(ISO/IEC - 7-5 Certified) SUMMER 8 EXAMINATION ject Name: Applied Mathematics Model wer ject Code: Important Instructions to eaminers: ) The answers should be eamined by key words and not as word-to-word as given in the model answer scheme. ) The model answer and the answer written by candidate may vary but the eaminer may try to assess the understanding level of the candidate. ) The language errors such as grammatical, spelling errors should not be given more Importance (Not applicable for subject English and Communication Skills. ) While assessing figures, eaminer may give credit for principal components indicated in the figure. The figures drawn by candidate and model answer may vary. The eaminer may give credit for any equivalent figure drawn. 5) Credits may be given step wise for numerical problems. In some cases, the assumed constant values may vary and there may be some difference in the candidate s answers and model answer. 6) In case of some questions credit may be given by judgement on part of eaminer of relevant answer based on candidate s understanding. 7) For programming language papers, credit may be given to any other program based on equivalent concept. 7 wer. Attempt any TEN of the following: a) b) Find the gradient of the curve y at y. at. y y at ----------------------------------------------------------------------------------------------------------------------- Divide into two parts such that their product is maimum. Let and y be two parts of y y Product P y Page /
(ISO/IEC - 7-5 Certified) SUMMER 8 EXAMINATION ject Name: Applied Mathematics Model wer ject Code: 7. b) c) P d P P dp Product is maimum. dp Let wer 5 and y 5 ------------------------------------------------------------------------------------------------------------ Evaluate: tan c tan c + d) Evaluate: log log log. d log log..log..log.log c Page /
(ISO/IEC - 7-5 Certified) SUMMER 8 EXAMINATION ject Name: Applied Mathematics Model wer ject Code: 7 wer. e) Evaluate: tan Let tan tan sec sec.tan tan sec.tan tan sec.tan log sec +c Put tan t sec.tan.tan dt = tdt log sec +c t = log sec +c tan = log sec +c f) ----------------------------------------------------------------------------------------------------------------------- Evaluate: A B A B Put A Put B c log log c log = log + Page /
= MAHARASHTRA STATE BOARD OF TECHNICAL EDUCATION (ISO/IEC - 7-5 Certified) SUMMER 8 EXAMINATION ject Name: Applied Mathematics Model wer ject Code: 7 wer. f) 9 TT.. 9 9 log c log c g) h) ------------------------------------------------------------------------------------------------------------ sin Evaluate: sin sin sin sin sin cos cos sec log sec tan c ----------------------------------------------------------------------------------------------------------------------- If a 8 a 8 find the value of 'a'. Page /
(ISO/IEC - 7-5 Certified) SUMMER 8 EXAMINATION ject Name: Applied Mathematics Model wer ject Code: 7 wer. h) i) j) 8 a a a 8 a 8 a 8 ------------------------------------------------------------------------------------------------------------ Find the area bounded y 9, to and the X-ais Area A 9 9 9 9 8 i.e. 8 b a y ---------------------------------------------------------------------------------------------------------------------- d y Find order and degree of the differential equation y d y y d y y th Taking 6 power on both sides, d y y Order Degree ----------------------------------------------------------------------------------------------------------------------- Page 5/
