MATHEMATICS Paper & Solutions

Transcription

1 CBSE-XII-8 EXAMINATION Series SGN MATHEMATICS Paper & Solutions SET- Code : 6/ Time : Hrs. Ma. Marks : General Instruction : (i) All questions are compulsor. (ii) The question paper consists of 9 questions divided into four section A, B, C and D. Sections 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. (iii) All questions in Section A are to be answered in one word, one sentence or as per the eact requirement of the question. (iv) There is no overall choice. However, internal choice has been provided in questions of four marks each and questions of si marks each. You have to attempt onl one of the alternatives in all such questions. (v) Use of calculators is not permitted. You ma ask for logarithmic tables, if required. Question numbers to carr mark each SECTION - A. Find the value of tan ( ) cot ( ) Sol. tan ( ) cot ( ) π π π π π π a If the matri A b is skew smmetric, find the values of 'a' and 'b'. Sol. a A b For skew smmetric matri A T A a b T b a b a a b b (On comparing LHS & RHS) a, CP Tower, Road No., IPIA, Kota (Raj.), Ph: 7- /

2 CBSE-XII-8 EXAMINATION. Find the magnitude of each of the two vectors a and b, having the same magnitude such that the angle between them is 6 and their scalar product is 9. Sol. Magnitude of two vectors a & b are same a b a b a b cos θ a a a cos 6 9 a 9 a b. If a * b denotes the larger of 'a' and 'b' and if a o b (a * b) +, then write the value of () o (), where * and o are binar operations. Sol. a o b (a * b) + o ( * ) + + SECTION - B Question numbers to carr marks each. Prove that : sin sin ( ),, Sol. sin sin ( ) Let sin θ sin θ sinθ sinθ sin θ sinθ Case I When sinθ π π θ 6 6 π π θ Also sinθ θ sin ( ) sin sin ( ), CP Tower, Road No., IPIA, Kota (Raj.), Ph: 7- /

3 CBSE-XII-8 EXAMINATION 6. Given A, compute A and show that A 9I A. 7 Sol. A 7 A adj A A 7 7 To Prove A 9I A LHS A 7 7 RHS 9I A LHS RHS 7. Differentiate tan + cos with respect to. sin Sol. tan + cos (Let) sin tan sin cos tan cot cos tan π tan π d, CP Tower, Road No., IPIA, Kota (Raj.), Ph: 7- /

4 CBSE-XII-8 EXAMINATION 8. The total cost C() associated with the production of units of an item is given b C() Find the marginal cost when units are produced, where b marginal cost we mean the instantaneous rate of change of total cost at an level of output. Sol. C() dc() marginal cost (MC) (.)( ). () + When MC. ( 9).( ) +. cos + sin 9. Evaluate : cos Sol. cos + sin cos cos + ( cos ) cos + cos cos + cos + sec + tan + C. Find the differential equation representing the famil of curves a e b+, where a and b are arbitrar constants. Sol. a e b+ Take log on both sides log log (a e b + ) log loga + log(e b + ) Differentiate both sides w.r.t. d d + (b + ) d b d d + d d + i.e. the required differential eq n., CP Tower, Road No., IPIA, Kota (Raj.), Ph: 7- /

5 CBSE-XII-8 EXAMINATION. If θ is the angle between two vectors î ĵ + kˆ and î ĵ + kˆ, find sin θ. Sol. a b a b cosθ a b (î ĵ + kˆ) (î ĵ + kˆ) cosθ ( )( ) cosθ cosθ sinθ cos θ A black and a red die are rolled together. Find the conditional probabilit of obtaining the sum 8, given that the red die resulted in a number less than. Sol. Let F : no. of red die is less than. E : sum of no is 8 E {(, 6) (, ) (, ) (, 6) (6, )} P(E) 6 F {(, ) (, ) (, )... (6, ) (, ) (, )... (6, ) (, ) (, )... (6, )} 8 P(F) 6 Also E F {(, ) (6, )} P(E F) 6 P(E F) P(E/F) P(F) 9 / 6 8/ 6 SECTION - C Question numbers to carr marks each. Using properties of determinants, prove that + + z + Sol. + + z + 9 (z + + z + z)., CP Tower, Road No., IPIA, Kota (Raj.), Ph: 7- /

