Important Instructions to the Examiners:

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1 (ISO/IEC Certified) Winter Eamination Subject & Code: Applied Maths (7) Model Answer Page No: /6 Important Instructions to the 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 the candidate and those in the model answer may vary. The eaminer may give credit for any equivalent figure drawn. ) 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 the model answer. 6) In case of some questions credit may be given by judgment 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.

2 (ISO/IEC Certified) Subject & Code: Applied Maths (7) Page No: /6 ) Attempt any TEN of the following: a) b) c) Find the point on the curve y y ² But given that, slope m 8 ( ) y ² the point is, at which slope is. Find the radius of curvature of the curve y log ( sin ) at. y log sin d y d y cos cot sin ec & cos at, cot and ec cos + + ρ d y Integrate w. r. t. of + cos + cos cos cos sin + c Note: In the solution of any integration problems, if the constant c is not added, mark may be deducted.

3 (ISO/IEC Certified) Subject & Code: Applied Maths (7) Page No: /6 ) d) cos ( log ) Evaluate Put log t cos( log ) dt costdt sin t + c ( ) sin log + c e) ( ) cos log I Put log t dt I cos tdt sin t + c ( ) sin log + c Evaluate ( + ) A B + ( + ) + + A + B Put + A + A Put +, + B B + ( + ) + + ( + ) + log log + + c +

4 (ISO/IEC Certified) Subject & Code: Applied Maths (7) Page No: /6 ) A B + ( + ) + + A + B f) For A + + For +, B + ( + ) + + ( + ) + log log + + c Evaluate tan + g) tan tan d tan tan tan + tan + tan log ( + ) + c Evaluate ( ) log log[ ] log[ ] log log log

5 (ISO/IEC Certified) Subject & Code: Applied Maths (7) Page No: /6 ) h) i) Find the area contained by the curve to. A b a y ( sin ) cos + cos + cos [ ] y sin + + from + + or.9 Find the order and degree of the D. E. Order. d y y. j) k) d y y d y y Degree Form a differential equation if y Asin + B cos. y Asin + B cos Acos Bsin d y Asin B cos + d y y ( Asin B cos ) + y From a pack of cards one is drawn at random. Find the probability of getting a king. m p n or.769

6 (ISO/IEC Certified) Subject & Code: Applied Maths (7) Page No: 6/6 ) l) n n S m n A m p n or.769 n n S C m n A C m p n or.769 An unbiased coin is tossed times. Find the probability of getting heads. p. q p. Here n C C ) a) p C p q n r n r r C or. 6 Attempt any FOUR of the following: Find the equation of the tangent and normal to the curve y y,. + + at + y + y + + y + y y + y ( + y ) 9 y 9 y + y 9 the slope of tangent at, ( ) is + ( ) 9 m

7 (ISO/IEC Certified) Subject & Code: Applied Maths (7) Page No: 7/6 ) the equation of tangent is y + y ( ) + y or + y the slope of normal m the equation of tangent is y + y + 6 ( ) y 76 or + y + 76 b) c) A beam is bent in the form of the curve y sin sin. Find the radius of curvature of the beam at this point at. y sin sin cos cos d y & sin + sin at, cos cos and d y sin + sin / + + ρ or or.9 d y Find the maimum and minimum values of Let y ³ 8 ² d y

8 (ISO/IEC Certified) Subject & Code: Applied Maths (7) Page No: 8/6 ) For stationary values, d) , 8 d y At, 6 6 < At, y has ma imum value and it is y ³ 8 ² d y At 8, > At 8, y has min imum value and it is y 8 ³ 8 8 ² tan Evaluate + tan tan tan + tan log sec + c sin tan cos + tan sin + cos cos sin cos + sin log cos + sin + c sin tan cos + tan sin + cos cos sin Put cos + sin t cos + sin + dt ( sin cos ) + t log t + c ( sin ) log cos + + c

9 (ISO/IEC Certified) Subject & Code: Applied Maths (7) Page No: 9/6 ) e) f) Evaluate sin ( ) sin sin sin cos cos + + c sin sin sin ( ) cos sin ( t ) dt t t + c Put cos t sin dt cos cos + c + Evaluate + + I ( ) ( )( + ) + A B C For + + A ( )( + ) ( )( + ) For, + + B 8 ( + ) ( + ) For +, + + C 8 ( ) ( + ) + / / 8 / / /8 / 8 I log + log ( ) log ( + ) + c 8 8

