izkn'kz iz'u&i= MODEL QUESTION PAPER ¼ mpp xf.kr ½ ( HIGHER - METHEMATICS ) le; % 3?kaVs

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1 izkn'kz iz'u&i MODEL QUESTION PAPER ¼ mpp f.kr ( HIGHER - METHEMATICS ) le; %?kavs d{kk & oha iw.kkzad % 00 Time : hours Class - XII th M.M. : 00 funsz'k %& % - lhkh iz'u vfuok;z gsa A - iz'u i esa fn;s ;s funsz'kksa dk lko/kkuh iwozd i<+dj iz'uksa ds mrrj nhft, A - iz'u Øekad & 06 ls rd esa vkarfjd fodyi fn;s ;s gsa A 4- iz'u Øekad & 06 ls rd izr;sd iz'u ij 4 vad vkoafvr gsa A 5- iz'u Øekad & ls 9 rd izr;sd iz'u ij 5 vad vkoafvr gsa A 6- iz'u Øekad & 0 ls rd izr;sd iz'u ij 6 vad vkoafvr gsa A Instructions :-. All questions are compulsory.. Read the Instructions of question paper carefully and write their answer.. Internal option are given in Question No. 06 to. 4. Question No. 06 to carry 4 marks each. 5. Question No. to 9 carry 5 marks each. 6. Question No. 0 to carry 6 marks each. ( SECTION-'A' ) ¼ [k.m&^v* Q.0 Choose the correct option. 5 lgh fodyi NkafV, A (i) d - dks vkaf'kd fhkuuksa esa fohkrd dhft, A d - + (a) + (b) - d (-) (+) (c) - (d) - d - + d - + Resolve d - d - + into partial fraction. (a) + (b) - d (-) (+) (c) - (d) - d - + d - + Cont...

2 (ii) (iii) (iv) (v) ;fn tan - - tan - y tan - z gks rks z dk eku gksk (a) -y (b) +y (c) -y (d) +y If tan - - tan - y tan - z then value of z is (a) -y (b) +y (c) -y (d) -y +y -y ;fn,d js[kk ffofhk; funsz'kkad v{kksa ds lkfk α, β, γ dks.k cukrh gs rks cosα + cosβ + cosγ dk eku gksk (a) - (b) - (c) (d) If a line makes angles α, β, γ with three dimentional co-ordinate ais then cosα + cosβ + cosγ is equal to... (a) - (b) - (c) (d) ;fn lery - 6y - z 7 vksj + y - kz 5,d nwljs ij yec gs rks k dk eku gksk (a) 0 (b) (c) (d) If the plane - 6y - z 7 and + y - kz 5 are perpendicular to each other then the value of k will be (a) 0 (b) (c) (d) funsz'kkad v{kksa ls ] o &4 vur% [k.m dkvus okys lery dk lehdj.k gksk & y z (a) (b) + y - z 4 4 (c) + y - z - (d) None of these 4 The equation of a plane which makes, and -4 intercepts from the co-ordinates ais is (a) + y - z 0 (b) + y - z 4 4 (c) + y - z - (d) None of these 4 Q.0 lr; vlr; dfku fyf[k, 5 (i) mu js[kkvksa ds chp dk dks.k ftudh fnd~dkst;k;sa ]4]5 vksj 4]&] 5 gsa] 0 0 gksk A (ii) o`rr r - (i + j + k) 5 dk dsunz (, -, ) gksk A (iii) lfn'k a dk lfn'k b ij iz{ksi a.b gksk A a -y -y Cont...

