This is the mathematics formulae book for class 12 students. Here, we have covered all the formulae needed to solve the questions of Mathematics.
Permutation
The total arrangement of total objects taken some or all at a time is called permutation.
The permutation of 'n' objects taken 'r' at a time is denoted by $^np_{r}$ and is defined by;
1. $^np_{r}=\frac{n!}{(n-r)!}$
2. Permutation of an object, all different;$$=\frac{n!}{p!\,q!\,r!}$$ where,
n=total objects
p=repetition of first kind
q=repetition of second kind
r=repetition of third kind
3. Circular permutation= $(n-1)!$
4. If clockwise and anticlockwise arrangement are not distinguishable [necklace of beads and beads into bracelet];
Total number of permutation=$\frac{(n-1)!}{2}$
Combination
The selection of total objects taken some or all at a time is called combination.
The combination of 'n objects taken 'r' at a time is denoted by $^nc_{r} \, or \, c(n,r)$ and defined by;
$$^nc_{r}=\frac{n!}{(n-r)!\times {r!}}$$
Binomial Theorem
The algebraic expression of two terms which are connected by the operation '+' or '-' is binomial.
To find the expansion of Binomial expression for higher studies, a formula is developed which is called Binomial Theorem.
Group Theory
Binary Operation:
The operation which connect the two numbers into single number is called binary operation.
Ex: $2,3\in{N}\, and\, 2+3=5\in{N}$, + is a binary operation on N
but, $2,3\in{N}\implies{2-3=-1\not\in{N}}$. So, - is not a binary operation on N.
Any rule which assign to each order pair of the element of $\mathbb{Z}$ of integer to the single element of Z is called binary operation on $\mathbb{Z}$.
In symbol, $f: z\times{z}+z$
Some binary operations are: $+,-,\times, \div, +_{4}, \times_{3},$etc.
Complex Number
A number of the form $z=a+ib=(a,b)$ where first part is real and second part is an imaginary parts.
Example: $4+5i$
Quadratic Equation
An equation of the form $ax^{2}+bx+c=0$ where $a\neq{0}$ and a,b,c are constants is called quadratic equation in x.
Example:
$4x^{2}+3x+5=0, \, x^{2}-3=0$, etc
Sequence and Series
An arrangement of numbers in a particular order is called sequence.
The sum of the elements of a sequence is called series.
Matrix Based System of Linear Equations
The system of linear equations is a set of two or more linear equations involving the same variable.
↠An equation which is in the form $ax+by+c$, where a,b, and c are real numbers, and a,b are not simultaneously zero is called linear equation in the two variables x and y.
↠A finite collection of linear equations in eh variables $x_{1},x_2,x_3,....,x_n$ is called a system of linear equations in these variables.
↠A set of values of the variables that satisfies all the equations of the system is said to be the solution of the system.
Note:
Consistent and inconsistent Systems:
consistent→at least one solution
inconsistent→otherwise
A pair of linear equations in two variables can be represented as:
$a_{1}x+b_{1}y+c_{1}=0$.....①
$a_{2}x+b_{2}y+c_{2}=0$.....②
1. If both the lines intersect at a point, then there exists a unique solution.
Algebrically, if $\frac{a_{1}}{a_{2}}\neq{\frac{b_{1}}{b_{2}}}$ then the system is consistent and independent.
2. Algebrically, when $\frac{a_{1}}{a_{2}}={\frac{b_{1}}{b_{2}}}={\frac{c_{1}}{c_{2}}}$ then the system is dependent and consistent.
3.Algebrically,if $\frac{a_{1}}{a_{2}}={\frac{b_{1}}{b_{2}}}\neq{\frac{c_{1}}{c_{2}}}$ then the system is said to be inconsistent and has no solution.
#Cramer's Rule:
*Row Equivalent Matrix
*Inverse Matrix Method
Inverse Circular Functions
Let $y=f(x)=sinx$ be a function then inverse function of sin is denoted by $y=sin^{-1}x$ (or Arc sinx) whose domain is [-1,1] and range is $[\frac{-\pi}{2},\frac{\pi}{2}]$.
