CBSE-XII-2013 EXAMINATION (MATHEMATICS) The value of determinant of skew symmetric matrix of odd order is always equal to zero.

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1 CBSE-XII- EXAMINATION (MATHEMATICS) Cod : 6/ Gnal Instuctions : (i) All qustions a compulso. (ii) Th qustion pap consists of 9 qustions dividd into th sctions A, B and C. Sction A compiss of qustions of on mak ach, Sction B compiss of qustions of fou maks ach and Sction C compiss of 7 qustions of si maks ach. (iii) All qustions in Sction A a to b answd in on wod, on sntnc o as p th act quimnt of th qustions. (iv) Th is no ovall choic, Howv, intnal choic has bn povidd in qustions of fou maks ach and qustions of si maks ach. You hav to attmpt onl on of th altnativs in all such qustions. (v) Us of calculatos is not pmittd. You ma ask fo logaithmic tabls, if quid.. Wit th pincipal valu of tan ( ) cot ( ). tan ( ) π/ cot ( ) π π/6 Hnc π/ (π π/6) π/. Wit th valu of tan sin cos. Q cos π 6 tan ( sin( π/6)) tan π sin tan tan π/. Fo what valu of, is th mati A a skw-smmtic mati? Th valu of dtminant of skw smmtic mati of odd od is alwas qual to zo. ( ) ( ) 6, CP Tow, Road No., IPIA, Kota (Raj.), Ph: 7- Wbsit : info@capointgoup.com /

2 CBSE-XII- EXAMINATION. If mati A and A ka, thn wit th valu of k. Givn A ka k k k k k k. Wit th diffntial quation psnting th famil of cuvs m, wh m is an abita constant. m...() diffntiating with spct to, w gt d/ m diffntial quation of cuv d 6. If A ij is th cofacto of th lmnt a ij of th dtminant 6, thn wit th valu of a A A ( ) + M wh M is th mino of a. A ( ) + 6 A 6 A (8 ) A a A () 7. P and Q a two points with position vctos a b and a + b spctivl. Wit th position vcto of point R which divids th lin sgmnt PQ in th atio : tnall. P.V. of P is a b P Q P.V. of Q is a + b Point R divids sgmnt PQ in atio : tnall. (P.V. of p) (P.V.of Q)() P.V. of R (a b)() (a + b)() a b P.V. of R P.V. of R b a R, CP Tow, Road No., IPIA, Kota (Raj.), Ph: 7- Wbsit : info@capointgoup.com /

3 CBSE-XII- EXAMINATION 8. Find, if fo a unit vcto a, ( a ) ( + a ). Givn a ( a ) ( + a ) a Find th lngth of th ppndicula dawn fom th oigin to th plan + 6z +. p a + b + cz + d a + b + c O(,, ) p p p 9 M + 6z + p 7 p. Th mon to b spnt fo th wlfa of th mplos of a fim is popotional to th at of chang of its total vnu (maginal vnu). If th total vnu (in ups) civd fom th sal of units of a poduct is givn b R() + 6 +, find th maginal vnu, whn, and wit which valu dos th quations indicat. R() dr MR whn MR Consid f : R + [, ) givn b f() +. Show that f is invtibl with th invs f of f givn b f (), wh R + is th st of all non-ngativ al numbs. f : R + [, ) f() + f '() > (on - on) As f() + Rag [, ) co-domain onto So f is invtibl. Futh : + ± As > so O f () f (), CP Tow, Road No., IPIA, Kota (Raj.), Ph: 7- Wbsit : info@capointgoup.com /

4 CBSE-XII- EXAMINATION. Show that : 7 tan sin Solv th following quation : cos(tan ) sin cot Lt sin θ thn sin θ Now tan sin tanθ tanθ If sin θ thn + tan θ 8 tan θ + tan θ tan θ 8 tan θ + 8 ± 6 tan θ 6 8 ± 8 ± 7 tan θ tan θ o 7 tan sin 7 Hnc povd. cos (tan ) LHS. lt tan θ tan θ cos θ + tan θ + Hnc cos (tan ) + R.H.S Lt cot θ cot θ thn sin θ + cot θ Now LHS RHS , CP Tow, Road No., IPIA, Kota (Raj.), Ph: 7- Wbsit : info@capointgoup.com /

