Mathematics set 3, Model papers of Madhya Pradesh Board of Secondary Education, XIIth Class

7/29/2019 Mathematics set 3, Model papers of Madhya Pradesh Board of Secondary Education, XIIth Class

1/32

(1)

I

'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


2/32

(2)

I

gk;j lsd.Mjh Ldwy ijh{kk& 2012&13HIGHER SECONDARY SCHOOL EXAMINATION-2012-13

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 R;sd 'u ds lkk vkarfjd fodYi fn,x, gSaA

(2) 'u&1 A, B, C, D, E oLrqfu"B 'u gSa R;sd dk 1 vad fu/kkZfjr gSA(3) 'u&2 ls 'u 8 rd ds 'u a ds R;sd 'u ds 4 vad fu/kkZfjr gSaA(4) 'u&9 ls 'u 15 rd ds 'u a esa R;sd 'u ds 5 vad fu/kkZfjr gSaA(5) 'u& 16 o 17 ds fu/kkZfjr vad 6 gSaA

Instructions(1) All question are compulsory internal choices are given in every question.(2) Question-1 A, B, C, D, E are objective type and 1 marks alloted to each

question.(3) Question-2 to 8 carries 4 marks each.(4) Question-9 to 15 carries 5 marks each.

(5) Question-16 and 17 carries 6 marks each.

oLrqfu"B 'u (Objective Type Questions)

'u&1 (A) fuEufYkf[kr 'u a ds lkk fn, x, fodYi a esa ls lgh fodYi pqudjviuh mkj&iqfLrdk esa fYkf[k,& 5

(i) tan11

2+ tan1

1

3= ................ gxk&

(a) tan11

6(b)

3

(c)4

(d)

6

(ii) rhu lfn'k t fkqt ds 'kh"kZ fcUnqv a ls fn"V ef/;dkv a ls fu/kkZfjr gSa] dk;x gS&(a) 0

H(b) 1

(c)1

3 (d) 1

(iii) ;fn aH

dk v'kwU; lfn'k ftldk ekikada gS rkkm dk ,d v'kwU; lfn'k


3/32

(3)

I

gSA rc ma ,dkad lfn'k gxkA

(a) m = + 1 (b) m = aH

(c) m =1

aH (d) 0

(iv) ,d pj fkT;k j[kus okY xYkkdkj xqCckjs dh fkT;k 4 lseh gSA blds vk;ruifjorZu dh nj gxkA

(a) 64 ?ku lseh@lsdUM (b) 32 lseh@lsdUM(c) 48 lseh@lsdUM (c) 72 lseh@lsdUM

(v) lfn'k a 2 i 3 j k + + vj 2 i j k ds chp d.k gS&

(a) 0 (b)4

(c)6 (d)

2

Q.-1 (A) Choose the correct choice from the follwoing multiple choicequestion 5

(i) tan11

2+ tan1

1

3= ...............

(a) tan11

6(b)

3

(c)4 (d)

6

(ii) The sum of three vectors from the vertices of triangle towardmedian are

(a) 0H

(b) 1

(c)1

3 (d) 1

(iii) If aH

is a non-zero vector whose resultant is a. m is an anothernon-zero vector. ma is a unit vector

(a) m = + 1 (b) m = aH

(c) m =1

aH (d) 0

(iv) A ballon has a variable radius of 4 cm. The rate of changing involume is(a) 64cm3/sec. (b) 32cm3/sec.(c) 48cm3/sec. (c) 72cm3/sec.


4/32

(4)

I

(v) The angle between the vector 2 i 3 j k + + and 2 i j k is

(a) 0 (b)4

(c) 6

(d) 2

'u&1 (B) fj Lkkua dh iwfrZ dhft,& 5

(i) ;fn1 A B

x(x 1) x x 1= +

+ + g ] r B =----------------------A

(ii) a . bH H

----------------------- tcfd , aH

o bH

ds chp dk d.k gSA

(iii) ;fn y = 1 + x +2 3x x

2 3+ + ........... rc

dy

dx= .........

(iv) 2 2a x dx dk eku ------------------ gSA(v) sec x dx dk eku -------------------- gSA

Q.-1 (B) Fill in the blanks 5

(i) If 1 A B

x(x 1) x x 1= +

+ + then B =----------------------.

(ii) a . bH H

----------------------- when , is angle between the vector aHand b

H-

(iii) If y = 1 + x +2 3x x

2 3+ + ........... then

dy

dx= .........