6 CBSE-XII-8 EXAMINATION + z + + z z z R R + R + R z z z + + z z + + z z + (z) z + z C C C & C C C z z + + z z (z + z + + z) ( + 9) 9(z + + z +z) RHS z. If ( + ) d, find. Sol. d π If a (θ sin θ) and a ( cos θ), find when θ. ( + ) ( + d d ) [ + ] + ( + ) + ( + d d ) + ( + d ) [ ( + )] d ( + ) ( + ) a(θ sinθ), a( cosθ), dθ a( cosθ) d a( + sinθ) dθ, CP Tower, Road No., IPIA, Kota (Raj.), Ph: 7-6 /

7 CBSE-XII-8 EXAMINATION d d / dθ / dθ asinθ a( cosθ) d sinπ / cos / ( θ π / ) π sin( π π / ) cos( π π / ) sin π / / + cosπ / + / d. If sin (sin ), prove that d + tan + cos. Sol. sin(sin) d cos(sin) cos. d cos(sin)( sin) + cos [ sin(sin)] d sin cos(sin) cos sin(sin) d d LHS + tan + cos sin cos(sin) cos sin (sin) + tan cos cos(sin) + cos sin (sin) sin sin cos(sin) + cos cos(sin) cos sin cos(sin) + sin cos(sin) RHS 6. Find the equations of the tangent and the normal, to the curve at the point (, ), where and >. Find the intervals in which the function f() (a) Strictl increasing, (b) Strictl decreasing. Sol () since (, ) lies on () () + 9 (, ) Diff. w.r. to + + is, CP Tower, Road No., IPIA, Kota (Raj.), Ph: 7-7 /

8 CBSE-XII-8 EXAMINATION d + 8 d 6 9 d 6 (,) 9 7 Eq n of tangent ( ) m( ) ( ) ( ) Eq n of normal ( ) ( ) m 7 ( ) ( ) f() + + f '() +. f '() + ( )( ) ( )( )( + ) + + increasing in interval (, ) (, ) decreasing in interval (, ) (, ) 7. An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantit 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 b nearb settled lower income families, for whom water will be provided, what kind of value is hidden in this question? Sol. Let the length, width & height of the open tank be, & units Volume Total surface area +, CP Tower, Road No., IPIA, Kota (Raj.), Ph: 7-8 /

9 CBSE-XII-8 EXAMINATION S V + ds V V d S 8V + 8V V + + > 8 Hence S is minimum when ie the depth (height) of the tank is half of the width cos 8. Find ( sin )( + sin ) cos Sol. ( sin )( + sin ) put sin t cos dt dt ( t)( + t ) A Bt + C + ( t)( + t ) t + t A( + t ) + (Bt + C)( t) put t A() t A Comparing coefficients of t & t t A + ( B) B A B t B C B C t + + dt t t + log( t) t + dt dt t + + t + log ( sin) + log (t + ) + tan t + C log ( sin) + log (sin + ) + tan (sin) + C, CP Tower, Road No., IPIA, Kota (Raj.), Ph: 7-9 /

10 CBSE-XII-8 EXAMINATION 9. Find the particular solution of the differential equation e tan + ( e ) sec d, given that π when. d π Find the particular solution of the differential equation + tan sin, given that when. Sol. e tan + ( e ) sec d e tan (e ) sec d d e tan e sec sec e sec sec d e tan sec sec e d tan tan d sec tan [ e ] sec d tan e tan t sec d dt dt e t e e u e du logt log u + log C log(tan) log(e )C tan C(e ) put π, tan π C( ) C tan (e ) d + (tan ) sin tan IF e log sec e sec..sec sec sin + C sin + C cos tan sec + C sec sec + C, CP Tower, Road No., IPIA, Kota (Raj.), Ph: 7- /

11 CBSE-XII-8 EXAMINATION. Let a î + ĵ kˆ, b î ĵ + kˆ and c and b and d. a. c î + ĵ kˆ. Find a vector d which is perpendicular to both Sol. a î + ĵ kˆ b î ĵ + kˆ c î + ĵ kˆ To find vector d, such that d c d b & d a Let d î + ĵ + zkˆ ( î + ĵ + zkˆ) (î + ĵ kˆ) + z...() ( î + ĵ + zkˆ) (î ĵ + kˆ) + z...() ( î + ĵ + zkˆ) (î + ĵ kˆ) + z...() eq n () + eq n () + z + z + z...() eq n () + eq n () + z 6 + z 8 + z 8 ( + z) 8 + z...() eq. () () z z + Put & z in () + z r d î + ĵ + zkˆ r 6 d î + ĵ + kˆ, CP Tower, Road No., IPIA, Kota (Raj.), Ph: 7- /