5 CBSE-XII- EXAMINATION. Using poptis of dtminants, pov th following : ( + ) ( + ) LHS Now, appl C C + C + C ( + ) (Taking common ( + ) fom C ) Now, appl R R R R R R ( + ) + ( + ) ( + ) ( + ) ( + ) 9 ( + ). Hnc povd.. If d ( + log ), pov that. log log...() Diffntiating w..t. d d log + d log + {fom () d log + + log d log (log + ) + log d ( + log ) log } + log, CP Tow, Road No., IPIA, Kota (Raj.), Ph: 7- Wbsit : info@capointgoup.com /

6 CBSE-XII- EXAMINATION. Diffntiat th following with spct to : sin + + (6) sin + + (6) sin + (6) sin (6) + (6) tan (6) d 6 + (6) d 6 log6 + (6) log6 + k k, if < 6. Find th valu of k, fo which f () is continuous at. +, if < If a cos θ and a sin θ, thn find th valu of d π at θ. 6 + k k, if < f () +, if < function f() is continuous at f() lim f () + + k k lim + k k lim + k + + k + lim ( + k) ( k) [ + k + k ] lim k + k + k k k k k, CP Tow, Road No., IPIA, Kota (Raj.), Ph: 7- Wbsit : info@capointgoup.com 6 /

7 CBSE-XII- EXAMINATION a cos θ and a sin θ a cos d θ sin θ and a sin θ cos θ dθ dθ d d dθ dθ d tan θ d dθ sc θ d sc θ ( a cos θ sin θ) d sc θ coscθ a d a π 7a θ 6 7. Evaluat : cos cos α cos cosα Evaluat : cos cosα cos cosα (cos ) (cos α ) cos cosα (cos cosα)(cos cosα) (cos cosα) (cos + cosα) (sin + cos α) + c + I + + I I ( + ) ( + ) I I + I + +, CP Tow, Road No., IPIA, Kota (Raj.), Ph: 7- Wbsit : info@capointgoup.com 7 /

8 In I lt + + t ( + ) t dt ( + ) tdt t I dt t t CBSE-XII- EXAMINATION Ι ( ) Now in I I log[( + ) + ( + ) ] + Now I I + I I log( ) + c 8. Evaluat : ( + ) I ( + ) I ( + ) Lt t dt dt I t(t + ) I dt t t + I {log t log(t + )} + c t I log + c t + I log + c + 9. Evaluat : π sin + I π + I π I π + + sin sin(π ) sin...(), CP Tow, Road No., IPIA, Kota (Raj.), Ph: 7- Wbsit : info@capointgoup.com 8 /

9 sin I π...() sin + Adding () & () w gt I π + + I [ ] π I π I π sin sin CBSE-XII- EXAMINATION. If a î ĵ + 7 kˆ and b î ĵ + λ kˆ, thn find th valu of λ, so that a + b and a b a ppndicula vctos. a î ĵ + 7 kˆ b î ĵ + λ kˆ a + b 6 î ĵ + (7 + λ) kˆ a b î ĵ + (7 λ) kˆ givn ( a + b ) and ( a b ) a ppndicula (a + b ) ( a b ) {6 î ĵ + (7 + λ) kˆ } { î ĵ + (7 λ) kˆ } 6( ) + ( ) + (7 + λ) (7 λ) + 9 λ λ λ ±. Show that th lins î + ĵ kˆ + λ(î+ ĵ + kˆ ) ; î ĵ + µ( î + ĵ + 6 kˆ ) a intscting. Hnc find thi point of intsction. Find th vcto quation of th plan though th points (,, ) and (,, ) and ppndicula to th plan + z. If th givn lins a intscting thn th shotst distanc btwn th lins is zo and also th hav sam common point î + ĵ kˆ + λ( î + ĵ + kˆ ) z + ( λ) (Lt) Lt P is (λ +, λ +, λ ) Also, î ĵ + µ( î + ĵ + 6 kˆ ) + z ( µ) (Lt) 6 Lt Q is (µ +, µ, 6µ) If lins a intscting thn P and Q will b sam. λ + µ +...() λ + µ...() λ 6µ...(), CP Tow, Road No., IPIA, Kota (Raj.), Ph: 7- Wbsit : info@capointgoup.com 9 /

