mathematics, set 2, model papers of madhya pradesh board of secondary education, xiith class

7/29/2019 Mathematics, set 2, Model papers of Madhya Pradesh Board of Secondary Education, XIIth Class

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'u&ik CYkw fUV

ijh{kk & gk;j lsds.Mjhd{kk&12 iw .kkd&100fo"k;& mPp xf.kr le;&3 ?k.Vk

- bdkbZ bdkbZ ij oLrq fu"B dq Ykfu/kkZfjr 'u vadokj 'u a dh la[;k 'u vad

1 vad 4 vad 5 vad 6 vad

1- vkaf'kd f 05 01 01 & & 01

2- izfrYke QYku 05 01 01 & & 01

3- lerYk T;kferh;

4- lerYk 15 04 & 01 01 025- ljYk js[kk ,oa xYkk

6- lfn'k

7- lfn'k a dk xq.kuQYk 15 04 & 01 01 02

8- lfn'k a dk fkfoeh;T;kferh; esa vuq;x

9- QYku] lhek rkk 05 & & 01 & 01

lkarR;10- vodYku 10 02 02 & & 02

11- dfBu vodYku

12- vodYku dk vuq;x 05 01 01 & & 01

13- lekdYku

14- dfBu lekdYku 15 05 & 02 & 02

15- fuf'pr lehdYku

16- vodYk lehdj.k 05 & & 01 & 01

17- lglaca/k 05 01 01 & & 01

18- lekJ;.k 05 01 01 & & 01

19- kf;drk 05 & & 01 & 01

20- vkafdd fof/k;k 05 05 & & & &

dqYk 100 25 07 07 02 16


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gk;j lsd.Mjh Ldwy ijh{kk&2012&2013HIGHER SECONDARY SCHOOL EXAMINATION

kn'kZ 'u&i=

Model Question Paper

mPp xf.krHIGHER MATHEMATICS

(Hindi and English Versions)

Time& 3 aVs Maximum Marks100funsZ'k&

(1) lh 'u gy djuk vfuok;Z gSA(2) 'u a ij vk/kkfjr vad muds lEeq[k n'kkZ, x, gSaA(3) 'u - 1 ls 5 rd oLrqfu"B 'u gSaA(4) 'u 6 ls 21 rd R;sd 'u esa vkarfjd fodYi fn, x, gSaAInstructions(1) All questions are compulsory to solve.(2) Marks have been indicated against each question(3) From Question No. 1 to five are objective type questions

(4) Internal options are given in question No. 6 to 21

[k.M&v Section-A oLrqfu"B 'u (Objective Type Questions)

'u&1 R;sd oLrqfu"B 'u esa fn, x, fodYia esa ls lgh mkj pqfu,&

(i)1

( 2)x x + ds vkaf'kd f gSa&

(a)1 1

1x x

+(b)

1 1

2 1x x+

+ +

(c)1 1 1

2 2x x

+ (d)

1 1

2 1x x

+ +

(ii) tan11

2

+ tan11

3 dk eku gS&

(a) tan11

6(b)

3

(d)4

(d)

6

(iii) fcUnq (4, 3, 5) dh Y v{k ls nwjh gS&

(a) 34 (b) 5

(c) 41 (d) 15


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(iv) ml lerYk dk lehdj.k t v{k a ls bdkbZ YkackbZ ds vUr% [k.M dkVrk gS]gS&(a) x + y + z = 0 (b) x + y + z = 1

(c) x + y + z = 3 (d) x + y + z = 1

(v) ;fn fdlh js[kk ds fnd~vuqikr 1, 3, 2 g r ml js[kk dh fnddT;k,

gSa&

(a)1 3 2

, ,14 14 14

(b)

1 2 3, ,

14 14 14

(c)1 3 2

, ,14 14 14

(d)

1 2 3, ,

14 14 14

Write the correct anwer from the given options provided in every objectivetype question

(i) Partial fractions of

1

( 2)x x + are

(a)1 1

1x x

+(b)

1 1

2 1x x+

+ +

(c)1 1 1

2 2x x

+ (d)

1 1

2 1x x

+ +

(ii) The value of tan11

2+ tan1

1

3 is

(a) tan1 16

(b)3

(d)4

(d)

6

(iii) Distance of the point (4, 3, 5) from Y axis is

(a) 34 (b) 5

(c) 41 (d) 15

(iv) Equation of a plane which cuts the unit intercepts with the coordinate axis

is(a) x + y + z = 0 (b) x + y + z = 1

(c) x + y + z = 3 (d) x + y + z = 1

(iv) If direction ratios of a line are 1, 3, 2 then direction consines of line

are

(a)1 3 2

, ,14 14 14

(b)

1 2 3, ,

14 14 14

(c)

1 3 2

, ,14 14 14

(d)

1 2 3

, ,14 14 14


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'u&2- fuEufYkf[kr dkua esa lR;/vlR; dku NkVdj viuh mkjiqfLrdk esafYkf[k,A

(i) fcUnqv a (1, 2, 3) vj (4, 5, 6) d feYkkus okYkh js[kk ds fnd~vuqikr 5,3, 9 gSaA

(ii) rhu vlekUrj] v'kwU; lfn'k lerYkh; gus ds fYk, mudk vfn'k fkd

xq.kuQYk 'kwU; grk gSA(iii) sin (cos1x) dk vodYku xq.kkad 'kwU; grk gSA(iv) sin x + cos x dk egke eku 2 gSA

(v) . .ii j j k k + + dk eku 3 gSAWrite True / False in the following statement

(i) The direction ratios of the line joining the points (1, 2, 3) and (4, 5, 6)

are 5, 3, 9

(ii) If three non parallel non zero vectos are coplaner than the scalar triple

product of them will be zero.

(iii) The differential Coefficient of sin (cos1x) is zero.

