Posts for Tag: HSC

Differentiation Formulas – HSC Math

In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus, the study of the area beneath a curve. if $y=f(x)$ is a function, then its first derivative, i.e. ${\dy}/{\dx}=f'(x)=\lim↙{h→0}{f(x+h)-f(x)}/h$ GeneralRead More

Functions – HSC Math

A function is a special relationship where each input has a single output. It is often written as $y=f(x)$ where $x$ is input and $y$ is the output value. Say $x_1,x_2∈D$ $f(x_1)=f(x_2)$ if $x_1=x_2$ or, $f(x_1)≠f(x_2)$ if $x_1≠x_2$ then the function is called a one-to-one function. Where D is the set of decimal numbers. ifRead More

Trigonometric Properties of a Triangle – HSC Math

Sine Rule $a/{Sin\ A}=b/{Sin\ B}=c/{Sin\ C}$ Cos Rule $cos\ A={b^2+c^2-a^2}/{2bc}$ $cos\ B={a^2+c^2-b^2}/{2ac}$ $cos\ C={a^2+b^2-c^2}/{2ab}$ $a=b\ cos\ C+c\ cos\ B$ $b=c\ cos\ A+a\ cos\ C$ $c=a\ cos\ B+b\ cos\ A$ Tan Rule $tan\ {B-C}/2={b-c}/{b+c}\ cot\ A/2$ $tan\ {A-B}/2={a-b}/{a+b}\ cot\ C/2$ $tan\ {C-A}/2={c-a}/{c+a}\ cot\ B/2$ $tan\ A/2=√{(s-b)(s-c)}/√{s(s-a)}={(s-b)(s-c)}/{∆}$ $tan\ B/2=√{(s-a)(s-c)}/√{s(s-c)}={(s-a)(s-c)}/{∆}$ $tan\ C/2=√{(s-a)(s-b)}/√{s(s-c)}={(s-a)(s-b)}/{∆}$ Here, $s={a+b+c}/2, ∆=$Area of that triangle.Read More

Trigonometric Formulas – HSC Math

In circular system, $θ°={\arc}/{\radius}=s/r$ or, $s=rθ$ $1 \radian=1 \rad=1^c=(180/π)^°$ Area of a circular sector (symbol ⌔) $⌔\AOB=1/2r^2θ$ [θ in radian] $sin\ θ=1/{cosec\ θ}; cos\ θ=1/{sec\ θ}; tan\ θ=1/{cot\ θ}$ $tan\ θ={sin \θ}/{cos\ θ}; cot\ θ={cos\ θ}/{sin\ θ}$ $sin^2\ θ+cos^2\ θ=1$ $sec^2\ θ-tan^2\ θ=1$ $cosec^2\ θ-cot^2\ θ=1$ $-1≤sin\ θ≤1$ $-1≤cos\ θ≤1$ $cosec\ θ≤-1, 1≤cosec\ θ$ $sec\Read More

Permutation and Combinations – HSC Math

$\n!=\n(\n-1)(\n-2) … … 3.2.1$ i. $^\n\P_\r=\n(\n-1)(\n-2) … … (n-r+1)$ ii. $^\n\P_\n=\n!$ iii. $^\n\P_\r={\n!}/{(\n-\r)!}$ $^\n\C_\r=^\n\C_{\n-\r}={\n!}/{{\n!}(\n-\r)!}$ $^\n\C_\r+^\n\C_{\r-1}=^{\n+1}\C_\r$ Number of combination of $\n$ different items taken at least one at a time is $2^n-1$. i. If $\n$ items are arranges in a cyclic order, number of permutation is $(\n-1)!$ ii. If teh cycle can be flipped, number ofRead More

Circle Formulas – HSC Math

With centre $(\h,\k)$ and radius $\r$ a circle is: $(\x-\h)^2 + (\y-\k)^2 = \r^2$ If centre $(\h,\k)$ and passes through $(\α,\β)$: $(\x-\h)^2 + (\y-\k)^2 = (\α-\h)^2 + (\β-\k)^2$ General equation of a circle: $\x^2+\y^2+2\gx+2\fy+\c=0$ where $(-\g,-\f)$ is centre and radius, $\r=\g^\2+\f^\2−\c$. (g, f and c are three constants) Position of $(\x_1,\y_1)$ with respect to aRead More

Straight Line Formulas – HSC Math

Cartesian and polar co-ordinates: x = r cosθ $\r = √{\x^2+\y^2}$ $\θ = \tan^{-1}\y/\x$ Distance formula for (x,y) (a,b) Distance = $√{(\x-\a)^2+(\y-\b)^2}$ for the polar, $√{(\r_1^2+\r_2^2-2\r_1\r_2\cos(\θ_1-\θ_2)}$ Division law: internal division, $({\m_1\x_2+\m_2\x_1}/{\m_1+\m_2},{\m_1\y_2+\m_2\y_1}/{\m_1+\m_2})$ external division, $({\m_1\x_2-\m_2\x_1}/{\m_1-\m_2},{\m_1\y_2-\m_2\y_1}/{\m_1-\m_2})$ Middle point: $({\x_1+\x_2}/2,{\y_1+y_2}/2)$ Centroid of triangle: $({\x_1+\x_2+\x_3}/3,{\y_1+y_2+y_3}/3)$ Equation of straight line: i. y = mx + c ii. y – y1Read More

Vector Formulas – HSC Math

$\A↖{→}.\B↖{→} = \AB \cosθ$ $∴ \A \cosθ={|\A↖{→}.\B↖{→}|}/\B, \Projection \of \A↖{→} \on \B↖{→}$ $∴ \B \cosθ={|\A↖{→}.\B↖{→}|}/\A, \Projection \of \B↖{→} \on \A↖{→}$ $\A↖{→}.\B↖{→} = \A_x\B_x+\A_y\B_y+A_z\B_z$ $|\A↖{→}\x\B↖{→}| = \AB \sinθ$ $[\A↖{→}\x\B↖{→}] = |\table \i↖{→}, \j↖{→}, \k↖{→};\A_x,\A_y,\A_z;\B_x,\B_y,\B_z|$ $\i↖{→}.\i↖{→} = \j↖{→}.\j↖{→}=\k↖{→}.\k↖{→}=1$ [$∵\cos0^o=1$] $\i↖{→}.\j↖{→} = \j↖{→}.\k↖{→}=\k↖{→}.\i↖{→}=0$ [$∵\cos90^o=0$] $\i↖{→}\x\i↖{→} = \j↖{→}\x\j↖{→}=\k↖{→}\x\k↖{→}=0↖{→}$ [$∵\sin0^o=0$] $\i↖{→}\x\j↖{→} = \k↖{→}$ $\j↖{→}\x\k↖{→} = \i↖{→}$ $\k↖{→}\x\i↖{→} = \j↖{→}$ $\a↖{→}Read More

Matrix and Determinant Formulas – HSC Math

Let A is a matrix with m row and n column. Then the dimension of a matrix A is m x n. Let A and B are matrices with dimension m x n and p x q respectively. Then multiplication of A and B, i.e. A.B is possible if and only if n=p. if $A=[\tableRead More