(ISO/IEC - 7-5 Certified) SUMMER 8 EXAMINATION ject Name: Applied Mathematics Model wer ject Code: 7 wer. (k) Form a D.E. if y a cos b d y d y d y y a cos b a sin b a cos b y y l) m) n) ---------------------------------------------------------------------------------------------------------------------- d y Verify that y sin is a solution of the differential equation 9y y sin cos d y 6sin d y 9 sin d y 9y d y 9y ----------------------------------------------------------------------------------------------------------------------- Find the probability of occurrence of the digit when an unbiased dice is thrown. S A S=,,,,5,6 n =6 A n = n A P A.667 n S 6 ------------------------------------------------------------------------------------------------------------- A coin is tossed times.what is the probability that appears an odd number of times? S HHH, HHT, HTH, THH, TTH, THT, HTT, TTT Page 6/
(ISO/IEC - 7-5 Certified) SUMMER 8 EXAMINATION ject Name: Applied Mathematics Model wer ject Code: 7 wer. n) n S n A.5 ns 8 8 A: head appears odd number of times A HHH,, TTH, THT, HTT n A p A n S 8 A: tail appears odd number of times n A.5 ns 8 A HHT, HTH, THH, TTT n A p A Note: If student has considered either head or tail and attempted to solve give appropriate marks. a) ---------------------------------------------------------------------------------------------------------------------- Attempt any FOUR of the following: Determine a and b such that slope of curve y a b at, is same as the slope of y y a b 6y a a a 6y y at, a a y Slope is slopes are equal. a a and b 6 + Page 7/
(ISO/IEC - 7-5 Certified) SUMMER 8 EXAMINATION ject Name: Applied Mathematics Model wer ject Code: 7 wer. b) c) Find the maimum and minimum value of the function y Let y 6 d y 6 6 Consider 6 6 6 or at 6 d y 6 y is minimum at 6 y min at d y ma 6 6 6 6 8 y is maimum at y 6 = 6 -------------------------------------------------------------------------------------------------------------------- Find radius of curvature of y log sin at y log sin d y d y.cos cot sin cos at cot ec cosec Page 8/
(ISO/IEC - 7-5 Certified) SUMMER 8 EXAMINATION ject Name: Applied Mathematics Model wer ject Code: 7 wer c) d) e) Radius of curvature is, i.e. Evaluate sin sin. sin c d y d sin sin sin. sin sin c ------------------------------------------------------------------------------------------------------------- cos Evaluate: sin sin sin Put sin t cos dt dt ( t)( t)( t) A B C consider ( t )( t )( t ) t t t t t A t t B t t C Put t A Put t B Put t Page 9/
(ISO/IEC - 7-5 Certified) SUMMER 8 EXAMINATION ject Name: Applied Mathematics Model wer ject Code: 7 wer. e) f) C ( t)( t)( t) t t t dt dt ( t)( t)( t) t t t log t log t log t c log sin log sin log sin c ----------------------------------------------------------------------------------------------------------------------- Evaluate: 8 8 89 9 TT. 9 ++ Page /
(ISO/IEC - 7-5 Certified) SUMMER 8 EXAMINATION ject Name: Applied Mathematics Model wer ject Code: 7 wer. f) tan c tan c dt t dt t dt t tan t c tan tan c Put t dt t c Page /
(ISO/IEC - 7-5 Certified) SUMMER 8 EXAMINATION ject Name: Applied Mathematics Model wer ject Code: 7 wer.. tan c tan tan c c ---------------------------------------------------------------------------------------------------- Solve any FOUR of the following: 6 a) Evaluate: TT.. tan tan ta n tan Page /
(ISO/IEC - 7-5 Certified) SUMMER 8 EXAMINATION ject Name: Applied Mathematics Model wer ject Code: 7 wer. a) b) ----------------------------------------------------------------------------------------------------------------------- Evaluate: 9 6 cos 9sec 6 9 tan 6 9 9 tan 6 9 6 cos / cos 9 6 cos cos / cos 9 6 cos cos cos sec sec sec sec 9 tan 5 Put tan t sec when to t to dt dt dt dt 9 t 5 9 t 5 t 5 dt 5 9 t 9 Page /
(ISO/IEC - 7-5 Certified) SUMMER 8 EXAMINATION ject Name: Applied Mathematics Model wer ject Code: 7 wer. b) c) dt 9 5 t t tan 5 5 t t tan tan 9 5 5 5 5 tan tan 5 5 5 5 tan tan 5 5 5 ------------------------------------------------------------------------------------------------------------ Find the area included between the curves y a and ay y a --------- ay y a n eq. a 6a a 6a a 6 a,a 6 b Area A y y a a a A a a Page /