(iv) 2 2a x dx ------------------(v) sec x dx --------------------

'u&1 (C)lR;@vlR; fYkf[k,& 5

(i) cot x dx = log sin x

(ii) lekdYku] vodYku dh frYke f;k gSA(iii) sin x dk x ds lkis{k lekdYku cos x grk gSA

(iv)lg&laca/kd xq.kkad dk eku lnSo /kukRed grk gSA

(v) ;fn byx = 1.2 rc bxy = 1.4 grk gSA

Q.-1 (C) True / False 5

(i) cot x dx = log sin x

(ii) Intigration is inverse process of differentiation

(iii) the integration of sin x, with respect to x is cos x.


5/32

(5)

I

(iv) The value of correlotion coefficient always positive

(v) If byx

= 1.2 then bxy

= 1.4.

'u&1 (D) Lra "A" ls Lra "B" dk tM+h feYkku dhft, & 5"A" "B"

(i) ddx

tan1 x (a) 82

(ii) fcUnqv a (2, 4, 5) o (2, 5, 4) ds chp (b) 21

1 x+dh nwjh gxh

(iii) fcUnqv s (4, 3, 5) dh XZ lerYk ls nwjh gxh (c) 10(iv) fcUnq (1, 2, 3) dh y&v{k ls YkEcor~ nwjh gxh (d) 13(v) ,d js[kk[k.M dk funsZ'kka{k a ij {i (3, 4, 12) (e) 3

gS rc js[kk dh YkEckbZ gxhQ.-1 (D)Match the column& 5

"A" "B"

(i)d

dxtan1 x (a) 82

(ii) The distance between the points (2, 4, 5) and (b) 21

1 x+(2, 5, 4)

(iii) The distance of points (4, 3, 5) from (c) 10the XZ plane

(iv) Parpendicular distance of points (1, 2, 3) (d) 13yaxis

(v) The projection of line segment of (3, 4, 12) (e) 3then length of line.

'u&1 (E) ,d okD; esa mkj fYkf[k,& 5

(i) fe;k fLkfr fof/k dk iqujko`fk lwk fYkf[k,A

(ii) U;wVu jslu fof/k dk lwk fYkf[k,A

(iii)leYkEc prqqZt fu;e dk lwk fYkf[k,A

(iv) flEilu ds fu;e dk lwk fYkf[k,A

(v) 0.5125 E 03 0.4021 E 02 dk xq.kuQYk D;k gxk\

Q1. (E) Write the answer of following questions in one sentance& 5

(i) If teration formula for fase position method.

(ii) Write the formula for Newton Raphson method.


6/32

(6)

I

(iii) Write the formula is Trapezoidal rule.

(iv) Write the formula is simpnson's rule

(vi) Find the value of0.5125 E 03 0.42021 E 02

'u&2 2x 5(x 1) (x 2)

+ ds vkaf'kd f a esa fo dhft,A 4

vkok

2

x 3

(x 2) (x 9)

++ d vkaf'kd f a esa fo dhft,A

Divide into partial fraction -2x 5

(x 1) (x 2)

+ or

Divide into partial fraction- 2x 3

(x 2) (x 9)

++

'u&3- fl) dhft, fd& 4tan1 1 + tan1 2 + tan1 3 =

vkok

fl) dhft, fd& cos14

5+ sin1

5

13= cos1

33

65

Prove that - tan1

+ tan1

2 + tan1

3 = or

Prove that cos14

5+ sin1

5

13= cos1

33

65'u&4- ke fl)kar ls sin 2x dk vodYku Kkr dhft,A 4

vkok;fn y = ex+e

x+.....g r fl) dhft, fd

dy y

dx 1 y= Differentiate by first principle of sin 2x

or

If y = ex+ex+.....

then prove thatdy y

dx 1 y=

'u&5- ;fn y log x log x log x ........= + + + 4

g r fl) dj fddy 1

dx x(2y 1)=


7/32

(7)

I

vkok

;fn y =sinx

1 cos x+g r

dy

dxdk eku Kkr djA

If y log x log x log x ........= + + + then prove thatdy 1

dx x(2y 1)

=

or

If y =sinx

1 cos x+then prove that value of

dy

dx.

'u&6- ,d d.k lehdj.k S = 5 et cos t ds vuqlkj xfr djrk gSA ;fn t =2

g r mldk osx o Roj.k Kkr dhft,A 4vkok

,d iRkj ij dh vj Qsadk tkrk gSA bldh xfr dk lehdj.k S = 490t 4.9 t2 gSA iRkj }kjk kIr vf/kdre pkbZ Kkr dhft,A tgk t oS e'k%lsdUM o ehVj esa gSA

Equation of a moving partical S = 5 et cos t. If t =2

then find its

velocity and accelaration.or

A stone is thrown upward and the equation of its velocity is S = 490t 4.9 t2 where, t and S are in second and meter respectively. Calculatethe maximum height adopted by stone.

