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Class 12 Ncert solutions Maths || 3d Geometry Class 12 Ncert solutions Maths || 3d Geometry Next Exercises Exercise 11.2 Exercise 11.3                                                               Exercise 11.1 1. If a line makes angles \(90^{o},135^{o},45^{o}\)with the x, y and z-axes respectively, find its direction cosines. Sol: Let l, m, n be the direction cosines of the line. \(\therefore\) l =\(\cos 90^{o}=0\), m=\(\cos 135^{o}=\cos (90^{o} + 45^{o})=-\sin 45^{0}=-\frac{1}{\sqrt2}\) and n =\(\cos 45^{o}=\frac{1}{\sqrt2}\) So, the direction cosines are 0,\(-\frac{1}{\sqrt2}\),\(\frac{1}{\sqrt2}\) 2. Find the direction cosines of a line which makes equal angles with the coordinate axes . Sol:  Let l,m and n be the direction cosines of the line. Since they make equal angles with the coordinate axes s...

                                    Class 12 Ncert Solutions Maths Differential Equations                                                                         Exercise 9.6 Previous Exercises Exercise 9.1 &9.2 Exercise 9.4 Exercise 9.5 For each of the differential equations given in Exercises 1 to 12, find the general solution: 1. \(\frac{dy}{dx}+ 2y=sinx\) Sol: Comparing the given differential equation with $$ \frac{dy}{dx} + Py =Q $$ we get P=2 and Q=sinx Now, I.F(integrating factor) =\(e^{\int P\ dx}=e^{\int 2dx}=e^{2x}\) Hence the solution of the equation is given by \begin{align} y(I.F)&=\int (Q\times I.F)\ dx + C\\ y.e^{2x} &=\int e^{2x}\sin x \ dx +C\tag 1\\ \end{align...

                                      Class 12 Ncert Solutions Maths Differential Equations                                                                      Exercise 9.5 Previous Exercises Exercise 9.1 &9.2 Exercise 9.4 Next Exercises Exercise 9.6 In each of the Exercises 1 to 10 , sow that the given differential equation is homogeneous and solve each of them. 1. \((x^2+xy)\) dy = \((x^2+y^2)\) dx Sol: The given equation can be written as follows \begin{align} \frac{dy}{dx}&=\frac{x^2+y^2}{x^2+xy}\\ \\ \frac{dy}{dx}&={x^2(1+\frac{y^2}{x^2})\over x^2(1+\frac{y}{x})}\\ \\ \frac{dy}{dx}&={(1+\frac{y^2}{x^2})\over (1+\frac{y}{x})}\tag 1\\ \\ &=g({y\over x})\\ \end{align...

Class 12 Ncert Solution Maths Differential Equations Previous Exercises Exercise 9.1 &9.2 Next Exercises Exercise 9.5 Exercise 9.6                                                                             Exercise 9.4 For each of the differential equations in Exercises 1 to 10, find the general solution: 1.\(\frac{dy}{dx}=\frac{1-\cos x}{1+ \cos x}\) Sol: We have \begin{aligned} \frac{dy}{dx}&=\frac{1-\cos x}{1+ \cos x}\\ &= \frac{\sin^2{x\over2} + \cos^2{x\over2} - \cos^2{x\over2} + \sin^2{x\over2}}{\sin^2{x\over2} + \cos^2{x\over2} +\cos^2{x\over2} - \sin^2{x\over2}}\\ &= \frac{\sin^2{x\over2}+ \sin^2{x\over2}}{ \cos^2{x\over2} +\cos^2{x\over2}}\\ &= \frac{2\sin^2{x\over2}}{ 2\cos^2{x\over2}}\\ &= \tan^2{x\over2}\\ \frac{dy}{dx}...

                                        Class 12 Ncert Solutions  Maths Differential Equations                                                                           Exercise 9.1 Next Exercises Exercise 9.1 &9.2 Exercise 9.4 Exercise 9.6 Determine order and degree(if defined) of differential equations given in Exercises 1 to 10. 1. \(\frac{d^4y}{dx^4}+ sin(y''')\) Ans: The order of the differential equation is 4 because the highest power of the highest derivative is 4.  The degree of the given equation is not defined since it is nota polynomial equation 2. \(y^{'} + 5y = 0\) Ans: The order of the equation is 1 and the degree of the equaion is also 1. 3. \(\left(\frac{d...