Á'u&i«k CYkw fáuv ijh{kk & gk;j lsds.mjh d{kk& fo"k;& mpp xf.kr Ø- bdkbz bdkbz ij olrqfu"b dqyk fu/kkzfjr Á'u vadokj Á'u a dh la[;k Á'u vad vad 4 vad 5 vad 6 vad - vkaf'kd fòé 05 0 0 & & 0 - izfryk e QYku 05 0 0 & & 0 - leryk T;kferh; iw.kk±d&00 le;&?k.vk 4- leryk 5 04 & 0 0 0 5- ljyk js[kk,oa x Ykk 6- lfn'k 7- lfn'k a dk xq.kuqyk 8- lfn'k a dk f«kfoeh; 5 04 & 0 0 0 T;kferh; esa vuqá; x 9- QYku] lhek r kk 05 & & 0 & 0 lkarr; 0- vodyku 0 0 0 & & 0 - dfbu vodyku - vodyku dk vuqá; x 05 0 0 & & 0 - lekdyku 4- dfbu lekdyku 5 5- fuf'pr lehdyku 05 & 0 & 0 6- vodyk lehdj.k 05 & & 0 & 0 7- lglaca/k 05 0 0 & & 0 8- lekj;.k 05 0 0 & & 0 9- Ákf;drk 05 & & 0 & 0 0- vkafdd fof/k;k 05 05 & & & & dqyk 00 5 07 07 0 6 ()
gk;j lsd.mjh Ldwy ijh{kk& 0& HGHER SECONDARY SCHOOL EXAMNATON-0- çkn'kz ç'u&i Model Question Paper mpp xf.kr HGHER MATHEMATCS (Hindi and English Versions) Time& ÄaVs Maximum Marks 00 funsz'k& () lòh ç'u gy djuk vfuok;z gsa ÁR;sd Á'u ds lk k vkarfjd fodyi fn, x, gsaa () Á'u& A, B, C, D, E olrqfu"b Á'u gsa ÁR;sd dk vad fu/kkzfjr gsa () Á'u& ls Á'u 8 rd ds Á'u a ds ÁR;sd Á'u ds 4 vad fu/kkzfjr gsaa (4) Á'u&9 ls Á'u 5 rd ds Á'u a esa ÁR;sd Á'u ds 5 vad fu/kkzfjr gsaa (5) Á'u& 6 o 7 ds fu/kkzfjr vad 6 gsaa nstructions () All question are compulsory internal choices are given in every question. () Question- A, B, C, D, E are objective type and marks alloted to each question. () Question- to 8 carries 4 marks each. (4) Question-9 to 5 carries 5 marks each. (5) Question-6 and 7 carries 6 marks each. olrqfu"b Á'u (Objective Type Questions) Á'u& (A) fueufykf[kr Á'u a ds lk k fn, x, fodyi a esa ls lgh fodyi pqudj viuh mùkj&iqflrdk esa fykf[k,& 5 (i) tan + tan... g xk& (a) tan 6 (c) 4 6 (ii) rhu lfn'k t f«kòqt ds 'kh"kz fcunqv a ls fn"v ef/;dkv a ls fu/kkzfjr gsa] dk ; x gs& (a) 0 r (b) (c) (d) (iii) ;fn a r dk v'kwu; lfn'k ftldk ekikad a gs r kk m dk,d v'kwu; lfn'k (b) (d) ()
gsa rc ma,dkad lfn'k g xka (a) m + (b) m a r (c) m a r (d) 0 (iv),d pj f«kt;k j[kus oky x Ykkdkj xqcckjs dh f«kt;k 4 lseh gsa blds vk;ru ifjorzu dh nj g xka (a) 64?ku lseh@lsdum (b) lseh@lsdum (c) 48 lseh@lsdum (c) 7 lseh@lsdum (v) lfn'k a i + j + k$ i $ $ j k $ ds chp d.k gs& (a) 0 (b) 4 (c) (d) 6 Q.- (A) Choose the correct choice from the follwoing multiple choice question 5 (i) tan + tan... (a) tan 6 (c) 4 (ii) The sum of three vectors from the vertices of triangle toward median are (a) 0 r (b) (c) (d) (iii) f a r is a non-zero vector whose resultant is a. m is an another non-zero vector. ma is a unit vector (a) m + (b) m a r (b) (d) (c) m a r (d) 0 (iv) A ballon has a variable radius of 4 cm. The rate of changing in volume is (a) 64 cm /sec. (b) cm /sec. (c) 48 cm /sec. (c) 7 cm /sec. 6 ()