'u&7- lg&laca/k xq.kkad Kkr dhft,&cov (x, y) = 2.25 , var (x) = 6.25 , var (y) = 20.25 4

vkokn pj jkf'k; a x o y ds e/; lg&laca/k xq.kkad r g r fl) dhft, fd&

r =

2 2 2x y x y

x y2

+

tgk x2 , y2 , xy2 , e'k% x, y rkk (x y) ds fopj.k xq.kkad gSAFind the correlation coefficient, if given that

cov (x, y) = 2.25 , var (x) = 6.25 , var (y) = 20.25or

If r is the correlation coefficient between x andy then show that

r =

2 2 2x y x y

x y2

+

wherex2

, y2

and 2

xy

are the variants of x, y and (x y) respec-tively.


8/32

(8)

I

'u&8-lekJ;.k js[kkv a2x 9y + 6 = 0 ,oax 2y + 1 = 0 ds fYk, lg&lacakdh x.kuk dhft,

vkokfuEukafdr lkj.kh }kjk XokfYk;j esa 100 #i, ewY; ds laxr ikYk esa lokZf/kd mfpr

ewY; vkdM+ a }kjk Kkr dhft,A

'kgj XokfYk;j ikYk vlr ewY; 70 75ekud fopYku 2-5 3-0

nu a uxj a esa oLrq ds ewY; a esa lg&laca/k xq.kkad 0-8 gSA

Given the regression lines as 2x 9y + 6 = 0 and x 2y + 1 = 0respectively calculate the correlation coefficient

orAn article cost Rs. 100 at Gwalior and the corresponding most appro-

priate value at Bhopal using the following dataCity Gwalior BhopalMean value 70 75Standard deviation 2.5 3.0

The correlation between the value of the two cities is 0.8

'u&9- mu lerYk a ds lehdj.k Kkr dhft, t lerYkx 2y + 2z = 3 ds lekarjgSa rkk ftudh fcUnq (1, 2, 3) ls YkkfEcd nwjh 1 gSA 5

vkok

fl) dhft, fd n lekUrj lerYk a 2x 2y + z + 3 = 0 vj 4x 4y +

2z + 5 = 0 ds chp dh nwjh1

6gSA

Q. 9. Find the equation of the plane which are parallel to the plane x 2y + 2z = 3 and whose perpendicular distance from the point (1, 2, 3) is1. 5

orShow that the distance between two parallel plane 2x 2y + z + 3 =

0 and 4x 4y + 2z + 5 = 0 is 16

.

'u&10- lfn'k a a 2i j k = +H

vj b 3i 4j 4k = H

dk vfn'k xq.kuQYk vj muds

chp dk d.k Kkr dhft,A 5vkok

lfn'k a 3i 4j 5k + + , 4i 3j 5k vj 7i j+ ls cus fkqt dh ifjeki Kkrdhft,A

Q.-10 Find the scaler product of vector

a 2i j k = +

H

and

b 3i 4j 4k =

H


9/32

(9)

I

and also find the angle between themor

Find the perimeter of a triangle made by vectors 3i 4j 5k + + , 4i 3j 5k

and 7i j+ respectively.

'u&11- fuEukafdr QYku ds lkarR; dh foospuk dhft,A 5

f(x) = 2x 1

x 1

++

, x = 1 ij

vkok

x

(3x 1) (4x 2)lim

(x 8) (x 1)

+ dk eku Kkr dhft,A

Clarify the continuity of followoing function

f(x) = 2x 1

x 1++ , at x = 1

or

Find the value ofx

(3x 1) (4x 2)lim

(x 8) (x 1)

+

'u&12- js[kk y = x ,oa ijoYk; y2 = 16x ls f?kjs {k dk {kQYk Kkr dhft,A5

vkok

ijoYk; y2

= 4x ,oa x2

= 4y ls f?kjs {k dk {kQYk Kkr dhft,AQ.12 Find the area included between the parabola y2 = 16x and the liney = x

orFind the area included between the parabola y2 = 4x and x2 = 4y

'u&13-dx

5 4 sin x dk lekdYku Kkr dhft,Avkok

/ 2

0

sin x dx4sin x cos x

=+ d fl) dhft,A

Find the integral coefficient ofdx

5 4 sin xor

Prove that/ 2

0

sin x dx

4sin x cos x

=

+'u&14- fl) dj fd QYku 5


10/32

(10)

