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26 cardsHigher Secondary Certificate Examination (Bangladesh)

HSC Higher Mathematics & Physics Concepts

Essential formulas, theorems, and definitions in Higher Mathematics and Physics for Bangladesh HSC Board Examinations.

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Condition for Two Non-Zero Vectors a and b to Be Orthogonal

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Condition for Two Non-Zero Vectors a and b to Be Orthogonal

Their scalar dot product must be zero: a . b = 0 (

∣ a ∣∣ b ∣ cos θ = 0

, so

θ = 9 0^{\circ}

Formula for the Determinant of a 2x2 Matrix [[a, b], [c, d]]

det = a d - b c

. I f

\det = 0$, the matrix is singular and has no inverse.

Equation of Tangent to Circle

x^{2} + y^{2} = r^{2}

at Point

(x_{1}, y_{1})

x x_{1} + y y_{1} = r^{2}

Eccentricity Values of Conic Sections (Circle, Parabola, Ellipse, Hyperbola)

Circle: e = 0; Parabola: e = 1; Ellipse: 0 < e < 1; Hyperbola: e > 1.

Relationship Between Foci, Semi-Major Axis a, and Semi-Minor Axis b of an Ellipse

b^{2} = a^{2} (1 - e^{2})

$ and the coordinates of the foci are $

(\pm a e, 0)

w h e n

a > b$.

Cube Roots of Unity Properties for

ω

ω^{3} = 1

an d

1 + ω + ω^{2} = 0

$. The two imaginary roots are complex conjugates of each other.

Derivative of Inverse Trigonometric Function

y = arcsin (x)

\frac{d y}{d x} = \frac{1}{1 - x ^{2}}

f or

|x| < 1$.

Derivative of

y = arctan (x)

\frac{d y}{d x} = \frac{1}{1 + x ^{2}}

Derivative of Exponential Function

y = a^{x}

(

a > 0, a \neq = 1

)

\frac{d y}{d x} = a^{x} ln (a)

Integration by Parts Formula

\int u \frac{d v}{d x} d x = uv - \int v \frac{d u}{d x} d x

(or

\int u dv = uv - \int v du$).

Fundamental Trigonometric Limit as

x \to 0

lim_{x \to 0} \frac{s i n x}{x} = 1

(w i t h

x$ measured in radians).

Projectile Maximum Range Formula on Flat Ground

R_{max} = \frac{u ^{2}}{g}

$ achieved at launch angle $

α = 4 5^{\circ}

Projectile Maximum Height Formula

H_{max} = \frac{u ^{2} s i n ^{2} α}{2 g}

Lami's Theorem for Three Coplanar Concurrent Forces in Equilibrium

\frac{P}{s i n α} = \frac{Q}{s i n β} = \frac{R}{s i n γ}

$ where each force is proportional to the sine of the angle between the other two forces.

Escape Velocity Formula from Earth's Surface

v_{e} = \frac{2 GM}{R} = 2 g R \approx 11.2 km/s

Variation of Gravitational Acceleration

g

with Depth

d

Below Earth's Surface

g_{d} = g (1 - \frac{d}{R})

. A tE a r t h^{'} sce n t er (

d = R

),

g = 0

Kepler's Third Law of Planetary Motion

T^{2} \propto a^{3}

$ (the square of orbital period is proportional to the cube of semi-major axis).

Stokes' Law for Viscous Drag Force on a Sphere

F = 6 π η r v

w h er e

\eta$ is dynamic viscosity, $r$ is sphere radius, and $v$ is velocity.

Bernoulli's Equation for Incompressible Streamline Fluid Flow

P + \frac{1}{2} ρ v^{2} + ρ g h = constant

$ (conservation of mechanical energy in fluid dynamics).

Excess Pressure Inside a Liquid Drop vs Soap Bubble

Liquid drop (1 free surface): $

Δ P = \frac{2 T}{R}

$; Soap bubble in air (2 free surfaces): $

Δ P = \frac{4 T}{R}

Carnot Heat Engine Efficiency Formula

η = 1 - \frac{T _{C}}{T _{H}} = \frac{T _{H} - T _{C}}{T _{H}}

$ where temperatures must be in Kelvin.

First Law of Thermodynamics Equation

d Q = d U + d W

$ (heat supplied equals change in internal energy plus work done by the system).

Young's Double Slit Fringe Width Formula

β = \frac{λ D}{d}

w h er e

\lambda

i s w a v e l e n g t h,

i s d i s t an ce t oscr ee n, an d

d$ is slit separation.

Brewster's Law for Complete Polarization by Reflection

tan (i_{p}) = n

w h er e

i_p$ is polarizing angle and $n$ is refractive index of medium.

Einstein's Photoelectric Equation

h f = Φ + K_{max} = h f_{0} + \frac{1}{2} m v_{max}^{2}

w h er e

\Phi$ is work function and $f_0$ is threshold frequency.

de Broglie Matter Wavelength Formula

λ = \frac{h}{p} = \frac{h}{m v}

w h er e

i s P l an c k^{'} sco n s t an t (

6.626 \times 10^{-34} \text{ J s}$).

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