(ISO/IEC - 7-5 Certified) SUMMER 8 EXAMINATION ject Name: Applied Mathematics Model wer ject Code: 7 wer. c) d) e) a A a a a A a a a a a A a 6a A or 5.a ------------------------------------------------------------------------------------------------------------- Solve sec tan y sec ytan sec tan y sec y tan sec tan sec tan y y sec sec y tan tan y sec tan sec y tan y log tan log tan y c ------------------------------------------------------------------------------------------------------------ Solve: sin y sin y sin y ---------() Put y v dv dv From () dv sin v + Page 5/
(ISO/IEC - 7-5 Certified) SUMMER 8 EXAMINATION ject Name: Applied Mathematics Model wer ject Code: 7 wer. e) dv sin v dv sin v dv sin v sin v dv sin v sin v sin v dv c sin v sin v dv c cos v sec v tan vsec vdv c tan v sec v c tan sec y y c dv sin v v dt t Put tan t, dv, sin v t t dt t c t t dt c t t dt c t t dt c t c c t c y tan dv sin v dv c cos v Page 6/
(ISO/IEC - 7-5 Certified) SUMMER 8 EXAMINATION ject Name: Applied Mathematics Model wer ject Code: 7 wer. e) f) dv c v cos sec v dv c v tan c y tan c ------------------------------------------------------------------------------------------------------------- Solve : y y y y Put y v dv v v dv v v log v dv v v v dv v v v dv v v v dv v v v dv v v v dv v v dv v log c Page 7/
(ISO/IEC - 7-5 Certified) SUMMER 8 EXAMINATION ject Name: Applied Mathematics Model wer ject Code: 7 wer.. f) a) b) y log log c ---------------------------------------------------------------------------------------------------- Attempt any FOUR of the following: 6 Evaluate: cos sin cos cos Let I sin cos I cos sin cos sin I cos sin add and cos sin I I sin cos cos sin I cos sin I sin cos I I I ----------------------------------------------------------------------------------------------------------------------- Evaluate: 5 cos Put tan t Page 8/
(ISO/IEC - 7-5 Certified) SUMMER 8 EXAMINATION ject Name: Applied Mathematics Model wer ject Code: 7 wer. b) c) t dt cos, t t when to t to I dt t t 5 t I 5 t t I 5 5 t t I I dt t 9 t tan t I dt dt dt tan I tan tan I tan I or 6 ------------------------------------------------------------------------------------------------------------ Find the area of the circle y 9 using integration. y 9 y 9 y 9 A b a y sin Page 9/
(ISO/IEC - 7-5 Certified) SUMMER 8 EXAMINATION ject Name: Applied Mathematics Model wer ject Code: 7 wer. c) d) e) sin sin 9 -------------------------------------------------------------------------------------------------------------- Solve: y y Divide by y Comparing with Py Q, P Q P I. F. e e e Solution is log y. =. c y c y. I. F.= QI. F. c y c ---------------------------------------------------------------------------------------------------------------------- Solve: y cos y log sin y y Let M y cos y, N log sin y M N sin y, sin y y M N y Page /
(ISO/IEC - 7-5 Certified) SUMMER 8 EXAMINATION ject Name: Applied Mathematics Model wer ject Code: 7 wer. e) f) D.E. is eact Solution is, M N c ycons tan t terms not containing ' ' y y cos y c ycons tant y y log cos y c ------------------------------------------------------------------------------------------------------------ Verify that y e d ² y ² msin ( ²) m y Consider y e msin msin d e msin msin e m my m y msin msin e msin is the solution of the differential equation d y m y d y m y d y m y d y m y Consider y e d msin e m my --------- Page /