(v) The angle between the vector i $ + j $ + k$ and i $ $ j k $ is (a) 0 (b) 4 (c) (d) 6 Á'u& (B) fjä L kku a dh iwfrz dhft,& 5 (i) ;fn r r (ii) a.b A B + x(x+ ) x x+ g ] r B ----------------------A ----------------------- tcfd θ, a r o b r ds chp dk d.k gsa (iii) ;fn y + x + x x + +... rc dy dx... (iv) a x dx dk eku ------------------ gsa (v) sec x dx dk eku -------------------- gsa Q.- (B) Fill in the blanks 5 A B (i) f + then B ----------------------. x(x+ ) x x+ r r r r (ii) a.b ----------------------- when θ, is angle between the vector a and b - (iii) f y + x + x x + +... then dy dx... (iv) a x dx ------------------ (v) sec x dx -------------------- Á'u& (C) lr;@vlr; fykf[k,& 5 (i) cot x dx log sin x (ii) lekdyku] vodyku dh ÁfrYk e ÁfØ;k gsa (iii) sin x dk x ds lkis{k lekdyku cos x g rk gsa (iv) lg&laca/kd xq.kkad dk eku lnso /kukred g rk gsa (v) ;fn b yx. rc b xy.4 g rk gsa Q.- (C) True / False 5 (i) cot x dx log sin x (ii) ntigration is inverse process of differentiation (iii) the integration of sin x, with respect to x is cos x. (4)
(iv) The value of correlotion coefficient always positive (v) f b yx. then b xy.4. Á'u& (D) LraÒ "A" ls LraÒ "B" dk t M+h feykku dhft, & 5 "A" "B" (i) d dx tan x (a) 8 + x (ii) fcunqv a (, 4, 5) o (, 5, 4) ds chp (b) dh nwjh g xh (iii) fcunqv s (4,, 5) dh XZ leryk ls nwjh g xh (c) 0 (iv) fcunq (,, ) dh y&v{k ls YkEcor~ nwjh g xh (d) (v),d js[kk[k.m dk funs Z'kka{k a ij Á{ i (, 4, ) (e) gs rc js[kk dh YkEckbZ g xh Q.- (D) Match the column& 5 "A" "B" (i) d dx tan x (a) 8 + x (ii) The distance between the points (, 4, 5) and (b) (, 5, 4) (iii) The distance of points (4,, 5) from (c) 0 the XZ plane (iv) Parpendicular distance of points (,, ) (d) y axis (v) The projection of line segment of (, 4, ) (e) then length of line. Á'u& (E),d okd; esa mùkj fykf[k,& 5 (i) fe ;k fl kfr fof/k dk iqujko`fùk lw«k fykf[k,a (ii) U;wVu js lu fof/k dk lw«k fykf[k,a (iii) leykec prqòqzt fu;e dk lw«k fykf[k,a (iv) fleilu ds fu;e dk lw«k fykf[k,a (v) 0.55 E 0 0.40 E 0 dk xq.kuqyk D;k g xk\ Q. (E) Write the answer of following questions in one sentance& 5 (i) f teration formula for fase position method. (ii) Write the formula for Newton Raphson method. (5)