3 (iv) lfn'k (i j) k dk eku k gksk A (v) lfn'k 6j - j - k dh fnd~ dkst;k;sa 6 ] -] - gksk A Write true or False (i) The angle between the lines whose direction ratios are, 4, 5 and 4, -, 5 is 0 0 (ii) The centre of sphere r - (i + j + k) 5 is (, -, ). (iii) The projection of a on b is a.b a (iv) The value of (i j) k is k. (v) Direction cosines of the vector 6j - j - k are 6, -, Q.0 fjdr LFkkuksa dh iwfrz dhft, 5 (i) (ii) (iii) (iv) 0 dk ds lkis{k vidyu q.kkad gksk A e dk nok vodyu gksk A f() sin + cos dk egrre eku gksk A ;fn cov(,y) - 8.5, var.() 8 vksj var.(y) 8 rc lg lecu/k q.kkad dk eku gksk A (v) ;fn nks lekj;.k q.kkad Øe'k% -a rfkk gks rks lg lecu/k q.kkad dk eku gksk A Fill in the blanks. (i) Differential coefficient of 0 w.r.t is... (ii) n th derivatives of e is... (iii) The maimum value of f() sin + cos... (iv) If cov(,y) - 8.5, var.() 8 and var.(y) 8 then coefficient of correlation is... (v) If two regression coefficients are -a and then coefficient of -a correlation is... Q.04 dkwye ^v* ds fy, dkwye ^c* ls pqudj lgh tksm+h cukb, A 5 Make the correct pair for column 'A' choosing from column 'B'. (A) (B) (i) (a) [ a - + a sin - ] -a a (ii) sec (b) log (cosec -cot) -a Cont...4

4 (iii) a - (c) log tan + π 4 (iv) cosec (d) [ a - + a log + a - ] (v) (e) log -a +a a +a (f) log (+ + a ) (g) -a Q.05,d okd; esa mrrj nhft, A 5 (i) (ii) (iii) (iv) (v) f() ds fy, fleilu fu;e fyf[k, A f(), n 7 ds fy, Vªsistks,My fu;e fyf[k, A 0.64 E E 05 dk eku fyf[k, A fleilu fu;e esa fo"ke layxud okys y ds q.kkad fyf[k, A lehdj.k ewy vurjky (, ) esa Øfed fohkkyu fof/k ls Kkr dhft, A Give the answer of the following in one sentence. (i) Write the Simpson's rule for f(). (ii) Write the trapezoidal formula for f(), n 7. (iii) Write the value of 0.64 E E 05 (iv) b a b a Write the coefficient of y's with odd subscripts, in Simpson's rule. (v) Find the root of the equation in the interval (,) by bisection method. ( SECTION-'B' ) ¼ [k.m&^c* ( SHORT ANSWER TYPE QUESTION ) ¼ y?kqmrrjh; iz'u ¼izR;sd iz'u 4 vad Q.06 + (+ ( -9) dks vkaf'kd fhkuuksa esa O;Dr dhft;s A 4 b a b a Resolve + (+ ( -9) into partial fractions. Cont...5

5 dks vkaf'kd fhkuuksa esa O;Dr dhft;s A + 7 Resolve into partial fractions Q.07 fl) dhft, fd cos [ tan - {sin (cos - )} ] - 4 Prove that cos [ tan - {sin (cos - )} ] - fl) dhft, fd sin sin sin Prove that sin sin sin Q.08 dk vodyu q.kkad izfke fl)kar ls dhft;s A 4 Find the differential coefficient of from first principal. ;fn y cot gks rks Kkr dhft;s A dy If y cot then find dy d Q.09 ;fn y a sin m + b cos m gks rks fl) dhft;s fd y + m y 0 4 d y If y a sin m + b cos m then prove that + m y 0 ;fn y... dy gks rks fl) dhft, fd... dy If y then prove that Q.0,d d.k fueukafdr fu;e ls freku gs s 5e -t cos t tc t π/ gks rks d.k dk os o Rpj.k D;k gksk A 4 A particle is moving with following rule s 5e -t cos t Find the velocity and acceleration of particle when t π/ π π y (-y log) y (-y log) Cont...6

6 ykhk Qyu p() }kjk fn;k tkrk gs rks egrre ykhk Kkr dhft;s A Profit function is given by p() then find maimum profit. Q. fueukafdr vkadm+ksa ds fy;s lg laca/k q.kkad Kkr dhft;s %& 4 ifr dh vk;q ifru dh vk;q Find the correlation coeff of the following data. Husband's age Wives age ;fn nks pj jkf'k;ksa vksj y dk lg laca/k q.kkad p gs rks fl) dhft;s fd p + y - -y If p be the correlation coeff of two variable and y then prove that p + y - -y y y Q. fueu vkadm+ksa esa y dk eku Kkr dhft, 4 Js.kh y e/; ekud fopyu.6.5 lg laca/k q.kkad P 0.99 Find the value of y whor Js.kh y e/; ekud fopyu.6.5 Coefficient of correlation P 0.99 Cont...7

