S.Y. Diploma : Sem. III Applied Mathematics Time : Hrs.] Prelim Question Paper Solution [Marks : Q. Attempt any TEN of the following : [] Q.(a) Find the gradient of the tangent of the curve y at 4. [] y / / the gradient at 4 is, 4 / Q.(b) Find the point on the curve y 6 where the tangent is parallel to the ais. [] y 6 4 6 But tangent is parallel to -ais. 4 6 y 6 9 9 the point is, Q.(c) Evaluate: tan cot [] tancot tan tancotcot tan cot (sec cosec ) sec cosec tan cot + c Q.(d) Evaluate e e 6 e e 6 Pute t e dt dt t 6 dt t 4 t 4 log c.4 t 4 e 4 log c 8 e 4 []
S.Y. Diploma Maths III Q.(e) Evaluate: e e. e e e e e e c [½ + ½ mark] d Q.(f) Evaluate log [] log log d log log log log log + c [] Q.(g) Evaluate : log( ) log 6 log log 4 log log 4 [] Q.(h) Find the area enclosed by y + (above the -ais) and and. [] y 5 or 6.667 [] Q.(i) Find the order and degree of the equation y. Order y y 9 Degree
Prelim Question Paper Solution Q.(j) If the coin is tossed three times then find the probability of getting eactly two tails. [] P p(tail) q p p() n C p r q nr r C 8 or.75 Q.(k) A bag contains 7 white balls, 5 black balls and 4 red balls. If two balls are drawn at [] random from the bag. Find the probability that both the balls are white. Total 7 + 5 + 4 6 n 6 C m 7 C p m n 7 4 or.75 Q.(l) Two cards are drawn at random from a well shuffled pack of 5 cards. Find the [] probability that the two cards drawn are a king and a queen of the same suit. n n(s) 5 C 6 m n(pair of King and Queen of same suit) 4 p m 4 n 6 or. 66 Q. Attempt any FOUR of the following : [6] Q.(a) Find the equation of tangent and normal to the curve 4 + 9y 4 at point (,). 4 + 9y 4 4.9.y or 8 8y 8y 8 8 8y 4 9y at (, ), the slope of tangent is m 4. 9. 9 the equation of tangent is 9 y 9y 8 + + 9y or 9y + at (, ), the slope of normal is m 9 the equation of tangent is 9 y
S.Y. Diploma Maths III y 4 9 9 9 y 5 or 9 + y + 5 Q.(b) Find the maimum and minimum value of y 8 96. Let y 8 96 6 96 6 6 For stationary values, 6 96 6 7, 6 + 7.46, 4.46 At.46, 6(.46) 6 49.476 < At.46, y has maimum value and it is y (.46) 8(.46) 96 (.46).485 At 4.46, 6(4.46) 6 49.476 > At 4.46, y has minimum value and it is Y (4.46) 8(4.46) 96(4.46) 9.485 Q.(c) A metal wire 6 cm long is bent to from a rectangle. Find its dimensions when its area is maimum. Let and y be the sides of rectangle. + y 6 or + y 8 y 8 But area A y (8 ) 8 da 8 da For stationary values, da 8 9 da At 9, < At 9, A has maimum value and the other side is y 8 9 Q.(d) Evaluate tan tan tan tan sec tan Put tant sec dt dt t 4
Prelim Question Paper Solution log t c t log tan c tan Q.(e) Evaluate: 9log Put log t 9 log dt dt 9 t dt t t tan c log tan c Q.(f) Evaluate Put t, dt t dt t c c Q. Attempt any FOUR of the following : [6] Q.(a) Evaluate /4 /4 sec d sec sec sec /4 /4 tan tan /4 tan log sec tan log sec log sec 4 4 4 log 4 or.49 Q.(b) Evaluate I sin log sin(log) t t Put log t, e, e dt t e sint dt 5
S.Y. Diploma Maths III sin t e dt e dt sin t dt dt sin t e t t e costdt t t d sin t e t t t d cos t e dt e dt cost dt dt sin t e t t t cos t e e sint dt sin t e t t t cost e e sin tdt sin t e t [coste t + I] sin t e t coste t I I + I sin te t coste t I e t (sin t cos t) t log e e sin t cos t sin log cos log c I Q.(c) Find the area bounded by the curve y and line y. Given y, y, y y A b a Q.(d) Evaluate.67 tan d tan tan tan tan tan tan tan tan tan tan tan tan 4 4 or.85 4 6
Prelim Question Paper Solution Q.(e) Solve: y sin y. y y sin Put y v or y v dv v dv v v + sin v dv sin v cosecv dv cosecv dv log(cosec v cot v) log + c [½+½ mark] y y logcosec cot log + c Q.(f) Obtain the differential equation if y Acos (log ) + Bsin (log ). y A cos (log ) + B sin (log ) A sin (log ) B cos(log ) A sin (log ) + B cos (log ) Acoslog Bsinlog A cos(log ) B sin (log ) Acos log y B sin log y Q.4 Attempt any FOUR of the following : [6] / cos Q.4(a) Evaluate : cos sin I I I / /4 / / cos cos sin cos cos sin sin sin cos cos sin cos sin Replace / sin cos & cos sin 7
S.Y. Diploma Maths III I / I I 4 5 Q.4(b) Evaluate 7 I 5 Replace 7 7 7, &7 5 7 I 7 5 I 7 7 5 I 5 I I 5 I Q.4(c) Find by integration the area of the circle + y a. + y a y a y a At y, a a, a b A 4 y a 4 a a a a 4 a sin a a a 4 sin sin a 4 a Q.4(d) Evaluate e y + 4 e y e y + 4 e y e e y + 4 e y (e + 4 ) e y y e (e + 4 ) 8