Trigonometric Equation and General Solution
An equation involving trigonometric functions of unknown angle is known as a trigonometric equation.
The value of unknown angle that satisfies the trigonometric equation is called solution of trigonometric equation.
principle solution=$0\leq{x}\leq{2\pi}$
general solution=$0\leq{x}\leq{n\pi}, \, n\in{Z}$
1. If $sin\theta=0\,then\,\theta={n\pi},\,n\in{Z}$
2. If $\cos\theta=0\,then\,\theta=\frac{(2n+1)}{2},\,n\in{Z}$
3. If $\tan\theta=0\,then\,\theta=n{\pi},\,n\in{Z}$
4. If $cos\theta=cos\alpha\,then\,\theta=2n{\pi}\pm\alpha,\,n\in{Z}$
5. If $sin\theta=sin\alpha\,then\,\theta=n{\pi}+(-1)^n\alpha\,or\,n{\pi}\pm\alpha$
6. If $tan\theta=tan\alpha\,then\,\theta=n{\pi}+\alpha\,for\,n\in{Z}$
7. If $sin^2\theta=sin^2\alpha,\,cos^2\theta=cos^2\alpha,\,tan^2\theta=tan^2\alpha\,then\,\theta=n{\pi}\pm\alpha$
Conic Section
A curve obtained as the intersection of the surface of a cone with a plane.
If e=1, the curve is parabola
If e<1, the curve is an ellipse
If e>1, the curve is hyperbola
↠Parabola is locus of all points which are equally spaced from a fixed line and a fixed point.
↠Eccentricity is the distance from any point on the conic section to its focus, divided by the perpendicular distance from that point to the nearest directrix.
Ellipse:
Ellipse is a locus of a point in a plane such that the sum of length from fixed point is always constant.
↠The fixed point S and S' are called Foci.
↠The length of the line passing through foci is called major axis. [AA'] is called major axis]
↠The line perpendicular to major axis and passing through centre is called minor axis.[YOY'] is called minor axis]
↠The intersection point of ellipse and major axis is called vertices of an ellipse. [A,A']
↠O is the centre of the ellipse.
↠AA' is also called length of major axis.
↠LL' and QQ' which is perpendicular to major axis and passing through focus is called length of latus rectum.
Hyperbola
The conic section having eccentricity greater than 1 is called hyperbola.
The hyperbola is a locus of a point in the plane such that difference of distance from a fixed point is always constant. i.e.$[PF^{1}-PF=C]$
Co-ordinate in space
Plane
A plane is a flat surface where any two points connected by a straight line would lie entirely.
The general equation of a first degree in x,y,z represent a plane. A general equation of first degree in x,y,z is; $A_{x}+B_{y}+C_{z}+D=0$ where A,B,C, and D are constant and A,B,C are non-zero.
Vector Product of two Vectors
Let $\vec{a}=(a_{1},a_{2},a_{3}) \,and\,\vec{b}=(b_{1},b_{2},b_{3})$ be two space vectors. Then the vector (or cross) product of $\vec{a} \,and\,\vec{b}$ and is denoted by $\vec{a}\times{\vec{b}}$ and is defined by;
$\vec{a}\times\vec{b}=(a_{1},a_{2},a_{3})\,\times\,(b_{1},b_{2},b_{3})$
Correlation
The relationship between two or more variable is called correlation.
Karl' Pearson's Coefficient of Correlation:
1. Actual Mean Method: $r=\frac{\sum{XY}}{\sqrt{\sum{X^2}}\cdot\,\sqrt{\sum{Y^2}}}$
2.Direct Method: $r=\frac{n\sum{XY}-\sum{x}\cdot\,\sum{Y}}{\sqrt{n\sum{X^2}-(\sum{X})^2}\cdot\,\sqrt{n\sum{Y^2}-(\sum{Y})^2}}$
3.Assumed Mean Method: $r=\frac{n\sum{uv}-\sum{u}\cdot\,\sum{v}}{\sqrt{n\sum{u^2}-(\sum{u})^2}\cdot\,\sqrt{n\sum{v^2}-(\sum{v})^2}}$
Properties:
- $-1\leq{r}\leq{1}$
- $r=r_{XY}=r_{YX}$ [symmetrical]
- $r=\pm\sqrt{b_{YX}\cdot\,b_{XY}}$ [r is geometric mean between two regressive coefficient]
Regression
Regression is a statistical method used in finance, investing, and other disciplines that attempts to determine the strength and character of the relationship between one dependent variable (usually denoted by y) and a series of other variables (known as independent variables).