(iii) Write the formula is Trapezoidal rule. (iv) Write the formula is simpnson's rule (vi) Find the value of 0.55 E 0 0.40 E 0 x + 5 Á'u& ds vkaf'kd fòé a esa foòä dhft,a 4 (x ) (x ) v kok x+ d vkaf'kd fòé a esa foòä dhft,a (x + ) (x 9) x + 5 Divide into partial fraction - (x ) (x ) or x+ Divide into partial fraction- (x + ) (x 9) Á'u&- fl) dhft, fd& 4 tan + tan + tan v kok fl) dhft, fd& cos 4 5 + 5 sin cos 65 Prove that - tan + tan + tan or Prove that cos 4 5 + 5 sin cos 65 Á'u&4- Á ke fl)kar ls sin x dk vodyku Kkr dhft,a 4 v kok ;fn y e x+ex+... g r fl) dhft, fd dy y dx y Differentiate by first principle of sin x or f y e x+ex+... then prove that dy y dx y Á'u&5- ;fn y log x + log x + log x +... 4 g r fl) dj fd dy dx x(y ) (6)
;fn y sin x cosx + g r dy dx f y log x log x log x... v kok dk eku Kkr dj A + + + then prove that dy dx x(y ) or sin x f y then prove that value of dy + cosx dx. Á'u&6-,d d.k lehdj.k S 5 e t cos t ds vuqlkj xfr djrk gsa ;fn t g r mldk osx o Roj.k Kkr dhft,a ¼4½ v kok,d ir kj Åij dh v j Qs adk tkrk gsa bldh xfr dk lehdj.k S 490 t 4.9 t gsa ir kj }kjk Ákr vf/kdre Å pkbz Kkr dhft,a tgk t o S Øe'k% lsdum o ehvj esa gsa Equation of a moving partical S 5 e t cos t. f t then find its velocity and accelaration. or A stone is thrown upward and the equation of its velocity is S 490 t 4.9 t where, t and S are in second and meter respectively. Calculate the maximum height adopted by stone. Á'u&7- lg&laca/k xq.kkad Kkr dhft,& cov (x, y).5, var (x) 6.5, var (y) 0.5 ¼4½ v kok n pj jkf'k; a x o y ds e/; lg&laca/k xq.kkad r g r fl) dhft, fd& x y x y σ +σ σ r σx σy tgk σ x, σ y, σ x y, Øe'k% x, y r kk (x y) ds fopj.k xq.kkad gsa Find the correlation coefficient, if given that cov (x, y).5, var (x) 6.5, var (y) 0.5 or f r is the correlation coefficient between x and y then show that x y x y σ +σ σ r σx σy where σ x, σ y and σ x y are the variants of x, y and (x y) respectively. (7)
Á'u&8- lekj;.k js[kkv a x 9y + 6 0,oa x y + 0 ds fyk, lg&lacaèk dh x.kuk dhft, v kok fueukafdr lkj.kh }kjk XokfYk;j esa 00 #i, ewy; ds laxr Ò ikyk esa lokzf/kd mfpr ewy; vk dm+ a }kjk Kkr dhft,a 'kgj XokfYk;j Ò ikyk v lr ewy; 70 75 ekud fopyku -5-0 n u a uxj a esa olrq ds ewy; a esa lg&laca/k xq.kkad 0-8 gsa Given the regression lines as x 9y + 6 0 and x y + 0 respectively calculate the correlation coefficient or An article cost Rs. 00 at Gwalior and the corresponding most appropriate value at Bhopal using the following data City Gwalior Bhopal Mean value 70 75 Standard deviation.5.0 The correlation between the value of the two cities is 0.8 Á'u&9- mu leryk a ds lehdj.k Kkr dhft, t leryk x y + z ds lekarj gs a r kk ftudh fcunq (,, ) ls YkkfEcd nwjh gsa ¼5½ v kok fl) dhft, fd n lekurj leryk a x y + z + 0 v j 4x 4y + z + 5 0 ds chp dh nwjh 6 gsa Q. 9. Find the equation of the plane which are parallel to the plane x y + z and whose perpendicular distance from the point (,, ) is. ¼5½ or Show that the distance between two parallel plane x y + z + 0 and 4x 4y + z + 5 0 is 6. r r Á'u&0- lfn'k a a i $ $ j+ k$ v j b i $ 4j $ 4k$ dk vfn'k xq.kuqyk v j muds chp dk d.k Kkr dhft,a ¼5½ v kok lfn'k a i $ + 4 $ j + 5k$, 4i $ j $ 5k $ v j 7i $ + $ j ls cus f«kòqt dh ifjeki Kkr dhft,a r r Q.-0 Find the scaler product of vector a i $ $ j+ k$ and b i $ 4j $ 4k$ (8)