Prelim Question Paper Solution e y y e e 4 e 4 c Q.4(e) Solve (y + y ) + ( + y + siny). (y + y ) + ( + y + sin y) M y + y M + y y N +y + siny N y the equation is eact. M N c yconstant termsfreefrom y y siny c y y cos y c or y + y cos y c Q.4(f) Find the area bounded by y and y 4. y and y 4 ( 4) + 6, 8 b A a 8 y y 4 8 4 8 4(8) 8 4() 8 or.667 Q.5 Attempt any FOUR of the following : [6] Q.5(a) A husband and wife appear in an interview for two vacancies in the same post. The probability of husband s selection is 7 and that of wife selection is. What is the 5 probaility that: () Both of them will be selected () None of them will be selected. P(H) P(H) 7 7 6 7 P P(W ) 5 4 5 P (Both selected) P(H & W) P(H) P(W) 7 5 5 or.86 9
S.Y. Diploma Maths III P (None is selected) p(h & W) P(H) P(W) 6 4 7 5 4 or.686 5 Q.5(b) Solve cos ( + y) cos( + y) Put + y v dv dv cos v dv + cos v dv cosv dv cosv dv v cos v sec dv v tan + c / tan y + c Q.5(c) A skilled typist, on routine work, kept a record of mistakes per day during working days. Fit a Poission distribution to the set of observations. 4 5 6 y 4 9 4 9 y y 4 9 9 4 84 6 4 9 6 5 5 6 6 67 67 mean m.89 [ marks] m e m p r! r.89 e.89 r! r
Prelim Question Paper Solution Q.5(d) Solve (4 y + ycos y) + ( 4 y + cos y) (4 y + y cos y) + ( 4 y + cos y) M 4 y + y cos y M y 8 y y sin y + cos y N 4 y + cos y N 8 y sin yy + cos y the equation is eact. M N c ycons tan t termsfreefrom 4 y y cos y c 4 siny 4 y y c or 4 y + sin y c 4 y Q.5(e) Evaluate : /. / / / / / / 5/ 5/ 7/ 5/ 7/ [½+ ½ mark] 5/ 7/ 4 5 or.4 Q.5(f) 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 7.4) n, p. M np m e m p r! r P(more than ) p(maimum ) [p() + p() + p()] e e e!!! [.5 +.76 +.76].5
S.Y. Diploma Maths III Q.6 Attempt any FOUR of the following : [6] Q.6(a) A coin is tossed and a die is rolled. Show that the events head and si are independent and mutually eclusive. Case I Consider the eperiment of two events A coin is tossed and a die is rolled are taken together. S H, H,, H, H,4 H,5 H,6 T, T, T, T, 4T, 5T, 6 n. Let A event of occurring head, H,, H,, H,, H, 4, H, 5, (H, 6) A m 6 P m n 6 Let B event of occurring si, B {(H, 6), (T, 6)} m P(B) m n 6 Now AB {(H, 6)} m The probability of happening head and si is P(A B) m n But P P(B) 6. P(AB) P P(B) the events are independent. But P(A B) [or also P(AB) P + P(B) P(AB) the events are not mutually eclusive. Case II Consider the eperiment of two events A coin is tossed and a dies is rolled are taken together and done eclusively. (i) the set of tossing coin is {H, T}. Consequently n. Now let A event of occurring head, then m and hence P m n. (ii) the set of rolling of die is {,,, 4, 5, 6}. Consequently n 6. Now let B event of occurring si, then m in this case and hence P(B) m n 6 Now here in this case AB and hence P(AB) shows that the events are mutually eclusive but the events are not independent as: P P(B) 6 PAB Q.6(b) If two dice are rolled simultaneously then find the probability that total is 6 or. n n(s) 6 6 A {(, 5), (, 4), (, ), (4, ), (5, ), (4, 6), (5, 5), (6, 4)} M n 8 p m 8 n 6 9 or.
Prelim Question Paper Solution Q.6(c) In a sample of cases the mean of a certain test is 4 and standard deviation is.5. Assuming the distribution to be normal. (Given : A(.8).88, A(.4).554, A((.6).445.) Find: (i) How many students score between and 5? (ii) How many students score above 8? Given 4.5 N 4 (i) z.5.8 z 5 4. 4.5 P ( 5) P(.8 z.4) P(.8 z ) + P( z.4) P( z.8) + P( z.4) P ( 5).88 +.554.445 No. of students N.P.445 44.5 i.e., 444 (ii) 8 4 z.5.6 P(8 ) P(.6 z).5 P( z.6).5.445.548 No. of students N. P.548 54.8 i.e., 55 Q.6(d) The probability that a man aged 65 will live to 75 is.65. What is the probability that out of men which are now 65, 7 will live to 75? p.65 q p.5 p() n Cpq r nr r 7 7 C (.65) (.5) 7 [ marks].5 Q.6(e) Divide 8 into two parts such that their product is maimum. Let, y be the numbers. But + y 8 i.e. y 8 To maimize, p y (8 ) p 8 dp 8 dp dp For stationary values, 8 or 8 4 dp At 4, < At 4, p has maimum value.
S.Y. Diploma Maths III Q.6(f) A problem is given to three students A, B, C whose chances of solving it are, 4 and respectively. What is the chance that the problem is solved? 4 P P P P(B) 4 P(B) P(B) 4 P(C) 4 P(C) P(C) 4 P p (the problem is solved) p (the problem is not solved by all A, B, C) p (ABC) 4 4 9 or.96 4