PRE-BOARD MATHEMATICS-1st (Held on 26 December 2017)

Transcription

1 P-B M 7-8 PRE-BOARD MATHEMATICS-st (Held n 6 Decemer 07) ANSWER KEY (FULL SYLLABUS) M.M : 80 Generl Instructins:. The questin pper cmprises f fur sectins, A, B, C & D.. All questins re cmpulsry.. Sectin A Q t 6 re mrk ech 4. Sectin B Q7 t re mrks ech. 5. Sectin C Q t re mrks ech. 6. Sectin D Q t 0 re 4 mrks ech. 7. D prper numering f Answer in cpy & Drw Net Digrms 8. Attempt All sectins & Q's in sequentil rder SECTION A Ans. Nn terminting. (the denmintr hs fctr 7 ls, except fctrs nd 5 i.e., q m 5 m ) r the prime fctriztin f q is nt in the frm m 5 n. c Ans. Infinite c Ans. x + 6x x + x + x x(x+) + (x+) 0 (x + ) (x + ) 0 x, Ans.4 DE.5 cm AD BD AD AB DE [By Thl s therem] BC DE BC cm Ans.5 A sec4a csec (A 0 ) Sec4A sec {90 (A 0 )} sec (0 A) 4A 0 A 5A 0 A Ans.6 V : V : V : : V : V : V πr h : πr : πr h πr h : πr h : πr h[h r] : : : : SECTION B Ans.7 Gretest numer f 6 digit is LCM (4, 5, 6) 60 Required numer Rememer when is divisile y Ans.8 5, d 0 + 0d Let n th term f the given A.P. e 0 mre thn its st term n (n )d (n ) 45 n 4 n 44. Ans.9 Let ssume men 75 75

2 Clss frequency (f i ) Mid vlue (x i ) D x i x i 75 f i d i Σf i 50 Σf i d i 50 75, Σf i d i 50, Σf i 50 fd i i Men x f 50 x 78 Ans.0 Ttl numer f elementry events Vwels 6 (i) Prility f selecting vwel 6 (ii) Prility f selecting cnstnt P (E) 6 Ans. Fr clliner pints Δ 0 re (Δ) 0 Pints re (, + c), (, c + ) nd (c, + ) p c ( + c) (c + ) ( + ) ( + c) 7 r (Δ) { (c + ) + ( + ) + c( + c)} { ( + c) + c (c + ) + ( + )} r (Δ) {c c + c c c c 0 0 r (Δ) 0 Hence the given pints re clliner. Ans. (i) Length f the rc θ 60 πr cm 6 (ii) Are f sectr frmed y the rc θ 60 πr SECTION C cm Ans. Let 5 e rtinl equl t 5 5

3 Squring n th sides ( 5 ) 5 is irrtinl 5 + is rtinl which is cntrdictin. Ans. 4 By lng divisin methd: Reminder x + x + On cmpring, Ans.5 Then given system f equtins: x + y 5 x + ky 5, k (i) c c 5 5 Fr unique slutins Fr n slutin k c c k 6 k k 6 nd k 6 which is nt pssile. Hence, there is n vlue f k fr which the given system f equtins hs n slutin. Ans.6 Let e the first term nd d the cmmn difference f the A.P ( + d) + ( + 6d) 6 + 8d 6 + 4d 4d (i) Agin 7 8 ( + d) ( + 6d) 8c (ii) Putting 4d in Eqn. (): ( 4d + d) ( 4d + 6d) 8 ( d) ( + d) 8

4 9 4d 8 4d 8 9 d 4 r d d ± Cse I: When d. Frm Eqn. (), 4d 4. Sn n [ + (n )d] S6 6 S S 6 76 Cse II : When d Frm Eqn. () 4d S 6 6 { + (6 ) d} S 6 0 Ans.7 Let the side f equilterl tringle Are Rdius r r 7.7 Are f sectrs θ 60 πr θ 60 π θ cm Required re Are f ΔABC re f sectrs cm