and also find the angle between them or Find the perimeter of a triangle made by vectors i $ + 4 $ j + 5k$, 4i $ j $ 5k $ and 7i $ + $ j respectively. Á'u&- fueukafdr QYku ds lkarr; dh foospuk dhft,a ¼5½ x+ f(x) x +, x ij v kok lim (x ) (4x ) x dk eku Kkr dhft,a (x + 8) (x ) Clarify the continuity of followoing function x+ f(x) x +, at x or Find the value of lim (x ) (4x ) x (x + 8) (x ) Á'u&- js[kk y x,oa ijoyk; y 6x ls f?kjs { «k dk { «kqyk Kkr dhft,a ¼5½ v kok ijoyk; y 4x,oa x 4y ls f?kjs { «k dk { «kqyk Kkr dhft,a Q. Find the area included between the parabola y 6x and the line y x or Find the area included between the parabola y 4x and x 4y Á'u&- / 0 dx 5 4sinx dk lekdyku Kkr dhft,a v kok sin x dx sin x + cos x 4 d fl) dhft,a Find the integral coefficient of Prove that / 0 Á'u&4- fl) dj fd QYku dx 5 4sinx or sin x dx sin x + cos x 4 ¼5½ (9)
y x + ax + bn + c vodyku lehdj.k v kok dy dx 6 dk,d gyk gsa vodyku lehdj.k dy dx + y ex d gyk dhft,a Q.4 Show that the function y x + ax + bx + c is a solution of differential equation dy dx 6. Solve the differential equation dy dx + y ex Á'u&5- ;fn,d Ykhi o"kz dk ;kn`fpnd p;u fd;k x;k g r blesa 5 jfookj g us dh Ákf;drk Kkr dhft,a v kok 5 iùk a dh QsaVh gqbz rk'k dh xïh esa ls iù fudky tkrs gsa] buds YkkYk ;k bôk g us dh Ák;fdrk Kkr dhft,a Q.5 Find the probability that a leap year, selected at random will contain 5 sundays or Two cards are drawn at random from a pack 5 cards. What is the probality that either both are red or both are acess. x y z Á'u&6- ljyk js[kkv a v j x y 4 z 5 ds chp dh 4 4 5 U;wure nwjh Kkr dhft,a v kok ml x Y dk lehdj.k Kkr dhft, t fcunqv a (,, 4) ] (, 5, ) v j (,, 0) ls g dj tkrk gs r kk ftldk dsuæ leryk x + y + z 0 ij fl kr gsa Q.6 Find the minimum distance between straight lines x y z 4 and x z y 4 z 5 4 5 or or Find the equation of the sphere which passes threw the points (,, 4) ] (, 5, ) and (,, 0) and whose centre lines on the plane x + y + z 0. (0)