Formulae:
1. Regression equation of Y on X.
$Y-\bar{Y}=b_{YX}(X-\bar{X}$
where, $b_{YX}=\frac{n\sum{XY}-\sum{X}\cdot\sum{Y}}{n\sum{X^2}-(\sum{X})^2}$
2. Regression equation of X on Y.
$X-\bar{X}=b_{XY}(Y-\bar{Y}$
where, $b_{XY}=\frac{n\sum{XY}-\sum{X}\cdot\sum{Y}}{n\sum{Y^2}-(\sum{Y})^2}$
3. Relation between r and b.
The correlation coefficient 'r' is the geometric mean of two regression coefficients.$$r=\sqrt{b_{YX}\cdot\,b_{XY}}$$
Also, the regression coefficients can be expressed in terms of r and $\sigma$ as,
$b_{XY}=r\frac{\sigma_{X}}{\sigma_{Y}}$ and $b_{YX}=r\frac{\sigma_{Y}}{\sigma_{X}}$
Note:
- Regression equation of Y on X:$$Y=a+bX$$$$\sum{Y}=na+b\sum{X}$$$$\sum{XY}=a\sum{X}+b\sum{X^2}$$
- Regression equation of X on Y:$$X=a+bY$$$$\sum{X}=na+b\sum{Y}$$$$\sum{XY}=a\sum{Y}+b\sum{Y^2}$$
[The product of two regression coefficient is less than or equal to 1.]
Rank Correlation
A rank correlation is the statistics which measures the relation between the ranking. A ranking is the assignment to first, second, third, etc. to different observation of the given variable.
Spearman's Rank Correlation Coefficient:
The coefficient of rank correlation is calculated by using the formula;
$R=1-\frac{6\sum{d^{2}}}{n(n^{2}-1)}$ or, $r=1-\frac{6\sum{d^{2}}}{n(n^{2}-1)}$
where,
r=R=Coefficient of rank correlation
n=number of pairs of observations
Rank correlation coefficient for repeated ranks;
$r=1-\frac{6\,[\sum{d^{2}}+\frac{m(m^2-1)}{12}]\,}{n(n^{2}-1)}$
Probability
The branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true.
Dependent Events:
Two or more than two events are said to be dependent events, if the occurrence of an event of one trial effect the occurrence of others in the subsequent trials.
Formulae:
$P(E)=\frac{Favourable\,no.\,of\,cases}{Total\,no.\,of\,cases}=\frac{n(E)}{n(S)}$
Properties of Probability:
- $p+q=1$
- $p(certain\,event)=1$
- $p(impossible\,event)=0$
- $0\le{p}\le{1}$
$P(A\cup{B})=P(A or B)=P(A)+P(B)-P(A\cap{B})$
$P(A\cap{B})=P(A\,and\,B)=P(A)\times\,P{B}$ where A and B are independent events
$P{(A\cup{B})^{c}}=1-P(A\cup{B})$
$P(\bar{A})=1-P(A)$
Conditional Probability:
Let $E_{1}$ and $E_2$ be two dependent events then the conditional probability of $E_2$ when $E_1$ is already occurred is denoted by $P(\frac{E_2}{E_1})$ and defined by;
$P(\frac{E_2}{E_1})=\frac{P(E_1\cap{E_2})}{P(E_1)}\,,\,where\,P(E_1)\neq{0}$
$P(\frac{E_1}{E_2})=\frac{P(E_1\cap{E_2})}{P(E_2)}\,,\,P(E_2)\neq{0}$ where $E_2$ is already occurred.
Binomial Distribution:
Bernoulli Process:
An experiment containing only two outcomes is called Bernoulli process. The two outcomes are called success and failure and it is denoted by P & Q.
Example: When a coin is tossed, then, there are two outcomes head or trial.