(iv) The maximum value of sin x + cos x is 2

(v) The value of . .ii j j k k + + is 3.'u&3- fj Lkkua dh iwfrZ dhft,&

(i) sin x dk nok vodYkt --------------- ------- grk gSA

(ii) Yxzkt loZlfedk ls ( ) ( ). .a b c d H H H KH

-----------------------

(iii) x

y d -------------------- lekJ;.k xq.kkad dgrs gSaA

(iv) ;fn 0.75 r < 1 g r pj a esa ----------------- lg&laca/k grk gSA

(v) nks v'kwU; lfn'k aH

vkSj bH

lekarj gksrs gSa ;fn vkSj ;fn---------------- -Fill in the Blanks

(i) The nth derivative of sin x is ----------------------

(ii) By Lagrang's inequality ( ) ( ). .a b c d H H H KH

= -----------------------

(iii) xy

is -------------------- regression Coefficient

(iv) If 0.75 r < 1 then ----------------- Co-relation in variables.

(v) Two non zero vectors aH and b

H are parallel if and only if -----------------

'u&4- [k.M v ds fYk, [k.M c esa ls lgh mkj pqudj tM+h cukb,AMatch the Column by choosing from section (B) for section (A)

[k.M v [k.M c(Section-A) (Section-B)

(i) tanx dx (a) sin1x

a+ c


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(ii) 21

1 x+ dx (b)1

asec1

x

a

+ c

(iii) 2 2

1

a x dx (c) cosec1 x

a+ c

(iv) 2 2

1

a x

(d) log (cos x) + c

(v) 2 21

a x

+ dx (e) tan

1x + c

(f) cos1x

a

+ c

(g)1

a cot1

x

a

+ c

'u&5- fuEufYkf[kr 'ua ds mkj ,d 'kCn/okD; esa fYkf[k,&(i) U;wVu jSQlu fof/k ls fdlh la[;k dk oxZewYk~ Kkr djus dk lwk fYkf[k,A(ii) vkafdd fofk;ksa ls lacaf/kr leyEc prHkZqt fu;e gsrq lw+= fyf[k,A(iii) vkafdd fofk;ksa ls lacaf/kr flEilu dk ,d frgkbZ fu;e fyf[k,A(iv) U;wVu jSQlu fof/k ls fdlh la[;ky dk ?kuewYk Kkr djus dh fof/k dk lwk

fYkf[k,A(v) 0.3542E05 + 0.2681 E05 dk eku fYkf[k,A

Write the anwer of each question in one word/sentence of the following(i) Write the formula for square root of a number by Newton Reiphsons

method.

(ii) Write the formula Simpson's rule related by numerical method.

(iii) Write one thrid rule of Simson's related to numerical method.

(iv) Write the formula for cube root of a number by Newtan's reiphsons

method.

(v) Write the value of 0.3542E05 + 0.2681 E05.

[k.M&c(Section-B)

vfrYk?kq mkjh; 'u (Very Short Answer Type Questions)'u&6- fuEu O;atda d vkaf'kd f esa O; dhft,A 4 vad

2

1

5 6x x +Solve following expression into partial Fractions

2

1

5 6x x +vkok (or)


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2 5

( 1) ( 2)

x

x x

+ d vkaf'kd f a esa O; dj&

2 5

( 1) ( 2)

x

x x

+ Solve in partial fractions

'u&7- fl) dhft, fd 4 vad

tan11

a b

ab

+ + tan

11

b c

bc

+ + tan

1

1

c a

ca

+ = 0

Prove that- tan11

a b

ab

+ + tan

11

b c

bc

+ + tan

1

1

c a

ca

+ = 0

vkok (or)

fl) dj fd sin13

5

+ cos112

13

= sin156

65

Prove that sin13

5+ cos1

12

13= sin1

56

65'u&8- ;fn y = log x + log x + log x .......... g r 4 vad

fl) dhft, fd1

(2 1)

dy

dx x y=

If y = log x + log x + log x .......... then prove

that

1

(2 1)

dy

dx x y=

vkok (or)

y =1 x

1 x

+

dk x ds lkis{k vodYku dhft,A

Differentiate y =1 x

1 x

+

with respect to x.

'u&9- log tan4 2

x + dkxds lkis{k vodYku djA 4 vad

Differentiate log tan4 2

x + with respect to x

vkok (or)

tanx dk ke fl)kar ds }kjk vodYk xq.kkad Kkr djA

Find differential coefficient of tanx by the first principle

'u&10- ,d d.k /okZ/kj ij dh vj Qsadk tkrk gSA xfr dk lehdj.kS = ut 4.9 t2 gSA 20 ehVj dh pkbZ ij igqpus ds fYk, d.k dk kjafd

osx Kkr djA 4 vad


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A particle is thrown vertically upwards. The law of motion is S = ut 4.9

t2. Find the initial velocity of the particle to reach the height of 20 metres

vkok (or)fl) dhft, QYku x3 3x2 + 3x + 7 dk eku fcUnq x = 1 ij u r mfPp"Bvj u gh fufEu"B gSA

Prove that function x3 3x2 + 3x + 7 neither have a maxima nor minima atx = 1

'u&11- fuEu vkdM+a ls lekJ;.k js[kkva ds lehdj.k Kkr djA 4 vad

x 2 4 6 8 10

y 6 5 4 3 2

Find the equation of regression of lines from the following data

x 2 4 6 8 10

y 6 5 4 3 2

vkok (or)fuEu vkdM+ a ds vk/kkj ijx dhy ij lekJ;.k js[kk dk lehdj.k Kkr djA;fn y = 90 r x dk eku Kkr djAJs.kh x ylekUrj ek/; 18 100ekud fopyu 14 20

x vj y esa lg laca/k xq.kkad = 0.8From the followoing data. Find the line of regression ofx on y and estimate

the value of x, if y = 90.Series x y

Arithmetic Mean 18 100

Standard Deviation 14 20

Coefficient of coreelation between x and y = 0.8'u&12- dkYkZ fi;lZu fof/k dk ;x dj fuEu vkdM+a ls lg&laca/k xq.kkadKkr dhft,A 4 vad

x 3 4 6 8 9

y 90 100 130 160 170

Find the coefficient of correlation between x and y by using Karl Pearson'smethod from following data

x 3 4 6 8 9

y 90 100 130 160 170

vkok (or)fl) dhft, fd lg&laca/k xq.kkad r dk eku 1 ls +1 ds chp grk gSAProve that the value of coefficent of correlation lines between 1 and +1

y?kq mkjh; 'u (Short Answer Type Questions)'u&13- ,d lerYk v{k a d fcUnqABC ij feYkrk gS] blls cus ABC dk


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dsUd (a, b, c) gSA fl) dhft, fd lerYk dk lehj.k 3x y z

a b c+ + = gSA

5 vad

A plane intercepts the coordinate axis at ABC respectively the centroid of

ABC is (a, b, c) then prove that the equation of plane is 3x y za b c

+ + = -

vkok (or)ml lerYk dk lehdj.k Kkr dj t ewYk fcUnq ls gdj tkrk gS vj lerYk ax + 2y z = 1 rkk 3x 4y + 2 = 5 ij Ykac gAFind the equation of a plane which passes through the origin and perpendicular

to the planes x + 2y z = 1 and 3x 4y + 2 = 5.