10 CBSE-XII- EXAMINATION Solv () & () λ + µ λ µ + µ µ µ Put µ...() λ 6( ) λ + λ 8 λ Put µ & λ in () λ + µ + + ( ) + fom λ thn P is (, 6, ) fom µ thn Q is (, 6, ) as P and Q a sam lins a intscting lins and thi point of intsction is (, 6, ). A(,, ) ; B(,, ) givn plan 9 A B AB OB OA AB î + ĵ + kˆ givn plan + z n î ĵ + kˆ Th quid plan is ppndicula to givn plan. Thfo n of quid plan will b ppndicula to n and AB. n ( n AB) n î ĵ + kˆ AB î + ĵ + kˆ n AB 8î 7ĵ kˆ quid plan is n a n ( 8 î 7 ĵ kˆ ) ( î + ĵ kˆ ) ( 8 î 7 ĵ kˆ ) ( 8 î 7 ĵ kˆ ) ( 8 î 7 ĵ kˆ ) z 9, CP Tow, Road No., IPIA, Kota (Raj.), Ph: 7- Wbsit : info@capointgoup.com /

11 CBSE-XII- EXAMINATION. Th pobabilitis of two studnts A and B coming to th school in tim a 7 and 7 spctivl. Assuming that th vnts, A coming in tim and B coming in tim a indpndnt, find th pobabilit of onl on of thm coming to th school in tim. Wit at last on advantag of coming to school in tim. If P(A com in school tim) /7 P(B com in school tim) /7 P(A not com in school tim) /7 P(B not com in school tim) /7 P(onl on of thm coming school in tim) P(A) P( B ) + P( A ) P(B) Find th aa of th gatst ctangl that can b inscibd in an llips + a b. 6 Find th quations of tangnts to th cuv 8, which pass though th point,. Givn llips is + a b b (a cos θ, b sin θ) a a b Aa of ctangl A a cos θ b sin θ A ab sin θ A ma. ab Lt a point (, ) 8 6 ' ' Tangnt ( ) It passing though,. 8 So 8 ±, CP Tow, Road No., IPIA, Kota (Raj.), Ph: 7- Wbsit : info@capointgoup.com /

12 CBSE-XII- EXAMINATION. Find th aa of th gion boundd b th paabola and. 6 O (, ) B C A Rquid aa [aa of OAB Aa of cuv OCBA] A ()() A A 6. Find th paticula solution of th diffntial quation (tan )d ( + ), givn that whn,. 6 (tan )d ( + ) tan d d + IF IF d + tan. IF Q.IFd + c tan + tan tan tan. d + c ( + ) Put tan t tan t dt + c I t II tan t t + (t ) ( ) c tan tan tan tan + c 6. Find th quation of th plan passing though th lin of intsction of th plans (î + ĵ) 6 and (î ĵ kˆ), whos ppndicula distanc fom oigin is unit. 6 Find th vcto quation of th lin passing though th point (,, ) and paalll to th plans (î ĵ kˆ) and (î + ĵ + kˆ ) 6. P is (î + ĵ) 6 P is + 6, CP Tow, Road No., IPIA, Kota (Raj.), Ph: 7- Wbsit : info@capointgoup.com /