Á'u&7- fl) dj & r r r r r r r r r a+ b b+ c c+ a a bc v kok 5 bdkbz dk,d cyk $ i j $ + k$ dh fn'kk esa dk;z djrk gsa r kk fcunq ( ) i $ j $ + k$ ls xqtjrk gsa bl cyk dk fcunq $ i + $ j + k$ ds ifnr% lfn'k vk?kw.kz Kkr dhft,a Q.7 Prove that r r r r r r r r r a+ b b+ c c+ a a bc or A force of 5 unit acts along the ( $ i j $ + k$ ) and passes through the points ( i $ j $ + k$ ). Find the vector moment of it about the points $ $ ( i + $ j + k) vkn'kz mùkj Q. (A) (i) (c) (ii) (a) 4 0 r (iii) (c) m r (iv) (a) 64 cm a /sec. (v) (d) Q. (B) r r (i) B (ii) a.b ab cosθ (iii) dy dx ex (iv) (v) log (sec x + tan x) Q. (C) (i) lr; (ii) lr; (iii) vlr; (iv) vlr; (v) vlr; Q. (D) (i) (b) (ii) (a) 8 + x (iii) (e) (iv) (d) x a x a x + sin a ()
(v) (c) 0 Q. (E) (i) x n+ x n (ii) x n+ x n (iii) b a xn xn f(x n) f(x n ) f(x n ) f'(x ) n f (x n ) f(x) dx h [(y o +y n ) + 4 (y + y +... + Y n ) b a (iv) f(x) dx h [(y o +y n ) + 4 (y + y +... + Y n ) (y + y 4 +... + Y n )] (v) 0.060765 E 0 mùkj&- ekuk fd x + 5 A + B (x ) (x ) x x ----------¼½ (x ) (x ) ls lehdj.k ¼½ ds n u a v j xq.kk djus ij& x + 5 A (x ) + B (x ) -----------¼½ ¼ vad½ x j[kus ij + 5 0 + B ( ) B 9 ¼ vad½ v j x j[kus ij + 5 A ( ) + 0 A 7 lehdj.k ¼½ es a A 7 v j B 9 j[kus ij& x + 5 7 9 + (x ) (x ) x x v kok dk mùkj& x+ x+ (x + ) (x 9) (x + ) (x + ) (x ) (x ) (x ) + ¼ vad½ ¼ vad½ ekuk A + B ---------¼½ (x + ) (x ) x+ x (x + ) (x ) ls lehdj.k ¼½ ds n u a v j xq.kk djus ij A (x ) + B (x+) ---------¼½ ¼ vad½ x j[kus ij& 0 + B ( + ) ()
B 5 x j[kus ij& A ( ) + 0 A 5 ¼ vad½ ¼ vad½ lehdj.k ¼½ es a A 5 v j B 5 j[kus ij (x ) (x ) + + x+ (x + ) (x 9) vr% mùkj&- ge tkurs gsa fd& + 5(x+ ) 5(x ) + 5(x+ ) 5(x ) tan x + tan y + tan z tan x+ y+ z xyz xy yz zx vc x, y, z j[kus ij tan ()+tan ()+tan () tan + + 6 tan 6 6 ¼ vad½ ¼ vad½ tan 0 0 tan (0) ¼ vad½ ;k 0 ¼ vad½ ijurq] rhu /kukred d.k a dk eku 'kwu; ugè g ldrka vr% tan ()+tan ()+tan () ¼ vad½ v kok dk mùkj& L.H.S. cos 4 5 5 sin sin 4 5 + sin 5 ¼ vad½ sin sin 6 + sin 5 5 5 + 5 sin ()
sin sin 5 5 9 + 5 69 5 5 4 + 5 5 6 0 sin + 65 65 sin 56 65 ¼ vad½ ¼ vad½ 56 cos 65 cos 65 ¼ vad½ mùkj&4- ekuk& f (x) sin x f (x + h) sin (x + h) sin (x + h) ¼ vad½ dy ge tkurs gsa fd& dx f(x+ h) f(x) lim h 0 h sin(x + h) sinx lim h 0 h v kok dk mùkj& fn;k x;k gs& log Y us ij x + h + x x + h x cos sin lim h 0 h cos(x + h)sin h lim h 0 h ¼ vad½ lim cos (x + h) lim sin h h 0 h 0 h cos (x + 0) cos x ¼ vad½ y e x+ex+... y e x+y log y log e x+y log y (x + y) log e ¼ vad½ (log e ) (4)
log y x + y log y y x x ds lkis{k vodyku djus ij& d (log y y) d dx dx (x) d dy log y dx dx dy dy ydx dx mùkj&5- fn;k x;k gs& ¼ vad½ ¼ vad½ ¼ vad½ dy dx y y y log x + log x + log x +... y log x + y ¼ vad½ y log x + y ¼ vad½ x ds lkis{k vodyku djus ij& y dy dx + dy ¼ vad½ x dx (y ) dy dx x dy dx ¼ vad½ x(y ) v kok dk mùkj& fn;k gs& y x ds lis{k vodyku djus ij& sin x cosx + d d dy dx ( + cos x) sin x sin x ( + cos x) dx dx ( + cos x) ¼ vad½ ( + cos x)cos x sin x( sin x) ( + cos x) cos x + cos x + sin x (+ cos x) ¼ vad½ (5)
