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Higher Mathematics Formulas

উচ্চতর গণিত

Higher mathematics formula bank covering Apollonius theorem, quadratic equations, coordinate geometry, binomial expansions, and vector algebra.

Other Subjects:PhysicsChemistryGeneral MathematicsHigher MathematicsInformation & Communication Technology (ICT)Biology

HSC (Paper 1) - Chapter 1: Matrices & Determinants

2 Formulas
HSCChapter 1 (Paper 1)

Inverse Matrix

বিপরীত ম্যাট্রিক্স • Matrices & Determinants

A−1=1det⁡(A)adj(A)A^{-1} = \frac{1}{\det(A)} \text{adj}(A)
A^-1 = (1)/(det(A)) adj(A)
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HSCChapter 1 (Paper 1)

Cramer's Rule

ক্র্যামারের নিয়ম • Matrices & Determinants

x=DxD,y=DyD,z=DzDx = \frac{D_x}{D}, \quad y = \frac{D_y}{D}, \quad z = \frac{D_z}{D}
x = (D_x)/(D), y = (D_y)/(D), z = (D_z)/(D)
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HSC (Paper 2) - Chapter 4: Polynomials & Quadratic Equations

2 Formulas
HSCChapter 4 (Paper 2)

Quadratic Formula

দ্বিঘাত সূত্র • Polynomials & Quadratic Equations

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
x = frac-b ± sqrt(b^2 - 4ac)2a
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HSCChapter 4 (Paper 2)

Sum & Product of Roots

মূলের যোগফল ও গুণফল • Polynomials & Quadratic Equations

α+β=−ba,αβ=ca\alpha + \beta = -\frac{b}{a}, \quad \alpha\beta = \frac{c}{a}
α + β = -(b)/(a), αβ = (c)/(a)
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HSC (Paper 1) - Chapter 3: Straight Lines

6 Formulas
HSCChapter 3 (Paper 1)

Internal & External Section Formula

অন্তর্বিভক্তিকরণ ও বহির্বিভক্তিকরণ সূত্র • Straight Lines

x=m1x2±m2x1m1±m2,y=m1y2±m2y1m1±m2x = \frac{m_1 x_2 \pm m_2 x_1}{m_1 \pm m_2}, \quad y = \frac{m_1 y_2 \pm m_2 y_1}{m_1 \pm m_2}
x = (m_1 x_2 ± m_2 x_1)/(m_1 ± m_2), y = (m_1 y_2 ± m_2 y_1)/(m_1 ± m_2)

💡 Condition: Coordinates dividing line segment between two points in ratio m1:m2 internally or externally.

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HSCChapter 3 (Paper 1)

Centroid & Area of Triangle

ত্রিভুজের ভরকেন্দ্র ও ক্ষেত্রফল • Straight Lines

G=(x1+x2+x33,y1+y2+y33),Δ=12∣x1(y2−y3)+x2(y3−y1)+x3(y1−y2)∣G = \left(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}\right), \quad \Delta = \frac{1}{2}|x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)|
G = ((x_1+x_2+x_3)/(3), (y_1+y_2+y_3)/(3)), Δ = (1)/(2)|x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)|

💡 Condition: Formulas for triangle centroid and enclosed polygon area via coordinate determinants.

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HSCChapter 3 (Paper 1)

Standard Forms of Straight Line

সরলরেখার আদর্শ সমীকরণসমূহ • Straight Lines

y=mx+c,y−y1=m(x−x1),xa+yb=1,xcos⁡α+ysin⁡α=py = mx + c, \quad y - y_1 = m(x - x_1), \quad \frac{x}{a} + \frac{y}{b} = 1, \quad x\cos\alpha + y\sin\alpha = p
y = mx + c, y - y_1 = m(x - x_1), (x)/(a) + (y)/(b) = 1, xcosα + ysinα = p

💡 Condition: Slope-intercept, point-slope, intercept, and normal forms of linear equations.

