Mathematics Formula Bank

📐 Algebra

Polynomial Definition

$$\ \text{} P(x) = a_n x^n + a_{n-1} x^{n-1} + \\dots + a_1 x + a_0 \ \text{}$$

A mathematical expression consisting of variables and coefficients combined using addition, subtraction, and multiplication.

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Value of a Polynomial

$$\ \text{} P(k) = a_n k^n + a_{n-1} k^{n-1} + \\dots + a_0 \ \text{}$$

The value obtained by substituting a real number k for the variable x in the polynomial P(x).

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Zero of a Polynomial

$$\ \text{} P(\\alpha) = 0 \\iff \\alpha \\text{ is a zero of } P(x) \ \text{}$$

A real number alpha is a zero of polynomial P(x) if and only if P(alpha) = 0.

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Linear Polynomial

$$\ \text{} P(x) = ax + b \\quad (a \\neq 0) \ \text{}$$

A polynomial of degree one, forming a straight line when graphed.

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Quadratic Polynomial

$$\ \text{} P(x) = ax^2 + bx + c \\quad (a \\neq 0) \ \text{}$$

A polynomial of degree two, forming a parabola when graphed.

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Degree of a Polynomial

$$\ \text{} \\text{Degree}(P(x)) = n \\quad (\\text{where } a_n \\neq 0) \ \text{}$$

The highest exponential power of the variable in any non-zero term of the polynomial.

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Factor Theorem

$$\ \text{} P(a) = 0 \\iff (x - a) \\text{ is a factor of } P(x) \ \text{}$$

A theorem linking factors and zeros of a polynomial: (x-a) is a factor if and only if P(a) = 0.

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a2 + b2

$$\ \text{} (a + b)^{2} - 2ab \ \text{}$$

Standard algebraic representation and mathematical formula for studying a2 + b2.

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a - b

$$\ \text{} a^{2} - 2ab + b^{2} \ \text{}$$

Standard algebraic representation and mathematical formula for studying a - b.

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a + b + c

$$\ \text{} a^{2} + b^{2} + c^{2} + 2ab + 2bc + 2ca \ \text{}$$

Standard algebraic representation and mathematical formula for studying a + b + c.

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a - b - c

$$\ \text{} a^{2} + b^{2} + c^{2} - 2ab + 2bc - 2ca \ \text{}$$

Standard algebraic representation and mathematical formula for studying a - b - c.

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a + b

$$\ \text{} a^{3} + 3a^{2}b + 3ab^{2} + b^{3} \ \text{}$$

Standard algebraic representation and mathematical formula for studying a + b.

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a3 - b3

$$\ \text{} (a - b)(a^{2} + ab + b^{2}) \ \text{}$$

Standard algebraic representation and mathematical formula for studying a3 - b3.

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a + b

$$\ \text{} a^{4} + 4a^{3}b + 6a^{2}b^{2} + 4ab^{3} + b^{4} \ \text{}$$

Standard algebraic representation and mathematical formula for studying a + b.

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a4 - b4

$$\ \text{} (a - b)(a + b)(a^{2} + b^{2}) \ \text{}$$

Standard algebraic representation and mathematical formula for studying a4 - b4.

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a5 - b5

$$\ \text{} (a - b)(a^{4} + a^{3}b + a^{2}b^{2} + ab^{3} + b^{4}) \ \text{}$$

Standard algebraic representation and mathematical formula for studying a5 - b5.

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x + y + z

$$\ \text{} x^{2} + y^{2} + z^{2} + 2xy + 2yz + 2xz \ \text{}$$

Standard algebraic representation and mathematical formula for studying x + y + z.

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x - y - z

$$\ \text{} x^{2} + y^{2} + z^{2} - 2xy + 2yz - 2xz \ \text{}$$

Standard algebraic representation and mathematical formula for studying x - y - z.

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x3 + y3 + z3 - 3xyz

$$\ \text{} (x + y + z)(x^{2} + y^{2} + z^{2} - xy - yz - xz) \ \text{}$$

Standard algebraic representation and mathematical formula for studying x3 + y3 + z3 - 3xyz.

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x2 + y2 + z2 - xy - yz - zx

$$\ \text{} 1/2 [(x - y)^{2} + (y - z)^{2} + (z - x)^{2}] \ \text{}$$

Standard algebraic representation and mathematical formula for studying x2 + y2 + z2 - xy - yz - zx.

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x2 + y2 + z2

$$\ \text{} x^{4} + y^{4} + z^{4} + 2(x^{2}y^{2} + y^{2}z^{2} + z^{2}x^{2}) + 2(x^{2}yz + xy^{2}z + xyz^{2}) \ \text{}$$

Standard algebraic representation and mathematical formula for studying x2 + y2 + z2.

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a + b + c

$$\ \text{} a^{3} + b^{3} + c^{3} + 3(a^{2}b + ab^{2} + b^{2}c + bc^{2} + c^{2}a + ac^{2}) + 6abc \ \text{}$$

Standard algebraic representation and mathematical formula for studying a + b + c.

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x + y

$$\ \text{} (x^{2} - y^{2})^{2} \ \text{}$$

Standard algebraic representation and mathematical formula for studying x + y.

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a + b

$$\ \text{} (a^{2} - b^{2})^{2} \ \text{}$$

Standard algebraic representation and mathematical formula for studying a + b.

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x + y + z

$$\ \text{} x^{3} + y^{3} + z^{3} + 3(x^{2}y + xy^{2} + y^{2}z + yz^{2} + z^{2}x + zx^{2}) + 6xyz \ \text{}$$

Standard algebraic representation and mathematical formula for studying x + y + z.

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a + b + c + d

$$\ \text{} a^{2} + b^{2} + c^{2} + d^{2} + 2(ab + ac + ad + bc + bd + cd) \ \text{}$$

Standard algebraic representation and mathematical formula for studying a + b + c + d.

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Linear Equation in One Variable

$$\ \text{} ax + b = 0 \\quad (a \\neq 0) \ \text{}$$

An algebraic equation of degree one involving a single variable.

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Solution of Linear Equation

$$\ \text{} x = -\\frac{b}{a} \ \text{}$$

The unique solution to the standard linear equation ax + b = 0.

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Onto Function (Surjection)

$$\ \text{} \\forall y \\in B, \\; \\exists x \\in A \\text{ such that } f(x) = y \ \text{}$$

A function where every element in the codomain has at least one pre-image in the domain.

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One-to-One Function (Injection)

$$\ \text{} f(x_1) = f(x_2) \\implies x_1 = x_2 \ \text{}$$

A function that maps distinct domain elements to distinct codomain elements.

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Bijective Function (Bijection)

$$\ \text{} f: A \\to B \\text{ is bijective } \\iff f \\text{ is injective and surjective} \ \text{}$$

A function that is both injective (one-to-one) and surjective (onto).

