Jr. INTER MODEL PAPER MATHS 30-Jan-16.pdf

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14 Ñ§ýlÅ

MýSÆý‡*²Ë$ l Ô¶æ°ÐéÆý‡… l f¯]lÐ]lÇ l 30 l 2016

Junior Inter Public Exam Model Papers MATHEMATICS, Paper-I (A) (English Version) Time: 3 Hours Max. Marks: 75 Section - A I. Very Short Answer Questions. Answer all Questions. Each Question carries "Two" marks. 10 × 2 = 20 M 1. Find the Domain of function 2. f: R → R and f (x ) =

x 2 − 3x

5. If 4 i +

prove that f(Tanθ) = Cos2θ 3. A certain book shop has 10 dozen chemistry books, 8 dozen physics books, 10 dozen economics books. Their selling prices are Rs. 80, Rs. 60 and Rs. 40 each respectively. Find the total amount that book shop will receive by selling all the books using matrix algebra. 1 4 0   4. If  −1 0 7  is a skew symmetric  − x −7 0

6. Let a = i + 2 j + 3k and b = 3i + j. Find the unit vector in the direction of a + b . 7. Let ar = ri + rj + kr and br = 2ir + 3jr + rk Find the projection vector of a on b and its magnitude. 8. Find the value of Sin 2 82 o

(Telugu Version) Time: 3 Hours Max. Marks: 75 Section - A I.

A† çÜÓËµ çÜÐ]l*«§é¯]l {ç³Ô¶æ²Ë$. A°² {ç³Ô¶æ²ËMýS$ çÜÐ]l*«§é¯]l… Æ>Ä¶æ$…yìl. {ç³† {ç³Ô¶æ²MýS$ Æð‡…yýl$ Ð]l*Æý‡$PË$. 10 × 2 = 20 M

1.

x 2 − 3x

{ç³§ólÔ¶æ… MýS¯]l$Vö¯]l…yìl.

2. f: R → R, f (x ) =

x = l og(3 + 10)

2 2 1

3.

4.

7. 8.

Ë$ çÜÐ]l*…™èlÆý‡…V> E…sôæ p ÑË$Ð]l G…™èl? a = i + 2 j + 3k, b = 3i + j AÆ‡¬™ól a + b ¨Ô¶æÌZ Ä¶r æÊ°sŒ r ræ çÜr¨Ô¶ræ¯]l$ rMýS¯]l$Vö¯] r lr…yìl. a = i + j + k, b = 2i + 3j + k AÆ‡¬™ól a òO ³ b Ñ„óSç³MýS… G…™èl? 1 1 Sin 82 − Sin 22 ÑË$Ð]l G…™èl? 2 2 2

o

2

o

9. f (x) = Sin4x + Cos4x, x ∈ R BÐ]lÆý‡¢¯]l… G…™èl? 10. Sinhx = 3 AÆ‡¬™ól x = l og(3 + 10) A° °Æý‡*-

AÆ‡¬™ól

°Æý‡*í³…^èl…yìl. JMýS ç³#çÜ¢MýS §ýl$M>×æ…ÌZ ç³¨ yýlf¯]lÏ Æý‡ÝëÄ¶æ$¯]l Ô>çÜˆ…, G°Ñ$¨ yýlf¯]lÏ ¿o†MýS Ô>çÜˆ…, ³ç ¨ yýlf¯]lÏ BÇ¦MýS Ô>çÜˆ ç³#çÜ¢M>Ë$ E¯é²Æ‡¬. Ðésìæ AÐ]l$ÃMýS… «§ýlÆý‡Ë$ Æý‡*. 80, Æý‡*. 60, Æý‡*. 40. D ç³#çÜ¢M>Ë¯]l$ AÑ$Ã™ól §ýl$M>×æ §éÆý‡$ËMýS$ G…™èl Ððl¬™èl¢… Ð]lçÜ$¢…§ø Ð]l*{†M> ç³§ýl®†ÌZ MýS¯]l$Vö¯]l…yìl. ÝûçÙtÐ]l Ð]l*{†MýS AÆ‡¬™ól x ÑË$Ð]l?

