All Class Alzebra – 50 Formulas (बीजगणित सूत्र)
1. Identities (समानताएँ)
(a + b)² = a² + 2ab + b²
(a - b)² = a² - 2ab + b²
(a + b)(a - b) = a² - b²
(x + a)(x + b) = x² + (a + b)x + ab
(x - a)(x - b) = x² - (a + b)x + ab
(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
(a + b)² + (a - b)² = 2(a² + b²)
(a + b)² - (a - b)² = 4ab
(x + y)² + (x - y)² = 2(x² + y²)
(a - b - c)² = a² + b² + c² - 2(ab + bc - ca)
2. Cubes (घन सूत्र)
(a + b)³ = a³ + b³ + 3ab(a + b)
(a - b)³ = a³ - b³ - 3ab(a - b)
a³ + b³ = (a + b)(a² - ab + b²)
a³ - b³ = (a - b)(a² + ab + b²)
(x + y + z)³ = x³ + y³ + z³ + 3(x + y)(y + z)(z + x)
(x + y)³ + (x - y)³ = 2(x³ + 3xy²)
(x + y)³ - (x - y)³ = 6x²y + 2y³
(a + b)³ + (a - b)³ = 2(a³ + 3ab²)
(a + b)³ - (a - b)³ = 6a²b + 2b³
(p + q)³ + (p - q)³ = 2(p³ + 3pq²)
3. Factorization (गुणनखंडन)
x² + (a + b)x + ab = (x + a)(x + b)
x² - (a + b)x + ab = (x - a)(x - b)
ax² + bx + c = a(x - α)(x - β)
x² - (m + n)x + mn = (x - m)(x - n)
x² - p² = (x - p)(x + p)
a² - b² = (a - b)(a + b)
x³ + y³ = (x + y)(x² - xy + y²)
x³ - y³ = (x - y)(x² + xy + y²)
x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - zx)
x⁴ - y⁴ = (x² - y²)(x² + y²)
Extra Useful Identities (अतिरिक्त उपयोगी समानताएँ)
(a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴
(a - b)⁴ = a⁴ - 4a³b + 6a²b² - 4ab³ + b⁴
(x + y)⁵ = x⁵ + 5x⁴y + 10x³y² + 10x²y³ + 5xy⁴ + y⁵
(x - y)⁵ = x⁵ - 5x⁴y + 10x³y² - 10x²y³ + 5xy⁴ - y⁵
(a + b)⁶ = a⁶ + 6a⁵b + 15a⁴b² + 20a³b³ + 15a²b⁴ + 6ab⁵ + b⁶
(a - b)⁶ = a⁶ - 6a⁵b + 15a⁴b² - 20a³b³ + 15a²b⁴ - 6ab⁵ + b⁶
All Class Geometry + Trigonometry Formulas (ज्यामिति + त्रिकोणमिति सूत्र)4/h2>
Geometry (ज्यामिति)
Triangle area = (1/2) × base × height (त्रिभुज क्षेत्रफल)
Heron’s formula = √[s(s-a)(s-b)(s-c)] (हीरोन सूत्र)
Pythagoras theorem = a² + b² = c² (पाइथागोरस प्रमेय)
Equilateral triangle area = (√3/4)a² (समभुज त्रिभुज क्षेत्रफल)
Rectangle area = l × b (आयत क्षेत्रफल)
Square area = a² (वर्ग क्षेत्रफल)
Parallelogram area = base × height (समांतर चतुर्भुज क्षेत्रफल)
Rhombus area = (1/2)d₁d₂ (समचतुर्भुज क्षेत्रफल)
Trapezium area = (1/2)(a+b)h (समलंब क्षेत्रफल)
Circle area = πr² (वृत्त क्षेत्रफल)
Circumference = 2πr (वृत्त परिधि)
Arc length = (θ/360) × 2πr (चाप लंबाई)
Sector area = (θ/360) × πr² (खंड क्षेत्रफल)
Sphere volume = (4/3)πr³ (गोला आयतन)
Sphere surface area = 4πr² (गोला पृष्ठीय क्षेत्रफल)
Cylinder volume = πr²h (बेलन आयतन)
Cylinder surface area = 2πr(h+r) (बेलन क्षेत्रफल)
Cone volume = (1/3)πr²h (शंकु आयतन)
Cone surface area = πr(l+r) (शंकु क्षेत्रफल)
Cube volume = a³ (घन आयतन)
Cube surface area = 6a² (घन पृष्ठीय क्षेत्रफल)
Cuboid volume = l × b × h (घनाभ आयतन)
Cuboid surface area = 2(lb+bh+hl) (घनाभ पृष्ठीय क्षेत्रफल)
Diagonal of cube = √3a (घन विकर्ण)
Diagonal of cuboid = √(l²+b²+h²) (घनाभ विकर्ण)
Trigonometry (त्रिकोणमिति)