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HSCChapter 3 (Paper 1)

Angle Between Two Lines & Conditions

দুটি সরলরেখার মধ্যবর্তী কোণ ও শর্ত • Straight Lines

tan⁡θ=±m1−m21+m1m2(m1=m2  ⟹  সমান্তরাল,  m1m2=−1  ⟹  লম্ব)\tan\theta = \pm \frac{m_1 - m_2}{1 + m_1 m_2} \quad (m_1 = m_2 \implies \text{সমান্তরাল}, \; m_1 m_2 = -1 \implies \text{লম্ব})
tanθ = ± (m_1 - m_2)/(1 + m_1 m_2) (m_1 = m_2 implies সমান্তরাল, ; m_1 m_2 = -1 implies লম্ব)

💡 Condition: Angle θ between lines with slopes m1 and m2; parallelism and perpendicularity criteria.

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HSCChapter 3 (Paper 1)

Perpendicular Distance from Point to Line

বিন্দু হতে সরলরেখার লম্ব দূরত্ব • Straight Lines

d=∣ax1+by1+c∣a2+b2d = \frac{|a x_1 + b y_1 + c|}{\sqrt{a^2 + b^2}}
d = (|a x_1 + b y_1 + c|)/(sqrt(a^2 + b^2))

💡 Condition: Shortest Euclidean perpendicular distance from point (x1, y1) to line ax + by + c = 0.

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HSCChapter 3 (Paper 1)

Distance Between Parallel Lines

দুটি সমান্তরাল সরলরেখার দূরত্ব • Straight Lines

d=∣c1−c2∣a2+b2d = \frac{|c_1 - c_2|}{\sqrt{a^2 + b^2}}
d = (|c_1 - c_2|)/(sqrt(a^2 + b^2))

💡 Condition: Perpendicular distance separating two parallel linear equations.

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HSC (Paper 1) - Chapter 4: Circles

3 Formulas
HSCChapter 4 (Paper 1)

General Circle Equation & Radius

বৃত্তের সাধারণ সমীকরণ ও কেন্দ্র-ব্যাসার্ধ • Circles

x2+y2+2gx+2fy+c=0  ⟹  Center: (−g,−f),r=g2+f2−cx^2 + y^2 + 2gx + 2fy + c = 0 \implies \text{Center: } (-g, -f), \quad r = \sqrt{g^2 + f^2 - c}
x^2 + y^2 + 2gx + 2fy + c = 0 implies Center: (-g, -f), r = sqrt(g^2 + f^2 - c)

💡 Condition: General form of circle with center coordinates (-g, -f) and radius r.

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HSCChapter 4 (Paper 1)

Intercepts on Coordinate Axes

অক্ষদ্বয় হতে কর্তিত অংশের দৈর্ঘ্য • Circles

X-ছেদক=2g2−c,Y-ছেদক=2f2−cX\text{-ছেদক} = 2\sqrt{g^2 - c}, \quad Y\text{-ছেদক} = 2\sqrt{f^2 - c}
X-ছেদক = 2sqrt(g^2 - c), Y-ছেদক = 2sqrt(f^2 - c)

💡 Condition: Length of intercepts cut off by the circle from coordinate axes.

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HSCChapter 4 (Paper 1)

Tangent Condition & Equation

স্পর্শকের শর্ত ও স্পর্শক সমীকরণ • Circles

c2=a2(1+m2),xx1+yy1=a2c^2 = a^2(1 + m^2), \quad x x_1 + y y_1 = a^2
c^2 = a^2(1 + m^2), x x_1 + y y_1 = a^2

💡 Condition: Condition of tangency for line y = mx + c and point equation of tangent.

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HSC (Paper 1) - Chapter 7: Trigonometric Ratios of Associated Angles

5 Formulas
HSCChapter 7 (Paper 1)

Compound Angle Addition Formulas

যৌগিক কোণের ত্রিকোণমিতিক সূত্র • Trigonometric Ratios of Associated Angles

sin⁡(A±B)=sin⁡Acos⁡B±cos⁡Asin⁡B,cos⁡(A±B)=cos⁡Acos⁡B∓sin⁡Asin⁡B\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B, \quad \cos(A \pm B) = \cos A \cos B \mp \sin A \sin B
sin(A ± B) = sin A cos B ± cos A sin B, cos(A ± B) = cos A cos B ∓ sin A sin B

💡 Condition: Angle addition and subtraction formulas for sine and cosine.