'u&14- ;fn ABC dk dsUd G g r fl) dhft, fd 5 vad

GA GB GC O+ + =

KKKH KKKH KKKH KH

If G be the centriod of a triangle ABC then prove that GA GB GC O+ + =KKKH KKKH KKKH KH

vkok (or)lfn'k a ds ;x dk lkgp;Z fu;e fYkf[k, ,oa mls fl) dhft,State and prove Associative law of vector addition

'u&15- ;fn f(x) = log1

1

x

x

+

g r fl) dj fd& 5 vad

f (a) + f (b) = f 1

a b

ab

+

+

If f(x) = log1

1

x

x

+

then prove that

f (a) + f (b) = f1

a b

ab

+ +

vkok (or)

30

tan sinlim

x

x x

x

dk eku Kkr djA

Evaluate 30

tan sinlim

x

x x

x

'u&16-5 4sin

dx

x+ dk eku Kkr dj& 5 vad

Evaluate5 4sin

dx

x+vkok (or)


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22 6 8

dx

x x+ + dk eku Kkr dj

Evaluate 22 6 8

dx

x x+ +

'u&17- nh?kZo`k2 2

2 2

x y

a b+ = 1 dk {kQYk Kkr djA 5 vad

Find the area of ellipse

2 2

2 2

x y

a b+ = 1

vkok (or)

fl) dhft, fd/ 2

0

sin

sin cos

x

x x

+ dx = 4

Prove that / 20

sinsin cos

xx x

+ dx = 4

'u&18- vodYk lehdj.k gYk dhft,A

sec2x tan y dx + sec2 y . tan x dy = 0

Solve the following differential equation

sec2x tan y dx + sec2 y . tan x dy = 0

vkok (or)

fuEu vodYk lehdj.k d gYk djA(1 + y2) dx = (tan1 y x) dy

Solve the following differential equation

(1 + y2) dx = (tan1 y x) dy

'u&19- fdlh 'u d gYk djus ds A ds frdwYk la;xkuqikr 4 : 3 rkkmlh 'u d gYk djus ds B ds vuwdwYk la;xkuqikr 7 : 5 gSaA ;fn nuagYk djus dh df'k'k djrs gSa 'u ds gYk gus dh kf;drk Kkr djA

5 vad

The odds against A solving a problem are 4 : 3 and odds in favour of Bsolving that problem are 7 : 5. What is the probability that the problem will

be solved if they both try.

vkok (or),d ikls d n ckj mNkYkk tkrk gSA 4 ls vf/kd vad vkuk lQYkrk ekuk tkrkgSA lQYkrv a dh la[;k dk kf;drk caVu Kkr dhft,AA die is thrown twice A number greater than 4 is taken success find the

probablity distribution of number of successes.


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nh?kZ mkjh; 'u (Long Answer Type Question)

'u&20- fl) dhft, fd js[kk,1 2 3

2 3 4

x y z = = vj.

2 3 4

3 4 5

x y z = =

lerYkh; gSaA bu js[kkva ds frPNsn fcUnq Kkr djA

6 vad

Prove that the lines1 2 3

2 3 4

x y z = = and

2 3 4

3 4 5

x y z = = are coplaner

also find the point of intersection of lines.

vkok (or)

ml xY dk lehdj.k Kkr dj t fcUnqv a (1, 3, 4)] (1, 5, 2) vj (1, 3, 0) ls gdj tkrk gS rkk ftldk dsU lerYk x + y + z = 0 ij fLkr gSAFind the equation of sphere which passes through the points (1, 3, 4),

(1 5, 2) and (1, 3, 0) whose centre lines on the plane x + y + z = 0

'u&21- fuEu js[kkva ds chp dh U;wure nwjh Kkr djA 6 vad

( )2= + + + +H

r i j k i j k

( )2 2 2= + + +H

r i j k i j k

Find the shortest distance between two following lines

( )2= + + + +H

r i j k i j k

( ) ( )2 2 2= + + +H

r i j k i j k

vkok (or)fuEu fcUnqv a ls gdj tkus okY lerYk dk lehj.k Kkr djA

2 6 6 , 3 10 9 + + i j k i j k vj 5 6 i kFind the equation of plane passing through the points.

2 6 6 , 3 10 9 + + i j k i j k and 5 6 i kvkn'kZ mkj

mkj&1

(i) (c)1 1 1

2 2x x

+ (ii) (c)

4

(iii) (c) 41 (iv) (b) x + y + z = 1

(v) (a)1 3 2

, ,14 14 14

mkj&2

(i) vlR; (ii) lR;(iii) vlR; (iv) vlR;(v) lR;


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mkj&3

(i) sin2

nx

+

(ii)ac aa

bc bd

H H H H

HH HKH

(iii) x dky ij (iv) mPpLrjh; /kukRed

(v) 0a b =H H

mkj&4- lgh tfM+;k&[k.M v [k.M c

(i) tanx dx (d) log (cos x) + c

(ii) 21

1 x+ dx (e) tan1 x + c

(iii)2 2

1

a xdx (a) sin1

x

a+ c

(iv) 2 2

1

a x

(c) cosec1 + x

ac

(v) 2 21

a x

+ dx (g)

1

acot1

x

a

+ c

mkj&5-

(i) xn+1 =

1 Nx

n2 xn

+

(ii)b

a

hf(x)dx

2= [yo + yn + 2 {y1 + y2 + ...........yn1}] ;gk h =

b a

n

(iii)b

a

hf(x)dx

3= [yo + 4 (y1 + y3 + ........yn1) + 2 (y2 + ........yn1) + yn]