(ISO/IEC - 7-5 Certified) SUMMER 8 EXAMINATION ject Name: Applied Mathematics Model wer ject Code: 7 wer. f) d y m Multiply by d y m d y m y from d y m y Note: If student has considered - at inde place and attempted to solve give appropriate marks 5. ------------------------------------------------------------------------------------------------------------- Attempt any FOUR of the following: 6 a) Two unbiased dice are thrown.find the probability that the sum of the numbers obtained on two dice is neither a multiple of nor a multiple of.,,,,,5, 6,,,,,5, 6,,,,,5, 6 5,5, 5,5, 5,55, 6 S {,,,,,5, 6 ns ( ) 6 6, 6, 6, 6, 6,5 6,6 } A multiple of or multiple of A n A {,,,,5,,,, 6,,,5, 6,,,5, 6 5, n A.6667 ns 6 5, 5, 5,5 6, 6, 6, 6,6 } P A A' neither a multiple of nor a multiple of P A P A'.667. Page /
(ISO/IEC - 7-5 Certified) SUMMER 8 EXAMINATION ject Name: Applied Mathematics Model wer ject Code: 7 wer 5. a) ns ( ) 6 A neither a multiple of nor a multiple of A n A,, 6,,5,,,,5, 5, 66,6,5 n A. ns 6 P A ns ( ) 6 A multiple of A {,,,5,,, 6,,,5,,, 65,5,5,5 6, 6, B multiple of B {,,5,,,, 6,,5 5, 5, 6, 6,6 } ABmultiple of or multiple of A 6,6 } B{,,,,5,,,, 6,,,5, 6,,,5, 6 5,5, 5,5,56, 6,6,6,6 } B B n A B ns 6.6667 B '.6667. n A p A A ' neither a multiple of nor a multiple of P A B P A ns ( ) 6 A not multiple of A {,,,6,,,5,,,6,,,5 5, 5, 5, 6 6, 6, 6,5 } B not multiple of B {,,,, 6,,,5, 6,,,,5,,,, 6 5, 5, 5,5 5, 6 6, 6, 6, 6,5 } ABneither a multiple of nor a multiple of AB,, 6,,5,,,,5, 5, 66,6,5 Page /
(ISO/IEC - 7-5 Certified) SUMMER 8 EXAMINATION ject Name: Applied Mathematics Model wer ject Code: 7 5. a) b) wer n A B n A B P A B. n S 6 ns ( ) 6 A multiple of A {,,,5,,, 6,,,5 n A n A 8 9 ns 6,,,6 5, 5, 5,5 6, 6, 6,6 } 8 p A B multiple of B {,,5,,,, 6, nb nb ns 6 6 n A B 6 B ns 6 6,5 5, 5, 6, 6, 6 } p B A B {,5,,, 5, 6, 6 } n A B 6 p A 8 6 p A B p A p B p A B.6667 6 6 6 6 p A B' p A B. 6 6 ----------------------------------------------------------------------------------------------------------------------- Probability that a bomb dropped from a plane hits a target is.. Two bombs can destroy a bridge,if in all 6 bombs are dropped,find probability that the bridge will be destroyed. Given p., n 6 and q p.6 n r nr Cr p q p r p at least two bombs are required to destroy bridge p r p p C..6..6.7667 6 6 6 6 C Page /
(ISO/IEC - 7-5 Certified) SUMMER 8 EXAMINATION ject Name: Applied Mathematics Model wer ject Code: 7 wer 5. c) d) e) If the probability of a bad reaction from a certain injection is.. Determine the chance that out of individuals more than two will get a bad reaction. (Given: e.78) p., n m np. p p p p more than p p p 5... () () () e e e!!!. ----------------------------------------------------------------------------------------------------------------------- sin a Evaluate: sin sin a sin sin.cos a cos.sin a sin sin.cos a cos.sin a sin sin cos a cot.sin a cos a. sin a log sin c ----------------------------------------------------------------------------------------------------------------------- Evaluate: log sin Let I log sin --------------------() a a a By property f f f a I logsin log sin log cos I log sin Page 5/
(ISO/IEC - 7-5 Certified) SUMMER 8 EXAMINATION ject Name: Applied Mathematics Model wer ject Code: 7 wer 5. e) I log sin.cos sin.cos I log sin I log I log sin log I log sin log Put t dt when to t to I log sin t log I I log I I I log log I I dt log log sin --------------------() Using Property I log sin a a f f a Page 6/
(ISO/IEC - 7-5 Certified) SUMMER 8 EXAMINATION ject Name: Applied Mathematics Model wer ject Code: 7 wer 5. e) I log cos ---------------- Add and log cos I I log sin I log sin.cos sin.cos I log sin I log I log sin log I log sin log dt Put t when to t to dt I log sin t log log I log sin tdt log I log sin log I log sin Using Property log I I log I Page 7/
(ISO/IEC - 7-5 Certified) SUMMER 8 EXAMINATION ject Name: Applied Mathematics Model wer ject Code: 7 wer 5. f) Evaluate: sin cos sin cos sin cos sin d sin sin. cos cos cos sin c cos sin c sin cos Put sin cos dt t sin t sin. sin sin sin t t dt d t t dt t dt t dt dt t t t t t sin t t t t sin t t t dt dt dt Page 8/