7 fopj vkadm+ksa ( i, y i ) esa ds fy, ek/; eku 45 gs A y dk ij lekj;.k q.kkad 4 vksj dk y ij lekj;.k q.kkad gs] Kkr dhft, () lg laca/k q.kkad (ii) ds fy, ekud fopyu ;fn y ds fy, ekud fopyu gs A In bileeriate data ( i, y i ) the mean value for is 45, coefficient of regression of y on is 4 and coefficient of regression of on y is 9 then find (i) coefficient of correlation (ii) mean deviation for if mean deviation for y is. Q. fl) dhft, fd,d?ku ds fod.kksaz ds chp dk dks.k cos - Prove that the angle between the diagonal of a cube is cos - gksrk gs A 5 ewy fcunq ls qtjus okys ml lery dk lehdj.k Kkr dhft, tks fueu leryksa ij yac gks %& + y - z,oa - 4y + z 5 Find the equation of plane which passes through origin and perpendicular to planes :- + y - z and - 4y + z 5 Q.4 fdlh ABC esa Hkqtk BC dk e/; fcunq D gs fl) dhft, fd 5 AB + AC AD In ABC ; D is the mid point of BC prove that AB + AC AD lfn'k fof/k ls ABC esa fl) dhft, fd a b cos C + c cos B In ABC prove by vector method a b cos C + c cos B Q.5 fl) dhft;s fd Qyu f(), <,, >, < fcunq ij vlarr r gs A 5 Prove that function f(), is not continuous at, > 9 Cont...8

8 eku Kkr dhft, %& lim o e -e - lim o e -e - then find the value of. Q.6 eku Kkr dhft, %& 5 +- Find the value of +5+7 dk ds lkis{k lekdyu Kkr dhft, A Find the value of cov.r. to π o +- cos + 4sin Q7 eku Kkr dhft, 5 Evaluate : π o cos + 4sin y oø nh?kz o`rr a + b dk laiw.kz {ksqy Kkr dhft, A y Find the total area of curve ellipse :- a + Q.8 gy dhft, %& 5 (+cos) dy (-cos) Solve :- (+cos) dy (-cos) Cont...9 b

9 aa gy dhft, %& dy e -y + e -y Solve :- dy e -y + e -y Q.9 ;fn,d yhi o"kz dk ;kn`fpnd p;u fd;k ;k gks rks bl o"kz esa 5 jfookj gksus dj izkf;drk Kkr dhft, A If a leap year is it to be select at random then find the probability having 5 Sunday in this year. ;fn A vksj B dksbz nks?kvuk, gks rks fl) dhft, %& P(AUB) P(A) + P(B) - P (A B) If 'A' and 'B' are any two events prove that :- P(AUB) P(A) + P(B) - P (A B) y- z+ - y-6 z- Q.0 fl) dhft, fd js[kk, rfkk 4 leryh; gsa a A bldk izfrpnsn fcunq Hkh Kkr dhft, A 6 y- z+ - y-6 z- Prove that the lines and 4 are coplanar. Find its intersecting points.,d ksys dh ft;k r gs tks ewy fcunq ls gksdj tkrh gs rfkk v{kksa dks A, B, C ij feyrh gs] fl) dhft, fd ABC ds dsunzd dk fcunq ifk,d ksyk 9 ( +y +z ) 4 r gksk A The radius of a sphere is 'r' Which passes through the origin and meet the ais at A, B, C. Prove the centroide of ABC is a locus of sphere 9 ( +y +z ) 4 r Cont...0