12 CBSE-XII-8 EXAMINATION. Find the shortest distance between the lines r ( î ĵ) + λ ( î + ĵ kˆ ) and r ( î ĵ + kˆ ) + μ ( î + ĵ kˆ ). r Sol. (î ĵ) + λ(î + ĵ kˆ ) r r a î ĵ b î + ĵ kˆ r (î ĵ + kˆ) + μ(î + ĵ kˆ) r r a î ĵ + kˆ, b î + ĵ kˆ r r r r (a a) (b b) S.D. r r b b b r b r î ĵ kˆ î( + ) ĵ( + 6) + kˆ( ) î ĵ ( î + ĵ + k) (î ĵ) S.D 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 eactl one 'tail', what is the probabilit that she threw,, or 6 with the die? Sol. Let E be the event that the girl. Gets or on the roll P(E ) 6 Let E be the event that the girl gets,, or 6 on the roll P(E ) 6 Let A be event that she obtained eactl one tails If she tossed a coin times & eactl tail shows then [HTH, HHT, THH} P(A/E ) /8 P(A/E ) / (If she tossed a coin onl once & eactl shows) P(E)P(A / E) P(E /A) P(E )P(A / E ) + P(E )P(A / E ) Two numbers 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. Sol. The first five positive integers are,,,, we select two positive numbers in was Out of these two no. are selected at random & let X denote larger of the two no. X can be,, or, CP Tower, Road No., IPIA, Kota (Raj.), Ph: 7- /

13 CBSE-XII-8 EXAMINATION P(X ) P(larger no. is ) {(, ) and (, )} P(X ) P(X ) 6 8 P(X ) 6 8 Mean E(X) Variance Question numbers to 9 carr 6 marks each SECTION - D. Let A { Z : }. Show that R {(a, b) : a, b A, a b is divisible b } 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 b f(), R is neither one-one nor onto. Also, + if g : R R is defined as g(), find fog(). Sol. R {(a, b) : a, b A, a b is divisible b } Refleivit : for an a A a a, which is divisible b (a, a) R. So, R is refleive. Smmetr : Let (a, b) R a b is divisible b b a is also divisible b So R is smmetr Transitive : Let (a, b) R & (b, c) R a b is divisible b a b λ a b ± λ...() b c is divisible b b c μ b c ±μ...() Add () & (), CP Tower, Road No., IPIA, Kota (Raj.), Ph: 7- /

14 CBSE-XII-8 EXAMINATION a b + b c ±(λ + μ) a c ±(λ + μ) (a, c) R So, Transitive Hence, R is refleive, Smmetr & Transitive so it is an equivalence relation Let be an element of A such that (, ) R, then is divisible b,, 8,.,, 9 Hence, the set of all element of A which are related to in {,, 9} f() + for one-one f() f() ( ) So not one-one for onto f() cannot be epress in so not onto As g() fog() f[g()] f( ) ( ) + +. If A, find A. Use it to solve the sstem of equations + z + z + z. Using elementar row transformations, find the inverse of the matri A 7, CP Tower, Road No., IPIA, Kota (Raj.), Ph: 7- /

15 / MATHEMATICS CBSE-XII-8 EXAMINATION, CP Tower, Road No., IPIA, Kota (Raj.), Ph: 7- Sol. A ( + ) +( 6 + ) +( ) A 6 + Now A ; A ; A A ; A 9; A A ; A ; A A ij 9 Adj A [A ij ] T 9 9 T A 9 9 A AdjA Now z A X B X A B z ; and z A IA (Inverse of matri) 7 A R R R R R + R A R R R R R R A