5 Ans.8 Steps f cnstructin:. Tke pint O n the pper nd drw circle f rdius OA 5 cm. Tking O s centre drw ngle AOB 0. Cnst Tw Tngent n pint A & B cs 0 cs 70 Ans.9 sec 50 ct cs 0 cs 90 0 sec 50 ct csec 58 ct 58 tn 4tn tn 7 tn45 tn 5 tn 77 + csec 58 ct 58 tn (90 58 ) 4tn tn 7 tn 45 tn (90 7 ) tn (90 ) cs 0 sin 0 sec 50 tn 50 + csec 58 ct 58 4tn tn 7 tn 45 tn 7 ct + (csec 58 ct 58 ) 4 (tn ct ) (tn 7 ct 7 ) tn Ans.0 In single thrw f die ne f the six numers, 6 mrked. Ttl pssile utcmes 6 (,,, 4, 5, 6) (i) Ttl fvurle utcmes (either r 6 s n utcmes) P (E ) P (multiple f ) 6 (ii) Ttl fvurle utcmes (Getting n even numer nd multiple f 6 s n utcme) P (E ) P (n even numer nd multiple f ) 6 (iii) Ttl fvurle utcmes 4 (n even r multiple f,, 4, 6) P (E ) P (n even r multiple f ) 4 6 Ans. x x x x x x x x x x ( + ) ( + ) x ( + + x) ( + ) [x ( + + x) + ] 0 x ( + + x) + 0 ( + ) 0 x + x + x + 0 x (x + ) + (x + ) 0 (x + ) (x + ) 0 x + 0 x r x + 0 x x,

6 Ans. Rte fencing Rs. 4/m Ttl cst f fencing Rs. 580 Length f the fence m Circumference f the field πr 0 r m Are f the field πr m 7 Cst f plughing the field Rs Rs. 95 SECTION D Ans. Let the numer f rws e x. nd numer f plnts in ech rw y Ttl numer f plnts xy. A.T.Q. xy (x ) (y + ) xy xy + x y r x y () Agin xy (x + ) (y ) xy xy x + y 6 r x y 6 () On slving () nd () y 9 Frm eqn () x 9 x x 4 Ttl plnts x y Ttl trees 6 (ii) Wrd prlem relting t liner equtins in tw vriles. (iii) Plnting mre trees will help t sve envirnment. (iv) Yes, such type f ctivities will help the sciety ctive in lrge scle nd crete wreness in them ut their envirnment. Ans.4 Let p x + y where CL x nd A y In Δ ABC nd Δ LOC CAB CLO 90 ΔCAB ~ ΔCLO (By AA similrity) C C cmmn CA AB CL LO p x h x ph () In ΔALO nd ΔACD ALO ACD 90 ΔALO ΔACD [AA Similrly] A A cmmn AL OL AC DC y h ph y p Adding () nd () ph ph x + y p

7 p ph h h h metres Ans.5 OPT 90, Applying Pythgrs in Δ OPT OT OP + PT 5 + PT PT () AP AE (sy) [Length f tngents drwn frm pint t circle re equl] AT ( x) cm OE AB OEA 90 nd AET 90 In Δ AET AT AE + ET ( x) x + ET 44 + x 4x x + ( 5) x x 64 4x 80 x 0 cm AE 0 cm Similrly, BE 0 AB AE + BE cm AB 0 cm Ans.6 sec θ + tn secθ + tn θcsecθ θ csec θ sec θ secθ [Multiplying nd dividing e sec θ] csθ sec θ + tn θ. sinθ Ans.7 + tn θ + tn θ.ctθ (sec θ tn ) + tn θ + tn θ ( + tn θ) / [+ ( )] / tn θ ( ) / In equilterl ΔAOB OA OB AB OA OB AB OA ( 0) + ( 0) OB x + y AB (x ) + (y ) x + y 6x y + OA OB AB OA OB x + y (i)

8 nd x + y x + y 6x y + 6x + y x + y 6 y 6 x (ii) Frm (i) nd (ii) x 6 x + x + (6 x) 6 x 6x 0 x 0, If x 0, y If x 6 y Hence the crdintes f third vertex B re (0, ) r (, ) Ans.8 Let AB h e the height f twer. Let the speed f the cr e 0m/min then CD Distnce trvelled y the cr in min. CD v meter (Distnce speed time) Suppse the cr tkes t minutes t rech the twer AB frm D. then DA vt In Δ DAB tn 45 AB DA h vt h vt (i) In Δ CAB tn 0 AB AC h vt v h vt + v (ii) Frm eqn (i) nd (ii) (vt) vt + v t t + ( ) t t 6 t 6 (.7 + ) 6.7 t 6.9 minutes t 6 min sec [ 0.9 min sec sec] t 6 minutes sec The cr will rech the twer frm D in 6 min sec.

9 Ans.9 Wges (in Rs.) N. f wrkers C.I. f c.f. Mre then 80 Mre then 90 Mre then 00 Mre then 0 Mre then 0 Mre then 0 Mre then (60 5) 7 (5 4) 0 (4 04) 44 (04 60) (60 9) 9 (9 0) (0 0) Mre then n 60 n 60, 80, medin clss 0 0, l 0, c.f 56, f 44, h 0 n l c. f medin h f medin Rs Ans. 0 Let r e the rdius f the hemisphericl wl nd h e the height f the cylinder r 7 h 6 cm The cpcity f the wl vlume f the cylinder + vlume f the hemisphere r r cm πr h r cm cm 498 cm cm