{ç³ÐólÔ>Ë$ Hï³ G…òÜsŒ fÐ]lçßæÆŠ‡ÌêÌŒæ ¯ðl{çßæ* sñæMýS²ÌêhMýSÌŒæ Ä¶æÊ°Ð]lÇÞsîæ,

M>MìS¯éyýl "Hï³ G…òÜsŒæ' {ç³MýSr¯]l Ñyýl$§ýlË ^ólíÜ…¨. ©° §éÓÆ> Æ>çÙ‰ÐéÅç³¢…V> E¯]l² ÑÑ«§ýl {ç³¿¶æ$™èlÓ, {ò³•ÐólsŒæ MýSâêÔ>ËÌZÏ C…f±Ç…VŠS, A{WMýSËaÆý‡ÌŒæ, Ððl$yìlMýSÌŒæMýS$ çÜ…º…«¨…_¯]l »êÅ_ËÆŠ‡ yìl{XÌZÏ {ç³ÐólÔ>Ë$ MýSÍµÝë¢Æý‡$. Ñ¿êV>Ë$: C…f±Ç…VŠS, ºÄñæ*sñæM>²Ëi, ½sñæMŠS (yðlÆ‡¬È sñæM>²Ëi), ½sñæMŠS (A{W C…f±Ç…VŠS), ½&¸ëÆý‡ÃïÜ, ½sñæMŠS (çœ#yŠl OòÜ¯ŒlÞ A…yŠl sñæM>²Ëi&G‹œG‹Üsîæ), ½GïÜÞ (MýSÐ]l$ÇÛÄ¶æ$ÌŒæ A{W. A…yŠl ¼h¯ðl‹Ü Ð]l*Æð‡Psìæ…VŠS&ïÜH A…yŠl ½G…), ½GïÜÞ (A{WMýSËaÆŠ‡), ½GïÜÞ (àÇtMýSËaÆŠ‡),

çÜÓËµ çÜÐ]l*«§é¯]l {ç³Ô¶æ²Ë$. HOÐðl¯é 5 {ç³Ô¶æ²ËMýS$ çÜÐ]l*«§é¯]l… Æ>Ä¶æ$…yìl. {ç³† {ç³Ô¶æ²MýS$ 4 Ð]l*Æý‡$PË$. 5 × 4 = 20 M

1 2 2

11. A = 2 1 2 2 2 1

AÆ‡¬™ól

in intercept form. 13. Find the volume of the tetrahedron whose vertices are (1, 2, 1), (3, 2, 5), (2, −1, 0) and (−1, 0, 1). 14. If A+B = π/4, then prove that (1+Tan A) (1+Tan B) = 2. 15. If Tan(π Cosx) = Cot(πSinx)

n(4n 2 + 6n − 1) = 3 20. Solve by matrix inverse method 5x – 6y + 4z = 15 7x + 4y – 3z = 19 2x + y + 6z = 46 21. Show that

16. Prove that 3  8  77  Sin   + Sin −1   = Sin −1   5  17   85  −1

A2–4A–5I = 0

A° °Æý‡*í³…^èl…yìl. 12. çÜ¨Ô> ç³§ýl®†ÌZ çÜÆý‡â¶æÆó‡Q A…™èlÆý‡ Q…yýl Æý‡*ç³… ax + by = 1 A° °Æý‡*í³…^èl…yìl. ½ÒGïÜÞ A…yŠl Hòßæ^Œ , ½G‹œGïÜÞ, G…½½G‹Ü, ½yîlG‹Ü, ½HGÐðl$Ã‹Ü, ½òßæ^ŒlGÐðl$Ã‹Ü, ½G¯ŒlOÐðlG‹Ü, ¸ëÆ>Ã&yìl. AÆý‡á™èl: C…rÈÃyìlÄñæ$sŒæ (O»ñæï³ïÜ, G…ï³ïÜ) E¡¢Æý‡~™èl. B¯ŒlOÌñæ¯Œl Çh[õÜtçÙ¯Œl: œí {ºÐ]lÇ 3 & Ð]l*Ça 21 ç³È„ýS ™ól¨: H{í³ÌŒæ 29 Ððl»ŒæOòÜsŒæ: www.apeamcet.org

¯]lÌêÞÆŠ‡ÌZ G…½H

¯]lÌêÞÆŠ Ä¶æÊ°Ð]lÇÞsîæ B‹œ Ìê, Oòßæ§ýlÆ>»ê§ŠlMýS$

^ðl…¨¯]l òÜ…rÆŠ‡ çœÆŠ‡ Ðól$¯ólgŒæÐðl$…sŒæ çÜtyîl‹Ü "G…½H' {´ù{V>ÐŒl$ÌZ {ç³ÐólÔ>ËMýS$ §ýlÆý‡RêçÜ$¢Ë$ MøÆý‡$™ø…¨. ïÜrÏ çÜ…QÅ: 60 Ñ¿êV>Ë$: Ð]l*Æð‡Psìæ…VŠS Ðól$¯ólgŒæÐðl$…sŒæ, Oòœ¯é°ÞÄ¶æ$ÌŒæ

b2

b c a =

c2

2ca − b2

a2

c a

b2

a2

2ab − c2

b

= (a3 + b3 + c3 – 3abc)2 22. Find the shortest distance between the skew lines r = (6 i + 2 j + 2k) + t( i − 2 j + 2k) and

