Engineering Mathematics

Transcription

1 F.Y. Diploma : Sem. II [AE/CD/CE/CH/CM/CO/CR/CS/CV/CW/DE/ED/EE/EI/EJ/EN/ EP/ET/EV/EX/FE/IC/IE/IF/IS/IU/ME/MH/MI/MU/PG/PS/PT] Engineering Mathematics Time: Hrs.] Pre Question Paper Solution [Marks : 00 Q. Attempt any TEN of the following : [0] Q.(a) If ( y) + i ( + y) 7, find, y. [] (A) ( y) + i( + y) 7 ( y) + i( + y) 7 + 0i y 7 and + y 0 y 7 + y y Q.(b) Epress in the form a + ib. (A) i i i i i i i i i i i i 5 or i 5 5 i i where a, b R, i. [] Q.(c) If f() show that f( ) f(). [] (A) f(-) () 5() f() () 5() f() f() Q.(d) State whether the function f() (A) f() e e e e f() f() is even. e e is even or odd. [] Q.(e) Evaluate. [] 0 sin (A) 0 sin 0 sin 0 sin

2 Vialankar : F.Y. Diploma Engg. Mathematics Q. (f) Evaluate (A) log log [] 0 0 log log Q.(g) If y e sin cos find d. [] (A) d d d d e sin cos e cos sin sin cos e d d d e sin sin e cos cos sin cos e e sin e cos e sincos e sin cos sin cos Y OR e sin cos e sin d d d e sin sin e d d e cos sin e e cos sin Q.(h) Find d if y log ( + ). [] (A) y log( + ) d d d + + Q. (i) Find if sin, y cos. [] d (A) sin, y cos d d cos and d sin d / d d / d sin cos tan

3 Pre Question Paper Solution Q.(j) If + y find d. [] (A) + y y 0 d y d d y Q.(k) Show that root of equation 5 0 lies between and. [] (A) f() 5 f() < 0 f() 6 > 0 Therefore the root lies between and Q. (l) Find first iteration by Jacobi s method : 0 + y + z, + 0y + z, + y + 0z 5 (A) 0 + y + z, + 0y + z, + y + 0z 5 [] y z 0 z y 0 5 y z 0 Now we start with : 0 0 y 0 z y z Q. Attempt any FOUR of the following : [6] Q.(a) If f() then show that f[f()]. [] (A) f() f f[f()] f

4 Vialankar : F.Y. Diploma Engg. Mathematics Q.(b) Epress the following number in polar form i. [] (A) r / 80 tan or / / or tan / z r(cos + isin) cos0 + isin0 or cos i sin Q.(c) Find all cube root of unity. [] (A) Let z + 0i a, b 0 r 0 0 tan 0 z r(cos + i sin) cos0 + isin0 cosk + isink / [cosk + isink] / k k cos isin For k 0, / cos(0) + isin(0) + 0 For k, / cos i sin i For k, / cos i sin i

5 Pre Question Paper Solution Q.(d) Simplify using De-Moivre s theorem : (A) θ θ cos5 isin5 5 cos θisin θ 7 7 θ θ cos isin cos θisin θ θ θ cos5 isin5 5 cos θisin θ 7 7 θ θ cos isin cos θisin θ cos θisin θ cos θisin θ cos θisin θ cos θisin θ cos θisin θ cos θisin θ cos θisin θ cos θisin θ cos isin cos isin cos isin [] Q.(e) If f() +, solve the equation f() f( ). [] (A) f() + f( ) ( ) () But f() f() or , or.5, 0.5 Q.(f) Simplify i + i 0 + i 50 + i 00. [] (A) i + i 0 + i 50 + i 00 + (i ) 5 + (i ) 5 + (i ) 5 + () 5 + () 5 + () Q. Attempt any FOUR of the following : [6] Q.(a) If f() a + b + and f(), f(), find a and b. [] (A) f() a + b + f() a() +b() + a + b + f() a() + b() + a + b + But f(), f() a + b + a + b + a + b a + b 8 a b 5

6 Vialankar : F.Y. Diploma Engg. Mathematics Q.(b) If f() log then prove that f (A) f log log log log log f() f(). [] Q.(c) Evaluate (A) 6 6. [] or 0.5 Q.(d) Evaluate (A) sin cos. [] tan sin cos tan sin cossin cos sin cos sincos sincos cos sin cos sin cos sin cos sin cos sin cos cos sin cos cos cos 6

7 Q.(e) Evaluate 5 (A) Pre Question Paper Solution. [] or.5 Q.(f) Evaluate (A) [] log log Q. Attempt any FOUR of the following : [6] Q.(a) If y sin ( ) find d. [] (A) Put sin y sin ( ) sin sin sin sin (sin) sin d. 7

