NEET Important Topic - Physics Mathematical Tools

Name of the Concept

Key Points of Concept

1. Quadratic equation

• A quadratic equation is a second-degree equation in which the highest power to which the variable is raised is to the power 2.

• The quadratic equation form is given below where a, b and c are constants

$ax^2+bx+c=0$

• The discriminant of the quadratic equation is given by,

$D=b^2-4ac$

• The solution of the quadratic equation is given by,

$x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$

• One of the applications of quadratic equations in physics is calculating the time when the ball kicked by reaches at a height H. Here, we will get two values of time t1 and t1 after solving the quadratic equations.

2. Binomial expansion

• The pattern of binomial expansion is given by the binomial theorem as follows

$(a+b)^n=a^n+na^{n-1}b^1+\dfrac{n(n-1)}{2\times1}a^{n-2}b^2+\dfrac{n(n-1)(n-2)}{3\times2\times1}a^{n-3}b^3...~...+b^n$

• Similarly, the expansion of (1+x)n is given by,

$(1+x)^n=1+nx^1+\dfrac{n(n-1)}{2\times1}x^2+\dfrac{n(n-1)(n-2)}{3\times2\times1}x^3...~...+x^n$

• If the values x is very small, we can neglect the higher powers of x and rewrite the binomial expansion as

$(1+x)^n=1+nx$

3. Important trigonometric identities.

• Trigonometric ratios has positive or negative values depending on the quadrant a given below

Image: Trigonometric function signs

• Some of the important trigonometric relations are given below.

$\sin^2\theta+\cos^2\theta=1$

$1+\tan^2\theta=\sec^2\theta$

$\sin(A+B)=\sin A\cos B+\cos A\sin B$

$\sin(A-B)=\sin A\cos B-\cos A\sin B$

$\cos(A+B)=\cos A\cos B-\sin A\sin B$

$\cos(A-B)=\cos A\cos B+\sin A\sin B$

$\tan(A+B)=\dfrac{\tan A+\tan B}{1-\tan A\tan B}$

4. Differentiation

• Let y be a function of x, then ratio of change of y for a very small change in x can be represented by 

$\dfrac{dy}{dx}=\lim_{\Delta x \rightarrow 0}\dfrac{\Delta y}{\Delta x}$

• One of the application of differentiation is finding the slope of y-x graph

$\text{slope of PQ}=\dfrac{dy}{dx}$

• Some of the differentiation of basic function is given below.

$\dfrac{d}{dx}(x^n)=nx^{n-1}$

$\dfrac{d}{dx}(x)=1$

$\dfrac{d}{dx}(\text{constant})=0$

$\dfrac{d}{dx}(\ln x)=\dfrac{1}{x}$

$\dfrac{d}{dx}(e^ x)=e^ x$

$\dfrac{d}{dx}(a^ x)=a^ x\ln e$

$\dfrac{d}{dx}(\sin x )=\cos x$

$\dfrac{d}{dx}(\cos x )=-\sin x$ 

5. Rules of differentiation

• Let f(x) and g(x) be two functions of x. The addition and subtraction rule of differentiation is given by

$\dfrac{d}{dx}(f(x)\pm g(x))=\dfrac{d}{dx}f(x)\pm \dfrac{d}{dx}g(x)$

• Constant rule:

$\dfrac{d}{dx}(kf(x))=k\dfrac{d}{dx}f(x)$

Where k is a constant

• Product rule: 

$\dfrac{d}{dx}(f(x)\cdot g(x))=f(x)\cdot\dfrac{d}{dx}g(x)+g(x)\cdot\dfrac{d}{dx}f(x)$

• Quotient rule:

$\dfrac{d}{dx}\left(\dfrac{f(x)}{g(x)}\right)=\dfrac{g(x)\cdot\dfrac{d}{dx}f(x)-f(x)\cdot\dfrac{d}{dx}g(x)}{\left(g(x)\right)^2}$

6. Applications of differentiation

• Velocity is the rate of change of displacement with respect to time

$v=\dfrac{dx}{dt}$

• The slope of the displacement time graph gives velocity.
  
  

Image: slope of displacement time graph 

• Acceleration is the rate of change of velocity with respect to time.

$a=\dfrac{dv}{dt}$

$a=\dfrac{d}{dt}\left(\dfrac{dx}{dt}\right )$

$a=\dfrac{d^2x}{dt^2}$

Image: slope of velocity time graph 

• The slope of the velocity time graph gives the acceleration.

7. Integration and its application

• Integration is basically the summation of infinite quantities, of infinitely small values.

• If f(x) is the derivative of g(x) with respect to x, then g(x) is the integral of f(x)

$f(x)=\dfrac{d}{dx}g(x)$

$\int f(x)dx=g(x)+c$

Where c is an arbitrary constant.

• We can calculate the area under the curve of a given graph using integration.

Image: Area under the curve 

• Some of integral of basic function is given below

$\int dx=x$

$\int x^ndx=\dfrac{x^{n+1}}{n+1}+c$

$\int \dfrac{1}{x}.dx=\ln x +c$

$\int e^xdx=e^x+c$

$\int a^xdx=\dfrac{a^x}{\ln a}+c$

8. Definite integration

• When an integral is defined between two definite limits a and b, it is said to be a definite integration.

$\int_{a}^{b}f(x)dx=[g(x)]^b_a=g(b)-g(a)$

Sl. No

Name of the Concept

Formula

1.

Trigonometric formulae

$\sin 2A=2\sin A\cos A$

$\cos 2A=2\cos^2A-1=1-2\sin^2A=\cos^2A-\sin^2A$

$\sin (A+B)+\sin(A-B)=2\sin A\cos B$

$\cos(A+B)+\cos(A-B)=2\cos A\cos B$

2.

Differentiation of trigonometric functions

$\dfrac{d}{dx}(\tan x)=\sec^2x$

$\dfrac{d}{dx}(\cot x)=-\csc^2x$

$\dfrac{d}{dx}(\sec x)=\sec x\tan x$

$\dfrac{d}{dx}(\csc x)=-\csc x\cot x$

3.

Logarithmic formulae

$\ln e=1$

$\ln 1=0$

$log_{10}10^x=x$

$\ln (mn)=\ln m+\ln n$

$\ln\left(\dfrac {m}{n}\right)=\ln m-\ln n$

4.

Integration formulae of trigonometric function

$\int\cos x.dx=\sin x+c$

$\int\sin x.dx=-\cos x+c$

$\int\sec^2 x.dx=\tan x+c$

$\int\csc^2 x.dx=-\cot x+c$

$\int\sec x\tan x.dx=\sec x+c$

$\int\csc x\cot x.dx=-\csc x+c$

5.

Conversion coefficients

$1\text{ light year}=9.46\times10^{15}~m$

$1A.U=1.496\times10^{11}~m$

$1~A=10^{-10}~m$

$1\text{ Fermi}=10^{-15}~m$

$1eV=1.6\times10^{-19}~J$

$1\text{ Horse power}=746~W$