10 Q. ml ksys dk lfn'k lehdj.k Kkr dhft, tks fcunqvksa A; (, -, 4) rfkk B; (-5, 6, -7) dks feykus okys js[kk[k.m dks O;kl ekudj [khapk ;k gs A ksys ds lehdj.k ds dkrrhz; :i dk fueu dhft, A Find the vector equation of a sphere which is drawn as a diameter meeting through segment points A (, -, 4) and B (-5, 6, -7). Drive is Cartesian equation. mu nks js[kkvksa ds chp dh U;wure nwjh Kkr dhft,] ftuds lfn'k lehdj.k %& r ( i + j ) + λ ( i - j + k) rfkk r (i + j - k ) + µ (i - 5j + k) gs A Find the shortested distance of the lines, whose vector equations are :- r ( i + j ) + λ ( i - j + k) and r (i + j - k ) + µ (i - 5j + k)

11 vkn'kz mrrj MODEL ANSWERS ¼ mpp f.kr ( HIGHER - METHEMATICS ) le; %?kavs d{kk & oha iw.kkzad % 00 Time : hours Class - XII th M.M. : 00 Q.0 dk gy A ( SECTION-'B' ) ¼ [k.m&^v* ¼dqy $$$$ 5 vad (vi) (c) - vad (vii) (c) vad (viii) (b) - vad (i) (a) 0 vad y z () (b) + - vad 4 Q.0 lr; vlr; dfku NkafV, A ¼dqy $$$$ 5 vad (vi) vlr; vad (vii) vlr; (viii) vlr; vad vad (i) vlr; vad () lr; vad Q.0 fjdr LFkkuksa dh iwfrz dhft, A ¼dqy $$$$ 5 vad (vi) 0 log0 vad (vii) e (viii) d (-) (+) -y +y e vad vad (i) vad () - vad Q.04 Column 'A' dk 'B' ls lgh tksm+h cukb;s A ¼dqy $$$$ 5 vad (i) (e) log -a vad -a a +a (ii) sec (c) log tan + π vad u Cont...

12 (iii) a - (a) [ a - + a sin - ] vad a (iv) cosec (b) log (cosec -cot) vad (v) (d) [ a - + a log + a ] vad +a Q.05,d okd; esa mrrj nhft, A ¼dqy $$$$ 5 vad (i) Q.06 dk gy h h [(y o +y n )+4(y +y +y y n- +(y +y y n- ) ] vad (ii) [(y 0 +y 7 ) + (y +y +...+y 6 ) ] tgk h vad (iii) 0.64 E 05 vad (iv) 4 vad (v).5 vad ( SECTION-'B' ) (+) (-) (+) (-) ¼ [k.m&^c* A B d + vad + (-) A(-) + B(+) (i) leh- (i) esa j[kus ij A (-) + B (+) + (+) ( -9) + (+) (+) (-) (+) (-) A(-) + B(+) (+) (-) b-a 7 A (0) + 5 B 5B or B 5 blh izdkj ut leh- (i) esa j[kus ij Cont...

13 A (-, -) + B (-+) A (-5) + B (0) - 5A ;k A (+) (-5) 5(+) + 5(+) Ans. ¼dqy $$$ 4 vad vfkok OR (i) leh- vad vc ekuk ( + 4) - ( + 8) (+4) - (+4) (-) (+4) vc ekuk A + B (ii) leh- (-) (+4) (+4) (-) (-) (+4) A(-) + B(+4) (+4) (-) 5+8 A(-) + B(+4) 5+8 (A+B) + (-A+4B) ds q.kkadksa dh rqyuk djus ij 5 A+B (iii) 8 -A + 4B (iv) leh- (iii),oa (iv) ls 0 A + B 8 -A + 4B 8 6B Cont...4

14 B Eq n (iii) ls 5 A+.'. A '. (+4) (-) (+4) + (-).' (+4) (-) Ans. ¼dqy $$$ 4 vad Q.07 dk gy L.H.S. cos [ tan - {sin (cos - )} ] cos [ tan - {sin (sin - - )} ] [ '.' cos - sin - - ] cos [ tan - - ] cos [ cos - - ] - R.H.S. sin sin sin sin sin sin π sin sin vad 65 ('.' ewy sin - + sin - y sin - ( -y + y - ) ds vuqlkj) sin sin - 65 sin sin sin Cont...5