cos x + ( + cos x) + cosx mùkj&6& fn;k gs s 5 e t cos t x ds lkis{k vodyku djus ij& ¼ vad½ v osx ds 5 [e t ( sin t) + cos t ( e t )] ¼ vad½ dt osx v 5 [ e t. sin t e t. cos t] 5e t (sin t + cos t) ¼ vad½ Roj.k a dv 5[e t (cos t sin t)+(sin t+cos t]( e t ) dt 5e t [cos t sin t sin t cos t] 5e t ( sin t) ¼ vad½ Roj.k a 0e t. sin t vc t ij d.k dk osx υ 5 e / sin + cos 5 e / ( + 0) 5 e / bdkbz v j Roj.k a 0 e / sin / 0 e / 0 e / bdkbz ¼ vad½ v kok dk mùkj& fn;k gs& s 490 t 4.9 t -----¼½ t ds lkis{k vodyku djus ij& osx υ ds 490 t 9.8 t ------¼½ ¼ vad½ dt vf/kdre Å pkbz ij osx 'kwu; g rk gsa vr% 490 9.8 t 0 t 490 ¼ vad½ 9.8 t 50 lsd.m vf/kdre Å pkbz s 490 50 4.9 (50) ¼ vad½ 4500 50 (6)
mùkj&7 lg&laca/k xq.kkad& v kok dk mùkj& ge tkurs gsa fd& r (x, y) r (x, y) 50 ehvj¼ vad½ cov (x, y) var(x) var(y).5 6.5 0.5.5 6.565.5.5 0. mùkj ls ¼ vad½ ¼ vad½ ¼ vad½ σ x y [ x y) (x y) ] x Σ ¼ vad½ [ x x) (y y) ] x Σ Σ x x) (y y) (x x)(y y x ¼ vad½ Σ(x x) + Σ(y y) x x Σ(x x)(y y) ¼ vad½ x σ x + σ y r σ x σ y Σ(x x) (y y) r n σx σy r σ x σ y σ x + σ y σ (x y) ¼ vad½ r x y (x y) σ +σ σ σ σ mùkj&8- lekj;.k js[kk& x 9y + 6 0 9y x + 6 x y 9 x + 6 9 y (7)
y 9 x + b yx ¼y dh x ij lekj;.k js[kk½ 9 iqu% x y + 0 x y b xy ¼x dh y ij lekj;.k js[kk½ r r xy bxy byx ¼ vad½ ¼ vad½ 9 4 9 ¼ vad½ b yx 9 σy r σ x 9 σy 9 σy 9 9 Qσ x 9 σ x Q ¼ vad½ v kok dk mùkj& ekuk XokfYk;j es a olrq ds ewy; d x r kk Ò ikyk es a y ls n'kkz;k tkrk gs r Á'ukuqlkj& x 70 y 75 σx.5 σy.0 r 0.8 ¼ vad½ y dh x ij lekj;.k js[kk dk lehdj.k σy y y r ( x x ) σx (8)
y 75 0.8.0 (x 70).5 ¼ vad½ y.4 (n 70) + 75.5 y 4 (00 70) + 75 5 ¼ vad½ 4 0 5 + 75 44 5 + 75 Œ[fn;k gs x 00] 8.8 + 75 0.80 #- ¼ vad½ mùkj&9- fn, x, leryk dk lehdj.k x y + z -------¼½ blds lekurj leryk dk lehdj.k x y + z + k 0 -----¼½ ¼ vad½ fcunq (,, ) ls leryk ij MkY x, YkEc dh YkEckbZ gs vr% + ( )() + + k + () + () ¼ vad½ 4 + 6 + k 9 + + k + k + k + k 0 ;k 6 ¼ vad½ k 0 j[kus ij leryk dk lehdj.k x y + z 0 r kk k 6 j[kus ij x y + z 6 0 ¼ vad½ v kok dk mùkj& fn, x, leryk a ds lehdj.k x y + z + 0 -----¼½ r kk 4x 4y + z + 5 0 x y + z + 5 0 ------¼½ ¼ vad½ (9)
n lekarj leryk ds chp dh nwjh d d d d a + b + c 5 + ( ) + () 9 ¼ vad½ ¼ vad½ 6 mùkj&0 fn;k gs a r i $ $ j + k$ b r i $ 4$ j 4k$ r r a.b i $ $ j + k $. i $ 4j $ 4k ;fn buds chp dk d.k θ gs r & lehdj.k ¼½ esa j[kus ij cos θ r a r b r b cos θ $ ( ) ( ) (0) ¼ vad½ [ + ( ) ( 4) + ( ) ( 4)] 6 + 4 4 6 a.b r r a. b Q i + j + k$ $ $ $ i± $ jk$ ki $ $ ¼ vad½ 0 r r ------¼½ ¼ vad½ + ( ) + () ¼ vad½ 6 () + ( 4) + ( 4) + 6+ 6 ¼ vad½ 4 6 6 4 6 4 ¼ vad½
