a + b)x + ab\)
• \((x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2zx\)
• \(x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx)\)