Formula for finding probability of 'r' success in n trials:
If n=number of trials
p= probability of success
q=probability of failure
r=probability of 'r' success
Then,
$p(r)=^nc_{r}p^r{r}q^{n-r}$
Formulae:
- $p(x=r)=^nc_{r}p^r{r}q^{n-r}$
- $p+q=1,\, p=1-q\,q=1-p$
- Mean of binomial distribution=$n\cdot{p}$
- $S.D.=\sqrt{npq}$
- $Variance=n\cdot{p}\cdot{q}$
- Binomial distribution=$(q+p)^{n}$
Derivative
The instantaneous rate of change of a quantity with respect to the other is called derivative.
Differentials:
1. $\Delta{y}=f(x+\Delta{x})-f(x)$ is the actual change in dependent variable y.
2. The differential of independent variable x, denoted by dx, is defined by $dx=\Delta{x}$
3. The differential of dependent variable y, denoted by $dy$ defined by $dy=f'(x)dx$; which is the approximate change in y.
4. Error=|Actual change-Approximate change|
5. Percentage error=$|\frac{\Delta{y}-dy}{y}|\times\,100$
Tangent and Normal:
Slope of tangent=$tan\theta=\frac{dy}{dx}$
Slope of normal=$\frac{-1}{slope\,of\,tangent}$
Equation of tangent$\implies\,y-y_{1}=(\frac{dy}{dx})_{x=x_{1},y=y_{1}}(x-x_{1}$
Equation of normal$\implies\,y-y_{1}=-(\frac{dx}{dy})_{x=x_{1},y=y_{1}}(x-x_{1}$
#The tangent to the curves are parallel to x-axis if $\frac{dy}{dx}=0$ [horizontal].
and, parallel to y-axis if $\frac{dx}{dy}=0$ [vertical] or $\frac{dy}{dx}$ is undefined.
L' Hospital's Rule for the form $\frac{0}{0}$
If $\phi{(x)}$ and $\psi{(x)}$ are two functions of x such that their derivative $\phi'(x)$ and $\psi'(x)$ are continuous at x=0 and if $\phi(a)=\psi(a)=0$ then;
$\lim_{x\to{a}}\frac{\phi(x)}{\psi(x)}=\lim_{x\to\,a}\frac{\phi'(x)}{\psi'(x)}=\lim_{x\to\,a}\frac{\phi'(a)}{\psi'(a)}$, provided that $\psi'(a)\neq\,0$.
Antiderivatives
Antiderivative is a function that reverses what the derivative does. It is the inverse of derivative.
1. $\int{x^ndx}=\frac{x^{n+1}}{n+1}+c$,$n\neq{-1}$
2. $\int{\frac{1}{x}dx}=ln|x|+x$, $x\neq{0}$
3. $\int{e^xdx}=e^x+c$
4. $\int{a^xdx}=\frac{a^x}{lna}+c$
5. $\int{sinxdx}=-cosx+c$
6. $\int{cosxdx}=sinx+c$
7. $\int{tanxdx}=lnsecx+c$
8. $\int{cotxdx}=lnsinx+c$
9. $\int{secxdx}=ln|secx+tanx|+c=ln|tan(\frac{\pi}{4}+\frac{x}{2})|+c$
10. $\int{cosecxdx}=ln|cosecx-cotx|+c=lntan\frac{x}{2}+c$
11. $\int{sec^2xdx}=tanx+c$
12. $\int{cosec^2xdx}=-cotx+c$
13. $\int{secx\cdot\,tanxdx}=secx+c$
14. $\int{cosecx\cdot\,cotxdx}=-cosecx+c$
15. $\int{\frac{1}{\sqrt{1-x^2}}dx}=sin^{-1}x+c$
16. $\int{\frac{1}{\sqrt{1+x^2}}dx}=tan^{-1}x+c$
17. $\int{\frac{1}{|x|\sqrt{x^2-1}}dx}=sec^{-1}x+c$