10 (ISO/IEC Certified) Subject & Code: Applied Maths (7) Page No: /6 ) + + I ( ) ( )( + ) + A B C A + + B + C Put + + A + + A Put, B + ) a) B 8 Put +, C C + / / 8 / / / 8 / 8 I log + log ( ) log ( + ) + c 8 8 Attempt any FOUR of the following. Evaluate ( ) + sin sin sin sin sin +

11 (ISO/IEC Certified) Subject & Code: Applied Maths (7) Page No: /6 ) + ( ) ( ) b) sin sin sin sin sin + 9 Evaluate Re place 9 I & + 9 I I I I [ ] I I I ( ) ( ) 9 + +

12 (ISO/IEC Certified) Subject & Code: Applied Maths (7) Page No: /6 ) I c) I I I [ ] I I Find the area of region included between the parabola y + and the line y +. y and y or, a A y y ( ) ( ) + + or. [ ] y and y + + or + +, ( ) ( ) A + +

13 (ISO/IEC Certified) Subject & Code: Applied Maths (7) Page No: /6 ) d) or. [ ] A or. + y + y + Solve the D. E. y y y + c + + y y + c + + y log ( + ) + log ( + y ) c y y y + c + + y Put + u and + y v du y dv du + dv c u v log ( u) + log ( v) c log ( + ) + log ( + y ) c M M ( + y ) y y N N y ( + ) y the equation is eact. M + N c y cons tant terms free from

14 (ISO/IEC Certified) Subject & Code: Applied Maths (7) Page No: /6 ) e) f) ( ) + y + y c tan y ( + y ) + c Solve the D. E. ( ) + y Put + y v dv + dv v dv v dv + v + v v v dv + v v dv + v v dv + v dv + v v v + c y tan ( y) c y y Solve the D. E. ( + y) y ( + y) y y y + y + y y Put v or y v dv v + dv + ( v) v v +.

15 (ISO/IEC Certified) Subject & Code: Applied Maths (7) Page No: /6 ) ) a) dv v v + + v dv v v + v dv v + v + v dv v + v dv v + dv v log v + v log + c y y log + log + c Attempt any FOUR of the following. / Evaluate / 6+ cot / / I /6+ cot /6 cos + sin Re place / / sin I sin cos /6 sin + cos & cos sin I / /6 cos cos + sin I / /6 / I I /6 [ ] cos + cos + / /6 6 6 I sin sin

16 (ISO/IEC Certified) Subject & Code: Applied Maths (7) Page No: 6/6 ) I / /6 / /6 + cot + cos sin / /6 sin sin + cos / /6 sin sin + cos I / /6 cos cos + sin b) I / /6 / I I /6 [ ] cos + cos + / /6 sin sin 6 6 I sin Evaluate + cos ² I sin + cos ² ( ) sin ( ) + cos ² ( ) ( ) sin + cos ² sin sin + cos ² + cos ² sin I + cos ²

17 (ISO/IEC Certified) Subject & Code: Applied Maths (7) Page No: 7/6 ) sin I + cos ² Put cos t sin dt t c) I I tan t dt + t² I tan tan I I Find the area of the circle + y 6 by integration. + y 6 y 6 y gives 6, A a y sin 6 + sin + sin [ ] Note: The above can also be solve on the same line as follows: b 6 6 a A y b A y 6 a and The Area 6.

18 (ISO/IEC Certified) Subject & Code: Applied Maths (7) Page No: 8/6 ) d) e) Solve the D. E. + + y tan cos P tan and Q cos y tan cos p tan logsec IF e e e sec y IF Q IF + c + y sec cos sec c y sec cos + c y sec sin + c y Solve the D. E. e + y + y + a + y + y M e + y + y M y + y y y N a + y + y N y + y the equation is eact. f) M + N c y cons tant terms free from ( ) y e + y + y + a c y a e + y + y + c log a y a or e + y + y + c log a d y Verify that y log is a solution of + y log d y +

19 (ISO/IEC Certified) Subject & Code: Applied Maths (7) Page No: 9/6 ) ) y log d y d y Attempt any FOUR of the following. a) A room has electric lamps. From a collection of electric bulbs of which only are good, are selected at random and put in the lamps. Find the probability that the room is lighted by at least one of the bulbs. Good + Bad n C p p ( room is lightened by at least one bulb) p p room is not lightened bad bulbs 89 9 C C or * Good + Bad n C p p p ( room is lightened by at least one bulb) ( G & B or G & B or G) C 89 9 C C + C C + C or * Note: As the use of Non-programmable Electronic Pocket Calculator is permissible, calculating the value directly without using the actual formula of n C as shown in (*) is allowed and thus no mark to be deducted. r