23. If A + B + C = π prove that

III. Long Answer Questions. Answer any 'Five' Questions. Each Question carries 'Seven' marks. 5 × 7 = 35 M 18. If f : A→B is bijective function then prove that fof–1 = IA and f–1of = IB

HÆý‡µyól sñæ{sêòßæ{yýl¯Œæ çœ$¯]l ç³ÇÐ]l*×æ… G…™èl? 14. A+B = π/4 AÆ‡¬™ól (1+Tan A) (1+Tan B) = 2 A° °Æý‡*í³…^èl…yìl. 15. Tan(π Cosx) = Cot(πSinx) AÆ‡¬™ól π 1  C os  x −  = ± A° °Æý‡*í³…^èl…yìl. 4 2 2  3

8

A B C + C os + C os 2 2 2 π−A π− B π− C C os C os = 4C os 4 4 4

C os

24. Show that

çÜÒ$MýSÆý‡×êË¯]l$ Ð]l*{†MýS ÑÌZÐ]l$ ç³§ýl®†ÌZ Ýë«¨…^èl…yìl. 2

2bc − a 2

c2

b2

21. b c a =

c2

2ca − b2

a2

c a

b2

a2

2ab − c2

a

b c b

= (a3 + b3 + c3 – 3abc)2 A°

a 2 + b2 + c2 17. CotA + CotB + CotC = 4Δ

°Æý‡*í³…^èl…yìl.

22. r = (6 i + 2 j + 2k) + t( i − 2 j + 2k), r = (−4 i − k) + s(3 i − 2 j − 2k)

A° °Æý‡*í³…^èl…yìl.

§ýl*Æý‡… MýS¯]l$Vö¯]l…yìl. 23. A + B + C = π AÆ‡¬™ól

Ë Ð]l$«§lý Å MýS°çÙt

A B C + C os + C os 2 2 2 π−A π− B π− C C os C os = 4C os 4 4 4 C os

Section-C III. ©Æý‡ƒ

çÜÐ]l*«§é¯]l {ç³Ô¶æ²Ë$. HOÐðl¯é 5 {ç³Ô¶æ²ËMýS$ çÜÐ]l*«§é¯éË$ Æ>Ä¶æ$…yìl. {ç³† {ç³Ô¶æ²MýS$ 7 Ð]l*Æý‡$PË$.

5 × 7 = 35 M 18. f : A→B ¨ÓVýS$×æ {ç³Ðól$Ä¶æ$… AÆ‡¬™ól fof–1 = IA f–1of = IB A° ^èl*ç³…yìl. 19. VýS×ìæ™é¯]l$VýSÐ]l$¯]l íÜ§é®…™èl… {ç³M>Æý‡… 1.3 + 3.5 + 5.7 + ... n ç³§éË$ n(4n + 6n − 1) 3

r1 r2 r3 1 1 + + = − bc ac ab r 2R

2x + y + 6z = 46

77

  −1   −1  −1  16. Sin   + Sin   = Sin   5  17   85 

=

c2

b c

r = (−4 i − k) + s(3 i − 2 j − 2k)

Section - C

2

2

2bc − a 2

a

π 1  Prove that C os  x −  = ± 4 2 2 

A° °Æý‡*í³…^èl…yìl.

Section-B II.

x y + = 1 is the equation of a straight line a b

13. (1, 2, 1), (3, 2, 5), (2, -1, 0), (-1, 0, 1)Ë™ø

2p j + pk, i + 2 j + 3k 3

í³…^èl…yìl.

f(Tanθ) = Cos2θ A°

1 4 0  −1 0 7     − x −7 0

6.

4i +

19. Prove by Mathematical Induction 1.3 + 3.5 + 5.7 + ... up to n terms

a 2 + b2 + c2 = 4Δ

II. Short Answer Questions. Answer any 'Five' Questions. Each Question carries 'Four' marks. 5 × 4 = 20 M 1 2 2 11. If A = 2 1 2 Then prove that

5.

A2 – 4A – 5I = 0. 12. Prove by vector method that

17. Prove that Cot A + Cot B + Cot C

Section-B

2

1− x 1 + x2

1 1 − Sin 2 22o 2 2

9. Find the period of f (x) = Sin4x + Cos4x for any x ∈ R. 10. If Sinh x = 3 then prove that

matrix, then find x.

MATHEMATICS Paper-I (A)

find p.

i + 2 j + 3k,

2

1− x then 1 + x2

2p j + pk is parallel to the vector 3

A° ^èl*ç³…yìl.