18. $\int{(ax+b)^n}dx=\frac{1}{a}\,\frac{(ax+b)^{n+1}}{n+1}+c$, $n\neq{-1}$
19. $\int{\frac{1}{ax+b}dx}=\frac{1}{a}\,ln|ax+b|+c$
20. $\int{(uv)dx}=u\int{vdx}-\int{[\frac{d}{dx}(u)\int{vdx}]dx}+c$
21. $\int{\frac{1}{x^2-a^2}dx}=\frac{1}{2a}ln|\frac{x-a}{x+a}|+c$
22. $\int{\frac{1}{a^2-x^2}dx}=\frac{1}{2a}ln|\frac{a+x}{a-x}|+c$
23. $\int{\frac{1}{x^2+a^2}dx}=\frac{1}{a}\,tan^{-1}\frac{x}{a}+c$
24. $\int{\frac{1}{\sqrt{a^2-x^2}}dx}=sin^{-1}\frac{x}{a}+c$
25. $\int{\frac{1}{\sqrt{x^2-a^2}}dx}=ln|x+\sqrt{x^2-a^2}|+c$
26. $\int{\frac{1}{\sqrt{a^2+x^2}}dx}=ln|x+\sqrt{x^2+a^2}|+c$
27. $\int{\sqrt{a^2-x^2}dx}=\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}sin^{-1}\frac{x}{a}+c$
28. $\int{\sqrt{x^2-a^2}dx}=\frac{x}{2}\sqrt{x^2-a^2}+\frac{a^2}{2}ln|x+\sqrt{x^2-a^2}|+c$
29. $\int{\sqrt{x^2+a^2}dx}=\frac{x}{2}\sqrt{x^2+a^2}+\frac{a^2}{2}ln|x+\sqrt{x^2+a^2}|+c$
30. $\int{sinhxdx}=coshx+c$
31. $\int{coshxdx}=sinhx+c$
32. $\int{tanhxdx}=lncoshx+c$
33. $\int{cothxdx}=lnsinhx+c$
34. $\int{sechxdx}=tan^{-1}|sinhx|+c=2tan^{-1}(tanh\frac{x}{2})+c$
35. $\int{cosechxdx}=lntanh\frac{x}{2}+c$
36. $\int{sech^2xdx}=tanhx+c$
37. $\int{cosech^2xdx}=-cothx+c$
38. $\int{sechx\cdot\,tanhxdx}=-sechx+c$
39. $\int{cosechx\cdot\,cothxdx}=-cosechx+c$
Integrals of Rational Fractions:
Factor in denominator↠↠↠↠↠Corresponding partial fractions
1. $(x+a)(x+b)$ ↠↠↠↠↠$\frac{A}{x+a}+\frac{B}{x+b}$
2. $(x+a)^2$↠↠↠↠↠$\frac{A}{x+a}+\frac{B}{x+a}^2$
3. $(x+b)(x+a)^2$↠↠↠↠↠$\frac{A}{x+b}+\frac{B}{x+a}+\frac{c}{x+a}^2$
4. $ax^2+bx+c$↠↠↠↠↠$\frac{Ax+B}{ax^2+bx+c}$ where A,B and C are constant
Differential Equations
An equation involving dependent variable, independent variable and derivable is called differential equation.
Example:
- $\frac{dy}{dx}=\frac{y}{x}$
- $\frac{d^2y}{dx^2}+\frac{dy}{dx}=2x$
- $(\frac{dy}{dx})^2+x+y=0$
Order and degree of differential equation:
The highest derivative occurring in a differential equation is called order of differential equation.
The power of greatest derivative of differential equation is called degree of differential equation.
Example:
- $(\frac{d^2y}{dx^2})^4+\frac{dy}{dx}+\frac{y}{x}=0$ $$\therefore\,Order\,of\,differential\,equation=2$$$$\therefore\,Degree\,of\,differential equation=4$$
- $[1+(\frac{dy}{dx})^{2}]^{3}=(\frac{d^3y}{dx^3})^{2}$ $$\therefore\,Order\,of\,differential\,equation=3$$$$\therefore\,Order\,of\,differential\,equation=2$$
Solution of differential equation:
Any relation between the variables, which is free from derivatives and satisfies the given differential equation is called solution of differential equation.
