Time allowed : 3 hours Maximum Marks : 100

Transcription

1 CBSE XII EXAMINATION-8 Series SGN SET- Roll No. Candiates must write code on the title page of the answer book Please check that this question paper contains printed pages. Code number given on the right hand side of the question paper should be written on the title page of the answer- book by the candidate. Please check that this question paper contains 9 questions. Please write down the Serial Number of the question before attempting it. minute time has been allotted to read this question paper. The question paper will be distributed at. a.m. From. a.m. to. a.m., the students will read the question paper only and will not write any answer on the answer-book during this period. MATHEMATICS & SOLUTION Time allowed : hours Maimum Marks : General Instructions : (i (ii (iii (iv (v All questions are compulsory. The question paper consists of 9 questions divided into four sections A, B, and C and D. Section A comprises of questions of one mark each, Section B comprises of 8 questions of two marks each, Section C comprises of questions of four marks each and Section D comprises of 6 questions of si marks each. All questions in Section A are to be answered in one word, one sentence of as per the eact requiremt of the question. There is no overall choice. However, internal choice has been provided in question of four marks each and questions of si marks each. You have to attempt only one of the alternatives in all such questions. Use of calculators is not permitted. You may ask for logarithmic tables, if required. P.T.O Corporate Office : CG Tower, A-6 &, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.- PSPD Head Office: C-, Jawahar Nagar, Main Road, Kota (Rajasthan- Contact : 7-, Toll Free : CIN: U8RJ7PLC9 PAGE #

2 CBSE XII EXAMINATION-8 Question number to carry mark each. SECTION A Q. Find the value of tan cot (. Ans. tan ( cot ( ( cot ( Q. Ans. a If the matri A is skew symmetric, find the values of a and b. b a A b for skew symmetric metri of A. a, b Q. Find the magnitude of each of the two vectors a and b, having the same magnitude such that the 9 angle between them is 6 and their scalar product is. Ans. a b a b 9, 6 9 a b cos 6 a a a 9 9 a Q. If a * b denotes the larger of a and b and if a b (a * b +, then write the value of ( (, where * and are binary operations. Ans. a b (a * b + ( ( ( * + + Question numbers to carry marks each. SECTION B Q. Prove that : sin sin (,, Corporate Office : CG Tower, A-6 &, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.- PSPD Head Office: C-, Jawahar Nagar, Main Road, Kota (Rajasthan- Contact : 7-, Toll Free : CIN: U8RJ7PLC9 P.T.O PAGE #

3 CBSE XII EXAMINATION-8 Ans. sin sin (, Let sin t...( sint R.H.S. sin ( sin [sint sin t] sin (sint t sin L.H.S. sin sin ( Q 6. Given A, compute A and show that A 9I A. 7 Ans 6. A A 7 7 adj A A A 7 A 7 L.H.S. A Now 9I A I A L.H.S. R.H.S cos Q 7. Differentiate tan with respect to. sin cos Ans 7. Let y tan sin y tan cos sin cos y tan cot Corporate Office : CG Tower, A-6 &, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.- PSPD Head Office: C-, Jawahar Nagar, Main Road, Kota (Rajasthan- Contact : 7-, Toll Free : CIN: U8RJ7PLC9 P.T.O PAGE #

4 CBSE XII EXAMINATION-8 So y tan tan y d Q 8. The total cost C( associated with the production of units of an item is given by C( Find the marginal cost when units are produced, where by marginal cost we mean the instantaneous rate of change of total cost at any level of output. dc( Ans 8. Cost (MC d {.. + dc( at.(. ( The required marginal cost is. Rs. ~. Rs. cos sin Q 9. Evaluate : cos Ans 9. cos sin cos sin sin cos } sec tan C Q. Ans. Find the differential equation represention the family of curves y a e b+, where a and b are arbitaray constants. y ae b+ b a e b by b y Again diff. w.r. to y d y / ( / y Corporate Office : CG Tower, A-6 &, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.- PSPD Head Office: C-, Jawahar Nagar, Main Road, Kota (Rajasthan- Contact : 7-, Toll Free : CIN: U8RJ7PLC9 P.T.O PAGE #

5 CBSE XII EXAMINATION-8 d y y Q. If is the angle between two vectors î ĵ kˆ and î ĵ kˆ, find sin. a b Ans. cos a b 9 9 cos 7 sin cos Q. Ans. sin 9 A black and a red die are rolled together. Find the conditional probability of obtaining the sum 8, given that the red die resulted in number less than. Red dice & Black dice die st Black die nd Red A { (, 6 (6, (,, (,, (, } B { (, (, (,, (, (, (6, (, (, (, (, (, (6, (, (, (, (, (, (6, } A B (6,, (, A n (A B P B n (B 8 9 SECTION C Question numbers to carry marks each. Q. Using properties of determinants, prove that y 9 (y y yz z z Ans. L.H.S. y z (C C C y z z (C C C Corporate Office : CG Tower, A-6 &, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.- PSPD Head Office: C-, Jawahar Nagar, Main Road, Kota (Rajasthan- Contact : 7-, Toll Free : CIN: U8RJ7PLC9 P.T.O PAGE #

