Solution to Social Media Post: 22-07-2021 : Geometry

 

Two chords AC and BD of a circle intersect each other at the point O. If two tangents drawn at the points A and B intersect at the point P and two tangents drawn at the points C and D intersect at the point Q, prove that angle P + angle Q = 2 angle BOC 

Inverse Laplace Transformation

 In mathematics, the Laplace transform, named after its inventor Pierre-Simon Laplace is an integral transform that converts a function of a real variable  (often time) to a function of a complex variable  (complex frequency). The transform has many applications in science and engineering because it is a tool for solving differential equations. In particular, it transforms differential equations into algebraic equations and convolution into multiplication.

The Laplace transform of a function f(t), defined for all real numbers t ≥ 0, is the function F(s), which is a unilateral transform defined by

${\displaystyle F(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt}$

(Eq.1)

where s is a complex number frequency parameter

${\displaystyle s=\sigma +i\omega }$, with real numbers σ and ω.

An alternate notation for the Laplace transform is ${\displaystyle {\mathcal {L}}\{f\}}$ instead of F.

Get solution of the following problems.

Evaluate $\mathcal{L} \{ \sin^2 at \}$ $$$$ Evaluate $\mathcal{L} \{ e^{-2t} ( 3\cos 6t - 5 \sin 6t) \}$ $$$$ Evaluate $\mathcal{L} \{ e^{-2t} ( 3\cos 6t - 5 \sin 6t) \}$ $$$$ Evaluate $\mathcal{L}^{-1} \{ \frac{\alpha}{s-2} \}$ $$$$ Evaluate $\mathcal{L}^{-1} \{ \frac{1}{(s-1)(s-2)} \}$ $$$$ Evaluate $\mathcal{L}^{-1} \{ \frac{3s-2}{s62-4s+20} \}$ $$$$ Evaluate $\mathcal{L}^{-1} \{ \frac{1+s^8}{s^9)} \}$ $$$$ Evaluate $\mathcal{L}^{-1} \{ \frac{s}{s^2+2}+ \frac{6s}{s^2-16}+\frac{3}{s-3} \}$ $$$$ Evaluate $\mathcal{L}^{-1} \{ \frac{1}{(s+2)^2(s-2)} \}$ $$$$ Evaluate $\mathcal{L}^{-1} \{ \frac{1}{s^2-6s+10} \}$ $$$$ Evaluate $\mathcal{L}^{-1} \{ \frac{3s+7}{s^2-2s-3} \}$ $$$$ Evaluate $\mathcal{L}^{-1} \{ \frac{1}{(s+a)^3} \}$ $$$$ Evaluate $\mathcal{L}^{-1} \{ \frac{s}{(s^2-a^2)^2} \}$ $$$$ Evaluate $\mathcal{L}^{-1} \{ \frac{s}{(s^2+a^2)^2} \}$ $$$$ Evaluate $\mathcal{L}^{-1} \{ \frac{s^2}{(s^2+2^2)^2} \}$ $$$$ Evaluate $\mathcal{L}^{-1} \{ \frac{1}{s^2+6s+13} \}$ $$$$ Evaluate $\mathcal{L}^{-1} \{ \frac{1}{(s^2+1)(s^2+4s+5)} \}$ $$$$ Evaluate $\mathcal{L}^{-1} \{ \frac{3s^2+10s+3}{(s^2+1)(s^3+4s62+5s+2)} \}$ $$$$ Evaluate $\mathcal{L}^{-1} \{ \frac{2s+7}{(s+3)^4} \}$ $$$$ Evaluate $\mathcal{L}^{-1} \{ \frac{s+3}{(s^2+4)^2} \}$ $$$$ Download the file below:

Geometry Solved Problems : Circles

Solved problems on circles for class X. Get solutions of problems shown in the post and many more. Visit the link Solved Problems: Circles

Trigonometry: Solved Problems

Solutions of Triangle is an important topic in the JEE Main and JEE Advanced and in class XI for CBSE and other state boards in India. This topic comprises various formulae and rules like the sine rule, cosine rule, tangent rule etc. Questions based on the application of these formulas are often asked in exams. Revising these problems will help students to remember them and easily solve other questions of similar type and also apply their learning to new problems:

Solved Problems from the Book of S.L Loney Plane Trigonometry: CLICK HERE

More Problems to follow soon :)