12 TH STD - MATHEMATICS

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1 e e e e e e e e e e a TH STD - MATHEMATICS HALF YEARLY EXAMINATION 7 b) d) ANSWER KEY ( ) ONE MARK QUESTIONS : X k n ( a d j I ) d) no soluion 5 d) d) has onl rivial soluion onl if rank of he oeffiien mari is equal o he number of unknowns 6 d) 7 b) 8 b) 9 9 ) skew d) s in d) ollinear b. n ) i sin ( ) d) (,) * ) a rg ( ) a rg ( ) a rg ( ) (m.a) 5 d) 9 o 6 a) 7 ) 8 b) 9 b) a), 8 a, m / m in 5 8 m / se d) everwhere onave downward a) verial angen b) d) u, u, u are oninuous 5 a) 6 d) 9 7 a) 8 b) f ( a ) f ( ) 9 b) lo g ) ) ), '' ' ( ) a a 6 d) C {} 7 a) all 8 d) 8 9 b) 5 ) 8 SIX MARK QUESTIONS : X 6 6 A A is inverible. B B is inverible. AB AB AB is inverible. adj A ; A adj B ; B adj(ab) ; AB AB adj(ab) () B A () From () and (), AB B A The orresponding mari equaion is Augmened mari is A, B ~ R R R 6 7 ~ R R R 5 I is in ehelon form and i has hree non-ero rows. ρ A, B and ρ A. ρ A, B ρ A Caresian form : λ(sa) An poin : λ, λ, λ This poin lie on he plane 7 λ λ λ 7 λ λ λ 9 7 λ The required poin is (6,, 5) a a a, b b is o boh a and b is o a b λ(a b) λ a b a, b, a, b, are oplanar. a The angle beween he line and plane is sinθ b n b n 6 6 b ; n 7 b n θ sin 6 sin 5 Le r osθ isinθ r osθ isinθ. r, arg θ ; r, arg θ a ) ( p) q [ o s ( ) i s in ( )] b) 5 b) a rg a rg a rg hp://

2 e e e e e e e e e e a 6 AB i 97 BC 9 7i 8 CA i 6 6 AB BC CA 97 Hene he given poins represen a righ angled riangle on he Argand plane. 7 lim sin lim as, lim sin os form Coninuous on [,] No differenial on (,) Rolls heorem anno applied o he given funion. 8 ' 9 f ( ) se o s e f ' ', f 6 f is no monooni on, Le f d f d () d Le 6, d., f 6 6 d (6)...8 f d 6. f (wo deimal) 5 C. F ( A B ) e 5 P. I 7, P. I, P. I General Soluion : a C. F P. I ( A B ) e 6 p q p q ~(p q) T T T F T F F T F T F T F F F T p q ~p ~q (~p)(~q) T T F F F T F F T T F T T F T F F T T T The las olumns are idenial. ~(p q) (~p)(~q) 5 STATEMENT:Le G be a group. a, b, G, Then a*b a* b (lef anellaion law) b*a *a b (Righ anellaion law) Proof : a*b a a a b a (a ) (a a) b (a a) e b e b b a a b a a a a b (a a ) (a a ) b e e b a 5 P X <. ; P X > 9. P < X < 9 [ P X < P X > 9 ]...8 Ou of 8 Sudens, number of sudens sored beween and 9 is sudens p 9/, q / Mean 5, variane 5 S.D 5 p /, n P( X ) (a) Equaion of angen a an poin on is Inerep on he aes are a, b 55 (b) Equaion of normal a on is Inerep on aes are p ; q ap bq sin ( sin ) d 6 sin d sin a d TEN MARK QUESTIONS : X 56 SOLUTION: Le, and be he number of oins in eah aegor Re., Rs. and Rs.5 respeivel. 5 Pu k, k R. k k ; B Cramer s rule, k ; 5k, k R 7 k 7 k The Soluion se (,, ) ( k, 7 k, k), k 7 possible soluions are (,,), (5,,5), (8,6,6) and (,,7) a a hp://