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HSCChapter 7 (Paper 1)

Product to Sum Transformations

গুণফল থেকে যোগফলে রূপান্তর • Trigonometric Ratios of Associated Angles

2sin⁡Acos⁡B=sin⁡(A+B)+sin⁡(A−B),2cos⁡Acos⁡B=cos⁡(A+B)+cos⁡(A−B)2\sin A \cos B = \sin(A+B) + \sin(A-B), \quad 2\cos A \cos B = \cos(A+B) + \cos(A-B)
2sin A cos B = sin(A+B) + sin(A-B), 2cos A cos B = cos(A+B) + cos(A-B)

💡 Condition: Transforms trigonometric products into sums/differences, essential in calculus integration.

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HSCChapter 7 (Paper 1)

Sum to Product Transformations

যোগফল থেকে গুণফলে রূপান্তর • Trigonometric Ratios of Associated Angles

sin⁡C+sin⁡D=2sin⁡C+D2cos⁡C−D2,cos⁡C+cos⁡D=2cos⁡C+D2cos⁡C−D2\sin C + \sin D = 2\sin\frac{C+D}{2}\cos\frac{C-D}{2}, \quad \cos C + \cos D = 2\cos\frac{C+D}{2}\cos\frac{C-D}{2}
sin C + sin D = 2sin(C+D)/(2)cos(C-D)/(2), cos C + cos D = 2cos(C+D)/(2)cos(C-D)/(2)

💡 Condition: Converts sums of trigonometric ratios into factorized products.

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HSCChapter 7 (Paper 1)

Multiple Angle Formulas (2A)

গুণিতক কোণের সূত্রাবলী • Trigonometric Ratios of Associated Angles

sin⁡2A=2sin⁡Acos⁡A=2tan⁡A1+tan⁡2A,cos⁡2A=cos⁡2A−sin⁡2A=1−2sin⁡2A\sin 2A = 2\sin A \cos A = \frac{2\tan A}{1+\tan^2 A}, \quad \cos 2A = \cos^2 A - \sin^2 A = 1 - 2\sin^2 A
sin 2A = 2sin A cos A = (2tan A)/(1+tan^2 A), cos 2A = cos^2 A - sin^2 A = 1 - 2sin^2 A

💡 Condition: Double angle identities expressing sin 2A and cos 2A in terms of angle A.

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HSCChapter 7 (Paper 1)

Sine & Cosine Rules of Triangle

ত্রিভুজের সাইন ও কোসাইন সূত্র • Trigonometric Ratios of Associated Angles

asin⁡A=bsin⁡B=csin⁡C=2R,cos⁡A=b2+c2−a22bc\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R, \quad \cos A = \frac{b^2 + c^2 - a^2}{2bc}
(a)/(sin A) = (b)/(sin B) = (c)/(sin C) = 2R, cos A = (b^2 + c^2 - a^2)/(2bc)

💡 Condition: Fundamental laws relating sides and angles in arbitrary triangles with circumradius R.

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HSC (Paper 1) - Chapter 9: Differentiation (Calculus I)

4 Formulas
HSCChapter 9 (Paper 1)

First Principles & Power Rule

মূল নিয়মে অন্তরজের সংজ্ঞা ও ঘাত সূত্র • Differentiation (Calculus I)

f′(x)=lim⁡h→0f(x+h)−f(x)h,ddx(xn)=nxn−1f'(x) = \lim_{h \to 0}\frac{f(x+h) - f(x)}{h}, \quad \frac{d}{dx}(x^n) = n x^{n-1}
f'(x) = lim_h to 0(f(x+h) - f(x))/(h), (d)/(dx)(x^n) = n x^n-1

💡 Condition: Limit definition of derivative from first principles and the polynomial power rule.

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HSCChapter 9 (Paper 1)

Product & Quotient Rules

গুণফল ও ভাগফলের ডেরিভেটিভ • Differentiation (Calculus I)

ddx(uv)=udvdx+vdudx,ddx(uv)=vdudx−udvdxv2\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}, \quad \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}
(d)/(dx)(uv) = u(dv)/(dx) + v(du)/(dx), (d)/(dx)((u)/(v)) = (vfracdu)/(dx) - u(dv)/(dx)v^2

💡 Condition: Differentiation rules for composite products and quotients of differentiable functions.