I

y = x3 + ax2 + bn + c vodYku lehdj.k3

3

d y

dx= 6 dk ,d gYk gSA

vkok

vodYku lehdj.k

dy

dx+ y = ex

d gYk dhft,AQ.14 Show that the function y = x3 + ax2 + bx + c is a solution ofdifferential equation

3

3

d y

dx= 6.

or

Solve the differential equationdy

dx+ y = ex

'u&15- ;fn ,d Ykhi o"kZ dk ;knfPNd p;u fd;k x;k g r blesa 53 jfookj gusdh kf;drk Kkr dhft,A

vkok52 ik a dh QsaVh gqbZ rk'k dh xh esa ls 2 i fudkY tkrs gSa] buds YkkYk ;kbk gus dh k;fdrk Kkr dhft,A

Q.15 Find the probability that a leap year, selected at random will contain53 sundays

or

Two cards are drawn at random from a pack 52 cards. What is theprobality that either both are red or both are acess.

'u&16- ljYk js[kkv ax 1 y 2 z 3

2 3 4

= = vj

x 2 y 4 z 5

3 4 5

= = ds chp dh

U;wure nwjh Kkr dhft,Avkok

ml xY dk lehdj.k Kkr dhft, t fcUnqv a (1, 3, 4) ] (1, 5, 2) vj(1, 3, 0) ls gdj tkrk gS rkk ftldk dsU lerYkx + y + z = 0 ij fLkrgSA

Q.16 Find the minimum distance between straight linesx 1 y 2 z 3

2 3 4

= =

andx z y 4 z 5

3 4 5

= =

orFind the equation of the sphere which passes threw the points (1, 3, 4) ] (1, 5, 2) and(1, 3, 0) and whose centre lines on the planex + y + z = 0.


11/32

(11)

I

'u&17- fl) dj &

a b b c c a 2 a b c + + + = H H H H H H H H H

vkok

15 bdkbZ dk ,d cYk i 2j 2k + dh fn'kk esa dk;Z djrk gSA rkk fcUnq

2i 2j 2k + ls xqtjrk gSA bl cYk dk fcUnq ( )i j k+ + ds ifnr% lfn'k vk?kw.kZKkr dhft,A

Q.17 Prove that

a b b c c a 2 a b c + + + = H H H H H H H H H

or

A force of 15 unit acts along the ( i 2j 2k + ) and passes through the

points (

2i 2 j 2k + ). Find the vector moment of it about the points( )i j k+ +

vkn'kZ mkjQ.1 (A)

(i) (c)4

(ii) (a) 0

H

(iii) (c) m =

1

| a |H

(iv) (a) 64cm3

/sec.

(v) (d)2

Q. 1 (B)

(i) B = 1 (ii) a . bH H

= ab cos

(iii)dy

dx= ex (iv)

22 2 1x a x

a x sin2 2 a

+

(v) log (sec x + tan x)Q. 1 (C)

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

Q. 1 (D)

(i) (b) 21

1 x+(ii) (a) 82

(iii) (e) 3 (iv) (d) 13


12/32

(12)

I

(v) (c) 10Q. 1 (E)

(i) xn+1 = xn1 n n 1

n n 1

x x

f (x ) f (x )

f (xn1)

(ii) xn+1 = xn n

n

f (x )f '(x )

(iii)b

a f(x) dx =h

2[(y

o+y

n) + 4 (y

1+ y

3+ ......... + Y

n1)

(iv)b

a f(x) dx =h

3[(y

o+y

n) + 4 (y

1+ y

3+ ......... + Y

n1)

2 (y2 + y4 + ......... + Yn2)](v) 0.20607625 E 01

mkj&2- ekuk fd

2x 5

(x 1) (x 2)

+ =

A B

x 1 x 2+

----------1

(x 1) (x 2) ls lehdj.k 1 ds nu a vj xq.kk djus ij&2x + 5 = A (x 2) + B (x1) -----------2 1 vad

x = 2 j[kus ij2 2 + 5 = 0 + B (2 1)

B = 9 1 vadvj x = 1 j[kus ij2 1 + 5 = A (1 2) + 0

A = 7 1 vadlehdj.k 1 esa A = 7 vj B = 9 j[kus ij&

2x 5

(x 1) (x 2)