8 Vialankar : F.Y. Diploma Engg. Mathematics Q.(b) Using first principle find derivative of f() a. [] (A) f() a f( + h) a + h d f h f h0 h h a a h0 h h a a h0 h h a a h0 h a log a Q.(c) If u and v are differentiable functions of and y u.v then prove that [] d dv du u v d d (A) Let y uv. Let be infinitesimal increment in and y, u, v be corresponding infinitesimal increments in y, u, v. y + y (u + u) (v + v) uv + uv + vu + uv y uv + uv + vu + uv y uv + uv + vu + uv uv uv + vu + uv As u and v are very very small. uv is negligible. y uv + vu y uv vu v u u v y v u u v 0 0 y v u u v d dv du u v d d Q.(d) Differentiate w.r.t, tan 5 6. [] 5 (A) Let y tan 6 tan Put tan A and tan B tana tanb y tan tanatanb tan tan A B A + B tan () + tan () d 9 8

9 Pre Question Paper Solution Q.(e) Find d if + y + y. [] (A) + y + y 6 y y 0 d d 6 y y d d 0 6 y y 0 d d 6 y y t Q. (f) If y tan t and t sin t t (A) y tan t Put t tan tan y tan tan tan tan y tan t t And sin t Put t tan tan sin sin (sin) tan tan t y d find d. [] Q.5 Attempt any FOUR of the following : [6] sin Q.5(a) Evaluate. [] (A) Put t as, t 0 sin sin t t0 t sin t t0 t sint t0 t sin t t0 t log log Q.5(b) Evaluate. [] log log (A) 5 9

10 Vialankar : F.Y. Diploma Engg. Mathematics Let + h or h as, h 0 log h log h0 h h log h0h h log h0 h log h0 log e log e /h /h/ Q.5c) Using Bisection method find the approimate root of + 0 ( iterations). (A) + 0 f() + f() f() the root is in (, )..5 f(.5) 0.75 the root is in (,.5)..5.5 f(.5) 0.88 the root is in (.5,.5) [] Q.5(d) Using False Position method find the root of 0 ( iterations only). [] (A) f() f() f() the root is in (, ) af b bf a.667 f(b) f a f(.667).05 the root is in (.667, )..78 f(.78) 0. the root is in (.78, )

11 Pre Question Paper Solution Q.5(e) Using Newton Raphson method find the root of 9 0 (carry out iterations). (A) 9 0 f() 9 f () f() 9 f() 5 f 9 f' 9 OR f f f 9 Start with 0, [] Q.5(f) Using Newton Raphson method find approimate value of 0 ( iterations). [] (A) Let f() 0 f() f() f() 6 f 0 (i) f 0 (ii) OR f f 0 f Start with 0, (i) (ii) Q.6 Attempt any FOUR of the following : [6] Q.6(a) If y sin 5 cos 5 show that + 5y 0. d [] (A) y sin 5 cos5 cos5 5 + sin 5 5 d

12 Vialankar : F.Y. Diploma Engg. Mathematics 5 cos5 + 5 sin 5 5 sin cos 5 d 5(sin 5 cos5) 5y 5y d 0 Q.6(b) If a ( sin ), y a ( cos ) find d and d (A) a( sin) d a(cos ) d y a( cos) d a(sin) d / d d / d at, a sin a cos sin cos sin d cos at. [] or.79 Q.6(c) Solve by Jacobi s method performing iterations : 0 + y z 7, + 0y z 8, y + 0z 5 (A) 0 + y z 7 + 0y z 8 y + 0z 5 [] 0 8 z 0 5 y 0 7 y z y z Starting with 0 0 y 0 z y 0.9 z.5.0 y z.0.00 y.00 z.00

13 Pre Question Paper Solution Q.6(d) Solve by Gauss-Seidal method ( iterations) 5 + y + z 8, + 0y z 9, 6y + 5z 5 8 y z 0 9 z 5 6y (A) y z [] Starting with 0 0 y 0 z 0. y 0.8 z y z y z Q.6(e) Solve by Gauss Eination method : + y + z, + y + z, + y + z (A) + y + z + y + z + y + z [] + 6y + 9z + y + z 5y + 7z and 6 + y + z 6 + 9y + z 7y + z 5y + 7z 9y + 7z 77 + y z 5y 08 Q.6(f) Solve by Gauss Seidal method ( iterations) 5 y 9, 5y + z, y 5z 6, Taking 0.5, y 0 0.5, z (A) 5 y 9 5 y + 0z 9 5y + z OR 5y + z y 5z y 5z 6 [] 9 y 5 z 5 6 y 5 y z

14 Vialankar : F.Y. Diploma Engg. Mathematics Starting with 0.5, y 0 0.5, z y.08 z y.006 z 0.999