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HSCChapter 9 (Paper 1)

Derivatives of Elementary Functions

মৌলিক ফাংশনসমূহের ডেরিভেটিভ • Differentiation (Calculus I)

ddx(sin⁡x)=cos⁡x,ddx(cos⁡x)=−sin⁡x,ddx(ex)=ex,ddx(ln⁡x)=1x\frac{d}{dx}(\sin x) = \cos x, \quad \frac{d}{dx}(\cos x) = -\sin x, \quad \frac{d}{dx}(e^x) = e^x, \quad \frac{d}{dx}(\ln x) = \frac{1}{x}
(d)/(dx)(sin x) = cos x, (d)/(dx)(cos x) = -sin x, (d)/(dx)(e^x) = e^x, (d)/(dx)(ln x) = (1)/(x)

💡 Condition: Standard derivative formulas for trigonometric and exponential functions.

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HSCChapter 9 (Paper 1)

Conditions for Maxima & Minima

গুরুমান ও লঘুমানের শর্তাবলী • Differentiation (Calculus I)

f′(x)=0;f′′(x)<0  ⟹  গুরুমান (Max),f′′(x)>0  ⟹  লঘুমান (Min)f'(x) = 0; \quad f''(x) < 0 \implies \text{গুরুমান (Max)}, \quad f''(x) > 0 \implies \text{লঘুমান (Min)}
f'(x) = 0; f''(x) < 0 implies গুরুমান (Max), f''(x) > 0 implies লঘুমান (Min)

💡 Condition: First and second derivative tests to identify local maximum and minimum turning points.

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HSC (Paper 1) - Chapter 10: Integration (Calculus II)

4 Formulas
HSCChapter 10 (Paper 1)

Standard Indefinite Integrals

আদর্শ অনির্দিষ্ট যোগজসমূহ • Integration (Calculus II)

∫xn dx=xn+1n+1+c  (n≠−1),∫1x dx=ln⁡∣x∣+c,∫eax dx=1aeax+c\int x^n \,dx = \frac{x^{n+1}}{n+1} + c \; (n \neq -1), \quad \int \frac{1}{x}\,dx = \ln|x| + c, \quad \int e^{ax}\,dx = \frac{1}{a}e^{ax} + c
int x^n ,dx = fracx^n+1n+1 + c ; (n neq -1), int (1)/(x),dx = ln|x| + c, int e^ax,dx = (1)/(a)e^ax + c

💡 Condition: Fundamental anti-derivative rules for polynomial, reciprocal, and exponential forms.

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HSCChapter 10 (Paper 1)

Trigonometric Integrals

ত্রিকোণমিতিক ফাংশনের যোগজ • Integration (Calculus II)

∫sin⁡x dx=−cos⁡x+c,∫cos⁡x dx=sin⁡x+c,∫sec⁡2x dx=tan⁡x+c\int \sin x \,dx = -\cos x + c, \quad \int \cos x \,dx = \sin x + c, \quad \int \sec^2 x \,dx = \tan x + c
int sin x ,dx = -cos x + c, int cos x ,dx = sin x + c, int sec^2 x ,dx = tan x + c

💡 Condition: Basic antiderivatives of common trigonometric functions.

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HSCChapter 10 (Paper 1)

Integration by Parts Formula

খণ্ডশ যোগজীকরণ (Integration by Parts) • Integration (Calculus II)

∫uv dx=u∫v dx−∫(dudx∫v dx)dx\int u v \,dx = u \int v \,dx - \int \left( \frac{du}{dx} \int v \,dx \right) dx
int u v ,dx = u int v ,dx - int ( (du)/(dx) int v ,dx ) dx

💡 Condition: Product integration formula based on LIATE priority rule.