iz'u&0 dk v kok dk mùkj& fn;k& ABC esa\ θ cos 6 4 mùkj& fn;k gs& fn;k gs& BC uuur i $ + 4$ j + 5k $ uuuur BC + 4 + 5 9+ 6+ 5 50 uuuur 5 --------¼½ ¼ vad½ BC CA uuur 4i $ j $ 5k $ CA uuur 4 + ( ) + ( 5) 6 + 9 + 5 50 5 -------¼½ ¼ vad½ BA uuur 7i $ + $ j BA uuur 50 7 + 5 ------¼½ ¼ vad½ uuur uuur uuur f«kòqt dh ifjeki AB + BC + CA fn;k gs& f (x) + x ij 5 + 5 + 5 5 ¼ vad½ x+ x ()
f () + + (i) f () ¼ vad½ (ii) f ( + 0) lim h 0 ( + h) ( + h) + lim h 0 ( h) + + + h lim + h + h h 0 ¼ vad½ (iii) f ( 0) lim h 0 f ( h) ( h) + lim h 0 ( h) + h lim h + h h 0 ¼ vad½ f ( 0) Œ f () f ( + 0) f ( 0) ¼ vad½ vr% fn;k x;k QYku x ij lkarr; gsa Á'u& dk v kok dk mùkj& fn;k x;k QYku x ds fyk, dk :i /kkj.k dj Y rk gs r kk va'k o gj d x ls Òkx nsus ij& ¼ vad½ lim (x ) (4x ) x o (x + 8) (x ) 4 x x lim 8 + x x x o ¼ vad½ ¼ vad½ () (4) () () ¼ vad½ ()
a mùkj& nh xbz js[kk y x ------¼½ fn;k x;k oø y 6x -----¼½ lehdj.k ¼½ es y x j[kus ij y 6y y 6 0 y (y 6) 0 y 0 ;k y 6 rc x 0 r y 0 x 6 r y 6 vr% js[kk o oø d ÁfrPNsn fcunq O (0, 0) o B (6, 6) g axs ¼ vad½ vòh"v { «kqyk 6 (y y ) dx 0 6 0 4 4 (4 x x) dx 6 6 xdx 0 0 xdx / 6 6 x x / ¼ vad½ / 6 0 [6 0] ¼ vad½ 8 64 56 ()
5 8 5 84 8 v kok dk mùkj& fn, x, ijoyk; dk lehdj.k gs& y 4x x 4y ¼ vad½ --------¼½ -------¼½ ¼ vad½ lehdj.k ¼½ o ¼½ d gyk djus ij& y 4 ( 4y) y 8 y y 4 64y y 4 64y 0 y (y 64) 0 y 0 y 4 ;fn y 0 x 0 y 4 x 4 js[kkafdr Òkx dk { «kqyk Kkr djuk gsa { «kqyk 4 0 (y y ) dx 4 4 4x 0 0 x dx 4 ¼ vad½ ¼ vad½ (4)
4 4 x dx x dx 0 4 0 4 / x x 4 0 [(4)/ 0] 4 8 64 6 4 0 4 [4 0] 6 6 vòh"v { «kqyk 6 [ ] oxz bdkbz ¼ vad½ oxz bdkbz mùkj& cos x/ ls Òkx nsus ij dx 5 4sinx dx x x x x 5 cos + sin 4 sin cos x sec dx x x 5 + tan 4tan x sec dx x x 5 tan 4tan + 5 ¼ vad½ tan x t j[kus ij (5)
sec x. sec x dx dt dx dt dt 5t 8t + 5 dt 5 8 t t + 5 dt 4 + 5 5 t. t dt 5 4 6 t + 5 5 dt 5 4 9 t + 5 5 dt 5 4 t + 5 5 tan 5 5 5t 4 tan 4 t 5 5 Q ¼ vad½ dx x tan x + a a a iz'u- dk v kok dk mùkj& 5tan 4 tan / 0 sin x dx sin x + cos ¼ vad½ --------¼½ (6)
a f(a)dx a 0 0 f(a x)dx / 0 sin x sin x + cos x ¼½ v j ¼½ t M+us ij& / 0 cos x cos x + sin x / + 0 / 0 dx [ ] / 0 sin x sin a + cosx dx cosx --------¼½ ¼ vad½ ¼ vad½ 4 mùkj&4 fn;k x;k lehdj.k gs& y x + ax + bx + c x ds lkis{k vodyku djus ij dy dx x + ax + b iqu% x ds lkis{k vodyku djus ij& (7) ¼ vad½ ¼ vad½ dy 6x + a ¼ vad½ dx iqu% x ds lkis{k vodyku djus ij dy 6 ¼ vad½ dx iz'u&4 dk v kok dk mùkj& dy fn;k x;k lehdj.k dx y ex ;g y ds,d jsf[kd vodyku lehdj.k gs- vr% bldh rqykuk O;kid lehdj.k dy + Py Q ls djus ij ¼ vad½ dx