6 CBSE XII EXAMINATION-8 y z z (R R R y z y z (Common from C & C y 9 z y z 9 ( ( y ( (yz 9 [y + z + yz + yz] Q. Ans. If ( + y y, find. If a ( sin and y a ( cos, find when. ( + y y ( y ( y y ( y ( y y y y y y y y y y y y y y y y y y a ( sin y a ( cos a [ cos] d a( ( sin d a sin d / d a sin / d a( cos sin cos sin Corporate Office : CG Tower, A-6 &, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.- PSPD Head Office: C-, Jawahar Nagar, Main Road, Kota (Rajasthan- Contact : 7-, Toll Free : CIN: U8RJ7PLC9 P.T.O PAGE # 6

7 CBSE XII EXAMINATION-8 cos cot sin when. then cot d y Q. If y sin (sin, prove that Ans. y sin(sin...( cos(sin cos...( + tan + y cos. d y cos(sin ( sin cos ( sin (sin cos d y sin cos(sin cos by equation ( & ( d y sin y cos cos d y tan y cos (sin(sin Q 6. Find the equations of the tangent and the normal, to the curve 6 + 9y at the point (,y, where and y >. Find the intervals in which the function f( is (a strictly increasing, (b strictly decreasing. Ans y...( at (, y 6 9y at 6 + 9y 9y 6 8 y 9 y ± y > y So, P ( y (, diff. w. r. to of eq n ( + 8y 8y 8y Corporate Office : CG Tower, A-6 &, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.- PSPD Head Office: C-, Jawahar Nagar, Main Road, Kota (Rajasthan- Contact : 7-, Toll Free : CIN: U8RJ7PLC9 PAGE # 7

8 CBSE XII EXAMINATION-8 slope of tangent at point (, y 6 9y equation of tangent at point ( y y y ( y ( 7 equation of Normal at point ( y y y ( 7 y ( f( df ( df ( for Increasing f ( > + > + + > ( ( ( > ( ( > ( ( ( + > Q 7. (, (, for decreasing f( < (, (, An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of material will be least when depth of the tank is half of its width. If the cost is to be borne by nearby settled lower income families, for whom water will be provided, what kind of value is hidden in this question? Corporate Office : CG Tower, A-6 &, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.- PSPD Head Office: C-, Jawahar Nagar, Main Road, Kota (Rajasthan- Contact : 7-, Toll Free : CIN: U8RJ7PLC9 PAGE # 8

9 CBSE XII EXAMINATION-8 Ans 7. Let the l, b, h of the open tank be.,, y Volume of tank V y, Surface area of open tank S + y S + S + ds ds V V V V d S 8V for all V y When y, S is minimum the depth of tank is half of its width. Water capacity depend on the investment amount is low. As the families are of low income base the values associated with this question is to design the tank of minimum cost so that maimum number of families can afford it and avail water supply. Obviously at is responsibility of one and all to think & work for better ment of poor. cos Q 8. Find : ( sin ( sin Ans 8. cos ( sin ( sin sin t cos dt ( t ( t A Bt C Now ( t ( t ( t t A(+ t + (Bt + C ( t at t A + A at t A + C + C C at t A + ( B + C + ( B + B + B Corporate Office : CG Tower, A-6 &, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.- PSPD Head Office: C-, Jawahar Nagar, Main Road, Kota (Rajasthan- Contact : 7-, Toll Free : CIN: U8RJ7PLC9 PAGE # 9

10 CBSE XII EXAMINATION-8 Q 9. B t dt t t t dt t (t t log t + log t tan t C log sin + log sin tan sin C Find the particular solution of the differential equation e tan y + ( e sec y, given that y when. Find the particular solution of the differential equation y tan sin, given that y when. Ans 9. e tany + ( e sec y e tany (e sec y e sec y e tany e sec y on integrating both side e tan y log e log tan y logc at, y log log + log C log C log e log tan y log e e tany e e tan y e tan y e + tan y y tan sin Py Q is a linear D.E. Corporate Office : CG Tower, A-6 &, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.- PSPD Head Office: C-, Jawahar Nagar, Main Road, Kota (Rajasthan- Contact : 7-, Toll Free : CIN: U8RJ7PLC9 PAGE #