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Constant Function

$$\ \text{} f(x) = c \\quad (\\forall x \\in A) \ \text{}$$

A function whose output value is the same constant for every input value.

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🏛️ Geometry & Vectors

2D Cartesian Coordinate Representation

$$\ \text{} (x, y) \ \text{}$$

An ordered pair defining a unique point in a 2D Cartesian coordinate system.

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Origin Coordinate

$$\ \text{} O(0, 0) \ \text{}$$

The geometric reference point where the abscissa and ordinate axes intersect.

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Cartesian Quadrants

$$\ \text{} \\text{Quadrant I: }(x>0, y>0), \\; \\text{II: }(x<0, y>0), \\; \\text{III: }(x<0, y<0), \\; \\text{IV: }(x>0, y<0) \ \text{}$$

The four regions of the Cartesian plane segmented by the analytical coordinate axes.

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Analytical Distance Formula

$$\ \text{} d = \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \ \text{}$$

Finds the exact Euclidean distance between two coordinate positions on a flat plane.

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Line Midpoint Formula

$$\ \text{} M\\left(\\frac{x_1 + x_2}{2}, \\frac{y_1 + y_2}{2}\\right) \ \text{}$$

Calculates coordinates of the exact center of a line segment joining two known endpoints.

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Complementary Angles Theorem

$$\ \text{} \\theta_1 + \\theta_2 = 90^\\circ \ \text{}$$

Two distinct plane geometric angles whose individual physical addition sums to exactly 90 degrees.

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Supplementary Angles Theorem

$$\ \text{} \\theta_1 + \\theta_2 = 180^\\circ \ \text{}$$

Two distinct plane geometric angles whose individual physical addition sums to exactly 180 degrees.

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Parallel Lines

$$\ \text{} दो रेखाएँ जो कभी एक-दूसरे को नहीं काटतीं, उन्हें समानांतर रेखाएँ कहते हैं। \ \text{}$$

Standard algebraic representation and mathematical formula for studying Parallel Lines.

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Intersecting Lines

$$\ \text{} दो रेखाएँ जो एक बिंदु पर मिलती हैं, उन्हें काटती रेखाएँ कहते हैं। \ \text{}$$

Standard trigonometric values and properties of angles.

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Alternate Angles

$$\ \text{} दो समानांतर रेखाओं को एक ट्रांसवर्सल रेखा द्वारा काटा जाता है, ट्रांसवर्सल रेखा से बने कोण आंतर कोण कहलाते हैं। \ \text{}$$

Standard algebraic representation and mathematical formula for studying Alternate Angles.

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30° + 60° = 90°

$$\ \text{} ां 30^\\circ और 60^\\circ समकुच कोण हैं, जिनका योग 90^\\circ \ \text{}$$

Standard algebraic representation and mathematical formula for studying 30° + 60° = 90°.

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110° + 70° = 180°

$$\ \text{} ां 110^\\circ और 70^\\circ पूरक कोण हैं, जिनका योग 180^\\circ \ \text{}$$

Standard algebraic representation and mathematical formula for studying 110° + 70° = 180°.

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⟶ रेखाएँ AB || CD

$$\ \text{} ⟶ ट्रांसवर्सल रेखा: EF \ \text{}$$

Standard algebraic representation and mathematical formula for studying ⟶ रेखाएँ AB || CD.

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आंतर कोण

$$\ \text{} \\Rightarrow \\angle 3 = \\angle 4 (अनुकूली कोण) \ \text{}$$

Standard algebraic representation and mathematical formula for studying आंतर कोण.

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⇒ AB ∩ CD = P

$$\ \text{} \\Rightarrow रेखाएँ AB और CD P पर मिलती हैं। \ \text{}$$

Standard algebraic representation and mathematical formula for studying ⇒ AB ∩ CD = P.

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1 rad

$$\ \text{} \\frac{180^\\circ}{\\pi} \\approx 57^\\circ 17'45" \ \text{}$$

Standard algebraic representation and mathematical formula for studying 1 rad.

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1°

$$\ \text{} \\frac{\\pi}{180}rad \\approx 0.017453 \ \text{}$$

Standard algebraic representation and mathematical formula for studying 1°.

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कर्ण

$$\ \text{} (\\text{p})^{2} + (\\text{b})^{2} \ \text{}$$

Standard algebraic representation and mathematical formula for studying कर्ण.

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COS ratio at 0°

$$\ \text{} \\cos(0^\\circ) = 1 \ \text{}$$

The exact numerical trigonometric value of the COS function for angle 0°.

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COS ratio at 30°

$$\ \text{} \\cos(30^\\circ) = \\frac{\\sqrt{3}}{2} \ \text{}$$

The exact numerical trigonometric value of the COS function for angle 30°.

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COS ratio at 45°

$$\ \text{} \\cos(45^\\circ) = \\frac{1}{\\sqrt{2}} \ \text{}$$

The exact numerical trigonometric value of the COS function for angle 45°.

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COS ratio at 60°

$$\ \text{} \\cos(60^\\circ) = \\frac{1}{2} \ \text{}$$

The exact numerical trigonometric value of the COS function for angle 60°.

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COS ratio at 90°

$$\ \text{} \\cos(90^\\circ) = 0 \ \text{}$$

The exact numerical trigonometric value of the COS function for angle 90°.

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Fundamental Pythagorean Identity

$$\ \text{} \\sin^2\\theta + \\cos^2\\theta = 1 \ \text{}$$

The core trigonometric identity relating sine and cosine squared.

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Secant-Tangent Identity

$$\ \text{} \\sec^2\\theta - \\tan^2\\theta = 1 \ \text{}$$

Pythagorean identity relating secant squared and tangent squared.

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Cosecant-Cotangent Identity

$$\ \text{} \\csc^2\\theta - \\cot^2\\theta = 1 \ \text{}$$

Pythagorean identity relating cosecant squared and cotangent squared.

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-θ

$$\ \text{} -csc( \\theta ) \ \text{}$$

Standard algebraic representation and mathematical formula for studying -θ.

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90 - θ

$$\ \text{} cos( \\theta ) \ \text{}$$

Standard algebraic representation and mathematical formula for studying 90 - θ.

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180 - θ

$$\ \text{} sin( \\theta ) \ \text{}$$

Standard algebraic representation and mathematical formula for studying 180 - θ.

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चतुर्भुज के कोणों का योग

$$\ \text{} 360^\\circ \ \text{}$$

Standard algebraic representation and mathematical formula for studying चतुर्भुज के कोणों का योग.

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सामान्य चतुर्भुज का क्षेत्रफल

$$\ \text{} \\text{b} \\times \ \text{}$$

Formula to calculate the area of the given geometric shape.

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आयत का क्षेत्रफल

$$\ \text{} \\times चौड़ाई \ \text{}$$

Formula to calculate the area of the given geometric shape.

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आयत का परिमाप

$$\ \text{} 2 \\times ( + चौड़ाई) \ \text{}$$

Formula to calculate the perimeter/circumference of the geometric boundary.