(iv) xn+1 = 2

12

3

+

n

n

Nx

x

mkj&6-gj ds xq.ku[kaM djus ij

x2 5x + 6 = x2 2x 3x + 6

= x (x 2) 3 (x 2)

= (x 2) (x 3) 1 vad

vr% 21

5 6x x +=

1

( 3) ( 2)x x

2

1

5 6x x + =1

( 3) ( 2)x x = ( 3) ( 2)A B

x x+ ekuk .........(I) 1 vad


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;k1

( 3) ( 2)x x =( 2) ( 3)

( 3) ( 2)

A x B x

x x

+

;k 1 = A (x 2) + B (x 3) ..........(II)lehdj.k II esa x 3 = 0 x = 3 j[kus ij

1 = A (3 2) + 0 1 vad A = 1blh dkj lehdj.k II esa x 2 = 0 x = 2 j[kus ij

1 = 0 + B (2 3)

;k 1 = B B = 1

vr% 21

5 6x x +=

1 1

3 2x x

1 vad

'u&6 dk vkok dk mkj

2 5

( 1) ( 2)

x

x x

+ = 1 2

A B

x x+

ekuk ........(I) 1 vad

;k2 5

( 1) ( 2)

x

x x

+ =

( 2) ( 1)

( 1) ( 2)

A x B x

x x

+

;k 2x + 5 = A (x 2) + B (x 1) .........(II) 1 vadlehdj.k II esa x 2 = 0 x = 2 j[kus ij

2 2 + 5 = 0 + B (2 1)

;k B = 9 1 vadlehdj.k II esa x 1 = 0 x = 1 j[kus ij2 1 + 5 = A (1 2) + 0

;k 7 = A A = 7vr% okafNr vkaf'kd f

2 5

( 1) ( 2)

x

x x

+ =

7 9

( 1) 2x x

+

1 vad

'u&7 dk mkjge tkurs gSa fd&

lwk tan1 x tan1 y = tan1( )

(1 )

x y

xy

+ .....(I) 1 vad

mi;qZ lwk esax = a ,oa y = b j[kus ij

tan1 a tan1 b = tan11

a b

ab

+ .....(II) 1 vad

blh dkj


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tan1 b tan1 c = tan11

b c

bc

+ .....(III)

,oa tan1 c tan1 a =1

c a

ca

+ ......(IV)

vr% tan11

a b

ab

+ + tan

11

b c

bc

+ + tan

11

c a

ca

+

= tan1 a tan1 b + tan1 b tan1 c

+ tan1 c tan1 a

= 0 1 vad

vr% Li"Vr% tan11

a b

ab

+ + tan

11

b c

bc

+ + tan

11

c a

ca

+ 1 vad

vkok

sin13

5+ cos1

12

13 = sin1

56

65 1 vad

ekuk cos112

13= ......(I)

rc cos =12

13led .k ABC esa

AC2 = AB2 + BC2

;k 132 = AB2 + (12)2

;k AB = 5

vr% sin =5

13 = sin1

5

13 ......(I)

Li"V% cos112

13= sin1

5

13lehdj.k (I) ls

vc L.H.S. sin13

5

+ cos112

13

= sin13

5+ sin1

5

13......(II)

ge tkurs gSa fd lwk&

sin1 x + sin1 y = sin12 2

1 1x y y x + 1 vad

vr% sin13

5+ cos1

12

13

= sin15

3+ sin1

513


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= sin1

2 23 5 5 3

1 15 13 13 5

+

1 vad

= sin1

3 12 5 4

5 13 13 5

+

= sin156

65

1 vad

Li"V% sin15

3+ cos1

12

13 = sin1

56

65

'u&8 dk mkj

fn;k gS& y = log log log .........x x x+ + ......(I)

;k y = logx y+ 1 vadnu a i{k a dk oxZ djus ij

y2 = log x + y

;k y2 y = log x 1 vadnu a i{k a dk x ds lkis{k vodYku djus ij

d

dx(y2 y) =

d

dx(log x) 1 vad

;k 2y dy dydx dx

= 1x

;k (2y 1)dy

dx=

1

x

dy

dx=

1

(2 1)x y 1 vad

vkok dk mkj&

fn;k gS y =1

1

x

x

+

nu a i{k a dk oxZ djus ij

y2 =1

1

x

x

+

........(I) 1 vad

leh- I dk nu a i{k a dk oxZ djus ij

d

dx(y2) =

1

1

d x

dx x

+ 1 vad

;k 2y dydx

= 21 1(1 )x x

x + + kxQYk lwk


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;k 2ydy

dx= 2

2

(1 )x

+

;kdy

dx=

1

(1 ) (1 )y x x

+ +

y dk va'k o gj esa xq.kk djus ij

;kdy

dx= 2(1 ) . (1 )

y

y x x

+ + 1 vad

;kdy

dx=

1. (1 ) . (1 )

1

y

xx x

x

+ + +

y2 =1

1

x

x

+

dy

dx = 2

1

1x 1 vad'u&9 dk mkj

gYk& ekuk fd y = log tan4 2

x + nu a i{k a dk x ds lkis{k vodYku djus ij

dy

dx=

d

dxlog tan

4 2

x +

tan4 2

x + = t ekuk 1 vad

dy

dx=

d

dx(log t)

=d

dtlog t

dt

dx .

dy dy dt

dx dt dx

= 3 1 vad

=1

tan

4 2

. +

d x

t dx

[

4 2

x+ = u Yus ij]

=1

.d

t dutan u

d

dx

4 2

x + 1 vad

=1

tsec2u .

d

dx 4 2

x +

=1

tan

4 2

x +

sec24 2

x + .1

2

u ,oa t ds eku okfil djus ij


16/33

(16)