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HSCChapter 10 (Paper 1)

Definite Integral & Area Under Curve

নির্দিষ্ট যোগজ দ্বারা ক্ষেত্রফল • Integration (Calculus II)

A=∫aby dx=∫abf(x) dxA = \int_{a}^{b} y \,dx = \int_{a}^{b} f(x) \,dx
A = int_a^b y ,dx = int_a^b f(x) ,dx

💡 Condition: Calculates the net geometric area bounded by a continuous curve between limits [a, b].

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HSC (Paper 2) - Chapter 3: Complex Numbers

3 Formulas
HSCChapter 3 (Paper 2)

Modulus & Argument of Complex Number

পরমমান ও আর্গুমেন্ট নির্ণয় • Complex Numbers

z=x+iy  ⟹  ∣z∣=x2+y2,arg⁡(z)=tan⁡−1(yx)z = x + iy \implies |z| = \sqrt{x^2 + y^2}, \quad \arg(z) = \tan^{-1}\left(\frac{y}{x}\right)
z = x + iy implies |z| = sqrt(x^2 + y^2), arg(z) = tan^-1((y)/(x))

💡 Condition: Modulus magnitude and phase argument of complex number z = x + iy.

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HSCChapter 3 (Paper 2)

Properties of Cube Roots of Unity

এককের কাল্পনিক ঘনমূলের বৈশিষ্ট্য • Complex Numbers

1+ω+ω2=0,ω3=1,ω3n+r=ωr1 + \omega + \omega^2 = 0, \quad \omega^3 = 1, \quad \omega^{3n+r} = \omega^r
1 + ω + ω^2 = 0, ω^3 = 1, ω^3n+r = ω^r

💡 Condition: Algebraic sum and cyclic power properties of complex cube roots of unity.

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HSCChapter 3 (Paper 2)

Euler's Formula & Polar Form

অয়লারের সূত্র ও পোলার রূপ • Complex Numbers

z=r(cos⁡θ+isin⁡θ)=reiθz = r(\cos\theta + i\sin\theta) = r e^{i\theta}
z = r(cosθ + isinθ) = r e^iθ

💡 Condition: Polar exponential representation connecting trigonometry and imaginary exponentiation.

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HSC (Paper 2) - Chapter 6: Conics (Parabola, Ellipse, Hyperbola)

3 Formulas
HSCChapter 6 (Paper 2)

Standard Parabola Formula

পরাবৃত্তের আদর্শ সমীকরণ ও উপাদান • Conics (Parabola, Ellipse, Hyperbola)

y2=4ax  ⟹  শীর্ষ: (0,0),  উপকেন্দ্র: (a,0),  নিয়ামক: x=−a,  উপকেন্দ্রিক লম্ব: 4ay^2 = 4ax \implies \text{শীর্ষ: } (0,0), \; \text{উপকেন্দ্র: } (a,0), \; \text{নিয়ামক: } x = -a, \; \text{উপকেন্দ্রিক লম্ব: } 4a
y^2 = 4ax implies শীর্ষ: (0,0), ; উপকেন্দ্র: (a,0), ; নিয়ামক: x = -a, ; উপকেন্দ্রিক লম্ব: 4a

💡 Condition: Parabola elements with eccentricity e = 1, focal coordinate, directrix, and latus rectum.

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HSCChapter 6 (Paper 2)

Ellipse Equation & Eccentricity

উপবৃত্তের সমীকরণ ও উৎকেন্দ্রিকতা • Conics (Parabola, Ellipse, Hyperbola)

x2a2+y2b2=1  (a>b)  ⟹  e=1−b2a2<1,উপকেন্দ্রদ্বয়: (±ae,0)\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \; (a > b) \implies e = \sqrt{1 - \frac{b^2}{a^2}} < 1, \quad \text{উপকেন্দ্রদ্বয়: } (\pm ae, 0)
(x^2)/(a^2) + (y^2)/(b^2) = 1 ; (a > b) implies e = sqrt(1 - (b^2)/(a^2)) < 1, উপকেন্দ্রদ্বয়: ( ± ae, 0)

💡 Condition: Ellipse formula with eccentricity e < 1, foci at (±ae, 0), and area πab.