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समांतर चतुर्भुज का परिमाप

$$\ \text{} 2 \\times (a + b) \ \text{}$$

Formula to calculate the perimeter/circumference of the geometric boundary.

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वर्ग का क्षेत्रफल

$$\ \text{} (पक्ष)^{2} \ \text{}$$

Formula to calculate the area of the given geometric shape.

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वर्ग का परिमाप

$$\ \text{} 4 \\times (पक्ष) \ \text{}$$

Formula to calculate the perimeter/circumference of the geometric boundary.

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Rhombus

$$\ \text{} \\frac{1}{2} \\times (वि\\text{h}_1 \\times वि\\text{h}_2) \ \text{}$$

Formula to calculate the area of the given geometric shape.

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वृत्त की परिधि

$$\ \text{} 2 \\pi r \ \text{}$$

Formula to calculate the perimeter/circumference of the geometric boundary.

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वृत्त का क्षेत्रफल

$$\ \text{} \\pi r^{2} \ \text{}$$

Formula to calculate the area of the given geometric shape.

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Diameter

$$\ \text{} 2r \ \text{}$$

Standard algebraic representation and mathematical formula for studying Diameter.

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Radius

$$\ \text{} \\frac{d}{2} \ \text{}$$

Standard algebraic representation and mathematical formula for studying Radius.

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Sector Area

$$\ \text{} \\frac{\\theta}{360} \\times \\pi r^{2} \ \text{}$$

Formula to calculate the area of the given geometric shape.

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वृत्त का अर्द्धव्यास

$$\ \text{} \\frac{C}{2 \\pi} \ \text{}$$

Standard algebraic representation and mathematical formula for studying वृत्त का अर्द्धव्यास.

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a_1 + b_1

$$\ \text{} सदिश योग \ \text{}$$

Standard algebraic representation and mathematical formula for studying a_1 + b_1.

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ka_1

$$\ \text{} सदिश का स्लर गुणन \ \text{}$$

Standard algebraic representation and mathematical formula for studying ka_1.

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a_1^2 + a_2^2 + a_3^2

$$\ \text{} सदिश का परिमाण \ \text{}$$

Standard algebraic representation and mathematical formula for studying a_1^2 + a_2^2 + a_3^2.

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A × B = |i j k| |a_1 a_2 a_3| |b_1 b_2 b_3|

$$\ \text{} क्रॉस प्रोडक्ट (डिटर्मिनेंट ) \ \text{}$$

Standard algebraic representation and mathematical formula for studying A × B = |i j k| |a_1 a_2 a_3| |b_1 b_2 b_3|.

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|A × B| = |A||B|sinθ

$$\ \text{} क्रॉस प्रोडक्ट का परिमाण \ \text{}$$

Standard trigonometric values and properties of angles.

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B × C

$$\ \text{} सदिश त्रिपथ गुणनफल \ \text{}$$

Standard algebraic representation and mathematical formula for studying B × C.

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x_2 – x_1

$$\ \text{} दो बिंदुओं बीच का सदिश \ \text{}$$

Standard algebraic representation and mathematical formula for studying x_2 – x_1.

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OP̅ = xi + yj + zk

$$\ \text{} स्थिति सदिश \ \text{}$$

Standard algebraic representation and mathematical formula for studying OP̅ = xi + yj + zk.

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Area of a Rectangleआयत का क्षेत्रफल

$$\ \text{} Length \\times Width \\times चौड़ाई \ \text{}$$

Formula to calculate the area of the given geometric shape.

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Perimeter of a Rectangleआयत का परिमाप

$$\ \text{} 2 \\times (Length + Width)2 \\times ( + चौड़ाई) \ \text{}$$

Formula to calculate the perimeter/circumference of the geometric boundary.

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Area of a Parallelogramसमांतर चतुर्भुज का क्षेत्रफल

$$\ \text{} Base \\times Height\\text{b} \\times \ \text{}$$

Formula to calculate the area of the given geometric shape.

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Perimeter of a Parallelogramसमांतर चतुर्भुज का परिमाप

$$\ \text{} 2 \\times (Base + Side)2 \\times (\\text{b} + पार्श्व) \ \text{}$$

Formula to calculate the perimeter/circumference of the geometric boundary.

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Area of a Sector of a Circleवृत्त के क्षेत्रफल का क्षेत्रफल

$$\ \text{} \\frac{1}{2} \\times Radius^{2} \\times \\theta (in radians) \\frac{1}{2} \\times r^{2} \\times \\theta (रेडियन में) \ \text{}$$

Formula to calculate the area of the given geometric shape.

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Area of a Segment of a Circleवृत्त के खंड का क्षेत्रफल

$$\ \text{} Area of Sector - Area of Triangle का - त्रिभुज का \ \text{}$$

Formula to calculate the area of the given geometric shape.

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Circumference of a Circleवृत्त का परिधि

$$\ \text{} 2 \\times \\pi \\times Radius2 \\times \\pi \\times r \ \text{}$$

Formula to calculate the perimeter/circumference of the geometric boundary.

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Surface Area of a Circleवृत्त का सतह क्षेत्रफल

$$\ \text{} \\pi \\times Diameter \\pi \\times d \ \text{}$$

Formula to calculate the area of the given geometric shape.

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Perimeter of an Ellipseअंडाकार का परिमाप

$$\ \text{} 2 \\times \\pi \\times sqrt((a^{2} + b^{2}) / 2)2 \\times \\pi \\times sqrt((a^{2} + b^{2}) / 2) \ \text{}$$

Formula to calculate the perimeter/circumference of the geometric boundary.

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Area of a Regular Polygonनियमित बहुभुज का क्षेत्रफल

$$\ \text{} \\frac{1}{2} \\times Perimeter \\times Apothem \\frac{1}{2} \\times परिमाप \\times अपोथेम \ \text{}$$

Formula to calculate the area of the given geometric shape.

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Perimeter of a Regular Polygonनियमित बहुभुज का परिमाप

$$\ \text{} Number of Sides \\times Sideपार्श्वों संख्या \\times पार्श्व \ \text{}$$

Formula to calculate the perimeter/circumference of the geometric boundary.

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Area of a Circular Ringवृत्ताकार वलय का क्षेत्रफल

$$\ \text{} - \ \text{}$$

Formula to calculate the area of the given geometric shape.

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Perimeter of a Circular Ringवृत्ताकार वलय का परिमाप

$$\ \text{} 2 \\pi (R + r)2 \\pi (R + r) \ \text{}$$

Formula to calculate the perimeter/circumference of the geometric boundary.

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Perimeter of a Sector of a Circleवृत्त के सेक्टर का परिमाप

$$\ \text{} 2r + ( \\theta /360)2 \\pi r2r + ( \\theta /360)2 \\pi r \ \text{}$$

Formula to calculate the perimeter/circumference of the geometric boundary.