13 CBSE-XII- EXAMINATION P is (î ĵ kˆ) P is z Equation of plan passing though intsction of P and P is P + λp ( + 6) + λ( z) ( + λ) + ( λ) + ( λ)z + ( 6) Its distanc fom (,, ) is. ( + λ) ( λ) + ( λ) 6 ( + λ) + ( λ) + ( λ) 6 + 9λ + 6λ λ 6λ + 6λ 6 6λ + 6λ 6 λ λ ± Hnc quid plan is Fo λ, ( + 6) + ( z) + z 6 Fo λ, ( + 6) ( z) + + z 6 P is (î ĵ + kˆ) n î ĵ + kˆ P is (î + ĵ + kˆ) 6 n î + ĵ + kˆ Th lin paalll to plan P & P will b ppndicula to n & n b ( n n ) n î ĵ + kˆ n î ĵ + kˆ n n î + ĵ + kˆ b î + ĵ + kˆ Point is (,, ) a î + ĵ + kˆ quid lin is a + λb (î + ĵ + kˆ) + λ ( î + ĵ + kˆ ) 7. In a hock match, both tams A and B scod sam numb of goals up to th nd of th gam, so to dcid th winn, th f askd both th captains to thow a di altnatl and dcidd that th tam, whos captain gts a si fist, will b dclad th winn. If th captain of tam A was askd to stat, find thi spctiv pobabilitis of winning th match and stat whth th dcision of th f was fai o not. 6 P(6 gt) /6 P(6 not gt) P ( 6gt) /6 P(A win) P(A gt 6) + P ( 6gt) P ( 6gt) P(6 gt) + P ( 6gt) P ( 6gt) P ( 6gt) P ( 6gt) P(6 gt) P(A win) , CP Tow, Road No., IPIA, Kota (Raj.), Ph: 7- Wbsit : info@capointgoup.com /

14 CBSE-XII- EXAMINATION a Q S Similal winning fo B P(B win) P(A win) A manufactu consids that mn and womn woks a quall fficint and so h pas thm at th sam at. H has and 7 units of woks (mal and fmal) and capital spctivl, which h uss to poduc two tps of goods A and B. To poduc on unit of A, woks and units of capital a quid whil woks and unit of capital is quid to poduc on unit of B. If A and B a picd at j and j p unit spctivl, how should h us his soucs to maimis th total vnu? Fom th abov as an LPP and solv gaphicall. Do ou ag with this viw of th manufactu that mn and womn woks a quall fficint and so should b paid at th sam at? 6 if z ma. + tp A tp B wok capital 7 Subjct to, Lt objct of tp A Objct of tp B (, 7) D (, ) C E(, 8) B (, ) A (, ) ( + ) ( + ) 7 pts. Coodinat Z ma. + O (, ) Z A 7 7, Z E (, 8) Z C (, ) Z maimum vnu 6., CP Tow, Road No., IPIA, Kota (Raj.), Ph: 7- Wbsit : info@capointgoup.com /

15 CBSE-XII- EXAMINATION 9. Th managmnt committ of a sidntial colon dcidd to awad som of its mmbs (sa ) fo honst, som (sa ) fo hlping oths and som oth (sa z) fo supvising th woks to kp th colon nat and clan. Th sum of all th awads is. Th tims th sum of awads fo coopation and supvision addd to two tims th numb of awads fo honst is. If th sum of th numb of awads fo honst and supvision is twic th numb of awads fo hlping oths, using mati mthod, find th numb of awads of ach catgo. Apat fom ths valu, naml, honst, coopation and supvision, suggst on mo valu which th managmnt of th colon must includ fo awads. 6 Givn + + z..() ( + z) +..() ( + z)..() + + z + + z + z z AX B A (AX) A (B) I X A B X A.B ( Adj.A) B X A A A ( + 6) ( ) + ( ) A A 9 (Adj. A) 7 9 (Adj A) B 7 9 (Adj.) B ( Adj.A) B X A 9 X X,, z. z, CP Tow, Road No., IPIA, Kota (Raj.), Ph: 7- Wbsit : info@capointgoup.com /