+ =

7 9

x 1 x 2 +

1 vad

vkok dk mkj&

2x 3

(x 2) (x 9)++ =

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

++ +

=1

(x 2) (x 3)+

ekuk1

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

x 2 x 3+

+ ---------1

(x + 2) (x 3) ls lehdj.k 1 ds nu a vj xq.kk djus ij1 = A (x 3) + B (x+2) ---------2 1 vad

x = 3 j[kus ij& 1 = 0 + B (3 + 2)


13/32

(13)

I

B =1

51 vad

x = 2 j[kus ij& 1 = A (23) + 0

A = 1

5

1 vad

lehdj.k 1 esa A = 1

5vj B =

1

5j[kus ij

1

(x 2) (x 3)+ + = 1 1

5 (x 2) 5 (x 3)+

+

vr% 2x 3

(x 2) (x 9)

++ =

1

5 (x 2)+ +1

5 (x 3) 1 vad

mkj&3- ge tkurs gSa fd&

tan1 x + tan1 y + tan1 z = tan1x y z xyz

1 xy yz zx

+ +

vc x = 1 , y = 2 , z = 3 j[kus ij

tan1 (1)+tan1 (2)+tan1 (3) = tan11 2 3 1 2 3

1 1 2 2 3 3 1

+ + 1 vad

= tan16 6

1 2 6 3

= tan10

10

= tan1 (0) 1 vad= ;k0 1 vad

ijUrq] rhu /kukRed d.k a dk eku 'kwU; ug g ldrkAvr% tan1 (1)+tan1 (2)+tan1 (3) = 1 vad

vkok dk mkj&

L.H.S. = cos14

5 + sin15

13

= sin14

15

2 + sin1 5

13

1 vad

= sin116

125

+ sin15

13

= sin1

3

5

+ sin1

5

13


14/32

(14)

I

= sin13 25 5 9

1 15 169 13 25

+

1 vad

= sin13 12 5 4

5 13 13 5

+ 1 vad

= sin136 20

65 65

+

= sin156

65

= cos12

561

65

= cos133

65 1 vad

mkj&4- ekuk& f (x) = sin 2xf (x + h) = sin 2 (x + h)

= sin (2x + 2h) 1 vad

ge tkurs gSa fd&dy

dx=

h 0

f (x h) f (x)lim

h

+

=h 0

sin(2x 2h) sin 2xlim

h+

=h 0

2x 2h 2x 2x 2h 2x2cos sin

2 2limh

+ + +

=h 0

2cos(2x h) sin hlim

h

+1 vad

= h 0lim 2 cos (2x + h) h 0limsin h

h

= 2 cos (2x + 0) 1

= 2 cos 2x 1 vadvkok dk mkj&

fn;k x;k gS& y = ex+ex+.....

y = ex+y 1 vadlog Yus ij log y = log ex+y

log y = (x + y) log e

(log e = 1)


15/32

(15)

I

log y = x + y 1 vadlog y y = x

x ds lkis{k vodYku djus ij&

d

dx(log y y) =

d

dx(x)

d

dxlog y

dy

dx= 1 1 vad

1 dy dy

y dx dx = 1 1 vad

dy

dx=

y

1 ymkj&5- fn;k x;k gS&

y = log x log x log x .......+ + +

y = log x y+ 1 vady2 = log x + y 1 vad

x ds lkis{k vodYku djus ij&

2ydy

dx=

1 dy

x dx+ 1 vad

(2y 1)dy

dx =

1

x

dy

dx=

1

x(2y 1) 1 vad

vkok dk mkj&

fn;k gS& y =sinx

1 cos x+x ds lis{k vodYku djus ij&

dy

dx=

2

d d(1 cos x) sin x sin x (1 cos x)dx dx

(1 cos x)

+ +

+1 vad

= 2(1 cos x)cos x sin x( sin x)

(1 cos x)

+ +

=

2 2

2

cos x cos x sin x

(1 cos x)

+ +

+1 vad


16/32

(16)

I

= 2cos x 1

(1 cos x)

++

=1

1 cos x+ 2 vad

mkj&6& fn;k gS s = 5 et cos tx ds lkis{k vodYku djus ij&

v osx =ds

dt= 5 [et (sin t) + cos t (et)] 1 vad

osx v = 5 [et . sin t et . cos t]

= 5et (sin t + cos t) 1 vad

Roj.k a =dv

dt= 5[et(cos tsin t)+(sin t+cos t](et)

= 5et [cos t sin t sin t cos t]

