# Snehal-RTMNU-Engg-MA-1Sem-win 16

## B.E. All Branches First Semester (C.B.S.) / B.E. (Fire Engineering) First Semester Applied Mathematics – I

NKT/KS/17/7196
Time : Three Hours
Max. Marks : 80
Notes : 1. All questions carry marks as indicated.
2. Solve Question 1 OR Questions No. 2.
3. Solve Question 3 OR Questions No. 4.
4. Solve Question 5 OR Questions No. 6.
5. Solve Question 7 OR Questions No. 8.
6. Solve Question 9 OR Questions No. 10.
7. Solve Question 11 OR Questions No. 12.
8. Assume suitable data whenever necessary.
9. Use of non programmable calculator is permitted.
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1. a) If y cos(msin -1x)  prove that  [6M]
(1-x2)yn+2 -(2n+1)xyn+1+(m2-n2)yn=0

b) Evaluate [6M]
i)  $\lim_{x\rightarrow 0}\left [ \frac{1}{sin^{2}x}-\frac{1}{x^{2}}\right ]$

ii) $\lim_{x\rightarrow\frac{\pi }{2} }(cosx)^{cosx}$

OR

2. a)A curve is given by x =a sin θ ; y = bcos2θ. Find the radius of curvature at θ=π/3 [6M]

b) Expand 2x3+7x2+x- 7 in powers of (x -2) . [6M]

3. a) If  u(x + y) = x2 + y2 Then prove that [6M]
$\left [ \frac{\partial u}{\partial x} -\frac{\partial u}{\partial y}\right ]^{2} =4\left [ 1-\frac{\partial u}{\partial x} -\frac{\partial u}{\partial y}\right ]$

b) If  $u=tan^{-1}\left [ \frac{x^{\frac{1}{2}}+y^{\frac{1}{2}}}{x^{\frac{1}{3}}-y^{\frac{1}{3}}} \right]$

find the value of [6M]

$x^{2}\frac{\partial^2 u}{\partial x^2}+2xy\frac{\partial^2 u}{\partial x \partial y} +y^{2}\frac{\partial^2 u}{\partial y^2}$

c) If u = f (2x -3y,3y-4z, 4z -2x)
Then prove that [6M]

$\frac{1}{2}\frac{\partial u}{\partial x}+\frac{1}{3}\frac{\partial u}{\partial y}+\frac{1}{4}\frac{\partial u}{\partial z}=0$

OR

4. a)If u=\frac{x+y}{x-y} & v=\frac{xy}{(x-y)^{2}} Then show that u & v are functionally related? If so find the relation between them.[6M]

b) Find the points on the surface z2=xy +1 nearest to the origin. [6M]

c) Obtain Taylor’s expansion of tan-1(y/x) about (1, 1) upto the third degree terms. [6M]

5. a) Find the inverse of matrix by partitioning [6M]

A= $\begin{bmatrix} 1& 2& 3\\ 1& 3&5 \\ 1& 4& 12 \end{bmatrix}$

b) Test the consistency and solve [6M]
5x +3y+7z = 4
3x + 26y+ 2z = 9
7x + 2y+10z = 5

OR

6. a) Determine the rank of the matrix : [6M]

$\begin{bmatrix} 1 & 2 & 3 & 0\\ 2 &4 &3 & 2\\ 3& 2 &1 & 3\\ 6& 8& 7 & 5 \end{bmatrix}$

b) Using adjoint method, solve [6M]
x +2y+ z = 7
x + 3z =11
2x -3y =1

7.a) $\frac{\mathrm{d} y}{\mathrm{d} x}-ytanx=3e^{-sinx}$ [4M]

b) $\frac{\mathrm{d} x}{\mathrm{d} y}=\frac{1+y^{2}+cos^{2}x}{ysin2x}$ [4M]

c)$\frac{\mathrm{d} y}{\mathrm{d} x}+ytan x=y^{3}secx$ [4M]

OR

8.a) $x^{2}\left ( \frac{\mathrm{d} y}{\mathrm{d} x}\right )^{2}+ytanx=y^{3}secx$ [3M]

b) Solve : y+ px =x4 p [3M]

c) The equation of electromotive force in terms of current i for an electrical circuit having
resistance R and condenser of capacity C in series is :  [6M]

$E=Ri+\int \frac{i}{c}dt$.Find the current i at any time t when E= Emsin wt .

9. a) Solve (D2-5D+6)y =e2x +cosx  [6M]

b) Solve by using variation of parameters method [6M]
$\frac{\mathrm{d} ^{2}y}{\mathrm{d} x}-4\frac{\mathrm{d} y}{\mathrm{d} x}+4y=\frac{e^{2x}}{x}$
c) Solve $x^{2}\frac{\mathrm{d} ^{2}y}{\mathrm{d} x^{2}}-2x\frac{\mathrm{d} y}{\mathrm{d} x}-4y=x^{2}+2log x$

OR

10. a) Solve $\frac{\mathrm{d} ^{2}y}{\mathrm{d} x^{2}}=e^{-2y}$ [6M]
Given that y=0, dy/dx =0 when x=0

b) Solve  [6M] $\frac{\mathrm{d} ^{2}x}{\mathrm{d}t^{2}}=b\frac{dy}{dt}$
and $\frac{\mathrm{d} ^{2}y}{\mathrm{d}t^{2}}=a-b\frac{\mathrm{d} x}{\mathrm{d} t}$
c) A body executes damped forced vibrations given by the equation [6M]
$\frac{\mathrm{d} ^{2}x}{\mathrm{d}t^{2}}+2k\frac{\mathrm{d} x}{\mathrm{d} t}+b^{2}x=e^{-kt}sinwt$
solve the equation when w2=b2-k2

11. a) If 2cosθ = x+1/x and   2cosΦ = y +1/y  then prove that xmyn +1/xmyn  =2cos(mθ +nΦ) [4M]

b) Find all the values of$\left [ \frac{1}{2}+\frac{\sqrt{3}}{2}i \right ]^{\frac{3/4}{}}$ and show that the continued product of all the values is 1. [4M]

OR

12. a) Solve using De – Moivre’s theorem [4M]
x7– x4+ x 3-1=0

b) Separate tan-1 (x +iy) into real and imaginary parts. [4M]

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