J

=

2sec1 4 2

2tan

4 2

+ +

x

x

vr% log tan4 2

d x

dx

+ =

2sec1 4 2

2tan

4 2

+ +

x

x1 vad

'u&9 dk vkok dk mkj

gYk& ekuk f (x) = tanx

f (x + h) = tan( )+x h

lwk ddx

f (x) =0

( ) ( )lim

+ h

f x h f xh

1 vad

tand

xdx

=0

limh

tan( ) tan+ x h xh

=0

limh

tan( ) tan tan( ) tan

tan( ) tan

x h x x h x

h x h x

+ + +

+ +1 vad

va'k o gj esa tan( )x h+ tanx dk xq.kk djus ij

=0

tan( ) tanlim

tan( ) tanh

x h x

x h x

+ + + 1 vad

=0

sinlim

2 cos ( ) cos .( tan ( ) tan ) + + +hh

h x h x x h x

= 0 0

1 sin 1lim lim

2 cos ( ) cos . tan ( ) tan

+ + +

h h

h

h x h x x h x

=1 1

2 cos . cos . tanx x x

=

21 sec

2 tan

x

x1 vad

'u&10 dk mkjgYk& fn;k gS fd S = ut 4.9 t2 .........(I)lehdj.k (I) d t ds lkis{k vodYku djus ij


17/33

(17)

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osxdS

dt= u 9.8 t ......(II) 1 vad

tc d.k egke pkbZ ij igqprk gS rc mldk osx 'kwU; gxkrc u 9.8 t = 0

;k t = 9.8u 1 vad

lehdj.k (I) ls 20 = u .9.8

u 4.9

2

9.8

u 1 vad

;k 20 =2 2

9.8 2 9.8

u u

;k u2 = 20 2 9.8;k u2 = 392

;k u = 392

u = 14 2 eh@ls- 1 vad'u&10 dk vkok dk mkj

ekuk y = x3 3x2 + 3x + 7 ......(I)x ds lkis{k vodYku djus ij&

rcdy

dx= 3x2 6x + 3 ......(II) 1 vad

iqu% x ds lkis{k vodYku djus ij2

2

d y

dx= 6x 6 .....(III) 1 vad

QYku ds mfPp"B ;k fufEu"B gus ds fYk,

dy

dx= 0

rc 3x2 6x + 3 = 0;k x2 2x + 1 = 0

;k (x 1)2 = 0x = 1

x = 1 ds fYk,2

2

d y

dx= 6 (1 1)

;k2

2

d y

dx= 0 vfuf'prrk 1 vad

lehdj.k (III) d x ds lkis{k vodYku djus ij

3

3d y

dx= 6 0 1 vad


18/33

(18)

J

vr% QYku dk x = 1 ij u r mfPp"B ,oa u gh fufEu"B eku gSA'u&11 dk mkj

gYk&x y x2 y2 xy

2 6 4 36 12

4 5 16 25 20

6 4 36 16 24

8 3 64 9 24

10 2 100 4 20

x = 30 y = 20 x2 = 220 y2 = 90 xy = 100

2 vadekuk fd y dh x ij lekJ;.k js[kk dk lehdj.k fuEukuqlkj gS&

y = na + bx;k 20 = 5a + 30b .....(I)

mi;qZ lkj.kh ls eku j[kus ij,oa xy = ax + bx2

;k 100 = 30a + 220 b .....(II) 1 vadlehdj.k (I) o lehdj.k (II) d gYk djus ij

a = 7

b =

1

2vr% y dh x ij lekJ;.k js[kk dk lehdj.k

y = 7 0.5 x 1 vad'u&11 dk vkok dk mkj&

gYk& ge tkurs gSa fd lekJ;.k xq.kkad x dky ij

bxy = x

y

1 vad

= 0.8

14

20= 0.56

x dh y ij lekJ;.k js[kk dk lehdj.kx x = bxy (y y ) 1 vad

;k x 18 = 0.56 (y 100) (x = 18 , y = 100);k x = 38 + 0.56 y 1 vadvc pwfd y = 90rc x = 38 + 0.56 90

;k x = 12.4 1 vad


19/33

(19)

J

'u& 12 dk mkj

gYk& X =x

n

=30

5

1 vad

= 6

blh dkj Y =650

5= 130

lg&laca/k xq.kkad gsrq lkj.kh

x y x x y y (xx ).(yy ) (xx )2 (y y )2

3 90 3 40 120 9 1600

4 100 2 30 60 4 900

6 130 0 0 0 0 0

8 160 2 30 60 4 900

9 170 3 40 120 9 1600

x = 30 y = 650 (xx ). (xx )2 (yy )2

(yy ) = 26 = 5000

= 360

1 vad

lg&laca/k xq.kkad r = 2 2( ) . ( )

( ) ( )

x x y y

x x y y

1 vad

=360

26 . 5000

=18

5 13

= 0.998

~ 1 ifjiw.kZ /kukRed lg&laca/k 1 vad'u&12 dk vkok dk mkj

ge tkurs gSa fd n pj a x oy ds chp lg&laca/k xq.kkad

r =2 2

( ) . ( )

( ) . ( )

x x y y

x x y y

1 vad

r =2 2

.

.

X Y

X Y

.......(I)

x x =X ,oa y y = Y fYk[kus ij


20/33

(20)

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fdUrq 'oktZ vlfedk (Schwart's Inequality) ls(XY)2 (X)2 (Y)2 1 vad

;k 2

2 2

( )

( ) ( )

XY

X Y

1 1 vad

;k2

.

XY

X Y

1

;k r2 1 lehdj.k (I) ls 1 vad 1 x + 1

Yk?kq mkjh; 'u'u&13 dk mkj

gYk& ekuk lerYk X ls ] Y v{k ls rkkZ v{k ls vUr%[k.M dkVrk gS rc lerYk dklehdj.k

1x y z

r+ + =

......(I)

lerYk X, Y rkkZv{k a d e'k% A, B rkkC fcUnqv a ij feYkrk gSA vr% fcUnqv a A, B, C dsfunsZ'kkad e'k% (, O, O)] (O O) rkk (O, O Y)g axsA

pwfd 'ukuqlkj ABC ds dsUd ds funsZ'kkad (a, b, c) gSaA

vr% x = 1 2 33

x x x+ +, y =

1 2 3

3

y y y+ +rkk z = 1 2 3

3

z z z+ +1 vad

a =0 0

3

+ +=

3

= 3a

b =0 0

3

+ +=

3

= 3b

c =0 0

3

+ + =

3

= 3c 1 vad

lehdj.k I esa , rkk ds eku j[kus ij

3 3 3

x y z

a b c+ + = 1

;kx y z

a b c+ + = 3

vr% okafNr lerYk dk lehdj.k

x y z

a b c+ + = 3

gxk 1 vad

1 vad


21/33

(21)