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HSCChapter 6 (Paper 2)

Hyperbola Equation & Eccentricity

অধিবৃত্তের সমীকরণ ও উৎকেন্দ্রিকতা • Conics (Parabola, Ellipse, Hyperbola)

x2a2−y2b2=1  ⟹  e=1+b2a2>1,উপকেন্দ্রদ্বয়: (±ae,0)\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \implies e = \sqrt{1 + \frac{b^2}{a^2}} > 1, \quad \text{উপকেন্দ্রদ্বয়: } (\pm ae, 0)
(x^2)/(a^2) - (y^2)/(b^2) = 1 implies e = sqrt(1 + (b^2)/(a^2)) > 1, উপকেন্দ্রদ্বয়: ( ± ae, 0)

💡 Condition: Hyperbola equation with eccentricity e > 1 and focal positions.

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HSC (Paper 2) - Chapter 8: Statics

2 Formulas
HSCChapter 8 (Paper 2)

Lami's Theorem

লামির উপপাদ্য • Statics

Psin⁡α=Qsin⁡β=Rsin⁡γ\frac{P}{\sin\alpha} = \frac{Q}{\sin\beta} = \frac{R}{\sin\gamma}
(P)/(sinα) = (Q)/(sinβ) = (R)/(sinγ)

💡 Condition: Equilibrium condition for three concurrent coplanar forces balancing each other.

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HSCChapter 8 (Paper 2)

Triangle Law of Forces

বলের ত্রিভুজ সূত্র • Statics

POA=QAB=RBO\frac{P}{OA} = \frac{Q}{AB} = \frac{R}{BO}
(P)/(OA) = (Q)/(AB) = (R)/(BO)

💡 Condition: Geometric equilibrium condition represented by cyclic sides of a closed triangle.

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SSC - Chapter 1: Set and Function

3 Formulas
SSCChapter 1

Subsets Count

উপসেট সংখ্যা • Set and Function

n(P(A))=2nn(P(A)) = 2^n
n(P(A)) = 2^n
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SSCChapter 1

Three-Set Union

তিন সেটের সংযোগ • Set and Function

n(A∪B∪C)=n(A)+n(B)+n(C)−n(A∩B)−n(B∩C)−n(C∩A)+n(A∩B∩C)n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(C \cap A) + n(A \cap B \cap C)
n(A cup B cup C) = n(A) + n(B) + n(C) - n(A cap B) - n(B cap C) - n(C cap A) + n(A cap B cap C)
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SSCChapter 1

Fractional Inverse

ভগ্নাংশ বিপরীত ফাংশন • Set and Function

f(x)=ax+bcx+d  ⟹  f−1(x)=−dx+bcx−af(x) = \frac{ax+b}{cx+d} \implies f^{-1}(x) = \frac{-dx+b}{cx-a}
f(x) = (ax+b)/(cx+d) implies f^-1(x) = (-dx+b)/(cx-a)
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SSC - Chapter 2: Algebraic Expressions

2 Formulas
SSCChapter 2

Remainder Theorem

ভাগশেষ উপপাদ্য • Algebraic Expressions

P(x)=(x−a)Q(x)+R  ⟹  R=P(a)P(x) = (x-a)Q(x) + R \implies R = P(a)
P(x) = (x-a)Q(x) + R implies R = P(a)
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SSCChapter 2

Cyclic Cubic Identity

চক্র-ক্রমিক ঘন অভেদ • Algebraic Expressions

a3+b3+c3−3abc=12(a+b+c)[(a−b)2+(b−c)2+(c−a)2]a^3+b^3+c^3-3abc = \frac{1}{2}(a+b+c)[(a-b)^2+(b-c)^2+(c-a)^2]
a^3+b^3+c^3-3abc = (1)/(2)(a+b+c)[(a-b)^2+(b-c)^2+(c-a)^2]
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SSC - Chapter 3: Geometry (Apollonius Theorem)

3 Formulas
SSCChapter 3

Apollonius's Theorem

এ্যাপোলোনিয়াসের উপপাদ্য • Geometry (Apollonius Theorem)

AB2+AC2=2(AD2+BD2)AB^2 + AC^2 = 2(AD^2 + BD^2)
AB^2 + AC^2 = 2(AD^2 + BD^2)
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SSCChapter 3

