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August 2015 T05/ T24/TES022/TML022/EE/20150813 APPLIED MATHEMATICS-2 Instructions Marks-80 Time: 3 Hrs 1. All questions are compulsory. 2. “Long Answer type Question (LAQ)” is a supply type question of 15 marks which required a typical answer of about 60-80 lines in bout 32-40 minutes. 3. “Sort Answer type Question (SAQ)” is a supply type question of 5 marks which required a typical answer of about 15-20 lines in bout 08-10 minutes. 4. Figures to the right indicate marks allotted to the questions. 5. Assume suitable data if necessary 6. Use a nonprogrammable type of scientific calculator is allowed Marks SECTION-A Q.1 A) Evaluate : ( + )+ ( − )− 10 . → B)Evaluate : 5 − → − OR Q.1 A) A box with a square base is to have an open top. The area of the material for making the box is 192sq.cm.What should be the dimensions in order that the volume is as large as possible? B)Differentiate w.r. to x : log(sin x +10x) Q.2 A) Evaluate : (i) 10 5 10 (ii) B)Evaluate : 5 ( + ) OR Q.2 A) Find the area under the following curves,lines and axis. (i) y=2x-x2 and below x-axis (ii) = − 15 , line x=3 and x-axis; (iii) y=4-x2, lines x=0,x=2 and x-axis Q.3 From first principles, prove that ( )=− Q.4 Find the derivatives of the following functions by using quotient and chain rules − 5 5 + Q.5 A) Solve the following differential equations (i) + y=e-x tan x (ii) (iii) 15 =y tan x-2sin x =( + ) OR Q.5 A) Solve the following differential equations : (i) y2 dx+(x 2-xy +y 2)dy=0 (ii) x2ydx-(x 3+y3)dy=0 (iii)x 2dy+y(x+y)dx=0 15 Q.6 A)Find Laplace Transform of the following functions (i) sin 5t + cos 3t 10 (ii) (t2+1)2 (iii) OR Q.6 A)Find Laplace Transform of the following functions (i) t sin2t 15 (ii) t2 cos at (iii) − Q.7 Form the differential equation for All parabolas with laths rectum '4a' and whose axes are parallel to x-axis. 5 Q.8 Solve the following differential equations by the Laplace Transform Method 5 ( + ) = ; = = =