15 sin sin sin - π { } ¼dqy vad Q.08 dk gy ekuk f() (i),oa f(+h) +h (ii) vodyu ds izfke fl)kar ls dy d lim h o f(+h) - f() h lim ( +h - ) h o h lim ( +h - ) h o h +h+ +h+ lim h o h ( +h) -( ) +h+ lim h o h + h - +h + lim h o h h +h + '.' y cot - lim h o +h + +h + + Ans. + + ¼dqy vad put tanθ.'. θ tan -.'. y cot - +tan θ + tanθ Cont...6

16 cot - cot - cot - sec θ + tanθ secθ + tanθ +cosθ sinθ θ cot - + cos - π sin θ. cos θ cot - θ cos. cos sin. cos θ θ θ cot - [ cat θ ] θ or θ. tan -.'. d cot - dy + + d tan - Q.09 dk gy Ans. (+ ) ¼dqy vad y a sin m + b cos m (i) leh- dy d.'. {a sin m + b cos m} dy d y d d a sin m + b.cos m a {cos m. m} + b {-sin m. m} m {a cos m - b sin m} (ii) d dy d {m (a cos m - b sin m) } Cont...7

17 d d m [ a. cos m - b. sin m ] m [-sin m. a.m - b cos m.m] m [a sin m + b cos m].'. d y -m [y] leh- (i) ls d y + m y 0 Ans. ¼dqy vad.'. y y ( ) y y y/ log y log y/ y log y log d d '.' (.log y) (y.log ) dy dy y. + (log) y y dy - log... y -y.log y dy y dy.'. Ans. (-y log ) ¼dqy vad Q.0 dk gy s 5e -t cos t (i) leh- y Kkr djuk gs %& (i) d.k dk os ds dt t π/ (ii) d.k dk Roj.k d s dt t π/ Cont...8

18 lehdj.k (i) dk t ds lkis{k vodyu djus ij t ds dt ds dt ds dt 5 [e-t. (-sint) + cos t. (-e -t )] -5 e -t sint - 5 e -t cost -5e -t (sint + cost) (ii) leh π j[kus ij π t -5e -π/ π π (sin + cos ) vr% t -5e -π/ (+0) t π -5e -π/ ij d.k dk os -5e -π/ gksk lehdj.k (ii) dk iqu% t ds lkis{k vodyu djus ij d s dt π d s dt t π/ -5 [e -t d d (sint + cost) + (sint + cost) e -t ] dt dt t -5 [e -t (cost - sint) + (sint + cost) (-e -t )] -5 [e -t cost - e -t sint - e -t sint - e -t cost] -5 (-e -t sint) 0e -t sint π j[kus ij 0 e -π/ sin 0e -π/ 0e -π/ π π vr% t ij d.k dk Roj.k 0e -π/ gksk Ans. ¼dqy vad Kkr gs %& ykhk Qyu p() Kkr djuk gs %& (i) egrre ykhk ds fy, fcunq (ii) vfhk"v fcunq ij egrre ykhk Cont...9

19 '.' vr% '.' p() (i) leh- ds lkis{k vodyu djus ij p () (ii) leh- iqu% ds lkis{k vodyu djus ij p () - 6 Qyu dks egrre vfkok U;wure gksus ds fy, p () 0 ;k ;k (iii) leh- 4 ;k 6 ;k ij P () -6 (-Ve) ij Qyu P() egrre gksk A vr% dk eku lehdj.k (i) esa j[kus ij Qyu dk egrre eku P vr% ij Qyu dk egrre eku 49 ¼dqy vad Q. dk gy Kkr djuk gs %& pj 4 5 pj y pj,oa y ds e?; lglaca/k Cont...0