16 CBSE-XII-8 EXAMINATION R R R A A 6. Using integration, find the area of the region in the first quadrant enclosed b the -ais, the line and the circle +. Sol. + + ( ) ( ) (, ) M(, ) O P A(, ) For coordinate of M put in + 6 ± M(, ) Required Area area of shaded region area of OMA area OMP + area MPA ( ( ) ) ( ) + sin 6 + ( ) ( ) + sin ( ) + sin π π 8 + ( () + 6 ) ( + 6 ) 8 + 8π 8 π π, CP Tower, Road No., IPIA, Kota (Raj.), Ph: 7-6 /

17 π/ sin + cos 7. Evaluate : 6 + 9sin Sol. Evaluate ( + + e ), as the limit of the sum. π / sin + cos 6 + 9sin π / π / sin + cos 6 + 9[ (sin cos ) ] sin + cos 9(sin cos) sin cos t (cos + sin ) dt 9t t 9 t dt + t log t 9 log log log log 8 CBSE-XII-8 EXAMINATION log log ( + + e ) Lim h[f() + f( + h) + f( + h) + + f( + (n )h)] h Lim h[( + + e) + (( + h) + ( + h) + e +h ) + (( + h) + ( + h) + e + h ) + ] h Lim h[ + e + ( + h + h + + h + e +h ) + ( + h + h + + 6h + e + h ) + ] h, CP Tower, Road No., IPIA, Kota (Raj.), Ph: 7-7 /

18 CBSE-XII-8 EXAMINATION Lim h[ + e + ( + h + h + e +h ) + ( + h + h + e + h ) + ] h Lim h[n + e( + e h + e h + ) + h [ + + ] + h[ + +.)] h Lim h (e h n + e e Lim h e h n + e e Lim n n Lim n n Lim n nh h e n + e e e n + e e hn h ) + h h n + 6 n n /n / n + n n(n )(n ) 6 n n n 6 n n n + 6 n n e (e ) / n e n n n n 8 + e(e ) + + n(n ) + h hn + n + n n n + n n e(e ) + e(e ) 8. Find the distance of the point (,, ) from the point of intersection of the line Sol. r î ĵ + kˆ + λ ( î + ĵ + kˆ ) and the plane r. ( î ĵ + kˆ ). Equation of line is r ( î ĵ + kˆ ) + λ( î + ĵ+ kˆ ) Equation of plane is r (î ĵ + kˆ ) Now combined equation [( î ĵ+ kˆ ) + λ( î + ĵ+ kˆ )]( î ĵ + kˆ ) [( + λ)î+ ( + λ) ĵ + ( + λ)kˆ ] (î ĵ + kˆ ) + λ ( + λ) + + λ λ + λ So, equation of line is r ( î ĵ + kˆ ) + ( î + ĵ+ kˆ ) r î ĵ + kˆ, CP Tower, Road No., IPIA, Kota (Raj.), Ph: 7-8 /

19 CBSE-XII-8 EXAMINATION Let the point of intersection be (,, z) So, r î + ĵ + kˆ,, z point of intersection is (,, ) Distance b/w (,, ) & (,, ) ( ) + ( + ) + ( ) A factor manufactures two tpes of screws A and B, each tpe requiring the use of two machines, an automatic and a hand-operated. It takes minutes on the automatic and 6 minutes on the hand-operated machine 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 to screws 'B'. Each machine is available for at most hours on an da. The manufacturer can sell a packet of screws 'A' at a profit of 7 paise and screws 'B' at a profit of `. Assuming that he can sell all the screws he manufactures, how man packets of each tpe should the factor owner produce in a da in order to maimize his profit? Formulate the above LPP and solve it graphicall and find the maimum profit. Sol. Let the number of package of screw A Number of packages of screw B Item Number Machine A Machine B Profit Screw A minutes 6 minutes To paise.7 Rs Screw B 6 minutes minutes Rs. Ma time Available hours min hours minutes Automated Machine Works for screw A min Works on screw B 6 min + 6 +, Now ma Z.7 + s.t , Hand operated machine Works on screw A 6 min Works on screw B min ,, CP Tower, Road No., IPIA, Kota (Raj.), Ph: 7-9 /

20 CBSE-XII-8 EXAMINATION (, 8) 8 (, ) 7 6 (, ) Corner Points (, ) (, ) (, ) Value of Z (, ) (6, ) Hence, profit will be maimum, if the compan produces, packages of screw A packages of screw B Maimum Profit Rs, CP Tower, Road No., IPIA, Kota (Raj.), Ph: 7- /