BE Biomedical Engineering सत्र १ (इंजीनियरिंग) - University of Mumbai Important Questions for Applied Mathematics 1

Prove that ${\displaystyle {\sin }^5\theta =\frac{1}{16}[\sin 5\theta -5\sin 3\theta +10\sin \theta ]}$

Expand ${\displaystyle 2x^3+7x^2+x-1}$ in powers of x - 2

Find all values of ${\displaystyle {(1+i)}^{\frac{1}{3}}}$ and show that their continued product is (1+ 𝒊 ).

By using De Moivre's Theorem obtain tan 5θ in terms of tan θ and show that ${\displaystyle 1-10{\tan }^2\left(\frac{\pi }{10}\right)+5{\tan }^4\left(\frac{\pi }{10}\right)=0}$.

If y=(x+√x²-1 ,Prove that

${\displaystyle (x^2-1)y_{n+2}+(2n+1)x{y}_{n+1}+(n^2-m^2)y_n=0}$

Find the n^th derivative of ${\displaystyle \frac{x^3}{(x+1)(x-2)}}$

Find the nth derivative of cos 5x.cos 3x.cos x.

If ${\displaystyle y=e^{{\tan }^{-1}x}}$.Prove that

${\displaystyle (1+x^2)y_{n+2}+[2(n+1)x-1]y_{n+1}+n(n+1)y_n=0}$

Evaluate ${\displaystyle \underset{x\to 0}{lim}\sin x\log x.}$

If U = ${\displaystyle e^{xyz}f\left(\frac{xy}{z}\right)}$ prove that ${\displaystyle x\frac{\partial u}{\partial x}+z\frac{\partial u}{\partial x}2xyzu}$ and ${\displaystyle y\frac{\partial u}{\partial x}+z\frac{\partial u}{\partial z}=2xyzu}$ and hence show that ${\displaystyle x\frac{{\partial }^{2}u}{\partial z\partial x}=y\frac{{\partial }^{2}u}{\partial z\partial y}}$

If y=sin[log(x²+2x+1)] then prove that (x+1)²y_n+2 +(2n +1)(x+ 1)y_n+1 + (n²+4)y_n=0.

Find nth derivative of ${\displaystyle \frac{1}{x^2+a^2.}}$

Prove that log ${\displaystyle \left[\tan (\frac{\pi }{4}+\frac{ix}{2})\right]=i.{\tan }^{-1}\left(\sinh x\right)}$

Obtain tan 5𝜽 in terms of tan 𝜽 & show that ${\displaystyle 1-10{\tan }^2 \frac{x}{10}+5{\tan }^4 \frac{x}{10}=0}$

If y=etan_1x. prove that ${\displaystyle (1+x^2)yn+2[2(n+1)x-1]y_n+1+n(n+1)y_n=0}$

If ${\displaystyle Z=x^2\tan -1\frac{y}{x}-y^2\tan -1\frac{x}{y}\partial }$ 

Prove that ${\displaystyle \frac{{\partial }^{z}z}{{\partial }_y{\partial }_x}=\frac{x^2-y^2}{x^2+y^2}}$

Find tanhx if 5sinhx-coshx = 5 

Separate into real and imaginary parts of cos${\displaystyle {}^{-1}\left(\frac{3i}{4}\right)}$ 

Considering only principal values separate into real and imaginary parts

${\displaystyle i^{\left(\log \right)(i+1)}}$

Show that ${\displaystyle i\log \left(\frac{x-i}{x+i}\right)=\pi -2\tan 6-1x}$