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Area of an Annulusअंगुलाकार का क्षेत्रफल

$$\ \text{} - \ \text{}$$

Formula to calculate the area of the given geometric shape.

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approx.

$$\ \text{} \\pi [3(a + b) - ((3a + b)(a + 3b))] \\pi [3(a + b) - ((3a + b)(a + 3b))] \ \text{}$$

Formula to calculate the perimeter/circumference of the geometric boundary.

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Area of an Irregular Polygonअनियमित बहुभुज का क्षेत्रफल

$$\ \text{} - \ \text{}$$

Formula to calculate the area of the given geometric shape.

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Perimeter of an Irregular Polygonअनियमित बहुभुज का परिमाप

$$\ \text{} \\Sigma ((x_{i+1} - x_{i})^{2} + (y_{i+1} - y_{i})^{2}) \\Sigma ((x_{i+1} - x_{i})^{2} + (y_{i+1} - y_{i})^{2}) \ \text{}$$

Formula to calculate the perimeter/circumference of the geometric boundary.

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Volume of a Cubeघन का आयतन

$$\ \text{} Side^{3}भुजा^{3} \ \text{}$$

Formula to calculate the three-dimensional volume of the shape.

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Surface Area of a Rectangular Prismआयताकार प्रिज्म का सतह क्षेत्रफल

$$\ \text{} 2 \\times (Length \\times Width + Width \\times Height + Height \\times Length)2 \\times ( \\times चौड़ाई + चौड़ाई \\times + \\times ) \ \text{}$$

Formula to calculate the area of the given geometric shape.

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Volume of a Rectangular Prismआयताकार प्रिज्म का आयतन

$$\ \text{} Length \\times Width \\times Height \\times चौड़ाई \\times \ \text{}$$

Formula to calculate the three-dimensional volume of the shape.

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Surface Area of a Triangular Prismत्रिभुजाकार प्रिज्म का सतह क्षेत्रफल

$$\ \text{} Base Area \\times Perimeter of Base \\times Height\\text{b} \\times \\text{b} परिमाप \\times \ \text{}$$

Formula to calculate the area of the given geometric shape.

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Volume of a Triangular Prismत्रिभुजाकार प्रिज्म का आयतन

$$\ \text{} \\frac{1}{2} \\times Base \\times Height \\times Length \\frac{1}{2} \\times \\text{b} \\times \\times \ \text{}$$

Formula to calculate the three-dimensional volume of the shape.

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Surface Area of a Hexagonal Prismषट्कोणीय प्रिज्म का सतह क्षेत्रफल

$$\ \text{} 6 \\times Side Length \\times Height + 2 \\times Base Area6 \\times भुजा \\times + 2 \\times \\text{b} \ \text{}$$

Formula to calculate the area of the given geometric shape.

Class 12 cbse up_board jee

Volume of a Hexagonal Prismषट्कोणीय प्रिज्म का आयतन

$$\ \text{} Base Area \\times Height\\text{b} \\times \ \text{}$$

Formula to calculate the three-dimensional volume of the shape.

Class 12 cbse up_board jee

Surface Area of a Octagonal Prismअष्टकोणीय प्रिज्म का सतह क्षेत्रफल

$$\ \text{} 8 \\times Side Length \\times Height + 2 \\times Base Area8 \\times भुजा \\times + 2 \\times \\text{b} \ \text{}$$

Formula to calculate the area of the given geometric shape.

Class 12 cbse up_board jee

Volume of a Octagonal Prismअष्टकोणीय प्रिज्म का आयतन

$$\ \text{} Base Area \\times Height\\text{b} \\times \ \text{}$$

Formula to calculate the three-dimensional volume of the shape.

Class 12 cbse up_board jee

Surface Area of an Ellipsoidदीर्घवृत्ताकार का सतह क्षेत्रफल

$$\ \text{} 4 \\times \\pi \\times ((a^{p} \\times b^{p} + a^{p} \\times c^{p} + b^{p} \\times c^{p})/3)^{1/p} 4 \\times \\pi \\times ((a^{p} \\times b^{p} + a^{p} \\times c^{p} + b^{p} \\times c^{p})/3)^{1/p} \ \text{}$$

Formula to calculate the area of the given geometric shape.

Class 12 cbse up_board jee

Volume of an Ellipsoidदीर्घवृत्ताकार का आयतन

$$\ \text{} \\frac{4}{3} \\times \\pi \\times a \\times b \\times c \\frac{4}{3} \\times \\pi \\times a \\times b \\times c \ \text{}$$

Formula to calculate the three-dimensional volume of the shape.

Class 12 cbse up_board jee

Surface Area of a Right Circular Coneसमचतुर्भुज शंकु का सतह क्षेत्रफल

$$\ \text{} \\pi \\times r \\times (r + l) \\pi \\times r \\times (r + l) \ \text{}$$

Formula to calculate the area of the given geometric shape.

Class 12 cbse up_board jee

Volume of a Right Circular Coneसमचतुर्भुज शंकु का आयतन

$$\ \text{} \\frac{1}{3} \\times \\pi \\times r^{2} \\times h \\frac{1}{3} \\times \\pi \\times r^{2} \\times h \ \text{}$$

Formula to calculate the three-dimensional volume of the shape.

Class 12 cbse up_board jee

Surface Area of a Hollow Sphereखोखले गोले का सतह क्षेत्रफल

$$\ \text{} 4 \\times \\pi \\times (R^{2} + r^{2})4 \\times \\pi \\times (R^{2} + r^{2}) \ \text{}$$

Formula to calculate the area of the given geometric shape.

Class 12 cbse up_board jee

Volume of a Hollow Sphereखोखले गोले का आयतन

$$\ \text{} ( \\frac{4}{3} ) \\times \\pi \\times (R^{3} - r^{3}) ( \\frac{4}{3} ) \\times \\pi \\times (R^{3} - r^{3}) \ \text{}$$

Formula to calculate the three-dimensional volume of the shape.

Class 12 cbse up_board jee

Surface Area of a Frustum of a Coneशंकु के फ्रस्ट्रम का सतह क्षेत्रफल

$$\ \text{} \\pi \\times (R + r) \\times (R - r)^{2} + h^{2} + \\pi \\times R^{2} + \\pi \\times r^{2} \\pi \\times (R + r) \\times (R - r)^{2} + h^{2} + \\pi \\times R^{2} + \\pi \\times r^{2} \ \text{}$$

Formula to calculate the area of the given geometric shape.

Class 12 cbse up_board jee

Surface Area of a Trapezoidal Prismसमलंब प्रिज्म का सतह क्षेत्रफल

$$\ \text{} h \\times (a + b) + l \\times (a + b + ((a - b)^{2} + h^{2}) + 2l \\times h h \\times (a + b) + l \\times (a + b + ((a - b)^{2} + h^{2}) ) + 2l \\times h \ \text{}$$

Formula to calculate the area of the given geometric shape.