Median Formula

মধ্যমা নির্ণয় সূত্র • Geometry (Apollonius Theorem)

da2=2b2+2c2−a24d_a^2 = \frac{2b^2 + 2c^2 - a^2}{4}
d_a^2 = (2b^2 + 2c^2 - a^2)/(4)
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SSCChapter 3

Sides-Medians Relation

বাহু ও মধ্যমার সম্পর্ক • Geometry (Apollonius Theorem)

3(a2+b2+c2)=4(da2+db2+dc2)3(a^2 + b^2 + c^2) = 4(d_a^2 + d_b^2 + d_c^2)
3(a^2 + b^2 + c^2) = 4(d_a^2 + d_b^2 + d_c^2)
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SSC - Chapter 5: Equations (Quadratics & Discriminants)

2 Formulas
SSCChapter 5

Quadratic Formula

দ্বিঘাত সমাধান সূত্র • Equations (Quadratics & Discriminants)

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
x = frac-b ± sqrt(b^2 - 4ac)2a
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SSCChapter 5

Discriminant Formula

নিশ্চায়ক সূত্র • Equations (Quadratics & Discriminants)

D=b2−4acD = b^2 - 4ac
D = b^2 - 4ac
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SSC - Chapter 6: Linear Inequalities

1 Formulas
SSCChapter 6

Inequality Boundary

রৈখিক অসমতা সীমা • Linear Inequalities

ax+b≤c  ⟹  x≤c−ba(a>0)ax + b \le c \implies x \le \frac{c-b}{a} \quad (a > 0)
ax + b le c implies x le (c-b)/(a) (a > 0)
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SSC - Chapter 7: Infinite Geometric Series

2 Formulas
SSCChapter 7

Infinite Sum

অসীমতক সমষ্টি • Infinite Geometric Series

S∞=a1−r(∣r∣<1)S_\infty = \frac{a}{1 - r} \quad (|r| < 1)
S_infty = (a)/(1 - r) (|r| < 1)
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SSCChapter 7

Recurring Decimal

পৌনঃপুনিক রূপান্তর • Infinite Geometric Series

0.a˙b˙=ab99,a.bc˙d˙=abcd−ab9900.\dot{a}\dot{b} = \frac{ab}{99}, \quad a.b\dot{c}\dot{d} = \frac{abcd - ab}{990}
0.dotadotb = (ab)/(99), a.bdotcdotd = (abcd - ab)/(990)
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SSC - Chapter 8: Trigonometry (Circular Measure)

2 Formulas
SSCChapter 8

Arc Length

বৃত্তচাপের দৈর্ঘ্য • Trigonometry (Circular Measure)

s=rθ(θ in radians)s = r\theta \quad (\theta \text{ in radians})
s = rθ (θ in radians)
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SSCChapter 8

Sector Area

বৃত্তকলার ক্ষেত্রফল • Trigonometry (Circular Measure)

A=12r2θA = \frac{1}{2}r^2\theta
A = (1)/(2)r^2θ
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SSC - Chapter 9: Exponential & Logarithmic Functions

1 Formulas
SSCChapter 9

Change of Base

ভিত্তি পরিবর্তন সূত্র • Exponential & Logarithmic Functions

log⁡ab=ln⁡bln⁡a\log_a b = \frac{\ln b}{\ln a}
log_a b = (ln b)/(ln a)
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SSC - Chapter 10: Binomial Expansion

2 Formulas
SSCChapter 10

Binomial Theorem

দ্বিপদী উপপাদ্য • Binomial Expansion

(x+y)n=∑r=0n(nr)xn−ryr(x + y)^n = \sum_{r=0}^n \binom{n}{r} x^{n-r} y^r
(x + y)^n = Σ_r=0^n binomnr x^n-r y^r
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SSCChapter 10

Combination Formula

সমাবেশ সূত্র • Binomial Expansion

(nr)=n!r!(n−r)!\binom{n}{r} = \frac{n!}{r!(n-r)!}
binomnr = (n!)/(r!(n-r)!)
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SSC - Chapter 11: Coordinate Geometry (Shoelace Formula)

