what will be slope of the line y=mx+1 tangent to parabola y^=4x?
Question : The line y =mx+1 is a tangent to parabola y^2 =4x if m=? Solution : Given parabola \[y^2\ =4x\] compare it with the form \[y^2=4ax\] we get \[a=1.\]…
A hemisphere rests in equilibrium on a sphere of equal radius . If the flat surface of the hemisphere rests on the sphere then this equilibrium is stable or unstable
Question : A hemisphere rests in equilibrium on a sphere of equal radius . If the flat surface of the hemisphere rests on the sphere then this equilibrium is stable…
how do you find other two roots of a cubic equation if one root of the equation is given
In the following question, how do you find other two roots of a cubic equation if one root of the equation is given Question : If one root of the…
If the roots of the equation x^3-5x^2-16x+80=0 are 4, -4 then third root of the equation is
Question : If the roots of the equation x^3-5x^2-16x+80=0 are 4, -4 then third root of the equation is Solution ; Two roots of the equation \[x^3-5x^2-16x+80=0\] \[α=4,β=-4\] Let third…
how to reduce the equation x^2(y-px)=p^2y to clairaut form and find it’s general solution
Question : Reduce the equation x^2(y-px)=p^2y to clairaut form and find it's general solution Solution : To Reduce the equation \[x^2\left(y-px\right)=p^2y\] \[y-px=\frac{y}{x^2}p^2\] \[x^2-\frac{x^3}{y}p=p^2 \] (1) Let \[X=x^2,\ Y=y^2\] Then \[\frac{dX}{dx}=2x,\frac{dY}{dx}=2y\frac{dy}{dx}\]…
The distinct eigen values of the matrix [1 1 0;1 1 0;0 0 0] are
Question :The distinct eigen values of the matrix [1 1 0;1 1 0;0 0 0] are (1) 0,1 (2) 1, -1 (3) 0,2 (4) 1, 2 Solution : Let the…
How to prove that F=(x^2-y^2)i-(2xy+y)j is a conservative force
Question : Prove that F=(x^2-y^2)i-(2xy+y)j is a conservative force Solution : To prove that Force\[\vec{F}=\left(x^2-y^2+x\right)\hat{i}-\left(2xy+y\right)\hat{j}\] is a conservative force. We shall find curl of F \[ curl\left(\vec{F}\right)=\nabla\times\vec{F}=\left(\hat{i}\frac{\partial}{\partial x}+\hat{j}\frac{\partial}{\partial y}\right)\times\left(\left(x^2-y^2+x\right)\hat{i}-\left(2xy+y\right)\hat{j}\right)\] \[=\left(\hat{i}\times\hat{i}\right)\frac{\partial\left(x^2-y^2+x\right)}{\partial…
how to Find curl of the vector F=(x^2-y^2)i-(2xy+y)j
Question : Find curl of the vector F=(x^2-y^2)i-(2xy+y)j Solution : Given vector \[\vec{F}=\left(x^2-y^2+x\right)\hat{i}-\left(2xy+y\right)\hat{j}\] Then curl of F is \[ curl\left(\vec{F}\right)=\nabla\times\vec{F}=\left(\hat{i}\frac{\partial}{\partial x}+\hat{j}\frac{\partial}{\partial y}\right)\times\left(\left(x^2-y^2+x\right)\hat{i}-\left(2xy+y\right)\hat{j}\right)\] \[=\left(\hat{i}\times\hat{i}\right)\frac{\partial\left(x^2-y^2+x\right)}{\partial x}-\left(\hat{i}\times\hat{j}\right)\frac{\partial\left(2xy+y\right)}{\partial x}+\left(\hat{j}\times\hat{i}\right)\frac{\partial\left(x^2-y^2+x\right)}{\partial y}-\left(\hat{j}\times\hat{j}\right)\frac{\partial\left(2xy+y\right)}{\partial y}\] Since \[\hat{i}\times\hat{i}=\hat{j}\times\hat{j}=0\ ,\…
The work done by the force F=(x^2-y^2)i-(2xy+y)j displacing a particle in the xy plane from (0, 0) to (1, 1) along the parabola is
Question : The work done by the force F=(x^2-y^2)i-(2xy+y)j displacing a particle in the xy plane from (0, 0) to (1, 1) along the parabola is Solution : Given force \[\vec{F}=\left(x^2-y^2+x\right)\hat{i}-\left(2xy+y\right)\hat{j}\]…
Solved find the sum of the series (1/n+1/(n+1)+1/(n+2)+…..+1/4n as limit n approaches infinity
Question : Find the sum of the series 1/(n+1)+1/(n+2)+…..+1/4n as limit n approaches infinity Solution : Let the sum of the series as \[L=\lim_{n\rightarrow\infty}{\left[\frac{1}{n}+\frac{1}{n+1}+\frac{1}{n+2}+\ldots..+\frac{1}{4n}\right]\ }\] \[L=\lim_{n\rightarrow\infty}{\left[\frac{1}{n}+\frac{1}{n+1}+\frac{1}{n+2}+\ldots..+\frac{1}{n+3n}\right]\ }\] \[=\lim_{n\rightarrow\infty}{\frac{1}{n}\left[\frac{n}{n}+\frac{n}{n+1}+\frac{n}{n+2}+\ldots..+\frac{n}{n+3n}\right]\ }\] \[=\lim_{n\rightarrow\infty}{\frac{1}{n}\left[\frac{1}{1+\frac{0}{n}}+\frac{1}{1+\frac{1}{n}}+\frac{1}{1+\frac{2}{n}}+\ldots..+\frac{1}{1+\frac{3n}{n}}\right]\…