Solve Math Problems Fast

Introduction to solve math problems fast:

In this article we are going to discuss about the solve math problems fast. In the math problems have different types of topics. The level math problems has some of the followings numbers sets, logic, real number systems, functions and their graphs, probability and Statistics and some topics from algebra and geometry. The math  problems is more importance for all topics. Here we will see the example problems for solve math problems fast. Is this topic Conditional Probabilities hard for you? Watch out for my coming posts.

Example Problems for Solve Math Problems Fast:
Solve math problems fast – Example: 1

Solve `\int_1^2\int_1^x xy^2dx dy`

Solution:

`I=\int_1^2 \ [ \int_1^x xy^2 dy \ ]dx`

`\int_1^x xy^2dy=\ [ \frac{xy^3}{3}\ ]_{y=1}^{x}=\frac{x^4}{3}-\frac{x}{3}`

Therefore,`I=\int_1^2 \ [ \frac{x^4}{3}-\frac{x}{3} \ ]dx`

` =\int_1^2\frac{x^4}{3}dx-\int_1^2\frac{x}{3}dx`

`=\ [ \frac{x^5}{15} \ ] _1^2- \ [ \frac{x^2}{6} \ ]_1^2`

`=\ [ \frac{32}{15}-\frac{1}{15} \ ]-\ [\frac{4}{6}-\frac{1}{6}\ ]`

` =\frac{31}{15}-\frac{3}{6}`

` =\frac{31}{15}-\frac{1}{2}`

`=\frac{62-15}{30}=\frac{47}{30}`

Solve math problems fast – Example: 2

Determine the unit vector perpendicular to both the vectors `2\bar{i}+\bar{j}+3\bar{k},\bar{i}-2\bar{j}+\bar{k}`

Solution:

Let a,b be the given vectors `2\bar{i}+\bar{j}+3\bar{k},\bar{i}-2\bar{j}+\bar{k}`

`\bar{a}\times\bar{b} =` `[[i,j,k],[2,1,3],[1,-2,1]]`

which is equal to `\bar{i}(1+6)-\bar{j}(2-3)+\bar{k}(-4-1)=7\bar{i}+\bar{j}-5\bar{k}`

Therefore unit vector perpendicular to both the vectors is `\pm\frac{\bar{a}\times\bar{b}}{|a||b|}=\pm\frac{7\bar{i}+\bar{j}-5\bar{k}}{\sqrt{49+1+25}}=\pm\frac{1}{5\sqrt{3}}(7\bar{i}+\bar{j}-5\bar{k})`

Solve math problems fast – Example: 3

Evaluate `\nabla^2(\frac{x}{r^2})`

Solution:

`\nabla^2(\frac{x}{r^2})=\sum \frac{\partial^2}{\partial x^2}[\frac{x}{r^2}] --(1)`

Now,`\frac{\partial}{\partial x}(\frac{x}{r^2})=\frac{1}{r^2}-\frac{2x}{r^3}\frac{\partial r}{\partial x}=\frac{1}{r^2}-\frac{2x}{r^3}[\frac{x}{r}]=\frac{1}{r^2}-\frac{2x^2}{r^4} --(2) [ r^2=x^2+y^2+z^2 and \frac{\partial r}{\partial x}=\frac{x}{r}]`

Therefore,`\frac{\partial^2}{\partial r^2}=\frac{\partial}{\partial x}[\frac{\partial}{\partial x}(\frac{x}{r^2})]=\frac{\partial}{\partial x}[\frac{1}{r^2}-\frac{2x^2}{r^4}],` by using (2)

`=-\frac{2}{r^3}\frac{\partial r}{\partial x}-[\frac{4x}{r^4}-\frac{8x^2}{r^5}\frac{\partial r}{\partial x}]=-\frac{2}{r^3}(\frac{x}{r})-\frac{4x}{r^4}+\frac{8x^2}{r^5}\frac{x}{r}`

Therefore,`\frac{\partial^2}{\partial x^2}[\frac{x}{r^2}]=\frac{8x^3}{r^6}-\frac{6x}{r^4} --(3)`

Now,`\frac{\partial}{\partial y}(\frac{x}{r^2})=-\frac{2x}{r^3}(\frac{y}{r})`

`\frac{\partial^2}{\partial y^2}=\frac{\partial}{\partial y}[-\frac{2xy}{r^4}]`

`=-2x[\frac{1}{r^4}-\frac{4y}{r^5}\frac{\partial r}{\partial y}]=-2x[\frac{1}{r^4}-\frac{4y^2}{r^6}]=\frac{8xy^2}{r^6}-\frac{2x}{r^4} --(4)`

Similarly,`\frac{\partial^2}{\partial z^2}[\frac{x}{r^2}]=\frac{8xz^2}{r^6}-\frac{2x}{r^4} --(5)`

Adding (3),(4),(5), we have

`\sum \frac{\partial^2}{\partial x^2}=\frac{8x}{r^6}[x^2+y^2+z^2]-\frac{10x}{r^4}=-\frac{2x}{r^4}`

`or \nabla^2[\frac{x}{r^2}]=-\frac{2x}{r^4},` by using (1)


I have recently faced lot of problem while learning Sum of Uniform Random Variables, But thank to online resources of math which helped me to learn myself easily on net.

Practice Problems for Solve Math Problems Fast:

1. Find a vector of the magnitude 3 and that which is the perpendicular to both of the vectors `3\bar{i}+\bar{j}-4\bar{k},6\bar{i}+5\bar{j}-2\bar{k}`

`Answer: \pm (2\bar{i}-2\bar{j}+\bar{k})`

2. Solve `\int_1^2\int_3^4 \frac{1}{(x+y)^2}dx dy`

`Answer: log\frac{25}{24}`