Separation of Variables:
If the equation $Mdx+Ndy=0$ can be put in the form $f_{1}(x)dx+f_{2}(y)dy=0$ then it can be solved by integrating each term separately. Thus, the solution of the above equation is $\int{f_{1}(x)dx}+\int{f_{2}(y)dy}=c$.
Homogenous differential equation:
A differential equation of the first order and first degree is said to be homogenous differential equation if $\frac{dy}{dx}=f{(\frac{y}{x})}$.
Example:
$\frac{dy}{dx}=\frac{x^2+y^2}{x^2-y^2}=\frac{x^2(1+\frac{y^2}{x^2})}{x^2(1-\frac{y^2}{x^2})}=\frac{1+\frac{y^2}{x^2}}{1-\frac{y^2}{x^2}}=f(\frac{y}{x})$
Process for solving homogenous differential equation:
- Put $y=vx$, where v is a function of x
- Find $\frac{dy}{dx}=\frac{d}{dx}(vx)=v\frac{dx}{dx}+x\frac{dv}{dx}=v+x\frac{dv}{dx}$
- Put the value of $\frac{dy}{dx}$ and $y=vx$ and solve by separated method.
Exact Differential Equation:
A differential equation of the form $Mdx+Ndy=0$ where M and N are function of x or y or both is called exact differential equation if; $\boxed{Mdx+Ndy=d(xy)}$
It means $\boxed{Mdx+Ndy}$ is an exact or perfect or total differential.
Process for solving exact differential equation:
$\implies\,Mdx+Ndy=0$
$\implies\,d(xy)=0$
Integrating
$\int{d(xy)}=\int{0dx} \, or\, \int{0dy}$
$\therefore\,xy=c$
Formulae:
1. $ydx+xdy=d(xy)$
2. $\frac{ydx-xdy}{y^2}=d(\frac{x}{y})$
3. $\frac{xdy-ydx}{x^2}=d(\frac{y}{x})$
4. $x^2dx=d(\frac{x^3}{3})$
5. $y^4dy=d(\frac{y^5}{5})$
6. $2xydy+y^2dx=d(xy^2)$
7. $\int{dx}=x+c$
8. $\frac{ydx-xdy}{x^2+y^2}=\frac{\frac{ydx-xdy}{y^2}}{1+(\frac{x}{y})^2}=d[tan^{-1}\frac{x}{y}]$
9. $\frac{xdy-ydx}{x^2+y^2}=\frac{\frac{xdy-ydx}{y^2}}{1+(\frac{y}{x})^2}=d[tan^{-1}\frac{y}{x}]$
Linear Differential Equation:
A differential equation of the form $\frac{dy}{dx}+Py=Q$, where P and Q are function of x alone or constant [but not of y] is called linear differential equation of first order.
Example: $\frac{dy}{dx}+x^{2}y=2x$
Process of solving differential equation:
- Given equation
- $\frac{dy}{dx}+Py=Q$ and then find P and Q.
- Find integrating factor (I.F.)=$e^{\int{Pdx}}$ and solve.
- The solution is $y\times{I.F.}=\int{Q\times(I.F.)dx+c}$
Bernoulli's Equation:
A differential equation of the form $\frac{dy}{dx}+Py=Qy^{n}$ where P and Q are function of x only or constant and n is a real number, it is called Bernoulli's equation.
Example: $\frac{dy}{dx}+3x^2y=Qy^4$
Solving Process: Dividing both sides by $y^n$ and putting $y^{1-n}=v$.
Dynamics
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space.
Dynamics means continuous and productive activity or change.