20 y u i - v i y-y u i v i u i v i y 5 u i 0 v i 0 u i 0 v i 0 u i v i n 5 y y 5 n 5 pj o y ds e/; lg leca/k q.kkad P(,y) nu i v i - u i.v i nu i - (u i ). nu i - 5 (-0) vr% pj o y ds e/; lgleca/k q.kkad - ¼dqy vad (i) pj,oa y ds e/; lg leca/k q.kkad P (ii) pj,oa y dk izlj.k q.kkad Øe'k%,oa y. fl) djuk gs %& P + y - -y.y '.' (-) vksj y n n (y-y) n

21 .'. -y [(-y) - (-y)] Cont... Q. dk gy [(-) - (y-y)] n (-) + (y-y) - (-) (y-y) n n n + y - P.y ;k P.y + y - -y ;k P 7.6 y 4.8 P 0.99 ¼dqy vad.6 y.5 Kkr djuk gs %& (i) y dh ij lekj;.k js[kk dk lehdj.k (ii) j[kus ij y dk eku Kkr djuk y dh ij lekj;.k js[kk dk lehdj.k y-y P (-).5 y (-7.6) y (-7.6) ( '.' fn;k gs) y y y y - -y.y (i) pj dk e/;eku 45 (ii) y dk ij lekj;.k q.kkad 4 ¼dqy vad (iii) dk y ij lekj;.k q.kkad /9 Kkr djuk gs %&

22 (i) pj o y ds e/; lg leca/k q.kkad (ii) tcfd ;fn y Cont (i) '.' lg leca/k q.kkad P by by 4 (ii) '.' by P. y y, P rfkk by dk eku j[kus ij Q. dk gy 4 y fn;k gs 4 ¼dqy vad ekuk?ku ds dksj dhy a?ku dk,d 'kh"kz O(o,o,o) C B gs OA, OB, OC funsz'kkad A s v{k X, Y, Z gs A fcunq O, A, B, C, A, B, C O A X o P ds funsz'kkad gs %& B C O (o,o,o), A (a,o,o), B (o,a,o), C (o,o,a) A (o,a,o), B (a,o,a), C (a,a,o), P (a,a,a) AL, BM, CN, OP?ku ds pkj fod.kz gsa fod.kz AA ds fnd- vuqikr o-a, a-o, a-o -a,a,a) " BB " " (a,-a,a) ekuk fod.kkzsa AA vksj BB ds chp dk dks.k θ gs rks a a + b b + c c cosθ a + a + a a + a + a (-a) (a) + (a) (-a) + (a) (a) a + a + a a + a + a -a - a + a a Y Z θ cos - 9

23 fn;k gs %& Q.4 dk gy leryksa ds leh- gs %& ¼dqy vad Cont y - z (i) - 4y + z (ii) ewy fcunq (o,o,o) ls gksdj tkus okys lery dk leh- a(-o) + b(y-o) + c(z-o) 0] a + by + cz o leh- (iii) leh- (i),oa (ii) ij yac gs A.a +.b -.c o (iii) a + b - c o (iv),oa a - 4b + c o (v) (iv),oa (v) ls & b c b -4 b 4 b c k (ekuk) 5 a k, b k, c 5k leh- (iii) esa a,b,c dk eku j[kus ij ABC dh Hkqtk BC dk e/; fcunq D gs A a -4 a - a a k + ky + 5kz o + y + 5z o ¼dqy vad.'. BD DC BD DC BD - DC O C c -0 c 0 A D B

24 BS + CD CD O (i) Cont ABC esa & AB + BD AD (ii) ACD esa & AC + CD AD (iii).'. Q.5 dk gy leh- (ii) + (iii) ls & AB + BD + AC + CD AD AB + AC + (BD + CD) AD AB + AC AD ABC esa & ¼dqy vad BC a, CA b, AB c BC a A aa CA b A ba AB c A ca fp esa & BC + CA + AB O. BC - CA - AB. a - b - c. - (b + c) (i) nksuksa i{kksa dks a ls q.kk djus ij & a. a - a (b + c) a - a b - a c. - ab cos (π - C) - ac cos (π - B). a ab cos C + ac cos B. a b cos C + c cos B fn;k gs & Qyu f(), <,, > ¼dqy vad fl) djuk gs fd f(), ij vlarr gs A B π-b π-a A C π-c