= 5et (2 sin t) 1 vad

Roj.k a = 10et . sin t

vc t =2

ij d.k dk osx

= 5 e/2 sin cos2 2

+

= 5 e/2 (1 + 0)= 5 e/2 bdkbZ

vj Roj.k a = 10 e/2 sin /2= 10 e/2 1

= 10 e/2 bdkbZ 1 vadvkok dk mkj&

fn;k gS& s = 490 t 4.9 t2 -----1t ds lkis{k vodYku djus ij&

osx =ds

dt= 490 t 9.8 t ------2 1 vad

vf/kdre pkbZ ij osx 'kwU; grk gSAvr% 490 9.8 t = 0

t =490

9.81 vad

t = 50lsd.M

vf/kdre pkbZ s = 490 50 4.9 (50)2 1 vad

= 24500 12250


17/32

(17)

I

= 12250 ehVj 1 vadmkj&7 lg&laca/k xq.kkad&

r (x , y) =cov (x,y)

var(x) var(y) ls 1 vad

r (x, y) = 2.256.25 20.25

1 vad

=2.25

126.5625

=2.25

11.25

2 vad

= 0.2 mkjvkok dk mkj&

ge tkurs gSa fd&2x y = [ ]

21x y) (x y)

x 1 vad

= [ ]21

x x) (y y)x

=21 x x) (y y) 2(x x) (y y

x

1 vad

=2 21 1

(x x) (y y)x x

+

12 (x x) (y y)

x 1 vad

= x2 +

y2 2 r

x

y

r =x y

(x x) (y y)

n

2 r xy = x2 + y2 2(xy) 1 vad

r =

2 2 2x y (x y)

x y2

+

mkj&8- lekJ;.k js[kk&2x 9y + 6 = 0

9y = 2x + 6

y =2

9

x +6

9


18/32

(18)

I

y =2

9x +

2

31 vad

byx

=2

9y dh x ij lekJ;.k js[kk

iqu% x 2y + 1 = 0x = 2y 1b

xy= 2 x dh y ij lekJ;.k js[kk 1 vad

r rxy = xy yxb b

=2

29

=4

9

=2

31 vad

byx =2

9

ry

x

=2

9

2 y

3 3

=

2

9

y =2 9

9 2

= 12x

x

9

3

= = 3

3 1 vad

vkok dk mkj&ekuk XokfYk;j esa oLrq ds ewY; d x rkk ikYk esa y ls n'kkZ;k tkrk gS r

'ukuqlkj& x = 70y = 75

x = 2.5y = 3.0

r = 0.8 1 vady dh x ij lekJ;.k js[kk dk lehdj.k

y y = r ( )y

x xx


19/32

(19)

I

y 75 = 0.83.0

2.5(x 70) 1 vad

y =2.4

2.5(n 70) + 75

y = 2425

(100 70) + 75 1 vad

=24 30

25

+ 75

=144

5+ 75 [fn;k gS x = 100]

= 28.8 + 75

= 103.80 #- 1 vad

mkj&9- fn, x, lerYk dk lehdj.kx 2y + 2z = 3 -------1

blds lekUrj lerYk dk lehdj.kx 2y + 2z + k = 0 -----2 1 vad

fcUnq (1, 2, 3) ls lerYk ij MkY x, YkEc dh YkEckbZ 1 gS vr%

1 = 2 2 21 1 ( 2)(2) 2 3 k

1 (2) (2)

+ + +

+ + 1 vad

1 =1 4 6 k

9

+ +

1 = +3 k

3

+

3 + k = + 3

k = + 3 3

k = 0 ;k 6 1 vad

k = 0 j[kus ij lerYk dk lehdj.k

x 2y + 2z = 0rkk k = 6 j[kus ij

x 2y + 2z 6 = 0 1 vadvkok dk mkj& fn, x, lerYk a ds lehdj.k

2x 2y + z + 3 = 0 -----1rkk

4x 4y + 2z + 5 = 0

2x 2y + z +5

2 = 0 ------2 1 vad


20/32

(20)

I

n lekarj lerYk ds chp dh nwjh

d =1 2

2 2 2

d d

a b c

+ +1 vad

d =2 2 2

53

2

2 ( 2) (1)

+ +1 vad

=

1

2

9=

1

2

3

=1

62 vad

mkj&10fn;k gS a

H

= 2i j k +

bH

= 3i 4j 4k

a . bH H

= ( ) ( )2i j k . 3i 4j 4k + = [2 3 + (1) (4) + (1) (4)]= 6 + 4 4

= 6

2 2 2i j k 1

i j k k i 0

+ + = = =

3

1 vad

;fn buds chp dk d.k gS r&

cos =a . b

| a | . | b |

H H

H H ------1 1 vad

| a |H

= 2 2 22 ( 1) (1)+ + 1 vad

= 6

| b |H = 2 2 2(3) ( 4) ( 4)+ +

| b |H

= 3 16 16+ + 1 vad

= 41lehdj.k 1 esa j[kus ij

cos =6

6 411 vad

= 641


21/32

(21)