Class 12 cbse up_board jee

Volume of a Trapezoidal Prismसमलंब प्रिज्म का आयतन

$$\ \text{} ( \\frac{1}{2} ) \\times (a + b) \\times h \\times l ( \\frac{1}{2} ) \\times (a + b) \\times h \\times l \ \text{}$$

Formula to calculate the three-dimensional volume of the shape.

Class 12 cbse up_board jee

🔺 Trigonometry

Sine Ratio (sin θ)

$$\ \text{} \\sin\\theta = \\frac{\\text{Perpendicular}}{\\text{Hypotenuse}} \ \text{}$$

Trigonometric ratio of perpendicular side over hypotenuse side.

Class 9 Class 10 cbse up_board

Cosine Ratio (cos θ)

$$\ \text{} \\cos\\theta = \\frac{\\text{Base}}{\\text{Hypotenuse}} \ \text{}$$

Trigonometric ratio of adjacent side (base) over hypotenuse side.

Class 9 Class 10 cbse up_board

Tangent Ratio (tan θ)

$$\ \text{} \\tan\\theta = \\frac{\\text{Perpendicular}}{\\text{Base}} \ \text{}$$

Trigonometric ratio of perpendicular side over adjacent side.

Class 9 Class 10 cbse up_board

Cosecant Ratio (csc θ)

$$\ \text{} \\csc\\theta = \\frac{\\text{Hypotenuse}}{\\text{Perpendicular}} \ \text{}$$

Trigonometric ratio of hypotenuse side over perpendicular side, reciprocal of sine.

Class 9 Class 10 cbse up_board

Secant Ratio (sec θ)

$$\ \text{} \\sec\\theta = \\frac{\\text{Hypotenuse}}{\\text{Base}} \ \text{}$$

Trigonometric ratio of hypotenuse side over adjacent side (base), reciprocal of cosine.

Class 9 Class 10 cbse up_board

Cotangent Ratio (cot θ)

$$\ \text{} \\cot\\theta = \\frac{\\text{Base}}{\\text{Perpendicular}} \ \text{}$$

Trigonometric ratio of adjacent side over perpendicular side, reciprocal of tangent.

Class 9 Class 10 cbse up_board

SIN ratio at 0°

$$\ \text{} \\sin(0^\\circ) = 0 \ \text{}$$

The exact numerical trigonometric value of the SIN function for angle 0°.

Class 9 Class 10 cbse up_board

SIN ratio at 30°

$$\ \text{} \\sin(30^\\circ) = \\frac{1}{2} \ \text{}$$

The exact numerical trigonometric value of the SIN function for angle 30°.

Class 9 Class 10 cbse up_board

SIN ratio at 45°

$$\ \text{} \\sin(45^\\circ) = \\frac{1}{\\sqrt{2}} \ \text{}$$

The exact numerical trigonometric value of the SIN function for angle 45°.

Class 9 Class 10 cbse up_board

SIN ratio at 60°

$$\ \text{} \\sin(60^\\circ) = \\frac{\\sqrt{3}}{2} \ \text{}$$

The exact numerical trigonometric value of the SIN function for angle 60°.

Class 9 Class 10 cbse up_board

SIN ratio at 90°

$$\ \text{} \\sin(90^\\circ) = 1 \ \text{}$$

The exact numerical trigonometric value of the SIN function for angle 90°.

Class 9 Class 10 cbse up_board

TAN ratio at 0°

$$\ \text{} \\tan(0^\\circ) = 0 \ \text{}$$

The exact numerical trigonometric value of the TAN function for angle 0°.

Class 9 Class 10 cbse up_board

TAN ratio at 30°

$$\ \text{} \\tan(30^\\circ) = \\frac{1}{\\sqrt{3}} \ \text{}$$

The exact numerical trigonometric value of the TAN function for angle 30°.

Class 9 Class 10 cbse up_board

TAN ratio at 45°

$$\ \text{} \\tan(45^\\circ) = 1 \ \text{}$$

The exact numerical trigonometric value of the TAN function for angle 45°.

Class 9 Class 10 cbse up_board

TAN ratio at 60°

$$\ \text{} \\tan(60^\\circ) = \\sqrt{3} \ \text{}$$

The exact numerical trigonometric value of the TAN function for angle 60°.

Class 9 Class 10 cbse up_board

TAN ratio at 90°

$$\ \text{} \\tan(90^\\circ) = \\infty \ \text{}$$

The exact numerical trigonometric value of the TAN function for angle 90°.

Class 9 Class 10 cbse up_board

SEC ratio at 0°

$$\ \text{} \\sec(0^\\circ) = 1 \ \text{}$$

The exact numerical trigonometric value of the SEC function for angle 0°.

Class 9 Class 10 cbse up_board

SEC ratio at 30°

$$\ \text{} \\sec(30^\\circ) = \\frac{2}{\\sqrt{3}} \ \text{}$$

The exact numerical trigonometric value of the SEC function for angle 30°.

Class 9 Class 10 cbse up_board

SEC ratio at 45°

$$\ \text{} \\sec(45^\\circ) = \\sqrt{2} \ \text{}$$

The exact numerical trigonometric value of the SEC function for angle 45°.

Class 9 Class 10 cbse up_board

SEC ratio at 60°

$$\ \text{} \\sec(60^\\circ) = 2 \ \text{}$$

The exact numerical trigonometric value of the SEC function for angle 60°.

Class 9 Class 10 cbse up_board

SEC ratio at 90°

$$\ \text{} \\sec(90^\\circ) = \\infty \ \text{}$$

The exact numerical trigonometric value of the SEC function for angle 90°.

Class 9 Class 10 cbse up_board

📦 Mensuration

Triangle Area (Base & Height)

$$\ \text{} A = \\frac{1}{2} \\times b \\times h \ \text{}$$

Core geometric area calculation for any triangle of known base size and perpendicular height.

Class 9 cbse up_board

Equilateral Triangle Area

$$\ \text{} A = \\frac{\\sqrt{3}}{4} a^2 \ \text{}$$

Simplified area equation specifically for triangles where all three side lengths are equal to a.

Class 9 cbse up_board

Area of a Semicircle

$$\ \text{} A = \\frac{1}{2} \\pi r^2 \ \text{}$$

Total enclosed area of exactly half a geometric circle of radius r.

Class 9 cbse up_board

Semicircle Perimeter

$$\ \text{} P = \\pi r + 2r \ \text{}$$

Total perimeter along the curved arc and diameter of a secure semicircle.

Class 9 cbse up_board

Cube Volume & Surface Area

$$\ \text{} V = a^3 \\quad \\text{and} \\quad A = 6a^2 \ \text{}$$

Three-dimensional properties of a regular hexahedron cubical prism of side length a.

