K L UNIVERSITYFRESHMAN ENGINEERING DEPARTMENT

A.Y. 2009-10.

SEMESTER - ITime: 3 hours Comprehensive Examination Subject: Applied Mathematics-IMax. Marks: 100 Date: Code: MATH 1013

Note: Answer question 1 and any five from the remaining. Write your answers sequentially. 10X2=20M

1. (a) State whether the differential equation is linear or non

linear. Justify your statement

(b) State the order and degree of the differential equation:

(c) Indicate whether the differential equation is homogeneous or not. If homogeneous what is the degree

(d) State whether the differential equation is exact

(e) Can be the integrating factor for the differential equation:

(f) Can the differential equation is solvable in its present form

(g) Can the orthogonal trajectory for the curve be . Justify

(h)Evaluate the particular integral for the differential equation

(i) A circuit consists of components LCR. The circuit is driven by the Electromotive force E. Express the current flowing in the system by a differential equation. Can the circuit be called as a linear circuit. If so then justify

(j) Under what conditions, Lagrange’s mean value theorem be converted into Rolle’s theorem. 2. (a) Find the current at any time t > 0 in a circuit having in series a constant electromotive force 40V, a resistor 10Ω, and an inductor of 0.2H and given that the initial current is zero. (8 M)

(b) If a substance cools from 370K to 330 in 10 minutes. When the temperature of the surrounding air is 290K, find the temperature of the substance after 40 minutes (8 M)

3. (a) A circuit consists of an inductance of 0.05 henrys, a resistance of 5 ohms and a condenser of capacitance of 4x10 – 4 farad. If Q = i = 0 when t = 0, find Q(t) and i(t) when there is a constant emf of 110 volts. (8 M)

(b) A bacterial population (B) is known to have a growth rate proportional to (B) itself. If between noon and 2PM the population triples, at what time, no controls being exerted, should (B) become 100 times what it was at noon.(8)

4. (a) Show that the family of parabolas are self orthogonal (8 M) (b) Radium decomposes at a rate proportional to the quantity of radium present. Suppose it is found that in 25 years approximately 1.1% of a certain quantity of radium has decomposed. Determine approximately how long it will take for one half of the original amount of radium to decompose. (8 M)

5. (a) Using the Laplace transform technique evaluate (8 M)

(b) Evaluate (8 M)

6. (a) Using the Laplace transforms technique solve the differential equation

with the conditions y(0) = 0 and y’ (0) = 1 (8 M)

(b) Find the dimensions of a rectangular box of maximum capacity whose surface area is given when (a) the box is open at the top and (b) when the top of the box is closed. (8 M)

7. (a) Can Rolle’s theorem can be applied for f(x) = in [-1 , 1]. Give justification. (8 M)

(b) Establish the identity: (8 M)

8. (a) Suppose a closed rectangular box has length twice its breadth and has a Constant volume V. Find the dimensions of the box requiring least surface area (sheet metal) (8 M)

(b) Establish the identity (8 M)

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