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1 .. 3 MATHEMATICS (4) CLASS XII 6-7 Marking Schm Sction A, R. Hnc, R is not rfliv. 3 k 8 3. sin cos R is th idntit lmnt for if a a aa R Sction B 5. Lt sc. Thn sc and for < -, [/] Givn prssion = cot ( cot ) [/] = cot (cot( )) sc as 6. Lt A b a skw-smmtric matri. Thn b dfinition A A [/] 7. th ( i, j) th lmnt of A th ( i, j) th lmnt of ( A) [/] th ( j, i) th lmnt of A th ( i, j) th lmnt of A [/] For th diagonal lmnts i j th ( i, i) th lmnt of A th ( i, i) th lmnt of A th ( i, i) th lmnt of A Hnc, th diagonal lmnts ar all zro. [/] 3 3 tan tan 3 tan 3 d Lt log, 4,. [/] [/] d.5 d Givn intgral d c as d ( ) d

2 b a b or, b a...(). b b d...() [/] d d d Substituting in (), ( a ) [/] d. a b a b a b b 6 [/] b 46 [/]. P( A B) P( A / B) [/] PB ( ) P( AB) PB ( ) P( A) P( B) P( A B) PB ( ) [/] [/] 7 [/] Sction C 3. A 5 [/] adja [+/] adja A A 5 [/] Givn sstm of quations is AX B, whr X, B [/] X A B , 5 5 [/] [/]

3 4. f ( h) f ( ) ( h) Lf ( ) lim lim h h h h [+/] f ( h) f ( ) 3 6( h) Rf ( ) lim lim 6 [+/] h h h h Lf ( ) Rf ( ), f is not diffrntiabl at sin( ) lim f( ) lim f( ) k 6 For th continuit of f( ) at OR 6 [] [/], f ( ) lim f ( ) k [/] 6 6 d 5. ap cos pt, bpsin pt dt dt bp sin pt b tan pt [/] d ap cos pt a d bp sc pt dt d a d bp sc pt b d ( a ) b a pa cos pt ( a ) d [+/] 6. Lt th normal b at (, ) (, ) d (, ) to th curv. d Th slop of th normal at Th quation of th normal is ( ) [/] Th point (, ) satisfis it ( )...() [/] Also,...() [/] Solving () and (), w gt 3 3, [/] Th rquird quation of th normal is 3 3

4 OR 3 3 f ( ) 4sin cos 4cos sin sin 4 f ( ) 4 In th intrval Sign of f () Conclusion Marks (, ) 4 (, ) 4 -v as 4 f is strictl dcrasing in [, 4 ] +v as 4 f is strictl incrasing in [ 4, ] 7. Incras in subscription chargs = Rs, Dcras in th numbr of subscribr =. Obviousl, is a whol numbr. [/] Incom is givn b = (5 )(3 + ). Lt us assum for th tim bing 5, R, [/] d d d d, d d [/] is maimum whn =, which is a whol numbr. Thrfor, sh must incras th subscription chargs b Rs to hav maimum incom. [/] Magazins contribut, a grat dal, to th dvlopmnt of our knowldg. Through valuabl and subtl critical and commntar articls on cultur, social civilization, nw lif stl w larn a lot of intrsting things. Through rading magazins, our mind and point of viw ar consolidatd and nrichd. dt 8. cos t sin d dt Th givn intgral ( t )( t 4) A B Put t, ( )( 4) 4 ( 4) A B( ), A B, 4 A B A, B 3 3 Th givn intgral dt dt tan tan t t c 3 ( t ) 3 ( t 4) 3 6 cos 3 6 [/] tan (cos ) tan c [/]

5 d tan d ( tan ) tan 9. ( sin cos ) (cos sin ) tan cos sin log (cos sin ) I. F. (cos sin ) [] (cos sin ) (cos sin ) (cos sin ) sin c OR W hav d ( ) d f ( ), hnc homognous [/] ( ) d dv v, v [/] v dv v v v log v log log c v log ( v) log c [/] v ( v) c A ( ) A, th gnral solution [/]. LHS a ( b a b b 3b c c a c b 3 c c) a ( b a) 3 a ( b c) a ( c a) a ( c b) as b b c c 3 a b c a c b 3 a b c a b c a b c. As :3: 4 5: :, th lins ar not paralll [/] An point on th first lin is (,3,4 a) An point on th scond lin is (5 4,, ) Lins will b skw, if, apart from bing non paralll, th do not intrsct. Thr must not ist a pair of valus of,, which satisf th thr quations simultanousl: 5 4,3, 4 a Solving th first two quations, w gt, Ths valus will not satisf th third quation if a 3 [/]. Lt E First ball drawn is whit, E First ball drawn is grn, A Scond ball drawn is whit