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vkok dk mkj&gYk& ewYk fcUnq ls gdj tkus okY lerYk dk lehdj.k

ax + by + cz = 0 ......(I) 1 vadlerYk (I) lerYk a ftuds lehdj.k uhps fn, x, gSa] ij Ykac gSA

x + 2y z = 1 ......(II)

,oa 3x 4y + 2 = 5 ......(III)vr% a + 2b c = 0 .....(IV)

3a 4b + c = 0 ......(V) 2 vadlehdj.k (IV) ,oa (V) d gYk djus ij

2 4 3 1 4

a b c

c= =

;k1 2 5

a b c= = = k ekuk 1 vad

vr% a = k , b = 2k ,oa c = 5klehdj.k (I) ls

kx + 2ky + 5kz = 0

;k x + 2y + 5z = 0;gh okafNr lerYk dk lehdj.k gSA 1 vad

'u&14 dk mkjgYk& ekuk ewYk fcUnqO ds lkis{kABC ds 'kh"kZA, B, C ds fLkfr lfn'k e'k%

, ,a b cH H H

gS rc , ,OA a OB b OC c= = =KKKH H KKKH H KKKH H

qtkv aBC, AC rkkAB ds e/; fcUnqe'k% D, E rkk F gSA ef/;dk,AD,BE rkk CF dk frPNsn fcUnq G ABC dk dsUd gSA 1 vad

D, BC dk e/; fcUnq gS2

b cOD

+=

H H

ABC dk dsUd G, AD d 2 : 1 ds vuqikr esa fokftr djrk gS 1 vad

OG =2( ) / 2 1

3

b c a+ +

=3+ +

b c a

=3

a b c+ +H H H

.....(I) 1 vad

GA = A fLkfr lfn'k & G dk fLkfr lfn'k

=2

3

a b c H H H

......(II)

blh dkj&

GBKH = B dk fLkfr lfn'k & G dk fLkfr lfn'k


22/33

(22)

J

GBKKKH

=2

3

b a c .....(III)

,oa GCKH

=2

3

c a c H H H

......(IV) 1 vad

lehdj.k II, III ,oa IV ls

G A GB GC + +KH KKKH KKKH

=2 2 2

3 3 3

a b c b a c c a b + +

H H H H H H H H H

=2 2 2

3

a b c b a c c a b + + H

vr% GA GB GC + + = O 1 vadvkok dk mkj&

gYk& ;fn ,a b rkk c rhu v'kwU; lfn'k gS rclfn'k a ds ;x ds lkgp;Z fu;ekuqlkj

( )a b c+ +H H H

= ( )a b c+ +H H H

miifk& fpk ds vuqlkj eku dk lfn'k ,a bH H

rkk cH

dkxt ds rYk esa gSA tgk O ewYkfcUnq gS fpk

ds vuqlkj ,a OA b AB= =H KKKH H KKKH

rkk c BO=H KKKH

cuk;k rkkOC d feYkk;k fcUnq OB rkk AC d h feYkk;kA

AOB esa lnf'k ;x dk fkqt fu;e Ykxkus ij& 1 vad

OBKKKH

= OA AB+KKKH KKKH

1 vad

= a b+H H

vc fkqt OBC esa lfn'k ;x dk fkqt fu;e Ykxkus ij

OCKKKH

= OB BC +KKKKH KKKH

= ( )a b c+ +H H H

.....(1) 1 vadblh dkj ABC esa lfn'k ;x dk fkqt fu;e Ykxkus ij&

ACKKKH

= AB BC+KKKKH KKKH

= b c+H H

1 vadrkk OAC esa lkfn'k ;x dk fkqt fu;e Ykxkus ij&

OCKKKH

= OA AC +KKKH KKKH

= ( )a b c+ +H H H

......(2)

vr% lehdj.k 1 vj 2 ls

( )a b c

+ +

H H H

= ( )a b c

+ +

H H H

;gh lfn'k a ds ;x dk lkgp;Z fu;e gSA

1 vad


23/33

(23)

J

'u 15 dk mkj&

gYk& fn;k gS f (x) = log1

1

x

x

+

......(1)

lehdj.k (1) esa x = a j[kus ij

f(a) = log 11

aa+ .....(2) 1 vad

lehdj.k (1) esa x = b j[kus ij

f (b) = log1

1

b

b

+

......(3) 1 vad

lehdj.k (2) vj (3) d tM+us ij

f (a) + f (b) = log1

1

+

a

a+ log

1

1

b

b

+

f (a) + f (b) = log (1 ) (1 )(1 ) (1 )

a ba b

+ +[logm + logn = log m n]

f (a) + f (b) = log1

1

a b ab

a b ab

++ + +

.......(4) 1 vad

lehdj.k (1) esa x =1

a b

ab

++

j[kus ij

f1a b

ab+ + = log

1 1

11

a b

aba b

ab

+ +

++

+

= log

1

11

1

+ +

+ + ++

ab a b

abab a b

ab

f 1

a b

ab

+ + = log

1

1

a a ab

a b ab

++ + + ......(5) 1 vad

lehdj.k (4) o (5) ls&

f (a) + f (b) = f1

a b

ab

+ + 1 vad

vkok dk gYk&

30

tan sinlim

x

x x

x

= 3

0

sinsin

coslim

x

xx

x

x

1 vad


24/33

(24)