20. 5x – 6y + 4z = 15, 7x + 4y – 3z = 19,

çÜÈÓòÜ‹Ü A…yŠl M>Åí³rÌŒæ Ð]l*Æð‡PsŒæÞ, çßæ*ÅÐ]l$¯Œl ÈÝùÆŠ‡Þ Ðól$¯ólgŒæÐðl$…sŒæ, C¯ø²ÐólçÙ¯Œl A…yŠl çÜõÜt¯] ¼Ísîæ Ðól$¯ólgŒæÐðl$…sŒæ, M>ÆöµÆó‡sŒæ VýSÐ]lÆð‡²¯ŒlÞ, MøÆŠ‡t Ðól$¯ólgŒæÐðl$…sŒæ, ¼h¯ðl‹Ü Æð‡VýS$ÅÌôæçÙ¯ŒlÞ. AÆý‡á™èl: MýS±çÜ… 50 Ô>™èl… Ð]l*Æý‡$PË™ø HO§ðl¯é »êÅ_ËÆŠ‡ yìl{X. M>ÅsŒæ&2015/gêsŒæ&2016/ïÜÐ]l*ÅsŒæ 2015/ iÐ]l*ÅsŒæ/ iBÆŠ‡D ÝùPÆŠ‡ E…yéÍ. D ÝùPÆŠ‡ Ìôæ°ÐéÆý‡$ ¯]lÌêÞÆŠ‡ Ðól$¯ólgŒæÐðl$…sŒæ G…{sñæ¯ŒlÞ sñæ‹Üt (G¯Œl&Ððl$sŒæ)&2016MýS$ §ýlÆý‡RêçÜ$¢ ^ólçÜ$MøÐéÍ. G…í³MýS: {VýS*‹³ yìlçÜPçÙ¯Œl, ç³Æý‡Þ¯]lÌŒæ C…rÆý‡*ÓÅ, AyìlÃçÙ¯Œl sñæ‹Üt ÝùPÆŠ‡ §éÓÆ>. B¯ŒlOÌñæ¯Œl Çh[õÜtçÙ¯ŒlMýS$ _Ð]lÇ ™ól¨: íœ{ºÐ]lÇ 29 G¯Œl&Ððl$sŒæ ™ó ¨: Ð]l*Ça 27 Ððl»ŒæOòÜsŒæ: www.cms.nalsar.ac.in

A° °Æý‡*í³…^èl…yìl. 24.

r1 r2 r3 1 1 + + = − bc ac ab r 2R

A° °Æý‡*í³…^èl…yìl. Prepared by M.N. Rao

Subject Expert, Sri Chaitanya Educational institutions

G¯ŒlïÜòßæ^ŒlG… gôæDD&2016

¯ólçÙ¯]lÌŒæ Mú°ÞÌŒæ çœÆŠ‡ çßZrÌŒæ Ðól$¯ólgŒæÐðl$…sŒæ A…yŠl

MóSrÇ…VŠS sñæM>²Ëi, C…¨Æ> V>…«© ¯ólçÙ¯]lÌŒæ Kò³¯Œl Ä¶æÊ°Ð]lÇÞsîæ (CVø²) çÜ…Ä¶æ¬MýS¢…V> "G¯ŒlïÜòßæ^ŒlG… gôæDD&2016' {ç³MýSr¯]l Ñyýl$§ýlË ^ólÔ>Æ‡¬. ©° §éÓÆ> §ólÔ¶æ…ÌZ° ÑÑ«§ýl {ç³¿¶æ$™èlÓ, {ò³•ÐólsŒ MýSâêÔ>ËÌZÏ "àíÜµsêÍsîæ A…yŠl çßZrÌŒæ AyìlÃ°[õÜtçÙ¯Œl' »êÅ_ËÆŠ‡ yìl{X {´ù{V>Ð]l¬ÌZ {ç³ÐólÔ>Ë$ MýSÍµÝë¢Æý‡$. AÆý‡á™èl: C…rÆŠ‡/ ™èl™èlÞÐ]l*¯]l E¡¢Æý‡~™èl. Ð]lÄ¶æ$çÜ$: þOÌñæ 1 ¯ésìæMìS 22 Hâ¶æÏMýS$ Ñ$…^èlMýS*yýl§ýl$. ÑÐ]lÆ>ËMýS$ Ððl»ŒæOòÜsŒæ ^èl*yýlÐ]l^èl$a. B¯ŒlOÌñæ¯Œl Çh[õÜtçÙ¯ŒlMýS$ _Ð]lÇ ™ól¨: H{í³ÌŒæ 11 Æ>™èl ç³È„ýS ™ól¨: H{í³ÌŒæ 30 Ððl»ŒæOòÜsŒæ: www.nchm.nic.in