MATHEMATICS. CBSE Board Exam with 8 SAMPLE PAPERS CHAPTER-WISE. Past years questions Practice Exercises Value, Exemplar, HOTS questions

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2 Clss Success Files MATHEMATICS CBSE Bord Em with 8 SAMPLE PAPERS CHAPTER-WISE Pst yers questions Prctice Eercises Vlue, Eemlr, HOTS questions Quick Revison Mteril for Prcticl Ems

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4 C ontents All Indi Bord-5 AI- AI- Delhi Bord-5 DB- DB-. Reltions nd Functions -8. Inverse Trigonometric Functions Mtrices Determinnts Continuity nd Differentibility Aliction of Derivtives Integrls Aliction of Integrls Differentil Equtions 6-3. Vector Algebr Three Dimensionl Geometry Liner Progrmming Probbility

5 SAMPLE PAPERS SAMPLE PAPER - SP- SP-3 SAMPLE PAPER - SP-4 SP-6 SAMPLE PAPER - 3 SP-7 SP-9 SAMPLE PAPER - 4 SP- SP- SAMPLE PAPER - 5 SP-3 SP-5 SAMPLE PAPER - 6 SP-6 SP-8 SAMPLE PAPER - 7 SP-9 SP- SAMPLE PAPER - 8 SP- SP-4 SOLUTION SP-5 SP-9

6 All Indi Bord 5 (i) (ii) (iii) (iv) (v) (vi) (vii) GENERAL INSTRUCTIONS All questions re comulsory. Plese check tht this Question Per contins 6 questions. Mrks for ech question re indicted ginst it. Question to 6 in Section-A re Very Short Answer Tye Questions crrying one mrk ech. Question 7 to 9 in Section-B re Long Answer I Tye Questions crrying 4 mrks ech. Question to 6 in Section-C re Long Answer II Tye Questions crrying 6 mrks ech. Plese write down the seril number of the Question before ttemting it. Section-A r r r b.. Write the vlue of r r. If $ i+ $ j - k$, b i$ + $ j + k$ r nd c 5i$ - 4µ j + 3k µ, r r r + b c then find the vlue of. 3. Write the direction rtios of the following line: 4. If y-4-3, 3 é 3 A, 5 - then write A. ë û 5. Find the differentil eqution reresenting the curve y c + c. 6. Write the integrting fctor of the following differentil eqution: ( + y ) (tn y ) Section-B 7. Using the roerties of determinnts, rove the following: + ( - ) ( + ) ( - ) ( - )( - ) ( + )( - ) 3 ( ) 6-8. If sin t ( + cot t) nd y bcos t ( cos t). show tht 9. Find: b tn. t d æ - - z-z cos ç dz - è z+ z ø. Find the derivtive of the following function f() cos é - + z sin ë + û w.r.t., t. Evlute: sin sin + cos Evlute : ( ) cos. To rise money for n orhnge, students of three schools A, B nd C orgnised n ehibition in their loclity, where they sold er bgs, scr-books nd stel sheets mde by them using recycled er, t the rte of Rs, Rs 5 nd Rs 5 er unit resectively. School A sold 5 er bgs, scr-books nd 34 stel sheets. School B sold er bgs, 5 scr-books nd 8 stel sheets while school C sold 6 er bgs, 8 scr-books nd 36 stel sheets. Using mtri, find the totl mount rised by ech school. By such ehibition, which vlues re generted in the students? 3. Prove tht: 4. If -æ - b -æcos + b tn ç tn cos b ç è + ø è+ bcos ø Solve the following for : -æ - -æ + tn ç + tn ç, <. è - 3ø è + 3ø 4 æ ç A 3, f i n d A 5 A + 6 I. ç - è ø 5. Show tht four oints A, B, C nd D whose osition vectors re 4i$ + 5 µ j + kµ,-µ j - kµ,3i$ + 9µ j + 4kµ nd 4 - $i + µ j + kµ resectively re colnr.

7 AI- 6. Show tht the following two lines re colnr: - + d y z d nd - d + d - b + c y -b z -b -c b-t b b+t Find the cute between the lne 5 4y + 7z 3 nd the y-is. 7. A nd B throw die lterntively till one of them gets number greter thn four nd wins the gme. If A strts the gme, wht is the robbility of B winning? A die is thrown three times. Events A nd B re defined s below: A: 5 on the first nd 6 on the second throw. B: 3 or 4 on the third throw. Find the robbility of B. given tht A hs lre occurred. 8. Evlute: ( cot + tn ) 9. Find: Section-C. Using integrtion, find the re of the region bounded by the lines y +, y.. Find the differentil eqution for ll the stright lines, which re t unit distnce from the origin. Show tht the differentil eqution y + 3y is homogeneous nd solve it.. Find the direction rtios of the norml to the lne, which sses through the oints (,,) nd (,,) nd mkes MATHEMATICS 3. If the function f : R R be defined by f () 3 nd g : R R by g() 3 + 5, then find the vlue of (fog) (). Let A Q Q, where Q is the set of ll rtionl numbers, nd be binry oerttion defined on A by (, b) * (c,d) (c, b + d), for ll (, b) (c,d)îa. Find (i) the identity element in A (ii) the invertible element of A. 4. If the function f() 3 9m + m +. where m > ttins its mimum nd minimum t nd q resectively such tht q. then find the vlue of m. 5. The ostmster of locl ost office wishes to hire etr helers during the Deewli seson, becuse of lrge increse in the volume of mil hndling nd delivery. Becuse of the limited office sce nd the budgetry conditions, the number of temorry helers must not eceed. According to st eerience, mn cn hndle 3 letters nd 8 ckges er dy, on the verge, nd womn cn hndle 4 letters nd 5 ckets er dy. The ostmster believes tht the dily volume of etr mil nd ckges will be no less thn 34 nd 68 resectively. A mn receives ` 5 dy nd womn receives ` dy. How mny men nd women helers should be hired to kee the y-roll t minimum? Formulte n LPP nd solve it grhiclly. 6. 4% students of college reside in hostel nd the remining reside outside. At the end of the yer, 5% of the hostelers got A grde while from outside students, only 3% got A grde in the emintion. At the end of the yer, student of the college ws chosen t rndom nd ws found to hve gotten A grde. Wht is the robbility tht the selected student ws hosteler? ngle 4 with the lne + y 3. Also find the eqution of the lne.