Formulae:
Motion in a straight line:
1. $a=\frac{v-u}{t}$
2. $v=u+at$
3. $s=ut+\frac{1}{2}at^{2}$
4. $v^2=u^2+2as$
5. (th second displacement)$s_{t}=u+\frac{(2t-1)a}{2}$
where,
a=acceleration
v=final velocity
u=initial velocity
s=displacement
t=time
Motion under gravity:
1. Vertically downward motion:
- $v=u+gt$
- $v^2=u^2-2gh$
- $h=ut+\frac{1}{2}gt^2$
- $h_{t}=u+\frac{g}{2}(2t-1)$
2. Vertically upward motion:
- $v=u-gt$
- $v^2=u^2-2gh$
- $h=ut-\frac{1}{2}gt^2$
- $h_{t}=u-\frac{g}{2}(2t-1)$
Motion down in an inclined plane:
1. Projected downward:
- $v^2=u^2+2gsin\alpha\cdot{l}$
- $v=u+gsin\alpha\cdot{t}$
- $l=ut+\frac{1}{2}gsin\alpha\cdot{t^2}$ $$Note: \boxed{a=gsin\theta\, and\, sin\theta=\frac{h}{l}}$$
2. Projected upward:
- $v^2=u^2-2gsin\alpha\cdot{l}$
- $v=u-gsin\alpha\cdot{t}$
- $l=ut-\frac{1}{2}gsin\alpha\cdot{t^2}$ [where l=length of inclines plane]
Newton's law of motion, impulse:
1. Momentum = mv
Change in momentum=mv-mu
Average rate of change in momentum(F)= $\frac{m(v-u)}{t}$
2. Upward Force: $R=m(a+g)$
3. Downward Force: $R=m(g-a)$
4. Impulsive Force: $F=\frac{d(mv)}{dt}$, as m is constant
5. Principle of Conservation of linear momentum;
$mv-MV=0$
$m_{1}v_{1}+m_{2}v_{2}=(m_{1}+m_{2})V$
Projectiles:
1. Equation of motion of a projectile;
For the vertical motion, $vsin\theta=usin\alpha-gt$
For the horizontal motion, $vcos\theta=ucos\alpha$
2. Time to reach the greatest height; $t=\frac{usin\alpha}{g}$
3. Time of Flight; $T=\frac{2usin\alpha}{g}$
4. Range; $R=\frac{u^{2}sin2\alpha}{g}$
5. Maximum horizontal Range=$\frac{u^{2}}{g}$
6. Greatest height; $H=\frac{u^{2}sin^{2}\alpha}{2g}$
Statics
Statics is the branch of mechanics that is concerned with the analysis of acting on physical system that do not experience an acceleration, but rather, are in static equilibrium with their environment.
Formulae:
1. Resultant of two forces acting at a point;
$R^2=P^2+Q^2+2PQcos\alpha$
$\theta=tan^{-1}(\frac{Qsin\alpha}{P+Qcos\alpha})$
2. Resultant of a given force in two given directions;
$P=\frac{Fsin\beta}{sin(\alpha+\beta)}$, $Q=\frac{Fsin\alpha}{sin(\alpha+\beta)}$
3. Resolved parts of a number of coplanar concurrent forces;
$Rcos\theta=P_{1}cos\alpha_{1}+P_{2}cos\alpha_{2}+P_{3}cos\alpha_{3}+.........=X$
$Rsin\theta=P_{1}sin\alpha_{1}+P_{2}sin\alpha_{2}+P_{3}sin\alpha_{3}+.........=Y$
$R=\sqrt{x^{2}+y^{2}}$
$\theta=tan^{-1}(\frac{y}{x})$
4. Triangle of forces:
If three forces acting at a point be represented in magnitude and direction by the sides of a triangle, taken in order, they are in equilibrium.
5. Converse of the Triangle Forces:
If there forces acting at a point be in equilibrium, then they can be represented in magnitude and direction by the three sides of a triangle taken in order.
6. Lami's Theorem:
If three forces acting at a point are in equilibrium, then each force is proportional to the sine of the angle between the other two.
7. Varignon's Theorem:
The algebraic sum of the moments of two forces about any point in their plane is equal to the moment of their resultant about the same point.
8. Moment of a Force: $\implies$F.OA=F.P
9. If P and Q are two like parallel forces then; $\boxed{\frac{P}{BC}=\frac{Q}{AB}=\frac{R}{AC}}$where R=P+Q.
10. If P and Q are unlike parallel forces; if P>Q then R=P-Q.
$\boxed{\frac{P}{BC}=\frac{Q}{AC}=\frac{R}{AB}}$
11, If P and Q are unlike parallel forces and if Q>p then R=Q-P.
$\boxed{\frac{P}{BC}=\frac{Q}{AC}=\frac{R}{AB}}$.png)
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