Class 9 cbse up_board

Rectangular Prism/Cuboid Volume

$$\ \text{} V = l \\times w \\times h \ \text{}$$

Volume enclosed by six rectangular faces with length, width and height values.

Class 9 cbse up_board

Total Surface Area of Cuboid

$$\ \text{} A = 2(lw + lh + wh) \ \text{}$$

Calculates combined geometric surface area of all six rectangular sides of an envelope.

Class 9 cbse up_board

Sphere Metrics (Volume & Surface)

$$\ \text{} V = \\frac{4}{3}\\pi r^3 \\quad \\text{and} \\quad S = 4\\pi r^2 \ \text{}$$

Calculations for total volume capacity and exterior surface area of a solid globe of radius r.

Class 9 cbse up_board

Hemisphere Metrics

$$\ \text{} V = \\frac{2}{3}\\pi r^3 \\quad \\text{and} \\quad A_{\\text{total}} = 3\\pi r^2 \ \text{}$$

Half sphere volume and total surface area including solid flat circular bottom surface.

Class 9 cbse up_board

Cone

$$\ \text{} पृष्ठ \ \text{}$$

Standard algebraic representation and mathematical formula for studying Cone.

Class 9 cbse up_board

Cylinder

$$\ \text{} पृष्ठ \ \text{}$$

Standard algebraic representation and mathematical formula for studying Cylinder.

Class 9 cbse up_board

📊 Statistics & Probability

Arithmetic Mean

$$\ \text{} \\bar{x} = \\frac{\\sum_{i=1}^{n} x_i}{n} \ \text{}$$

Simplest average calculated by summing up raw values and dividing by count.

Class 9 Class 10 cbse up_board

Weighted Grouped Mean

$$\ \text{} \\bar{x} = \\frac{\\sum f_i x_i}{\\sum f_i} \ \text{}$$

Finds mean for values summarized in grouped frequency statistics classes.

Class 9 Class 10 cbse up_board

Median of Grouped Frequency Data

$$\ \text{} \\text{Median} = l + \\left(\\frac{\\frac{N}{2} - CF}{f}\\right)h \ \text{}$$

Finds positional central value of a grouped distribution with cumulative frequency boundary.

Class 9 Class 10 cbse up_board

Mode of Grouped frequency Data

$$\ \text{} \\text{Mode} = l + \\left(\\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\\right)h \ \text{}$$

Finds peak value corresponding to the maximum observation density class intervals.

Class 9 Class 10 cbse up_board

Classical Probability Formula

$$\ \text{} P(E) = \\frac{n(E)}{n(S)} \ \text{}$$

Ratio of cardinal size of event outcomes and complete sample space elements.

Class 9 cbse up_board

E

$$\ \text{} \\le \ \text{}$$

Standard algebraic representation and mathematical formula for studying E.

Class 9 cbse up_board

Step Deviation Method

$$\ \text{} \\bar{x} = a + h \\cdot \\left(\\frac{\\sum f_i u_i}{\\sum f_i}\\right) \\quad \\text{where } u_i = \\frac{x_i - a}{h} \ \text{}$$

Shortened computation for grouped mean using custom assumed mean offset.

Class 10 cbse up_board

Population Variance

$$\ \text{} \\sigma^2 = \\frac{\\sum (x_i - \\bar{x})^2}{N} \ \text{}$$

Measures dispersion along statistical data points around their mean standard offset.

Class 10 cbse up_board

Standard Deviation (SD)

$$\ \text{} \\sigma = \\sqrt{\\frac{\\sum (x_i - \\bar{x})^2}{N}} \ \text{}$$

The absolute square root of variance metric, showing average physical dispersion offset.

Class 10 cbse up_board

Coefficient of Variation

$$\ \text{} \\text{C.V.} = \\left(\\frac{\\sigma}{\\bar{x}}\\right) \\times 100 \ \text{}$$

Normalized relative statistical dispersion measure expressed as a percentage of mean.

Class 10 cbse up_board

Skewness

$$\ \text{} \\frac{\\Sigma (x - \\mu )^{3}}{n \\times \\sigma ^{3}} \ \text{}$$

Standard algebraic representation and mathematical formula for studying Skewness.

Class 10 cbse up_board

Kurtosis

$$\ \text{} \\frac{\\Sigma (x - \\mu )^{4}}{n \\times \\sigma ^{4}} \ \text{}$$

Standard algebraic representation and mathematical formula for studying Kurtosis.

Class 10 cbse up_board

Correlation

$$\ \text{} r = \\frac{\\Sigma (x - x̄)(y - ȳ)}{\\sqrt{( \\Sigma (x - x̄)^{2} \\times \\Sigma (y - ȳ)^{2})}} \ \text{}$$

Standard algebraic representation and mathematical formula for studying Correlation.

Class 10 cbse up_board

Regression

$$\ \text{} y = a + bx \ \text{}$$

Standard algebraic representation and mathematical formula for studying Regression.

Class 10 cbse up_board

Least Square Method

$$\ \text{} \\frac{\\Sigma y}{\\Sigma x} , \\frac{\\Sigma xy}{\\Sigma x^{2}} \ \text{}$$

Standard algebraic representation and mathematical formula for studying Least Square Method.

Class 10 cbse up_board

Probability of an event

$$\ \text{} P(E) = \\frac{सफल घटनाओं संख्या}{कुल घटनाओं संख्या} \ \text{}$$

Standard algebraic representation and mathematical formula for studying Probability of an event.

Class 10 cbse up_board

Probability of complementary events

$$\ \text{} P(E') = 1 - P(E) \ \text{}$$

Standard algebraic representation and mathematical formula for studying Probability of complementary events.

Class 10 cbse up_board

Addition Rule

$$\ \text{} P(A \\cup B) = P(A) + P(B) - P(A \\cap B) \ \text{}$$

Standard algebraic representation and mathematical formula for studying Addition Rule.

Class 10 cbse up_board

Multiplication Rule

$$\ \text{} P(A \\cap B) = P(A) \\times P(B|A) \ \text{}$$

Standard algebraic representation and mathematical formula for studying Multiplication Rule.

Class 10 cbse up_board

Conditional Probability

$$\ \text{} P(A|B) = \\frac{P(A \\cap B)}{P(B)} \ \text{}$$

Standard algebraic representation and mathematical formula for studying Conditional Probability.

Class 10 cbse up_board

Bayes Theorem

$$\ \text{} P(A|B) = \\frac{P(B|A) \\times P(A)}{P(B)} \ \text{}$$

Standard algebraic representation and mathematical formula for studying Bayes Theorem.

Class 10 cbse up_board

Total Probability Theorem

$$\ \text{} P(B) = \\Sigma P(B|A_{i}) \\times P(A_{i}) \ \text{}$$

Standard algebraic representation and mathematical formula for studying Total Probability Theorem.

Class 10 cbse up_board

Bernoulli's Trial

$$\ \text{} P(X = k) = \\frac{nCk \\times p^{k} \\times (1 - p)^{n-k}}{n} \ \text{}$$

Standard algebraic representation and mathematical formula for studying Bernoulli's Trial.