8 All Indi Bord-5 r. Let: i$ µ µ + j+ k 3 r b b µ µ i $ + b j + b3k r r r \ b µ µ µ µ ( µ µ i $ + j + 3k ) é( bi$ + b j + b3k ) $ ( i + j + 3k ) ë û 3 b b b 3 c c c 3 (b 3 b 3 ) (b 3 b 3 ) + (b b ) 3 3 b b 3 3 b + b b 3 b Alternte Method: r r b is vector erenciculr to both r nd b r. r r r r r r \ b ^ nd b ^ b r r r r r r Þ. b b cos9 r r r. ( + b) c µ µ µ µ µ µ ( i$ + j - k + i$ + j + k ) ( 5i$ - 4j + 3k ) µ µ µ ( 3i$ + 3j ) ( 5i$ - 4j + 3k ) ( 4) The eqution of the given line cn be rewritten s: y -4 z Thus, the given line hs direction rtios roortionl to, 3,. é 3 A 5 - ë û 3 \ A ¹ 5 - So, A is non-singulr mtri. Therefore, it is invertible. Now, T é- -5 é- -3 \ dja -3-5 ë û ë û We know A dj A A - é- -3 \ A (-9 ) -5 ë û é ë9 9 û SOLUTIONS AI-3 5. We hve y c + c...(i) Differentiting both side of (i) with resect to, we get c Substituting æ y + ç è ø c in (i), we get æ Þç + -y è ø This is the differentil eqution, which is reresenting the given curve. 6. ( + y ) (tn y ) Þ + y tn - y - Þ + y + tn - y - tn y Þ + y y ( + ) ( + ) \ Integrting fctor (IF) + y e e tn y + 7. ( - ) ( + ) 3 ( - ) ( -)( - ) ( + )( -) + ( -) 3 ( -) ( -) -( - ) ( + ) [Tking out ( ) common from R 3 ] - + ( -) 3 -( -) - [Alying C C C nd C C C 3 ] ( ) ( )[( + ) + 4 ] [Ending long R ] 6 ( )

9 AI-4 8. sin t ( + cos t) Þ sint + sint cost Þ sint + sin4t Differentiting both sides w.r.t. t.we get, cost + cos4t 4 Þ cos + cos4 ( t t) ( t t ) Þ cos + cos - Þ cos + cos - Now, ( t )( t ) y bcost -cost Þ y bcost -bcos t Differentiting both sides w.r.t. t, we get -b sint +b cost sint Þ -b sint+ 4bcos.sin t t Þ bsint cos - We know 9. y ( t ) bsint ( cost -) ( cost + )( cost - ) bsint b sin tcost b ( cost + ) tnt cos t æ - - z - z cos ç - + èz z ø - æ z - cos ç + èz ø Let z tn q - æ tn q- \ y cos ç tn q+ è ø - æ -tn q ç cos - ç + tn q è ø [cos - - -cos - ] - cos ( cos q ) ( ) MATHEMATICS -q -tn - Differentiting both sides w.r.t. z, we hve dz - + z - d æ - z -z - \ cos dz ç + èz z ø + z. - - f cos é + sin + ë û f - é + f ë û Let cos sin nd Now, - é + sin ë û cos f é - z cos cos æ + - ç è ø ë û + - ' Þ f nd f () Tking log on both sides, we get log f () log ' Þ f log + f Þ ' f log f + ' Þ f f log+ ' Þ f log+ Q + f f f ' ' + Q f ' f f At ( + ) ( + ) ( log ) f () - + log+ - +

10 All Indi Bord-5. Let Then, sin I...() sin cos + æ sin ç - è ø I æ sin - è ç ø cos - + é f f ( - ) ë û cos I...() cos sin + Adding () nd (), we get sin cos I + sin cos cos sin + + I I [ ] I I 4 sin sin cos \ + 4 ï cos ( ) í f ì cos ( ), for < 3 cos, for ï - ( ) < ïî cos cos cos \ + - Integrting both integrls on right hnd side, we get cos ( ) 3 é sin cos é sin cos ë û ë û AI-5 éæ éæ 3 æ (- ) ç æ ç ç ç + - ç è ø ç ç ç ç ç ëè ø û ëè ø è øû School 3. Article A B C Per Bgs 5 6 Scrbooks 5 8 Pstel Sheets The number of rticles sold by ech school cn be written in the mtri form s follows: X é ë û The rte of ech rticle cn be written in the mtri form s follows: Y [ 5 5] The mount collected by ech school is given by YX YX é [ ] ë [ ] Thus, the mount collected by schools A, B nd C re ` 85, ` 85 nd ` 97. resectively \ Totl mount collected ` ( ) `,65 The sitution highlights the helful nture of the students. æ - - b tn tn + b ç è ø ì æ b ü ï - - tn ï -ï ç b è + ø ï cos í ý ï æ -b ï ï+ ç tn ï + b î è ø þ û é æ Q tn cos ç ë è+ øû

11 CBSE-Bord Success Files Clss Mthemtics with 8 Smle Pers 3rd Edition 3% OFF Publisher : Dish Publiction ISBN : Author : Dish Publiction Tye the URL : htt:// Get this ebook