J

= 30

1sin 1

coslim

x

xx

x

=

( )30

sin 1 coslim

cosx

x x

x x

= 20

tan 1 coslim

x

x x

x x

1 vad

= 20 0

tan 1 coslim lim

x x

x x

x x

= 1 20

(1 cos ) (1 cos )lim

(1 cos )x

x x

x x

++ 0

tanlim 1

x

x

x

=

1 vad

=

2

20

1 coslim

(1 cos )x

x

x x

+

=

2

20

1 sinlim

1 cosx

x

xx +

=0

sin sinlim

. (1 cos )x

x x

x x x

+ 1 vad

=0

00

sin sinlim

limlim (1 cos )

x

xx

x x

x x

x

+ 0sin

lim 1x

x

x

= 3

=1 1

1cos

=1

1 1+

=1

21 vad

'u&16 dk mkj&

ekuk I =5 4 sin

dx

x+

I =

2

2 tan25 4

1 tan2

dx

x

x

+

+

2

tan2sin

1 tan2

x

xx

= +

3


25/33

(25)

J

I =2

2

5 1 tan 8 tan2 2

1 tan

2

dx

x x

x

+ +

+

I =

2

2

1 tan2

5 5tan 8tan2 2

xdx

x x

+

+ + 1 vad

I =

2

2

sec2

5tan 8tan 5

2 2

+ +

xdx

x x tan2

x= tj[kus ij

sec22

x dx = 2dt

1 vad

I = 22

5 8 5

dt

t t+ +

=2

5

2 815

dt

t t+ +

=2

5 2

2 4 4 162( ) 15 5 25

dt

t t + + +

1 vad

=2

5

24 9

5 25

dt

t + +

j[kus ij 1 vad

t +4

5= u

dt = du

I =2

2

2

5 3

5

+

du

u


26/33

(26)

J

=1

1

2 3 tan35 5

5

u

= 12 5tan3 3

u

=1

45( )

2 5tan3 3

+

t

=12 (5 )tan

3 3

+ t u

I =1 (5tan 4)2 2tan

3 3

x

+

1 vad

'u&16 dk vkok dk mkj&

22 6 8

dx

x x+ + = 21

2 3 4

dx

x x+ + 1 vad

=22

1

23 72 2

dx

x + +

1 vad

=1

31 2tan

7 72

2 2

x

+

2 vad

=11 2 3tan

7 7

x +

1 vad

'u&17 dk mkj&gYk& nh?kZ o`k dk lehdj.k

2 2

2 2

x y

a b+ = 1

2

2

y

b= 1

2

2

x

a

y2 = b22 2

2a x

a

1 vad


27/33

(27)

J

y =2 2b

a xa

vc ewYk fcUnq O ds fYk, x = O rkk fcUnqA ds fYk, x = a gS vkkZr~ {kQYkAOB ds fYk, x dk eku O ls a rd fopfjr grk gSA vr% lekdYku lhek, x =0 ls x = a gxh

nh?kZok dk {kQYk = 4 {kQYk OAB dk {kQYk 1 vad

= 40

aydx

= 42 2

0

a ba x dx

a lehdj.k 1 ls

=

22 2 1

0

4sin

2 2

+

ab x a x

a xa a

=2 2 2 1

0

2sin

+

ab x

x a x aa a

1 vad

=2 2 2 1 2 12

sin 0 sin 0b a

a a a a aa a

+

=2 1 2 12 0 sin (1) sin 0

ba a

a

+ 1 vad

=22 0

2b aa

= ab oxZ bdkbZ 1 vad

'u&17 dk vkok dk mkj&

ekuk I = 20

sin

sin cos

xdx

x x

+

= 20

sin

2sin cos

2 2

x

dxx x

+ 1 vad

I = 20

cos

cos sin

xdx

x x

+lehdj.k 1 o 2 ls tM+us ij

I + I = 20

sin cos

sin cos sin cos

x xdx

x x x x

+

+ +

2 vad


28/33

(28)

J

2 I = 20

sin cos

sin cos

x xdx

x x

++

2I = 20

dx

2I = [ ] / 20x

1 vad

2I =2

O

I =4

1 vad

'u&18 dk mkj&fn;k x;k lehdj.k gS&

sec2

x tan y dx + sec2

y tan x dy = 0 sec2 x tan y dx = sec2 x tan x dy

2sec

tan

xdx

x=

2sec

tan

ydy

y 1 vad

2sec

tan

xdx

x =2

sec

tan

ydy

y ...........(1) 1 vad

ekuk tan x = t 1 vadrkk tan y = 4rc sec2 x dx = dt 1 vadrkk sec2y dy = du

vr% lehdj.k 1 lsdt

t = 4du

log t = log u + log c log t + log u = log c log (t u) = log c t u = c 1 vad

tan x tan y = c'u&18 dk vkok dk mkj&

gYk& fn;k x;k vodYku lehdj.k gS& (1 + y2) dx = (tan1 y x) dy 1 vad

(1 + y2)dx

dy= tan1 y x

(1 + y2)dx

dy+ x = tan1 y


29/33

(29)

J

21

1

dxx

dy y

+ + =

1

2

tan

1

y

y

+;g lehdj.k x esa js[kh; vodYk lehdj.k gSA

vr% bldh rqYkuk

dx

dy + Px = Q ls djus ij 1 vad

P = 21

1 y+Q =

1

2

tan

1

y

y

+ 1 vad

lekdYku xq.kkad (I.F.) =Pdy

e 1 vad

=

1

21dy

ye +

= etan1

y

vr% x (I.F.) = Q . (I.F.) dy + c ls vh"V gYk gxk&

x . etan1y =

1

2

tan

1

y

y

+ . etan1y dy + c

x . etan1y = . .tt e dt c+[tan1 y = tekuk 2

1

dy

y+

= dt rc]

nk, i{k esa t d ke QYku ekudj [k.M'k% lekdYku djus ij&

x . etan1y = t . et 1. et dt + C x . etan1y = t . et et + c x . etan1y = tan1 y etan1y etan1y + c 1 vad

x = tan1 y 1 + c etan1y

'u&19 dk mkj&?kVuk A dk frdwYk la;xkuqikr = 4 : 3

P (A) =3 3

4 3 7=

+1 vad

?kVuk B dk vuqdwYku la;xkuqikr = 7 : 5

P (B) =7 7

7 5 12=

+1 vad

pwfdA vjB Lorak ?kVuk, gSaP (A B) = P (A) . P (B)

=

3 7 1

7 12 4 = 1 vad


30/33

(30)

