QUESTION PAPER CODE 65/2/1 EXPECTED ANSWERS/VALUE POINTS SECTION - A SECTION - B. f (x) f (y) = w = codomain. sin sin 5

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1 -.. { 8, 7} QUESTON AER ODE 6// EXETED ANSWERS/VALUE ONTS SETON - A 6.. Mrks.. k 7 6. tn ot 7. log or log 8.. Let, W. 6 i j 8 k. os SETON - B f nd both re even, f () f () f nd both re odd, f () f () f is odd nd is even i.e., ( ) is even, ( ) is odd f () f () Similrl for is even nd is odd. f is one one Rnge of f { f ( ), f ( ), f ( )...} {,,,,...} w odomin f is onto, Hene f is invertible f : W W f ( ), is odd, is even m. os os sin sin m m

2 , 6 stisf so not does OR 8 tn 8 tn 7 tn S. H. L. m 8 tn tn m 8 tn tn / m ot tn 8 8 tn. operting H.S. L. ( ) m

3 ( ) R R operting R R R R m Epnding long, ( ) ( ) ( ) m d d. sin θ b os θ, os θ b sin θ dθ dθ d d osθ b sinθ sinθ b osθ m d d d d d d d d. Tking log on both sedes m log n log (m n) log ( ) m n d d m n d d d d n m n m n m m d d n m n m d d ( ) ( ) m 6. f ( ) 6, let, Δ. m f ( ) f ( Δ) f ( ) Δ f ( Δ) ( Δ) f ( ) f ( )

4 f (.) (.) f () f () (.) ().6 OR f () 6 6 ( ) ( ) f (),, r m ve ve ve ve f () >, (,) U (, ) Stritl inresing f () <, (, ) U (,) Stritl deresing 7. ut osθ d sin θ dθ m θ osθ ( sinθ) dθ θ osθ dθ sinθ m { θ sinθ sinθ dθ { θ sin θ. sin θ dθ } θ sin θ os θ m os OR 7 ( ) d ( ) d m

5 7 ( ) d d m 7 ( ) log m 8 8. ( ) d ( )d d d m d d log utting, we get, log d. ntegrting ftor e ot e log sin sin m Solution is. sin os sin m sin sin d os sin m Here, Solution is sin os 6

6 Here ( b), ( b ), ( ) re oplnr, ( b) {( b ) ( ) } m m ( b) ( b b ) ( b ) ( b ) ( ) b ( b ) m ( b ) b ( ) b { ( b ) } b ( b ) Q, b, re oplner Similrl onverse prt n lso be proved. OR b i j k, b j k Let ( b) ( b) i j k i j k 6 i 6 j 6 k m. i j, b i j k i j k, b i j k i k m 7

7 b i j b i j 7 k k m b b ( ) ( b b ) 7 S D ( ) ( b b ) b b m. X be the number of red rds drwn X,, ( X ) 6 7 ( X ) 6 6 ( X ) m 7 ( X ) Men pi i 7 or. 7 8

8 SETON. Here or AX B A ( ) ( ) ( ) X A B oftors re : A A A A A A A A A,, One more vlue like puntulit, honest et m. Let V volume, S Totl surfe re r rdius, h height v V r r h h...(i) m S r rh

9 v v S r r r r r m ds dr v r r ds dr v r m d s dr v 6 > m v t r, Totl surfe re is minimum utting V r in (i) r r h r h. tn ( ) d se tn se tn tn m tn tn d d se tn se tn ( sin ) d ( se tn tn ) sin d os ( se tn se ) d ( se tn ) m sin os os sin m ( ) ( ) m

10 6. orret figure m Are of shded region ( ) d m ( ) sin m units sq. m 7. Let eqution of plne through (,, ) with dr s of perpendiulr s, b nd is ( ) b ( ) ( ) m The plne is to nd 8 b nd b k b k, b k, k Eqution of the plne is k ( ) k ( ) k ( ) 7 m Distne of plne from (,, ) is

11 d 7 6 m OR Line through A (,, ) nd B (,, ) is λ Generl point on the line is ( λ, λ, λ ) ( λ ) λ λ λ oint of intersetion is (,, ) m d () ( ) ( ) 6 m 8. Let the number of lmps nd shdes mnuftured be nd respetivel L... is Mimise Z Subjet to < < >, > m For orret grph m Verties of fesible region re (, ), A(6, ), B(, ) (, ) (A), (B) 6, () For m rof no. of lmps No. of shdes m Mimum rofit Rs. 6

12 . Let E : Sooter driver is hosen E : r driver is hosen E : Truk driver is hosen A : erson meets with n ident m 6 ( E ), ( E ), ( E ) ( A )., ( A )., ( A ). E E E m (.) E A 6 (.) (.) (.) m E E E or A A A m 7 OR Let E be the event drwing dimond rd n, p, q ( E) ( E)

13 () (i) () (ii) ( ) (iii)