3 Formulas
SSCChapter 11

Distance Formula

দূরত্ব সূত্র • Coordinate Geometry (Shoelace Formula)

d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
d = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)
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SSCChapter 11

Slope Formula

ঢাল সূত্র • Coordinate Geometry (Shoelace Formula)

m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}
m = (y_2 - y_1)/(x_2 - x_1)
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SSCChapter 11

Shoelace Formula

Shoelace ক্ষেত্রফল • Coordinate Geometry (Shoelace Formula)

Δ=12∣(x1y2+x2y3+… )−(y1x2+y2x3+… )∣\Delta = \frac{1}{2} |(x_1 y_2 + x_2 y_3 + \dots) - (y_1 x_2 + y_2 x_3 + \dots)|
Δ = (1)/(2) |(x_1 y_2 + x_2 y_3 + dots) - (y_1 x_2 + y_2 x_3 + dots)|
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SSC - Chapter 12: Planar Vectors

1 Formulas
SSCChapter 12

Vector Magnitude & Direction

ভেক্টরের মান ও দিক • Planar Vectors

∣v⃗∣=x2+y2,θ=tan⁡−1(yx)|\vec{v}| = \sqrt{x^2 + y^2}, \quad \theta = \tan^{-1}\left(\frac{y}{x}\right)
|v| = sqrt(x^2 + y^2), θ = tan^-1((y)/(x))
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SSC - Chapter 13: Solid Geometry (3D Shapes)

2 Formulas
SSCChapter 13

Cone Volume & Area

কোণকের আয়তন ও তল • Solid Geometry (3D Shapes)

l=r2+h2,  V=13πr2h,  A=πrll = \sqrt{r^2+h^2}, \; V = \frac{1}{3}\pi r^2 h, \; A = \pi r l
l = sqrt(r^2+h^2), ; V = (1)/(3)π r^2 h, ; A = π r l
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SSCChapter 13

Sphere Volume & Area

গোলকের আয়তন ও ক্ষেত্রফল • Solid Geometry (3D Shapes)

V=43πr3,  A=4πr2V = \frac{4}{3}\pi r^3, \; A = 4\pi r^2
V = (4)/(3)π r^3, ; A = 4π r^2
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SSC - Chapter 14: Probability

1 Formulas
SSCChapter 14

Probability Formula

সম্ভাবনা সূত্র • Probability

P(E)=n(E)n(S),0≤P(E)≤1P(E) = \frac{n(E)}{n(S)}, \quad 0 \le P(E) \le 1
P(E) = (n(E))/(n(S)), 0 le P(E) le 1
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❓ Higher Mathematics Board Examination FAQs

Common questions on formula application and exam problem-solving rules

Q1: What is Apollonius theorem and how does it determine triangle medians?

For △ABC\triangle ABC with median ADAD, Apollonius' theorem states: AB2+AC2=2(AD2+BD2)AB^2 + AC^2 = 2(AD^2 + BD^2). Median length from sides a,b,ca, b, c is AD=122b2+2c2−a2AD = \frac{1}{2}\sqrt{2b^2 + 2c^2 - a^2}.

Q2: How does the discriminant ($b^2 - 4ac$) determine quadratic root characteristics?

For ax2+bx+c=0ax^2 + bx + c = 0, discriminant D=b2−4acD = b^2 - 4ac. (1) D>0D > 0 and a perfect square: real, unequal, rational. (2) D>0D > 0 non-square: real, unequal, irrational. (3) D=0D = 0: real and identical (x=−b/2ax = -b/2a). (4) D<0D < 0: conjugate imaginary roots.

Q3: What are the distance and slope formulas for two Cartesian points?

Given points A(x1,y1)A(x_1, y_1) and B(x2,y2)B(x_2, y_2), distance is d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} and slope is m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}.

Q4: What is the general term formula for binomial expansion?

In (a+x)n(a + x)^n, the (r+1)(r+1)-th general term is Tr+1=(nr)an−rxrT_{r+1} = \binom{n}{r} a^{n-r} x^r, where (nr)=n!r!(n−r)!\binom{n}{r} = \frac{n!}{r!(n-r)!}.
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