SAMPLE QUESTION PAPER MATHEMATICS (4) CLASS XII 6-7 Tim allowd : 3 hours Maimum Marks : Gnral Instructions: (i) All qustions ar compulsor. (ii) This qustion papr contains 9 qustions. (iii) Qustion - 4 in Sction A ar vr short-answr tp qustions carring mark ach. (iv) Qustion 5- in Sction B ar short-answr tp qustions carring marks ach. (v) Qustion 3-3 in Sction C ar long-answr-i tp qustions carring 4 marks ach. (vi) Qustion 4-9 in Sction D ar long-answr-ii tp qustions carring 6 marks ach. Qustions to 4 carr mark ach. Sction-A. Stat th rason wh th Rlation rfliv. R a b a b, : on th st R of ral numbrs is not. If A is a squar matri of ordr 3 and A k A, thn find th valu of k. 3. If a and b ar two nonzro vctors such that a b a b, thn find th angl btwn a and b. 4. If is a binar opration on th st R of ral numbrs dfind b ab a b, thn find th idntit lmnt for th binar opration. Qustions 5 to carr marks ach. 5. Simplif cot for. Sction-B 6. Prov that th diagonal lmnts of a skw smmtric matri ar all zros. 7. If 5 6 6 6 tan,, d 3 thn prov that. d 4 9 8. If changs from 4 to 4., thn find th approimat chang in log. 9. Find d.. Obtain th diffrntial quation of th famil of circls passing through th points a,and a,.
. If a b 6, a b 4 and a, thn find b.. If P( A), P( B), P( A B), thn find P( A / B ). 5 3 5 Sction-C Qustions 3 to 3 carr 4 marks ach. 3. If A, thn using A, solv th following sstm of quations:,. 4. Discuss th diffrntiabilit of th function, f( ) at 3 6,. For what valu of k is th following function continuous at? 6 3 sin cos, 6 f( ) 6 k, 6 5. If asin pt, bcos pt, thn show that 6. Find th quation of th normal to th curv d a b. d, which passs through th point (, ). Sparat th intrval, into subintrvals in which th function 4 4 f ( ) sin cos is strictl incrasing or strictl dcrasing. 7. A magazin sllr has 5 subscribrs and collcts annual subscription chargs of Rs.3 pr subscribr. Sh proposs to incras th annual subscription chargs and it is blivd that for vr incras of R, on subscribr will discontinu. What incras will bring maimum incom to hr? Mak appropriat assumptions in ordr to appl drivativs to rach th solution. Writ on important rol of magazins in our livs. 8. Find sin cos cos 4 d.
tan d d d. 9. Find th gnral solution of th diffrntial quation Solv th following diffrntial quation: d d.. Prov that a {( b c) ( a b 3 c)} a b c.. Find th valus of a so that th following lins ar skw: z a 4, z. 3 4 5. A bag contains 4 grn and 6 whit balls. Two balls ar drawn on b on without rplacmnt. If th scond ball drawn is whit, what is th probabilit that th first ball drawn is also whit? 3. Two cards ar drawn succssivl with rplacmnt from a wll shuffld pack of 5 cards. Find th probabilit distribution of th numbr of diamond cards drawn. Also, find th man and th varianc of th distribution. Sction-D Qustions 4 to 9 carr 6 marks ach. 4. Lt f :, Rb a function dfind b f ( ) 9 6 5. Prov that f is not invrtibl. Modif, onl th codomain of f to mak f invrtibl and thn find its invrs. Lt b a binar opration dfind on Q Q b a b c d ac b ad,,,, whr Q is th st of rational numbrs. Dtrmin, whthr is commutativ and associativ. Find th idntit lmnt for and th invrtibl lmnts ofq Q. 5. Using proprtis of dtrminants, prov that a b c c c b c 3 a a a b c. a b b c a b