Class 10 cbse up_board

Binomial Distribution

$$\ \text{} P(X = k) = \\frac{nCk \\times p^{k} \\times (1 - p)^{n-k}}{n} \ \text{}$$

Standard algebraic representation and mathematical formula for studying Binomial Distribution.

Class 10 cbse up_board

Poisson Distribution

$$\ \text{} P(X = k) = \\frac{\\lambda ^{k} \\times e^{- \\lambda}}{k!} \ \text{}$$

Standard algebraic representation and mathematical formula for studying Poisson Distribution.

Class 10 cbse up_board

Normal Distribution

$$\ \text{} P(X = x) = \\frac{1}{\\sigma \\sqrt{2 \\pi}} \\times e^{-(x - \\mu )^{2} / 2 \\sigma ^{2}} \ \text{}$$

Standard algebraic representation and mathematical formula for studying Normal Distribution.

Class 10 cbse up_board

Expected Value

$$\ \text{} E(X) = \\Sigma x \\times P(x) \ \text{}$$

Standard algebraic representation and mathematical formula for studying Expected Value.

Class 10 cbse up_board

5 - 15 = -10

$$\ \text{} (-10)^{2} = 100 \ \text{}$$

Standard algebraic representation and mathematical formula for studying 5 - 15 = -10.

Class 12 cbse up_board jee

10 - 15 = -5

$$\ \text{} (-5)^{2} = 25 \ \text{}$$

Standard algebraic representation and mathematical formula for studying 10 - 15 = -5.

Class 12 cbse up_board jee

15 - 15 = 0

$$\ \text{} 0^{2} = 0 \ \text{}$$

Standard algebraic representation and mathematical formula for studying 15 - 15 = 0.

Class 12 cbse up_board jee

20 - 15 = 5

$$\ \text{} 5^{2} = 25 \ \text{}$$

Standard algebraic representation and mathematical formula for studying 20 - 15 = 5.

Class 12 cbse up_board jee

25 - 15 = 10

$$\ \text{} 10^{2} = 100 \ \text{}$$

Standard algebraic representation and mathematical formula for studying 25 - 15 = 10.

Class 12 cbse up_board jee

5

$$\ \text{} 2 − 5 = −3 \ \text{}$$

Standard algebraic representation and mathematical formula for studying 5.

Class 12 cbse up_board jee

2.5

$$\ \text{} (1 \\times 2.5) = 2.5 \ \text{}$$

Standard algebraic representation and mathematical formula for studying 2.5.

Class 12 cbse up_board jee

7.5

$$\ \text{} (3 \\times 7.5) = 22.5 \ \text{}$$

Standard algebraic representation and mathematical formula for studying 7.5.

Class 12 cbse up_board jee

xi − x̄

$$\ \text{} 50 \ \text{}$$

Standard algebraic representation and mathematical formula for studying xi − x̄.

Class 12 cbse up_board jee

−3

$$\ \text{} (−3)^{2} = 9 \ \text{}$$

Standard algebraic representation and mathematical formula for studying −3.

Class 12 cbse up_board jee

−1

$$\ \text{} (−1)^{2} = 1 \ \text{}$$

Standard algebraic representation and mathematical formula for studying −1.

Class 12 cbse up_board jee

1

$$\ \text{} (1)^{2} = 1 \ \text{}$$

Standard algebraic representation and mathematical formula for studying 1.

Class 12 cbse up_board jee

3

$$\ \text{} (3)^{2} = 9 \ \text{}$$

Standard algebraic representation and mathematical formula for studying 3.

Class 12 cbse up_board jee

xi − x̄

$$\ \text{} 20 \ \text{}$$

Standard algebraic representation and mathematical formula for studying xi − x̄.

Class 12 cbse up_board jee

yi − ȳ

$$\ \text{} 125 \ \text{}$$

Standard algebraic representation and mathematical formula for studying yi − ȳ.

Class 12 cbse up_board jee

10 − 18 = −8

$$\ \text{} (−8)^{2} = 64 \ \text{}$$

Standard algebraic representation and mathematical formula for studying 10 − 18 = −8.

Class 12 cbse up_board jee

12 − 18 = −6

$$\ \text{} (−6)^{2} = 36 \ \text{}$$

Standard algebraic representation and mathematical formula for studying 12 − 18 = −6.

Class 12 cbse up_board jee

23 − 18 = 5

$$\ \text{} (5)^{2} = 25 \ \text{}$$

Standard algebraic representation and mathematical formula for studying 23 − 18 = 5.

Class 12 cbse up_board jee

16 − 18 = −2

$$\ \text{} (−2)^{2} = 4 \ \text{}$$

Standard algebraic representation and mathematical formula for studying 16 − 18 = −2.

Class 12 cbse up_board jee

21 − 18 = 3

$$\ \text{} (3)^{2} = 9 \ \text{}$$

Standard algebraic representation and mathematical formula for studying 21 − 18 = 3.

Class 12 cbse up_board jee

🔢 Arithmetic & Number System

Union of Sets

$$\ \text{} A \\cup B = \\{x \\mid x \\in A \\text{ or } x \\in B\\} \ \text{}$$

Combined set containing all distinct elements that are in set A, set B, or both.

Class 11 cbse up_board jee

📈 Calculus

Direct Limit of Power Monomial

$$\ \text{} \\lim_{x \\to a} x^n = a^n \ \text{}$$

Evaluates standard continuous monomial function target directly at evaluation origin a.

Class 11 cbse up_board jee

Standard Power Limit Theorem

$$\ \text{} \\lim_{x \\to a} \\frac{x^n - a^n}{x - a} = n a^{n-1} \ \text{}$$

Essential limit formula used to prove the power rule derivative for polynomials.

Class 11 cbse up_board jee

Binomial Expansion Limit Quotient

$$\ \text{} \\lim_{x \\to 0} \\frac{(1+x)^n - 1}{x} = n \ \text{}$$

Basic continuous derivative equivalent of the standard binomial expansion when fraction variable goes to zero.

Class 11 cbse up_board jee

x + y - z

$$\ \text{} lim x + lim y - lim z \ \text{}$$

Standard algebraic representation and mathematical formula for studying x + y - z.

Class 11 Class 12 cbse up_board jee

xyz

$$\ \text{} lim x \\times lim y \\times lim z \ \text{}$$

Standard algebraic representation and mathematical formula for studying xyz.

Class 11 Class 12 cbse up_board jee

xy

$$\ \text{} \\frac{lim x}{lim y} \ \text{}$$

Standard algebraic representation and mathematical formula for studying xy.

Class 11 Class 12 cbse up_board jee

Fundamental Squeeze Limit

$$\ \text{} \\lim_{x \\to 0} \\frac{\\sin x}{x} = 1 \ \text{}$$

Standard limit of sine quotient evaluated as variable goes to 0.

