Question Paper: Applied Mathematics 1 Question Paper - May 18 - First Year Engineering (Semester 1) - Mumbai University (MU)
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## Applied Mathematics 1 - May 18

### First Year Engineering (Semester 1)

Total marks: 80
Total time: 3 Hours
INSTRUCTIONS
(1) Question 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Draw neat diagrams wherever necessary.

1(a) If $\tan \frac{x}{2} = \tan h\frac{u}{2}$, show that $u = \log[\tan(\frac{\pi}{4}+\frac{x}{2})]$
(3 marks) 00

1(b) Prove that the following matrix is orthogonal & hence find $A^{-1}$

$A = \frac{1}{3} .\matrix {-2 & 1 & 2 \\ 2 & 2 & 1 \\ 1 & -2 & 2}$

(3 marks) 00

1(c) State eulers theorem on homogeneous function of two variables & if $u = \frac{x+y}{x^2 + y^2}$ then evaluate $x \frac{\delta u}{\delta x} + y\frac{\delta u}{\delta y}$ .
(3 marks) 00

1(d) If $u = r^2 \cos2\theta, v = r^2 \sin2\theta. Find \frac{\delta(u, v)}{\delta(r,\theta)}$
(3 marks) 00

1(e) Find the $n^{th}$ derivative of $\cos5x.\cos3x.\cos x$
(4 marks) 00

1(f) Evaluate: $\lim_{x \rightarrow 0} (\frac{2x+1}{x+1})^{\frac{1}{x}}$
(4 marks) 00

2(a) Solve $x^4 - x^3 + x^2 - x +1 = 0$
(6 marks) 00

2(b) If $y = e^{\tan^{-1}x}$ . Prove that $(1+x^2)y_{n+2} + [2(n+1)x - 1] y_{n+1} + n(n+1)y_n = 0.$
(6 marks) 00

2(c) Examine the function $f(x,y) = xy(3-x-y)$ for extremes values & also find maximum and minimum values of $f(x,y)$
(8 marks) 00

3(a) Investigate for what values of $\lambda$ and $\mu$ the equations x+y+z = 6; x+2y+3z = 10; x+2y+$\lambda$z = $\mu$ have

• no solution
• a unique solution
• infinite number of solution
(6 marks) 00

3(b) If $u = f(\frac{y-x}{xy},\frac{z-x}{xz})$ , show that $x^2 \frac{\delta u}{\delta y} + y^2 \frac{\delta u}{\delta y} + z^2 \frac{\delta u}{\delta y} = 0$
(6 marks) 00

3(c) prove that $\log(\frac{a+ib}{a-ib}) = 2i\tan^{-1}(\frac{b}{a})$ & $\cos[i\log{(\frac{a+ib}{a-ib})}] = \frac{a^2 - b^2}{a^2 + b^2}$
(8 marks) 00

4(a) If $u = \sin^{-1}(\frac{x+y}{\sqrt x + \sqrt y})$ , prove that $x^2u_{xx} + 2xy u_{xy} + y^2 u_{yy} = \frac{-\sin u \cos 2u}{4 \cos^3u}$
(6 marks) 00

4(b) using encoding matrix $\matrix{1& 1 \\ 0&1}$ ; encode & decode the message "ALL IS WELL"
(6 marks) 00

4(c) Solve the following equations by Gauss Seidal method:

$10x_1 + x_2 + x_3 = 12$

$2x_1 + 10x_2 + x_3 = 13$

$2x_1 + 2x_2 + 10x_3 = 14$

(8 marks) 00

5(a) If $u = e^{xyz} f(\frac{xy}{z})$ where, $f(\frac{xy}{z})$ is an arbitrary function of $\frac{xy}{z}$, prove that:

$x\frac{\delta u}{\delta x} + z\frac{\delta u}{\delta z} = y\frac{\delta y}{\delta x} + z\frac{\delta u}{\delta z} = 2xyz.u$

(6 marks) 00

5(b) prove that $\sin^5 \theta = \frac{1}{16}(sin5\theta - 5\sin3\theta + 10\sin\theta)$
(6 marks) 00

5(c)

1. Prove that $\log(secx) = \frac{1}{2} x^2 + \frac{1}{12} x^4 + ..........................$
2. Expand $(2x^3 + 7x^2 +x - 1)$ in powers of (x-2)
(8 marks) 00

6(a) Prove that $\sin^{-1}(cosec\theta) = \frac{\pi}{2} + i\log(\cot\frac{\theta}{2})$
(6 marks) 00

6(b) Find the non - singular matrices P&Q such that A = $\matrix {1&2&3&2\\2&3&5&1\\1&3&4&5}$ is reduced to normal form. Also, find its rank.
(6 marks) 00

6(c) Obtain the root of $x^3 - x - 1 = 0$ by Regula Falsi Method (Take three iterations)
(8 marks) 00