6 Th rquird probabilit, b Bas Thorm, = P( E) P( A / E) P( E / A) P( E ) P( A / E ) P( E ) P( A / E ) [] 3. Lt X dnot th random variabl. X=,, n =, p = ¼, q = ¾ [/] i Total Marks p i 3 9 C C C 4 6 ip i 6/6 /6 / [+/] p 6/6 4/6 5/8 [/] i i Man = p i i [/] p p [/] Varianc = i i i i 5 3 [/] Sction D 4.,, (3 ) () Rang f = 5, codomain f, hnc, f is not onto and hnc, not invrtibl [] Lt us tak th modifid codomain f = 5, [/] Lt us now chck whthr f is on-on. Lt,, such that f ( ) f ( ) Hnc, f is on-on. Sinc, with th modifid codomain = th Rang f, f is both on-on and onto, hnc invrtibl. 6 From () abov, for an [+/] 5,,(3 ) f : 5,,, f ( ) 3 Lt a, b, c, dq Q. Thn b + ad ma not b qual to d + cb. W find that OR,,3,5,,3,,7,5 Hnc, is not commutativ.

7 Lt a b c d f QQ a bc d f ac b ad acf a b c d f,,,,,,(,, ),,, (, ), ) Hnc, is associativ., QQ is th idntit lmnt for if,,,,,, a b a b a b a b QQi.., a, b a, b a a, b i.., a a, b b a b, (, ) = (, ) satisfis ths quations. Hnc, (, ) is th idntit lmnt for [] c, d QQ is th invrs of a, b QQ if b c, d a, b a, b c, d,, i..,( ac, b ad) ( ca, d cb) (,) c, d. Th a a b invrs of a, bqq, a is (, ) [] a a 5. LHS = abc a b c c a b c a b b c a [/] a b c a b c c abc ( b c a )( b c a ) ( b c a )( b c a ) c a b c a ( b c a) a ( C C C3, C C C3) a b c c abc ( b c a) a abc ( b c a) ( b c a) c a a b c c abc abc ( a) ( c) ca ( b c a) a ( R3 R3 ( R R )) ac bc c c abc ( ba ca a ) a abcca ( ac) ( ca) ca [/] [/] ac bc c c abc a ( ba ca) a ( C C C3, C C C3) abcca ca

8 a b c c a b c c a a ( b c) a ( C C C3, C C C3) abcca abc 3 ab ac b bc ac ( a b c) [/] b OR Givn quation pq q pq q pq pr pq pr pq p q q r q pr q pr pq pr pq pr pq p q q r ( R R R ) q pr pq pr pq pr pq p q q r q pr p q r q r pq p q q r q pr q q q r ( C C C3) q p q q r q pr q rq pq q q pr q r p q pr ( ) ( )( ) (i.., p, q, r q ar in GP) or q r p =(i.., is a root of th quation q r p = 6. Figur [ Marks] Solving 5, w gt ( ) 5,

9 Th rquird ara = th shadd ara = ( 5 ( )) d ( 5 ( )) d [] ( 5 d d ( ( )) d 5 5sin 5 [+ ½ ] 5 (sin sin ) sq units [/] ( )sin( )cos( ) sin cos I d d 4 4 sin cos 4 4 sin ( ) cos ( ) I, I ( )cos sin d 4 4 cos sin ( )cos sin d [/], 4 4 cos sin 4 4 cos sin cos sin tan sc cosc cot I ( )[ d d] ( )[ d d] cos sin cos sin tan cot [] I ( )[ ] 4 t dt p dp substituting tan t, cot p tan sc d dt, cot cosc d dp I ( )[tan t] ( )[tan p] I 6 [/] OR 4 n ( ), ( ) lim ( ), 4 n, h r f f d h f rh nh n n n rh rh f ( rh) rh, f ( rh) h r r n n h nh ( ) h h [] 4 8 nh h h f d nh n, h h ( ) lim [ ] h h h h lim[4 h ] 8 h

10 iˆ ˆj kˆ 8. n b ˆ ˆ ˆ b 5i 7 j k 3 [] Th quation of th plan is r n (5iˆ 7 ˆj kˆ ) (4iˆ 3 ˆj kˆ ), i.., r (5iˆ 7 ˆj kˆ ) Th position vctor of an point on th givn lin is ( ) iˆ ( 3 ) ˆj ( 9 ) kˆ W hav ( )5 (3 )7 ( 9 ) [/] Th position vctor of th rquird point is ˆj 8kˆ [/] 9. Lt kg of Food b mid with kg of Food. Thn to minimiz th cost, C = subjct to th following constraints: 8,,, [] Graph [] At C Marks (, 8) Rs 56 (,4) Rs 38 (.) Rs 5 In th half plan < 38, thr is no point common with th fasibl rgion. Hnc, th minimum cost is Rs