Class 11 cbse up_board jee

Tangent Derivative Limit Quotient

$$\ \text{} \\lim_{x \\to 0} \\frac{\\tan x}{x} = 1 \ \text{}$$

Standard trigonometric limit of tangent division quotient as variable goes to 0.

Class 11 cbse up_board jee

Trigonometric Quadratic Squeeze Limit

$$\ \text{} \\lim_{x \\to 0} \\frac{1 - \\cos x}{x^2} = \\frac{1}{2} \ \text{}$$

Calculus quadratic limit showing deceleration rate of cosine difference near zero.

Class 11 cbse up_board jee

ax

$$\ \text{} \\frac{a}{b} \ \text{}$$

Standard trigonometric values and properties of angles.

Class 11 cbse up_board jee

Hyperbolic Limit at Infinity

$$\ \text{} \\lim_{x \\to \\infty} \\frac{1}{x} = 0 \ \text{}$$

Standard property showing rational hyperbolic fractions converge cleanly to absolute zero.

Class 11 cbse up_board jee

n > 0

$$\ \text{} 0 \ \text{}$$

Standard algebraic representation and mathematical formula for studying n > 0.

Class 11 cbse up_board jee

limx→∞ xx + 1

$$\ \text{} 1 \ \text{}$$

Standard algebraic representation and mathematical formula for studying limx→∞ xx + 1.

Class 11 cbse up_board jee

limx→∞ axn + ...bxn + ...

$$\ \text{} \\frac{a}{b} \ \text{}$$

Standard algebraic representation and mathematical formula for studying limx→∞ axn + ...bxn + ....

Class 11 cbse up_board jee

x

$$\ \text{} f(x) निरंतर है (Continuous at x = a) \ \text{}$$

Standard algebraic representation and mathematical formula for studying x.

Class 11 Class 12 cbse up_board jee

a

$$\ \text{} f(x) निरंतर है \ \text{}$$

Standard algebraic representation and mathematical formula for studying a.

Class 11 Class 12 cbse up_board jee

LHL

$$\ \text{} lim_{x \\rightarrow a⁻} f(x) \ \text{}$$

Standard algebraic representation and mathematical formula for studying LHL.

Class 11 cbse up_board jee

RHL

$$\ \text{} lim_{x \\rightarrow a⁺} f(x) \ \text{}$$

Standard algebraic representation and mathematical formula for studying RHL.

Class 11 cbse up_board jee

यदि LHL = RHL

$$\ \text{} limit मौजूद (exists) \ \text{}$$

Standard algebraic representation and mathematical formula for studying यदि LHL = RHL.

Class 11 cbse up_board jee

lim ex

$$\ \text{} \\infty \ \text{}$$

Standard algebraic representation and mathematical formula for studying lim ex.

Class 11 Class 12 cbse up_board jee

lim e-x

$$\ \text{} 0 \ \text{}$$

Standard algebraic representation and mathematical formula for studying lim e-x.

Class 11 Class 12 cbse up_board jee

x → 0⁺

$$\ \text{} - \\infty \ \text{}$$

Standard algebraic representation and mathematical formula for studying x → 0⁺.

Class 11 cbse up_board jee

x → 0, n > 0

$$\ \text{} 0 \ \text{}$$

Standard algebraic representation and mathematical formula for studying x → 0, n > 0.

Class 11 cbse up_board jee

lim xx!

$$\ \text{} 0 \ \text{}$$

Standard algebraic representation and mathematical formula for studying lim xx!.

Class 11 cbse up_board jee

1 + kx

$$\ \text{} e^{k}, (e = 2.71) \ \text{}$$

Standard algebraic representation and mathematical formula for studying 1 + kx.

Class 11 Class 12 cbse up_board jee

1 - 1x

$$\ \text{} \\frac{1}{e} \ \text{}$$

Standard algebraic representation and mathematical formula for studying 1 - 1x.

Class 11 Class 12 cbse up_board jee

xx e-x

$$\ \text{} 2 \\pi \ \text{}$$

Standard algebraic representation and mathematical formula for studying xx e-x.

Class 11 cbse up_board jee

1 + 1x

$$\ \text{} log_{a} e \ \text{}$$

Standard algebraic representation and mathematical formula for studying 1 + 1x.

Class 11 Class 12 cbse up_board jee

1+x

$$\ \text{} 1 \ \text{}$$

Standard algebraic representation and mathematical formula for studying 1+x.

Class 11 cbse up_board jee

lim xsin x

$$\ \text{} 1 \ \text{}$$

Standard trigonometric values and properties of angles.

Class 11 Class 12 cbse up_board jee

lim xtan x

$$\ \text{} 1 \ \text{}$$

Standard trigonometric values and properties of angles.

Class 11 Class 12 cbse up_board jee

lim 1 - cos xx

$$\ \text{} 0 \ \text{}$$

Standard trigonometric values and properties of angles.

Class 11 Class 12 cbse up_board jee

lim 1 - cos xx2

$$\ \text{} \\frac{1}{2} \ \text{}$$

Standard trigonometric values and properties of angles.

Class 11 Class 12 cbse up_board jee

θ → 0

$$\ \text{} 0 \ \text{}$$

Standard algebraic representation and mathematical formula for studying θ → 0.

Class 11 cbse up_board jee

180°

$$\ \text{} 0 \ \text{}$$

Standard algebraic representation and mathematical formula for studying 180°.

Class 11 cbse up_board jee

lim ln x

$$\ \text{} - \\infty \ \text{}$$

Standard algebraic representation and mathematical formula for studying lim ln x.

Class 12 cbse up_board jee

lim xn

$$\ \text{} 0 (n>0) \ \text{}$$

Standard algebraic representation and mathematical formula for studying lim xn.

Class 12 cbse up_board jee

x!

$$\ \text{} e \ \text{}$$

Standard algebraic representation and mathematical formula for studying x!.

Class 12 cbse up_board jee

2πx xx ex

$$\ \text{} e \ \text{}$$

Standard algebraic representation and mathematical formula for studying 2πx xx ex.

Class 12 cbse up_board jee

2π

$$\ \text{} (2 \\pi ) \ \text{}$$

Standard algebraic representation and mathematical formula for studying 2π.

Class 12 cbse up_board jee

1 + x

$$\ \text{} 1 \ \text{}$$

Standard algebraic representation and mathematical formula for studying 1 + x.

Class 12 cbse up_board jee

lim arcsin xx

$$\ \text{} 1 \ \text{}$$

Standard trigonometric values and properties of angles.

Class 12 cbse up_board jee

lim arctan xx

$$\ \text{} 1 \ \text{}$$

Standard trigonometric values and properties of angles.

Class 12 cbse up_board jee

arccos x

$$\ \text{} 2 \ \text{}$$

Standard trigonometric values and properties of angles.

Class 12 cbse up_board jee