J

'u gYk gus dh k;fdrkP (A B) = P (A) + P (B) P (A B) 1 vad

=3 1 1

7 12 4+

= 36 49 2184+ = 6484 1 vad

=16

21'u&19 dk vkok dk mkj&

n (S) = 4, 5, 6

4 ls vf/kd vad vkus dh kf;drk

P (A) =2 1

6 3

= 1 vad

4 ls vf/kd u vkus dh kf;drk

P (A) = 1 1 2

3 3= 1 vad

;fn X ;knfPNd pj gS rc x = 0, 1, 2 dbZ lQYkrk ug

P0 =P (x = 0) = P (AA ) =2 2 4

3 3 9 = 1 vad

P1 = P (x = 1) = P (AA ) + P (AA )

= 1 2 2 1 43 3 3 3 9

= 1 vad

P2 = P (x = 2) = P (AA) =1 1 1

3 3 9 = 1 vad

x 0 1 2

P14

9

4

9

1

91 vad

'u&20 dk mkj

gYk& js[kkv a ds lehdj.k&1 2 3

2 3 4

= =

x z z= r ekuk .......(I)

,oa2 3 4

3 4 5

= =

x y z= r1 ekuk .....(II)

lehdj.k (I) ls&x = 1 + 2r .......(a)

y = 2 + 3r .......(b)

z = 3 + 4r ........(c) 1 vad


31/33

(31)

J

blh dkj lehdj.k (II) ls&x = 2 + 3r1 .......(a)

y = 3 + 4r1 .......(b)

z = 3 + 4r1 ........(c) 1 vadjs[kk (I) o (II) frPNsnh js[kk, gSa&

1 + 2r = 2 + 3r1;k 2r 3r1 1 = 0 .....(III)blh dkj 3r 4r1 1 = 0 ......(VI),oa 4r 5r1 1 = 0 .....(V) 1 vadlehdj.k III, IV o V d gYk djus ij&

3 4

r

=

1 1

3 2 8 9

r=

+ +

;k 1r

=1 1

1 1

r

= r = 1 ,oa r1 = 1 1 vadr o r1 ds eku lehdj.k (V) d larq"V djrs gSa vr% nh xbZ js[kk, lerYkh;

gSaA x = 1 + 2r leh- (a) esa r dk eku j[kus ij

x = 1 2

x = 1

blh dkj y = 1

,oa z = 1 2 vadvr% fPNsn fcUnq (1 , 1 , 1) gxkA'u&20 dk vkok dk mkj&

ekuk fd xY dk lehdj.kx2 + y2 + z2 + 2 ux + 2 vy + 2wz + d = 0 ......(1)

pwfd fcUnq (1, 3, 4) , (1, 5 , 2) rkk (1, 3 , 0) xY ij fLkr gS

vr% 12 + (3)2 + 42 + 2u 1 + 2 v (3) + 2 4 + d = 012 + (5)2 + 22 + 2u 1 + 2 v (5) + 2 . 2 + d = 0

12 + (3)2 + (O)2 + 2u 1 + 2 v (3) + 2 . 0 + d = 01 vad

gYk djus ij2u 6v + 8 + d = 26 .......(2)

2u 10 v + 4 + d = 30 ......(3)2u 6v + d = 10 ......(4) 1 vad

pwfd xY (1) dk dsU lerYk x + y + z = 0 ij fLkr gSA u + v + w = 0 ......(5) 1 vad

lehdj.k 4 & lehdj.k 2w = 2


32/33

(32)

J

lehdj.k 5 & lehdj.k 34v + 4 w = 20

v = 3

lehdj.k (5) lsu = (v + w)

u = 1 2 vadu, v, ds eku a d lehdj.k 5 esa j[kus ij

2 18 + d = 10

d = 0

vr% lehdj.k I lsx2 + y2 + z2 2x + 6y 4z +10 = 10 1 vad

;gh xY dk lehdj.k gSA'u&21 dk mkj&

fn, x, lehdj.k a ls Li"V gS fd&1a =

2i j k+ +

1b = i j k+ + 1 vad

2a = 2 i j k

2b = 2 2i j k+ + 1 vad

2 1a aKKH KKH

= ( ) ( )2 2 + + i j k i j k

2 1a aKKH KKH

= 3 2i j k + 1 vad

1 2b bKH KKH

=

1 1 1

2 1 2

i j k

= (2 1) (2 2) (1 2)i j k + +

= 3 3i k+ 1 vad

U;wure nwjh =2 1 1 2

1 2

.| |

a a b bb b

1 vad

=

( ) ( )3 2 3 39 9

i j k i k + +

=3 6

3 2

=

3

2 1 vad


33/33

dsoYk ifj.kke Yus ij nwjh _.kkRed ug grh'u&21 dk vkok dk mkj&

ekuk fd fn, x, fcUnq A , B o C rkk P dbZ vU; fcUnq gS ftldk fLkfrlfn'k r gSA 1 vad

AP = P dk fLkfr lfn'k & A dk fLkfr lfn'k

;k AP = ( )2 6 6r i j k + 1 vadAB = B dk fLkfr lfn'k & A dk fLkfr lnf'k

= ( ) ( )3 10 9 2 6 6i j k i j k + +

= 4 3i j k + 1 vad

ACKKKH

= C dk fLkfr lfn'k & A dk fLkfr lfn'k

=

( )

( )5 6 2 6 6i k i j k

+

= 3 6i j 1 vadpwfd , ,AP AC AB

KKKH KKKH KKKHlerYkh; gSa

, , KKKH KKKH KKKHAP AB AC = 0

{ } { }( 2 6 6 ) . ( 4 3 ) ( 3 6 )r i j k i j k i j + + = 0 ......(I)

fdUrq { }( 4 3 ) ( 3 6 )i j k i j +

=

1 4 3

3 6 0

i j k

= (0 18) (0 9) (6 12)i j k + +

= 18 9 18i j k + + 1 vadvr% lehdj.k (I) ls&

(2 6 6 ) .[ 9 (2 2 )]r i j k i j k + + + = 0

;k (2 2 )r i j k +4+612 = 0

;k (2 2 )r i j k = 2 1 vad;gh lerYk dk lehdj.k gSA

mathematics, set 2, model papers of madhya pradesh board of secondary education, xiith class

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