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Expland \[ \frac{x}{e^x-1}\ \] in powers of x. hence prove that \[ \frac{x}{2}\left[ \frac{e^x+1}{e^x-1}\right]=1+\frac{1}{12}x^2-\frac{1}{720}x^4+\ ..... \]
engineering mathematics updated 5.1 years ago by teamques10 ★ 70k
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Using De-Moivr's theorem, prove that \[ \frac{\sin 6\theta}{\sin\theta}=16\cos^4\theta -16\cos^2\theta +3\ \]
engineering mathematics updated 5.1 years ago by teamques10 ★ 70k
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Using Gauss-seidel method, solve the following system of equations upto 3rd iteration. 5x-y=9 -x+5y-z=4 -y+5z=-6
engineering mathematics updated 5.1 years ago by teamques10 ★ 70k
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\[Show\ that\ \frac{\partial{}\left(u,v\right)}{\partial{}\left(x,y\right)}=6\ r^3\sin{2\ }\theta{}\\ given\ that\ u=x^2-y^2,\ v=2x^2-y^2\ \&\ x=r\cos{\theta{}},\ y=r\sin{\theta{}}.\]
engineering mathematics updated 5.1 years ago by teamques10 ★ 70k
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\[Prove\ that\tan{\left(i\log{\left(\frac{a-ib}{a+ib}\right)}\right)}=\frac{2ab}{a^2-b^2}\]
engineering mathematics updated 5.1 years ago by teamques10 ★ 70k
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\[ If\ cosec \left( \frac{\pi}{4}+ix\right)=u+iv\ \ prove\ that \ \left(u^2+v^2\right) ^2=2 \left(u^2-v^2 \right) \]
engineering mathematics updated 5.1 years ago by teamques10 ★ 70k
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If y=sinpx+cospx prove that \[y_n={P^n\left[1+{\left(-1\right)}^n\sin{\ }\right(2px)]}^{\frac{1}{2}}\]
engineering mathematics updated 5.1 years ago by teamques10 ★ 70k
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\[ prove\ that\ sech^{-1}\ \left(\sin\theta\right)=\log \left(\cot \frac{\theta}{2}\right) \]
engineering mathematics updated 5.1 years ago by teamques10 ★ 70k
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\[ If \ y = \dfrac {\sin^{-1}x}{\sqrt{1-x^2}} \] then prove that (1-x2)yn+1 - (2n+1)xyn-n2yn-1=0
engineering mathematics updated 5.1 years ago by teamques10 ★ 70k
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Solve the following equations by Gauss-seidal method 27x + 6y-z=85 6x+15y+2z=72 x+y+54z=110
engineering mathematics updated 5.1 years ago by teamques10 ★ 70k
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\[ if \ x=uv, \ y=\dfrac {u}{v} \] prove that J.J=1
engineering mathematics updated 5.1 years ago by teamques10 ★ 70k
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Find the stationary values x3+3xy2-3x2-3y2+4
engineering mathematics updated 5.1 years ago by teamques10 ★ 70k
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State and prove Eulers theorem for a homogeneous function in two variables: Hence verify the Eulers theorem for \[ u = \dfrac {\sqrt{xy}}{\sqrt{x+} \sqrt{y}} \]
engineering mathematics updated 5.1 years ago by teamques10 ★ 70k
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Reduce the matrix to normal form and find its rank where, \[ A= \begin{bmatrix} 1 &-1 &3 &6 \\1 &3 &-3 &-4 \\5 &3 &3 &11 \end{bmatrix} \]
engineering mathematics updated 5.1 years ago by teamques10 ★ 70k
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Prove that \[\log \sec x = \dfrac {1}{2}x^2 + \dfrac {1}{1z}x^2 + \dfrac {1}{45}x^6 \cdots \ \cdots \]
engineering mathematics updated 5.1 years ago by teamques10 ★ 70k
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if y=sin [log (x2+2x+1)] prove that (x+1)2 yn+2 + (2n+1 (x+1) )yn+1+ (n2+4)yn=0
engineering mathematics updated 5.1 years ago by teamques10 ★ 70k
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By using De Moivre's theorem Express \[ \dfrac {\sin 7\theta }{\sin \theta} \] in powers of sinθ only.
engineering mathematics updated 5.1 years ago by teamques10 ★ 70k
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Solve by crouts methods the system of equations 3x+2y+7z=4 2x+3y+z=5 3x+4y+z=7.
engineering mathematics updated 5.1 years ago by teamques10 ★ 70k
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If iii... ? =A+i B, prove that \[ \tan \left ( \dfrac {\pi A}{2} \right )= \dfrac {B}{A} \ and \ A^2 + B^2 = e^{-\pi B} \]
engineering mathematics updated 5.1 years ago by teamques10 ★ 70k
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Reduce the matrix A to normal form and hence find its rank where \[ A=\begin{bmatrix}0 &1 &-3 &-1 \\1 &0 &4 &3 \\3 &1 &0 &2 \\1 &1 &-2 &0 \end{bmatrix} \]
engineering mathematics updated 5.1 years ago by teamques10 ★ 70k
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\[ P.T.\ sinhx = X + \dfrac {x^3}{3!} + \dfrac {x^5}{5!} + \dfrac {x^7}{7!}+ \]
engineering mathematics updated 5.1 years ago by teamques10 ★ 70k
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\[ \text {if } x =\cos \Big [ \log \big (y^{1/m} \big) \Big ] \] then prove that (1-x2)yn+2-(2n+1)xyn+1-(m2+n2)yn = 0
engineering mathematics updated 5.1 years ago by teamques10 ★ 70k
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If cos6θ+sin6θ = α cos 4θ+β then prove that α+β=1.
engineering mathematics updated 5.1 years ago by teamques10 ★ 70k
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Apply Crout's method to solve \[ \left\{\begin{matrix} x-y + 2z=2 \ \ \\ 3x+2y-3z=2 \\ 4x -4y+2z=2 \end{matrix}\right. \]
engineering mathematics updated 5.1 years ago by teamques10 ★ 70k
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If u=xyz, v=x2+y2+x2, w=x+y+z then prove that \[ \dfrac {\partial x}{\partial u} = \dfrac {1}{(x-y)(x-z)} \]
engineering mathematics updated 5.1 years ago by teamques10 ★ 70k
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