I

= cos16

41

iz'u&10 dk vkok dk mkj&fn;k& ABC esa\

BCKKKH

= 3i 4 j 5k + +

BCKKKKH

= 2 2 23 4 5+ +

= 9 16 25+ += 50

BCKKKKH

= 5 2 --------1 1 vad

fn;k gS& CAKKKH

= 4i 3j 5k

CAKKKH

= 2 2 24 ( 3) ( 5)+ +

= 16 9 25+ +

= 50= 5 2 -------2 1 vad

fn;k gS& BAKKKH

= 7i j+

BAKKKH

= 2 27 1+= 50

= 5 2 ------3 1 vad

fkqt dh ifjeki = AB BC CA+ +

KKKH KKKH KKKH

= 5 2 + 5 2 + 5 2

= 15 2 2 vad

mkj&11

fn;k gS& f (x) = 2x 1

x 1

++

x = 1 ij


22/32

(22)

I

f (1) =1 1

1 1

++

(i) f (1) = 1 1 vad

(ii) f (1 + 0) =h 0lim (1 + h)

= 2h 0

(1 h) 1lim

(1 h) 1+ ++ +

= 2h 0

2 hlim

2 2h h

++ +

=2

2

= 1 1 vad

(iii) f (1 0) = h 0lim f (1 h)

= 2h 0

(1 h) 1lim

(1 h) 1

+ +

= 2h 0

2 hlim

2 2h h

+

1 vad

=2

2

f (1 0) = 1 f (1) = f (1 + 0) = f (1 0) 2 vadvr% fn;k x;k QYku x = 1 ij lkarR; gSA

'u&11 dk vkok dk mkj&

fn;k x;k QYku x ds fYk,

dk :i /kkj.k dj Yrk gS rkk va'k o

gj d x2 ls kx nsus ij& 1 vad

=x o

(3x 1) (4x 2)lim

(x 8) (x 1)

+ 1 vad

= x o

1 23 4

x xlim

8 11 1

x x

+

2 vad

=(3) (4)

(1) (1)

= 12 1 vad


23/32

(23)

I

mkj&12 nh xbZ js[kky = x ------1

fn;k x;k o y2 = 16x -----2lehdj.k 2 esa y = x j[kus ij

y2 = 16y

y2 16 = 0y (y 16) = 0

y = 0 ;k y = 16rc x = 0 r y = 0

x = 16 r y = 16vr% js[kk o o d frPNsn fcUnq

O (0, 0)o B (16, 16) g axs

2 vad

vh"V {kQYk =16

1 20(y y ) dx 1 vad

=16

0(4 x x) dx

= 416 16

0 0x dx x dx

= 4

16 163/ 2 2x x

3/ 2 2

= 2 3/ 22 116 0

3 2 [16

2 0] 1 vad

=

8

3 64

1

2 256


24/32

(24)

I

=512

3 128

=512 384

3

1 vad

= 1283

vkok dk mkj&fn, x, ijoYk; dk lehdj.k gS&

y2 = 4x --------1x2 = 4y -------2

1 vad

lehdj.k 1 o 2 d gYk djus ij&

y2 = 4 ( )4y 1 vad

y2 = 8 y

y4 = 64y

y4 64y = 0

y (y3

64) = 0y = 0

y = 4

;fn y = 0 x = 0y = 4 x = 4

js[kkafdr kx dk {kQYk Kkr djuk gSA 1 vad

{kQYk =4

0 (y1 y2) dx

=

2

4 40 0

x4x dx4


25/32

(25)

I

=4 4 2

0 0

12 x dx x dx

4

=

443/ 2 3

00

x 1 x2

3 4 32

= 2 2

3[(4)3/2 0]

1

4 3[43 0]

=4

3 8

1

12 64

=32 16

3 3

=16

3[2 1]

=16

3oxZ bdkbZ 2 vad

vh"V {kQYk =16

3oxZ bdkbZ

mkj&13

I =dx

5 4 sin x

I =2 2

dx

x x x x5 cos sin 4 2sin cos

2 2 2 2

+

cos2 x/2 ls kx nsus ij

I =

2

2

x

sec dx2x x

5 1 tan 4 tan2 2

+

I =

2

2

xsec dx

2x x

5 tan 4tan 52 2

+ 2 vad

tanx

2 = t j[kus ij


26/32

(26)

I

sec2x

2.

