JEE Physics Mathematical Tools Notes Set 02

Algebra Tools

Quadratic \( ax^2+bx+c=0 \) has roots \( x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \). Useful identities: \( (a\pm b)^2=a^2\pm2ab+b^2 \), \( a^2-b^2=(a-b)(a+b) \). For small x, binomial approximation: \( (1+x)^n\approx1+nx \), and \( e^x\approx1+x \), \( \ln(1+x)\approx x \).

Logarithms

\( \log(ab)=\log a+\log b \), \( \log\frac ab=\log a-\log b \), \( \log a^n=n\log a \), \( \ln x=2.303\log_{10}x \).

Trigonometry

Radian measure: \( \pi\ \text{rad}=180^\circ \). Basic identities: \( \sin^2\theta+\cos^2\theta=1 \), \( \sin2\theta=2\sin\theta\cos\theta \), \( \cos2\theta=\cos^2\theta-\sin^2\theta \). Sum formulae: \( \sin(A\pm B)=\sin A\cos B\pm\cos A\sin B \). Small angles: \( \sin\theta\approx\theta \), \( \cos\theta\approx1-\theta^2/2 \), \( \tan\theta\approx\theta \) (in radian).

Triangles, Area and Volume

Sine rule: \( \frac a{\sin A}=\frac b{\sin B}=\frac c{\sin C} \). Cosine rule: \( c^2=a^2+b^2-2ab\cos C \). Circle: area \( \pi r^2 \), circumference \( 2\pi r \). Sphere: surface \( 4\pi r^2 \), volume \( \tfrac43\pi r^3 \). Cylinder volume \( \pi r^2h \), cone volume \( \tfrac13\pi r^2h \).

Differentiation

\( \frac{d}{dx}x^n=nx^{n-1} \), \( \frac{d}{dx}\sin x=\cos x \), \( \frac{d}{dx}\cos x=-\sin x \), \( \frac{d}{dx}e^x=e^x \), \( \frac{d}{dx}\ln x=\frac1x \). Product rule \( (uv)'=u'v+uv' \), chain rule \( \frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx} \). A maximum or minimum of y occurs where \( \frac{dy}{dx}=0 \); it is a maximum if \( \frac{d^2y}{dx^2}<0 \).

Integration

\( \int x^ndx=\frac{x^{n+1}}{n+1}+C \) (n not equal to -1), \( \int\frac{dx}x=\ln x+C \), \( \int\sin x\,dx=-\cos x+C \), \( \int\cos x\,dx=\sin x+C \), \( \int e^xdx=e^x+C \). A definite integral gives the area under the curve between the limits.

Worked Example

Find the maximum value of \( y=6x-x^2 \).
Solution: \( y'=6-2x=0 \) gives x = 3 and \( y''=-2<0 \), so the maximum is \( y=18-9=9 \).