# Engineering Mathematics-III-Engineering-Pune University-May2019

**S.E. (Comp/IT) (II Semester) EXAMINATION, 2019**

**ENGINEERING MATHEMATICS—III**

**(2015 PATTERN)**

**Time: Three Hours Maximum Marks: 60**

N.B. :— (i) Neat diagrams must be drawn wherever necessary.

(ii) Figures to the right indicate full marks.

(iii) Use of an electronic pocket calculator is allowed.

(iv) Assume suitable data, if necessary.

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1. (a) Solve any two differential equations : **[8]**

(i)

(ii)

(iii), by using the method of variation

of parameters.

(b) Solve the integral equation : **[4]
**

**Or**

2. (a) An inductor of 0.25 henries, with negligible resistance, a capacitor of 0.04 farads are connected in series and having an alternating voltage [12 sin 6t]. Find the current and charge

at any time t. **[4]**

(b) Solve any one of the following : **[4]**

(i) Obtain z[4^{k}e^{–6k}], k ≥ 0.

(ii) Obtain

(c) Solve the difference equation :** [4]**

f(k + 2) – 7f(k + 1) + 12f(k) = 0

where f(0) = 0, f(1) = 3, k ≥ 0.

3. (a) The first four moments of a distribution about the value 5 are 2, 20, 40, and 50. Obtain the first four central moments,β_{1}and β_{2}.**[4]**

(b) Fit a straight line of the form Y = aX + b to the following data by the least square method : **[4]**

X | 1 | 3 | 4 | 5 | 6 | 8 |

Y | -3 | 1 | 3 | 5 | 7 | 11 |

(c) A riddle is given to three students whose probabilities of solving

it are 1/2 ,1/3 and 1/4 respectively.Find the probability that the riddle is solved,

[4]

**Or**

4. (a) In a sample of 1,000 cases, the mean of a certain examination is 14 and standard deviation is 2.5. Assuming the distribution to be normal. Find the number of students scoring between

12 and 15.** [4]**

[Given: Z1= 0.4, A1= 0.1554, Z2= 0.8, A2= 0.2881]

(b) During working hours, on an average 3 phone calls are coming into a company within an hour. Using Poisson distribution,find the probability that during a particular working hour, there

will be at the most one phone call. **[4]**

(c) For bivariate data, the regression equation of Y on X is 8x – 10y = – 66 and the regression equation of X on Y is 40x – 18y = 214. Find the mean values of X and Y. Also, find the correlation coefficient between X and Y.** [4]**

5. (a) Find the directional derivative of Φ= xy^{2} + yz^{2} + zx^{2} at(1, 1, 1) along the line 2(x – 2) = y + 1 = z – 1. **[4]**

(b) Find constants a, b, c so that** [4]**

is irrotational. **[4]**

(c) Find the workdone by the force

in taking a particle from (0, 0, 0) to (1, 2, 1). **[5]**

**Or**

6. (a) Show that (anyone) : [4]

(i)

(ii)

(b) Find the directional derivative of Φ = 4xz^{3} – 3x^{2}y^{2 }at (2, – 1, 2) along the tangent to the curve

x = e^{t}cos t, y = e^{t} sint ,z=e^{t} and t=0. **[4]**

(c) Find the workdone by, in taking a particle from (0, 0, 0) to (2, 4, 0) along the parabola y = x^{2}, z = 0. **[5]**

7. (a) Determine the analytic function f(z) = u + iv if u =2x – x^{3} + 3xy2. **[4]**

(b) Find the bilinear transformation that maps to points z = – i, 0, i into the points W = 1, 0, ∞.**[4]**

(c) Evaluate where c is the circle |z| = 3. **[5]**

**Or**

8. (a) Determine the analytic function f(z) = u + iv if u = 3x^{2}y + 2x^{2} – y^{3} – 2y^{2}. **[4]**

(b) Find image of the circle |z – 2i| = 2, under the mapping

w =1/2· **[4]**

(c) Evaluate

where c is the circle |z| = 3/2·** [5]**