11 CBSE XII EXAMINATION-8 P tan I.f. tan e logsec logsec I.f. sec e e y. I.f (I.f Q y.sec y.sec y sec sec.sin tan.sec sec + C at, y + C C Particular Sol n of D.E. y sec sec Q. Let a î ĵ kˆ, b î ĵ kˆ and c î ĵ kˆ. Find a vector d which is perpendicular to both c and b and d a. Ans. a î ĵ kˆ b î ĵ kˆ c î ĵ kˆ d (c b c b î ĵ kˆ Q. î ( ĵ (6 kˆ î 6 ĵ kˆ d ( î 6 ĵ kˆ given d a ( î 6 ĵ kˆ (î ĵ kˆ ( 8 + (7 8 ( 6 d (î 6 ĵ kˆ Find the shortest distance between the lines r (î ĵ (î ĵ kˆ and r (î ĵ kˆ (î ĵ kˆ. Corporate Office : CG Tower, A-6 &, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.- PSPD Head Office: C-, Jawahar Nagar, Main Road, Kota (Rajasthan- Contact : 7-, Toll Free : CIN: U8RJ7PLC9 PAGE #

12 CBSE XII EXAMINATION-8 Ans. r î ĵ (î ĵ kˆ r î ĵ kˆ (î ĵ kˆ a a (î ĵ kˆ (î ĵ a a î kˆ (a a (b b S.D. b b î ĵ kˆ b b î ĵ kˆ b b î ( ĵ ( 6 kˆ( Q. Ans. S.D. S.D. ( î kˆ (î ĵ Suppose a girl throws a die. If she gets or, she tosses a coin three times and notes the number of tails. If she gets,, or 6, she tosses a coin once and notes whether a head or tail is obtained. If she obtained eactly one tail, what is the probability that she threw,, or 6 with the die? E Getting or in single throw of a die. E,, or 6 in single throw of a die. A getting eactly one tail. P(E P(A E P (A E, P(E 6 6 C 8 Req. Prof. P(E A P(E P(A E P(E P(A E P(E P(A E 8 8 Corporate Office : CG Tower, A-6 &, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.- PSPD Head Office: C-, Jawahar Nagar, Main Road, Kota (Rajasthan- Contact : 7-, Toll Free : CIN: U8RJ7PLC9 PAGE #

13 CBSE XII EXAMINATION-8 Q. Two number are selected at random (without replacement from the first five positive integers. Let X denote the larger of the two numbers obtained. Find the mean and variance of X. Ans. We observe that X can take values,,, P(X P(X P(X P(X 6 8 Xi p i X i p i X i p i 6 8 p i X i p i X i Mean ( X p i X i 8 Var(X p i X i ( p i X i 7 Var(X 7 6 Question number to 9 carry 6 marks each. Q. SECTION D Let A { Z : }. Show that R {(a, b : a, b A, a b is divisible by } is an equivalence relation. Find the set of all elements related to. Also write the equivalence class []. Show that the function f : R R defined by f(, R is neither one-one nor onto. Also, if g : R R is defined as g(, find fog(. Ans. A { z : } {,,,...} R { (a, b : a b is multiple of } Refleivity :- Let a element a, a A a a, which is multiple of (a, a R Corporate Office : CG Tower, A-6 &, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.- PSPD Head Office: C-, Jawahar Nagar, Main Road, Kota (Rajasthan- Contact : 7-, Toll Free : CIN: U8RJ7PLC9 PAGE #

14 CBSE XII EXAMINATION-8 (iii So R is refleive Symmetry :- Let (a, b R, a, b A Now (a, b R, a b is multiple of a b, N b a (b, a R So R is symmetric. Transitivity Let (a, b & (b, c R a, b, c A a b & b c,n a b ± ( b c ± ( Eq n ( and ( a c ± ± a c is multiple of a c k (a, b and (b, c R (a, c R So R is transitive Hence R is equivalance relation Let be an element of A Such that (, R. then k,, 8,,, 8,,, 9 Now set of all elements Related to is {,, 9} equivalence class [] {, 6, } f( for one one f( f(y y y y + y + y y y + y y (y + y ( y ( y y & y y ( y It is many-one not one-one for onto or into Let f( y Corporate Office : CG Tower, A-6 &, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.- PSPD Head Office: C-, Jawahar Nagar, Main Road, Kota (Rajasthan- Contact : 7-, Toll Free : CIN: U8RJ7PLC9 PAGE #

15 CBSE XII EXAMINATION-8 y y + y y + y y y R, y R but at y R So It is into not onto g ( fog f(g( g ( ( Q. If A find A. Use it to solve the system of equations y + z + y z + y z. Using elementary row transformations, find the inverse of the matri A 7 Ans. A A ( + + ( ( A 6 + (unique Sol. Cofactor C 9 adj A C T 9 Corporate Office : CG Tower, A-6 &, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.- PSPD Head Office: C-, Jawahar Nagar, Main Road, Kota (Rajasthan- Contact : 7-, Toll Free : CIN: U8RJ7PLC9 PAGE #