(ISO/IEC - 7-5 Certified) SUMMER 8 EXAMINATION ject Name: Applied Mathematics Model wer ject Code: 7 wer 5. f) t sin t t dt t t sin t t t t t t sin t t t sin t t sin sin t t t c t sin t dt dt t t sin t sin t c sin sin sin sin sin sin sin c ----------------------------------------------------------------------------------------------------------------------- 6. a) Attempt any FOUR of the following: It is given that mean and variance of a binomial distribution are and / respectively what is the probability of obtaining i ii Eactly two successes. Less than two successes. Given mean np variance = npq q q p 6 np n n 6 i Eactly two successes 6 p 6C.9 Page 9/
(ISO/IEC - 7-5 Certified) SUMMER 8 EXAMINATION ject Name: Applied Mathematics Model wer ject Code: 7 wer 6. a) b) c) ( ii) Less than two success 6 6 p p 6C 6C.5 ----------------------------------------------------------------------------------------------------------------------- A card is drawn from a pack of cards numbered to find the probability of drawing a number which is a square. n S A which is square =,,9,6, 5,6, 9, 6,8, n A n A p A o r. n S ----------------------------------------------------------------------------------------------------------------------- Divide into two parts so that the product of the square of the one and the cube of the other may be the greatest possible. Let two parts of be and y y y P y P P P d P 5 dp 6 5 8 dp Put 6 5 d P 6 5,, at 8 576 Product is greatest when is divided into two parts and 8 Page /
(ISO/IEC - 7-5 Certified) SUMMER 8 EXAMINATION ject Name: Applied Mathematics Model wer ject Code: 7 wer 6. c) d) y y P y d P P P 8 6 P 8 6 5 dp 6 6 5 6 7 7 dp Let 6 6 5 6 6 5, 8, at 8 d P 6 7 8 7 8 8 576 Product is greatest when is divided into two parts 8 and ---------------------------------------------------------------------------------------------------------- Find the equation of tangent to the curve, y t, when t t t, y t t t and dt t dt t dt t t dt t at t, 5 at t, and y and slope m 5 + Page /
(ISO/IEC - 7-5 Certified) SUMMER 8 EXAMINATION ject Name: Applied Mathematics Model wer ject Code: 7 wer 6. d) e) equation is, y y m y 5 y 5 y8 5 y ----------------------------------------------------------------------------------------------------------------------- Given p A p B and p A B Evaluate : (i) p A / B (ii) p B / A (iii) p A B ' (iv) p A / B ' p( A B) p A p B p A B p A B p A B p A B (i) p A / B pb p A B (ii) p B / A p A (iii) p A B ' p A p A B 6 (iv) p A / B ' p A B ' 6 6 6 p B ' p B Page /
(ISO/IEC - 7-5 Certified) SUMMER 8 EXAMINATION ject Name: Applied Mathematics Model wer ject Code: 7 wer 6. f) In a certain eamination 5 students appeared. Mean score is 68 with S. D. 8. Find the number of students scoring i less than 5, ii more than 6. Given Area between z to z.5 is. 878 Area between z to z is. Given 68 8 N 5 5 68 i) z.5 8 p Less than 5 Aless than.5.5 Az to z.5.5.878. of students Np 5. 6. ie.., 6 6 68 ii) z 8 p More than 6 Amore than A to z z.5..5.8 of students N p 5.8.65 ie.., ------------------------------------------------------------------------------------------------------------- Important Note In the solution of the question paper, wherever possible all the possible alternative methods of solution are given for the sake of convenience. Still student may follow a method other than the given herein. In such case, first see whether the method falls within the scope of the curriculum, and then only give appropriate marks in accordance with the scheme of marking. ------------------------------------------------------------------------------------------------------------- Page /