20 (ISO/IEC Certified) Subject & Code: Applied Maths (7) Page No: /6 ) b) If % of the bolts produced by a machine are defective. Find the probability that out of bolts drawn i) One is defective. ii) At most two are defective. p. q.8 Here n i) p n r n r r C p q C.96 (.) (.8) c) (.) (.8) (.) (.8) (.) (.8) ii) p p + p + p C C C p p( at least ) p p + (.) (.8) C (.) (.8) C A bo contains red, white, black balls. Two balls are drawn at random. Find the probability that they are not of the same colour. Balls ( not of same colour) n n S C m n n(rw or WB or RB) + + m p n 9 or.69

21 (ISO/IEC Certified) Subject & Code: Applied Maths (7) Page No: /6 ) Balls + + ( not of same colour ) n n S C 9...(*) p p p(rw or WB or RB) C 8 C C + C C + C C or.69...(**) d) Note: If the step (*) is not written and directly the step (**) is written, full marks to be given. Evaluate cos Put tan t dt t, cos + t + t dt + t cos t + t dt + t ( + t ) ( t ) + t dt + t + t dt 9 t dt dt or ( ) 9 t t t t log c or log + t c t + tan log c + tan +

22 (ISO/IEC Certified) Subject & Code: Applied Maths (7) Page No: /6 ) e) f) / Evaluate log ( + ) / / I log + ( tan ) / / / / tan / log tan + / tan log + + tan / log + tan ( ) log log + tan log log + tan log I I log log log / [ ] / log I log 8 y Solve the D. E. + y y + y + y y Put t y dt y dt + t dt t P and Q

23 (ISO/IEC Certified) Subject & Code: Applied Maths (7) Page No: /6 ) 6) a) p log( ) IF e e e e the solution is, t IF Q IF + c log t c + t log + c log + c y Attempt any FOUR of the following. A If P ( A ), P ( B ') and P, find P( A B) and B B P A. P( B) P( B ') b) ( B) A P A P B P B A P( A B) P P( B) B B P( A B) / P A P ( A ) / Using Poisson distribution, find the probability that the ace of spades will be drawn from a pack of well shuffled cards at least one in consecutive trials. n p m np p p ( at least one) p or p at most e!.866

24 (ISO/IEC Certified) Subject & Code: Applied Maths (7) Page No: /6 6) c) In a sample of cases, the mean of certain test is and standard deviation is.. Assuming the distribution to be normal, find i) How many students score between and? ii) How many students score above 8? Given A(.8).88, A(.). and A(.6). Given, σ. z σ i) z.8. z.. p A z (.8.)...(*) (.8 ) (.) (.8) (.) A z + A z A z + A z of students.. 8 ii) z.6. p A( z.6) A( z) A( z.6). A( z.6)...8 of students.8.8 (*) Note: As the area under Standard Normal Curve represents probability, many authors use symbol of probability p instead of the symbol of area A i. e., instead of writing A(.8 z.) it is also written as p (.8 z.).

25 (ISO/IEC Certified) Subject & Code: Applied Maths (7) Page No: /6 6) d) A manufacturer can sell items at price of Rs. ( ) each. The cost of producing items in Rs. is + +. How many items must be sold so that his profit is maimum. ( ) Selling price Cost price + + But, profit selling price cost price ( ) ( ) p + + p + d p dp + dp For stationary values, e) + 8 d p At 8, < At 8, p is maimum. Thus the manufacturer can sell maimum 8 items. Find the equation of the tangents to the curve y, where it cuts -ais. y The curve cuts -ais, y ², ( ) ( ) and (, ) is the points on -ais are,, the slope of tangent at m ( ) the equation of tangent is y + y or + y +

26 (ISO/IEC Certified) Subject & Code: Applied Maths (7) Page No: 6/6 6) ( ) the slope of tangent at, is m ( ) the equation of tangent is y y or y f) Find the area of the region lying between the parabolas y a and ay. y a, ay a Also y a and y a a 6a ( a ) 6, a b a A y y a a a or also a a a a / a a / ( a) a ( a) a 6 a 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.

Answer. Find the gradient of the curve y x at x 4

Answer. Find the gradient of the curve y x at x 4 (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