P, Q e x lekdyku xq.kkad.f. e Pdx e Pdx e x ¼ vad½ vr% y (.F.) Q (.F.) dx + c y e x ex. e x dx + c y e x ex dx + c y e x n e + c ¼ vad½ y ex + c e x ¼ vad½ mùkj&5 Ykhi o"kz esa dqyk fnu a dh la[;k 66 g rh gs v kkzr~ jfookj lfgr 5 iw.kz lrkg rd fnu vfrfjä g axsa ;s fueufykf[kr g ldrs gsa& ¼ vad½ ¼½ l eokj] eaxykokj ¼½ eaxykokj] cq/kokj ¼½ cq/kokj] xq#okj ¼4½ xq:okj] 'kqøokj ¼5½ 'kqøokj] 'kfuokj ¼6½ 'kfuokj] jfookj ¼7½ jfookj] l eokj ¼ vad½ bu lòh leleòkoh fl kfr; a esa ls vafre n fl kfr;k jfookj g us ds vuqdwyk gs] vr% & vfò"v Ákf;drk?kVuk d s vudq Ywk fl kfr; adh la[;k dyq klaòkforfl kfr; adh l[a;k ¼ vad½ iqu% Ákf;drk 7 iz'u&5 dk v kok dk mùkj& n u a iùk a ds YkkYk g us dh Ákf;drk P (A) 6C 5 C 6 5 5 5 ¼ vad½ ¼ vad½ (8)
5 6 5 5 6 n u a iùk a ds bôk g us dh Ákf;drk & 4C P (B) 5 C 4 5 5 6 6 blh Ádkj n YkkYk bôs g us dh Ákf;drk & P (A B) 6 vhkh"v izkf;drk P (A B) P (A) + P (B) P (A B) mùkj&6 nh xbz ljyk js[kk, vksj x x rc U;wure nwjh SD 5 + 6 6 6 6 0 6 55 ¼ vad½ ¼ vad½ ¼ vad½ ¼ vad½ y z 4 -------¼½ y 4 z 5 4 5 ------¼½ α α β β γ γ a b c a b c (bc b c ) ¼ vad½ (9)
SD 4 5 4 4 5 ( 5 4 4) + ( 5 4) + ( 4 ) ¼ vd½ SD SD SD 4 4 5 ( ) + ( ) + ( ) [(5 6) (0 ) + (8 9)] 6 + 4 6 ¼ vad½ ¼ vad½ SD 6 ¼ vad½ iz'u&6 dk dk v kok dk mùkj& ekuk x Y dk lehdj.k& x + y + z + ux + vy +wz + d 0 --------¼½ fcunq (,, 4) ] (, 5, ) v j (,, 0) bl ij fl kr gsa vr% +9+6+u 6v+8w+d 0 ;k 6+u 6v+8w+d 0 --------¼½ +5+4+u 0v+4w+d 0 ;k 0 + u 0v + 4w + d 0...() +9+0+u 6v+0+d 0 ;k 0 + u 6v + d 0...(4) ¼ vad½ lehdj.k ¼½ esa ls lehdj.k ¼½ d?kvkus ij& 4v + 4w 4 0 ;k v + w 0 ------¼5½ ¼ vad½ lehdj.k ¼½ esa ls lehdj.k ¼4½?kVkus ij 8w + 6 0 ;k w ¼6½ ¼ vad½ lehdj.k ¼5½ esa j[kus ij& v 0 ;k v ------¼7½ ¼ vad½ x Y dk dsuæ ( u, v, w) leryk x + y + z 0 ij fl kr gs] blfyk, u v w 0 u + v + w 0 --------¼8½ v v j w j[kus ij (0)
a u + 0 u ¼ vad½ lehdj.k ¼4½ es a u o v ds eku j[kus ij 8 + d + 0 0 ;k d 0 lehdj.k ¼½ es u, v, w, d ds eku j[kus ij x Y dk lehdj.k& x + y + z + x + 6y 4z + 0 0 ¼ vad½ mùkj& 7 r r r r r r L.H.S. a+ b,b+ c,c+ a r r r r r r ( a+ b ). b+ c c+ a r r r r r r r r r r ( a+ b ). b c+ b a+ c c+ c a r r r r r r r r ( a+ b ). b c+ b a+ c a ¼ vad½ r r r r r r r r ( a+ b ). b c a b+ c a r r r r r r r r r a. ( b c) a ( a b) + a( c a) ¼ vad½ rrr r rr rr r rrr abc a.ab + a.c.a + bbc rrr r rr bab + b ca ¼ vad½ rrr r rr abc 0 + 0 + 0 0 + bca ¼ vad½ r r r r r r abc + a b c ¼ vad½ rrr abc v kok dk mùkj& lfn'k $ i j $ + k$ dh fn'kk esa ekud lfn'k fn;k gqvk lfn'k cyk F r 5 $ i j $ + k$ + 4+ 4 $ i j $ + k$ $ i j $ + k$ ¼ vad½ ()
( ) 5 $ i j $ + k$ ¼ vad½ ekuk lfn'k $ i+ $ j+ k$ r kk i $ j $ + k $ Øe'k% fcunq O o P d fu:fir djrs gsaa rc& i $ j $ + k$ $ i + $ j + k ¼ vad½ cyk F r uuur r OP r $ ( ) ( ) $ i j $ + k$ dk fcunq O ds ifjr% vk?kw.kz r r r F $ i j $ + k$ $ i j $ + k 5 $ ( ) ( ) $ i $ j k$ ( ) ¼ vad½ 4 4i $ $ j + k$ ¼ vad½ ()