3 e e e e e e e e e e a 57 a b i j k d i j k a b d 5i j k () a b d a b d i j k a b a b d i j k From () and (), a b d a b d a b d 5 i j k () a b d a b d 58 VECTOR FORM: a i j k, u i j k and v i j k The equaion of he plane is r a su v r i j k s i j k (i j k) CARTESIAN FORM:,,,,, l, m, n,, and l, m, n,, The equaion of he plane is l m n l m n The roos of he equaion are p±iq Le α p iq and β p iq ; α β iq q o θ p α q o θ i q os θ sin θ i q sin θ os θ isin θ ( α) n qn sin n (os θ isin θ)n θ ( α) n qn sin n (os nθ isin nθ) θ Similarl, ( β) n q n sin n θ ( α) n ( β) n a (os nθ isin nθ) q n (isin nθ) α β (iq ) sin n θ ( α) n ( β) n qn sinnθ α β sin n θ 6 Consider he parabola is open downward. Equaion of he parabola is a. I passes hrough he poin B(, -.5) a(.5) a 9 9 The poin, 7.5 lies on 9 a The waer srikes he ground m beond his verial line. a 6 Le F be he posiion of he Sun. Given e.6 and a 6 million miles. The shores disane F A CA CF a ae a e million miles The longes disane F A CF CA a ae a e million miles 6 Cener of ellipse enre of hperbola Lengh of LR 9/ b 9 a ie., a e 5 b Conjugae ais Conjugae ais parallel o Y ais Transverse ais is X ais ( ) ( ) 6 9 a b a ( ) ( ) aos θ, asin θ d dθ aos d θsinθ ; dθ asin θosθ; d asin θosθ sinθ d aos θsinθ osθ Slope of he angen a θ is sinθ osθ Slope of he normal a θ is osθ sinθ Equaion of he normal a θ is asin θ osθ ( sinθ aos θ) sinθ asin θ osθ aos θ osθ sinθ a os θ sin θ osθ sin a os θ sin θ os θ sin θ osθ sinθ aosθ 6 Le and be he lengh and breadh of he prined area. 8 8 Dimensions of he poser area are 8 and. Area of he poser A 8 A A () 8 8 A 6 8 For ma/min, A ±6 a 6 [ anno be ve] When 6, A () > When 6, A is minimum When 6,. Dimensions of he poser are m and 6 m. hp://

4 e e e e e e e e e e a 65 R.H.S. is no a homogeneous funion. Define f anu f is a homogeneous funion of degree. B Euler s heorem, anu f f f anu anu se u u se u u sinu osu u u sinu 66 a sin ; a( os) Pu os os, d d a( os); d d d d Surfae area d aos b a d asin d d d d d a os aos d 6a os udu Pu u, d du 6a os d a os udu a b reduion formula. 6a square unis. 67 The poin of inerseion of 9 and 6 are (,) and (, ). Required area an be solved abou -ais. Required area ( )d means from 9 means from 6 R.A. 6 d 6 8 square unis. 68 Le T be he emperaure of he offee a an ime. T S B Newon s law of ooling, 5 dt T 5 sine S 5 C d dt k(t 5) T 5? 5 d e k When, T 5 e 85 T 5 85e k a a When 5, T e 5k 5 85e 5k e 5k 5 85 When, T 5 85 e 5k T 5 85e k T 8.8 The required emperaure afer a furher inerval of 5 minues is P X <. ; P X > 9. P < X < 9 [ P X < P X > 9 ]...8 Ou of 8 Sudens, number of sudens sored beween and 9 is sudens WITH REPLACEMENT: X an ake he values as,,,. P(suess)P S 7 ; P(Failure) P F 7 P X 6 P X 8 P X P X 7 The required probabili disribuion is X P X p i, p i WITHOUT REPLACEMENT: P X C C 7C 5 5 P X C C 6 7C P X C C 7C 5 5 P X C C 7C (a) Le he ubi polnomial be f. Sine i aains maimum a and a minimum a. d d a a a and. d k d d k d d k ( ) d k When - and k ----() When and k -----() Solving () and () we ge, k and The ubi polnomial is. hp:// a

5 e e e e a 7 (b) Le M C / Closure aiom: Le, M,, M M Closure aiom is rue. Assoiaive aiom: Comple mulipliaion is alwas assoiaive. Assoiaive aiom is rue. (iii) Ideni aiom: Le M and suh ha M. The ideni elemen M. The ideni aiom is rue. (iv) Inverse aiom: Le M, were. Also M suh ha a The inverse elemen of is M The inverse aiom is rue. M, is a group. a a a a e PREPARED BY Tr. S SELVARAJ MS., BEd., MA., COA., VICE PRINCIPAL MGM MATRIC. HR. SEC. SCHOOL, KRISHNAGIRI DISTRICT. e e e e e hp://