16 CBSE XII EXAMINATION-8 Corporate Office : CG Tower, A-6 &, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.- PSPD Head Office: C-, Jawahar Nagar, Main Road, Kota (Rajasthan- Contact : 7-, Toll Free : CIN: U8RJ7PLC9 PAGE # 6 A A adj(a A 9 AX B X A B B. A adj.a X 9 X X z y, y, z A IA A 7 R R R ; R R + R A R R R A R R R ; R R R A BA

17 CBSE XII EXAMINATION-8 B A Q 6. Using integration, find the area of the region in the first quadrant enclosed by the -ais, the line y and the circle + y. Ans 6. line y...( circle + y... ( the area of the region OBMO y [ ] 8 unit Again the area of the region BMAB y ( ( sin 6sin ( 6 sin ( 8 unit So required area OBAMO 8 + ( 8. Ans. / sin cos Q 7. Evaluate : 6 9sin Evaluate : ( e, as the limit of the sum. / (sin cos Ans 7. (i 6 9sin / (sin cos 6 9 { ( sin} / (sin cos 9 (cos sin Put cos sin t (sin + cos dt Lower limit t (cos sin Upper limit t cos sin Corporate Office : CG Tower, A-6 &, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.- PSPD Head Office: C-, Jawahar Nagar, Main Road, Kota (Rajasthan- Contact : 7-, Toll Free : CIN: U8RJ7PLC9 PAGE # 7

18 CBSE XII EXAMINATION-8 dt 9t dt (t dt ( (t t log ( t (ii ( e b a h log log log f( lin h[f(a + f(a+h + f(a+h f(a + (n h] where h b a h n n Now I e lim h lim h f( f( h f( h... f( (n h h n 8 ( h ( h I lim h [ { ( e } {( h ( h e } {( h ( h e } h {(+(n h} + { + (n h} + e {+(n h} ] h [ { ( h ( h... { (n h} + {+( + h + ( + h (+(n h } + {e e e... e } ] lim h h [{n + h ( (n } + h { (n } h h ( (nh + { n + h ( (n } + e { + e h + e h +...+e (n h }] h n n(n h.n(n (n n(n (e lim n h. n h e h h 6 h e nh n(n h n(n (n e lim n h. e h h 6 h e Put h n lim h n n(n n n n n(n (n e lim e 6 n h h e h h nh Corporate Office : CG Tower, A-6 &, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.- PSPD Head Office: C-, Jawahar Nagar, Main Road, Kota (Rajasthan- Contact : 7-, Toll Free : CIN: U8RJ7PLC9 PAGE # 8

19 CBSE XII EXAMINATION-8 Q 8. Put nh n n lim 8 n n 8 n n ee 6 lim 8 ee n n 8 ee 8 8 ee 6 ee Ans. n n n Find the distance of the point (,, from the point of intersection of the line r î ĵ kˆ (î ĵ kˆ and the plane r (î ĵ kˆ. r î ĵ kˆ î ĵ kˆ... ( Ans 8. y z Cartesion form Gen. point at line P +,, + Given plane eq. r. î ĵ kˆ... ( Let P is intersection point of the ( & plane so P will satisfying the equation of plane ( kˆ î ĵ kˆ So, ĵ + Point P (,, distance between (,, & (, ; is unit Q 9. Ans 9. A factory manufactures two types of screws A and B, each type requiring the use of two machines, an automatic and a hand-operated. It takes minurtes on the automatic and 6 minutes on the handoperated machines to manufacture a packet of screws A while it takes 6 minutes on the automatic and minutes on the hand-operated machine to manufacture a packet of screws B. Each machine is available for at most hours on any day. The manufacturer can sell a packet of screws A at a profit of 7 paise and screws B at a profit of Rs.. Assuming that he can sell all the screws he manufactures, how many packets of each type should the factory owner produce in a day in order to maimize his profit? Formulate the above LPP and solve it graphically and find the maimum profit. Let number of screw of type A number of screws of type B y + 6y + y 6 + y + y 8, y Corporate Office : CG Tower, A-6 &, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.- PSPD Head Office: C-, Jawahar Nagar, Main Road, Kota (Rajasthan- Contact : 7-, Toll Free : CIN: U8RJ7PLC9 PAGE # 9

20 CBSE XII EXAMINATION-8 Ma. Z.7 + (y Z.7 + y So maimum. profit Rs. occurs at, y Corporate Office : CG Tower, A-6 &, IPIA, Near City Mall, Jhalawar Road, Kota (Raj.- PSPD Head Office: C-, Jawahar Nagar, Main Road, Kota (Rajasthan- Contact : 7-, Toll Free : CIN: U8RJ7PLC9 PAGE #