sin θ = opposite/hypotenuse (साइन)
cos θ = adjacent/hypotenuse (कोसाइन)
tan θ = opposite/adjacent (टैन्जेंट)
cot θ = adjacent/opposite (कोटैन्जेंट)
sec θ = hypotenuse/adjacent (सेकेंट)
csc θ = hypotenuse/opposite (कोसेकेंट)
Identity: sin²θ + cos²θ = 1
Identity: 1 + tan²θ = sec²θ
Identity: 1 + cot²θ = csc²θ
sin(A+B) = sinA cosB + cosA sinB
cos(A+B) = cosA cosB - sinA sinB
tan(A+B) = (tanA + tanB)/(1 - tanA tanB)
sin(A-B) = sinA cosB - cosA sinB
cos(A-B) = cosA cosB + sinA sinB
tan(A-B) = (tanA - tanB)/(1 + tanA tanB)
sin2A = 2sinA cosA
cos2A = cos²A - sin²A
tan2A = (2tanA)/(1 - tan²A)
sin3A = 3sinA - 4sin³A
cos3A = 4cos³A - 3cosA
tan3A = (3tanA - tan³A)/(1 - 3tan²A)
sin²A = (1 - cos2A)/2
cos²A = (1 + cos2A)/2
tan²A = (1 - cos2A)/(1 + cos2A)
sin⁴A + cos⁴A = 1 - 2sin²A cos²A
sin⁶A + cos⁶A = 1 - 3sin²A cos²A
Triangle area = (1/2) × base × height (त्रिभुज क्षेत्रफल)
Heron’s formula = √[s(s-a)(s-b)(s-c)] (हीरोन सूत्र)
Pythagoras theorem = a² + b² = c² (पाइथागोरस प्रमेय)
Equilateral triangle area = (√3/4)a² (समभुज त्रिभुज क्षेत्रफल)
Rectangle area = l × b (आयत क्षेत्रफल)
Square area = a² (वर्ग क्षेत्रफल)
Parallelogram area = base × height (समांतर चतुर्भुज क्षेत्रफल)
Rhombus area = (1/2)d₁d₂ (समचतुर्भुज क्षेत्रफल)
Trapezium area = (1/2)(a+b)h (समलंब क्षेत्रफल)
Circle area = πr² (वृत्त क्षेत्रफल)
Circumference = 2πr (वृत्त परिधि)
Arc length = (θ/360) × 2πr (चाप लंबाई)
Sector area = (θ/360) × πr² (खंड क्षेत्रफल)
Sphere volume = (4/3)πr³ (गोला आयतन)
Sphere surface area = 4πr² (गोला पृष्ठीय क्षेत्रफल)
Cylinder volume = πr²h (बेलन आयतन)
Cylinder surface area = 2πr(h+r) (बेलन क्षेत्रफल)
Cone volume = (1/3)πr²h (शंकु आयतन)
Cone surface area = πr(l+r) (शंकु क्षेत्रफल)
Cube volume = a³ (घन आयतन)
Cube surface area = 6a² (घन पृष्ठीय क्षेत्रफल)
Cuboid volume = l × b × h (घनाभ आयतन)
Cuboid surface area = 2(lb+bh+hl) (घनाभ पृष्ठीय क्षेत्रफल)
Diagonal of cube = √3a (घन विकर्ण)
Diagonal of cuboid = √(l²+b²+h²) (घनाभ विकर्ण)
sin θ = opposite/hypotenuse (साइन)
cos θ = adjacent/hypotenuse (कोसाइन)
tan θ = opposite/adjacent (टैन्जेंट)
cot θ = adjacent/opposite (कोटैन्जेंट)
sec θ = hypotenuse/adjacent (सेकेंट)
csc θ = hypotenuse/opposite (कोसेकेंट)
Identity: sin²θ + cos²θ = 1
Identity: 1 + tan²θ = sec²θ
Identity: 1 + cot²θ = csc²θ
sin(A+B) = sinA cosB + cosA sinB
cos(A+B) = cosA cosB - sinA sinB
tan(A+B) = (tanA + tanB)/(1 - tanA tanB)
sin(A-B) = sinA cosB - cosA sinB
cos(A-B) = cosA cosB + sinA sinB
tan(A-B) = (tanA - tanB)/(1 + tanA tanB)
sin2A = 2sinA cosA
cos2A = cos²A - sin²A
tan2A = (2tanA)/(1 - tan²A)
sin3A = 3sinA - 4sin³A
cos3A = 4cos³A - 3cosA
tan3A = (3tanA - tan³A)/(1 - 3tan²A)
sin²A = (1 - cos2A)/2
cos²A = (1 + cos2A)/2
tan²A = (1 - cos2A)/(1 + cos2A)
sin⁴A + cos⁴A = 1 - 2sin²A cos²A
sin⁶A + cos⁶A = 1 - 3sin²A cos²A