25 R.H.L. f (+h) lim Rf (+h) o + (+h) (+o) lim lim LHL - f() (-h) o - Cont lim Lf (-h) (-h) o - 9. f ()..'. Rf (+h) Lf (-h) f () vr% f(), ij vlarr gs A lim o lim + e -e - ¼dqy vad lim oo oo o lim oo o lim oo o. ¼dqy vad Q.6 dk gy +- [ + - ]

26 Cont [ ] 6 6 { ( + ) - ( ) } 4 4 ( + ) - ( ) [ + t ysus ij dt. ] t - ( ) dt. 4. log 4 + log log + - log (+) ¼dqy vad +5+7 t - t a ( ) [ ] 6 6

27 Cont ( + ) ( + ) - ( ) t dt. t - ( 59 ) 6 Q7 dk gy d 59 tan d tan - 6t d 59 d 59 d 59 π o tan - tan - 59 t d ( + ) cos + 4sin π ¼dqy vad ekuk I (cos / + sin / ) + 4 sin / cos o / π o cos / - sin / + 8 sin / cos / π sec / -tan / + 8 tan o / - π o sec / tan / - 4 tan / -

28 t tan / j[kus ij Cont...8 dt sec /. tc o, t o rfkk tc / rks t, o o dt dt.'. I - - t - 4t - t - t o dt d log 5 + t - ( 5) - (t-) t + d log - log d log d + 5 log Ans. 5-5 ¼dqy vad fn;k oø,oa y v{k ds lefer gs vr% oø dk leiw.kz {ksqy {ks OAB dk 4 quk gksk A + y ls a b b y a - a A oø ij P(,y) fy;k X ogha PM dk {ksqy y gs A o ij o rfkk A ij a, a vhkh"v {ksqy 4 y. o (-a,o) a 4 b a -. o a a 4b b a -. a o a Y B P (,y) O M B (o,-b) A (a,o)

29 Cont...9 4b a - + sin - a a 4b a a -a + sin sin - o a 4b. sin - ab. π π ab oz bdkbz a ¼dqy vad Q.8 dk gy (+cos) dy (-cos) dy -cos +cos lekdyu djus ij dy -cos + c +cos dy e -y + e -y sin / + c cos / tan / + c. y (sec / - ) + c. y tan / - + c Ans. ¼dqy vad + e y dy (e + ). e y dy e e y a e y e + + c e y e + + c. ¼dqy vad Q.9 dk gy ge tkurs gsa fd yhi o"kz esa 66 fnu gksrs gsa vr% blesa 5 iw.kz lirkg rfkk 'ks"k fnu cprs gsa A a a a o a

30 bu nks fnuksa ds fy;s fueu lkr lahkkouk, gks ldrh gsa%& (i) (ii) (iii) lkseokj rfkk eayokj eayokj rfkk cq/kokj cq/kokj rfkk q:okj Cont...0 (iv) q:okj rfkk 'kqøokj (v) (vi) 'kqøokj rfkk 'kfuokj 'kfuokj rfkk jfookj (vii) jfookj rfkk lkseokj vr% 'ks"k nks fnuksa esa,d jfookj gksus dh vuqdwy flfkfr;k (vi, vii) gsa A vr% 'ks"k nks fnuksaa esa,d jfookj gksus dh?kvuk ds vuqdwy gksus fd flfkfr dh lahkkouk izkf;drk dqy lahko ifj.kkeksa dh la[;k 7 ¼dqy vad ekuk fd izfrn'kz lef"v 5 ds nks mileqpp; A rfkk B gs rfkk n(a), n(b), n(aub), n(a B) Øe'k% leqpp;ksa A,B, AUB, A B esa vo;oksa dh la[;k dks n'kkzrs gsa %& leqpp; fl)kur ls %& n(aub) n(a) + n(b) - n (A B) nksuksa vksj n(s) ls Hkk nsus ij Q.0 dk gy n(aub) n(s) n(a) + n(b) - n(a B) n(s) n(s) n(s) izkf;drk dh ifjhkk"kk ls %& P(AUB) P(A) + P(B) - P(A B) nh bz js[kkvksa ds lehdj.k gsa %& A B ¼dqy vad y- z+ - y-6 z-,oa (i) - y-y z-z l m n S