p q p q If p, q and q r q r, p q q r thn, using proprtis of dtrminants, prov that at last on of th following statmnts is tru: (a) p, q, r ar in G. P., (b) is a root of th quation p q r. 6. Using intgration, find th ara of th rgion boundd b th curvs 5 and. 7. Evaluat th following: sin cos. 4 4 d sin cos Evaluat 4 ( ) d as th limit of a sum. 8. Find th quation of th plan through th point (4, -3, ) and prpndicular to th lin of intrsction of th plans + z 3 = and -3z =. Find th point of intrsction of th lin r iˆ ˆj kˆ ( iˆ 3 ˆj 9 kˆ) and th plan obtaind abov. 9. In a mid-da mal programm, an NGO wants to provid vitamin rich dit to th studnts of an MCD school.th ditician of th NGO wishs to mi two tps of food in such a wa that vitamin contnts of th mitur contains at last 8 units of vitamin A and units of vitamin C. Food contains units pr kg of vitamin A and unit pr kg of vitamin C. Food contains unit pr Kg of vitamin A and units pr kg of vitamin C. It costs Rs 5 pr kg to purchas Food and Rs 7 pr kg to purchas Food. Formulat th problm as LPP and solv it graphicall for th minimum cost of such a mitur? --------
MATHEMATICS (4) CLASS XII 6-7 Marking Schm Sction A.. 3, R. Hnc, R is not rfliv. 3 k 8 3. sin cos 45 4. 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 d 3 d 9 4 8. Lt log, 4,. [/] d [/] d d d.5 d 4 4 9. Givn intgral d c as d ( ) d
b a b or, b a...(). d d d b b d...() [/] d 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 3 5 4 5 3 4, 5 5 [/] [/]
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 6 6 6 6 f( ) k 6 For th continuit of f( ) at 6 [] [/], f ( ) lim f ( ) k [/] 6 6 d d 5. ap cos pt, bpsin pt dt dt d 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 d (, ) to th curv. d 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
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, 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 [/]
d tan d ( tan ) d d tan 9. d ( sin cos ) (cos sin ) d tan cos sin log (cos sin ) I. F. (cos sin ) [] (cos sin ) (cos sin ) d (cos sin ) sin c W hav d ( ) d d f ( ), hnc homognous [/] d ( ) d dv v, v d d [/] v d 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
Th rquird probabilit, b Bas Thorm, = P( E) P( A / E) P( E / A) P( E ) P( A / E ) P( E ) P( A / E ) 6 5 9 5 6 5 4 6 9 9 9 3. Lt X dnot th random variabl. X=,, n =, p = ¼, q = ¾ [/] i Total Marks [] p i 3 9 C 4 6 3 6 4 4 6 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 [/] 8 4 8 Sction D 4.,, 9 6 5 (3 ) 6 5...() 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 ( ) 3 6 3 6 3 3 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 ) 6 3 6 f : 5,,, f ( ) 3 Lt a, b, c, dq Q. Thn b + ad ma not b qual to d + cb. W find that,,3,5,,3,,7,5 Hnc, is not commutativ.
Lt a b c d f QQ a bc d f ac b ad acf a b c d f,,,,,,(,, ),,, (, ), ) Hnc, is associativ., Q Q 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 Q Q is th invrs of a, b Q Q 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
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 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,
Th rquird ara = th shadd ara = ( 5 ( )) d ( 5 ( )) d [] ( 5 d d ( ( )) d 5 5sin 5 [+ ½ ] 5 (sin sin ) sq units [/] 5 5 7. ( )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 [] 4 4 4 4 4 4 4 4 I ( )[ ] 4 t dt p dp substituting tan t, cot p tan sc d dt, cot cosc d dp I ( )[tan t] ( )[tan p] 4 4 8 I 6 [/] 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 8 8 4 h h h lim[4 h ] 8 h
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 = 5 + 7 subjct to th following constraints: 8,,, [] Graph [] At C Marks (, 8) Rs 56 (,4) Rs 38 (.) Rs 5 In th half plan 5 + 7 < 38, thr is no point common with th fasibl rgion. Hnc, th minimum cost is Rs 38. ------