31 leh- (i) dh rqyuk ls djus ij Cont (, y, z ) (0,, -) rfkk (, y, z ) (, 6, ) (l, m, n ) (,, ) rfkk (l, m, n ) (,, 4) nksuksa js[kk, leryh; gksus dh 'krz & - y -y z -z l m n 0 l m n (8-9) - 4 (4-6) + 6 (-4) (-) - 4 (-) + 6 (-) vr% nh bz js[kk, leryh; gksah A y- z+ leh- (i) ls r (ekuk) r, y r +, z r -,,oa r- r+-6 izfrpnsn fcunq r (r, r+, r-) r-- 4 (, +, - ) (, 6, ) ¼dqy vad ewy fcunq ls gksdj tkus okys ksys dk lehdj.k gs

32 + y + z + u + vy + wz (i) ksyk (i) v{kksa dks A,B,C ij feyrk gs A '. A,B,C ds funsz'kkad A(a,o,o), B(o,b,o), C(o,o,c) leh- (i) ls %& a + ua 0 u - a / b + vb 0 v - b / Cont... c + wc 0 w - c / leh- (i) esa u,v,w dk eku j[kus ij + y + z - a - by - cz 0 bl ksys dh ft;k r u + v + w - d r a + b + c (ii) ekuk fhkqt ABC dk dsunzd (α, β, γ,) gs a+o+o o+b+o a, b, c a α, b β, c γ leh- (ii) ls & Q. dk gy A 4r 9α + 9β + γ 4r 9(α +β +γ ). ¼dqy vad A,oa B ds funsz'kkad A(,-,4),oa B(-5,6,-7) gs A.'. A ds flfkfr lfn'k ekuk a i - j + 4k,oa B ds flfkfr lfn'k b -5i + 6j - 7k gs A AB dks O;kl ekudj [khaps ;s ksys dk leh- gs a b a +b +c c A (,-,4) o+o+c (r - a). (r - b) 0. B (-5,6,-7)

33 [ r - (i - j + 4k) ]. [ r - (-5i + 6j - 7k) ] 0 [ r - i - j + 4k ]. [ r + 5i - 6j + 7k ] (i) Cont (i) esa r i + yj + k j[kus ij & [i + yj + zk - i + j - 4k]. [i + yj + zk + 5i - 6j + 7k] 0 [(-) i + (y+) j + (z-4) k]. [(+5) i + (y-6) j + (+7) k] y - y z + z y + z + - y + z (ii) leh- (ii) ksys dk dkrhz; lehdj.k gs A Li"V gs fd ksys dk dsunz (- /, /, - / ) ft;k ¼dqy vad ;fn nks js[kkvksa ds lfn'k lehdj.k r a + tb rfkk r a + tb gks rks bu nksuksa js[kkvksa ds chp dh U;wure nwjh (a -a ). (b b ) d b b tgk a i + j a i + j - k a -a i + j - k - i - j i - k b i - j + k b i - 5j + k b b i j k - -5 i - j - 7k.'. (i - k). (i-j-7k) d i - j - 7k

34 ¼dqy vad

izkn'kz iz'u&i= MODEL QUESTION PAPER mpp xf.kr HIGHER - METHEMATICS le; % 3?kaVs Time : 3 hours Class - XII th M.M. : 100

izkn'kz iz'u&i= MODEL QUESTION PAPER mpp xf.kr HIGHER - METHEMATICS le; % 3?kaVs Time : 3 hours Class - XII th M.M. : 100 izkn'kz iz'u&i MODEL QUESTION PAPER mpp xf.kr HIGHER - METHEMATICS le; % 3?kaVs d{kk - iw.kkzad % 00 Time : 3 hours Class - XII th M.M. : 00 funsz'k %& - lhkh iz'u vfuok;z gsa A - iz'u i esa fn;s x;s funsz'k