1

2dx = dt

sec2x

2dx = 2 dt

= 2 2dt5t 8t 5 +

=2

2 dt

85t t 1

5 +

=2

2 dt

45t 2. t 1

5 +

= 22 dt

5 4 16t 1

5 25

+

= 22 dt

5 4 9t

5 25

+

= 2 22 dt

5 4 3t

5 5

+

2 vad

=2 3

5 5 tan1

4t

53

5

2 2

dx 1 xtan

a ax a

=

+3

=2

3tan1

5t 4

3

=2

3tan1

5tan 42

3

1 vad

iz'u-13 dk vkok dk mkj&

I =

/ 2

0

sinx

dxsin x cos

+ --------1


27/32

(27)

I

a

0f(a)dx =

a

0f (a x)dx

I =/ 2

0

sin x2

sin x cos x2 2

+

I =/ 2

0

cosx

cos x sin x

+ --------2 2 vad1 vj 2 tM+us ij&

2I =/ 2

0

sin x cosxdx

sin a cos x

++

2 I =/ 2

0dx

2I = [ ]/ 2

0

2 vad

2I =2

I =4

1 vad

mkj&14fn;k x;k lehdj.k gS& y = x3 + ax2 + bx + cx ds lkis{k vodYku djus ij

dy

dx= 3x2 + 2ax + b 1 vad

iqu% x ds lkis{k vodYku djus ij&2

2

d y

dx= 6x + 2a 2 vad

iqu% x ds lkis{k vodYku djus ij3

2

d y

dx= 6 2 vad

iz'u&14 dk vkok dk mkj&

fn;k x;k lehdj.kdy

dx y = ex

;g y ds ,d jSf[kd vodYku lehdj.k gS- vr% bldh rqYkuk O;kid lehdj.k

dydx

+ Py = Q ls djus ij 1 vad


28/32

(28)

I

P = 1 , Q = ex

lekdYku xq.kkad I.F. =Pdx

e

=Pdx

e= ex 2 vad

vr% y (I.F.) = Q (I.F.) dx + cy ex = ex . ex dx + cy ex = e2x dx + c

y ex =2ne

2+ c 1 vad

y =

1

2 ex

+ c ex

1 vadmkj&15

Ykhi o"kZ esa dqYk fnu a dh la[;k 366 grh gS vkkZr~ jfookj lfgr 52 iw.kZ lIrkgrd 2 fnu vfrfj g axsA ;s fuEufYkf[kr g ldrs gSa& 1 vad

1 leokj] eaxYkokj2 eaxYkokj] cq/kokj3 cq /kokj] xq#okj4 xq:okj] 'kqokj

5 'kq okj] 'kfuokj6 'kfuokj] jfookj7 jfookj] leokj 2 vadbu lh lelEkoh fLkfr; a esa ls vafre n fLkfr;k jfookj gus ds vuqdwYk gS]

vr% &

vf"V kf;drk =?kVuk d s vudq Yw k fLkfr; a dh la[;k

dYq k lakfor fLkfr; a dh l[a ;k1 vad

iqu% kf;drk =2

71 vad

iz'u&15 dk vkok dk mkj&nu a ik a ds YkkYk gus dh kf;drk

P (A) =2

2

26C

52 C1 vad

=

26 25

2 152 51

2 1


29/32

(29)

I

=13 25

26 51

1 vad

=325

1326

nu a ik a ds bk gus dh kf;drk &P (B) =

2

2

4 C

52 C

=

4 3

2 152 51

2 1

=6

1326 1 vadblh dkj n YkkYk bs gus dh kf;drk &

P (A B) =1

1326

vHkh"V izkf;drk P (A B) = P (A) + P (B) P (A B) 1 vad

=325 6 1

1326 1326 1326+

=

330

1326

=55

2211 vad

mkj&16nh xbZ ljYk js[kk,

x 1

2

=

y 2 z 3

3 4

= -------1

vkSjx 2

3

=

y 4 z 5

4 5

= ------2

rc U;wure nwjh SD =

1 1 1

1 1 1

2 2 2

21 2 2 1

a b c

a b c

(b c b c )

1 vad


30/32

(30)

I

SD =2 2 2

2 1 4 2 5 3

2 3 4

3 4 5

(3 5 4 4) (2 5 3 4) (2 4 3 3)

+ + 1 vd

SD =2 2 2

1 2 2

2 3 4

3 4 5

( 1) ( 2) ( 1) + + 1 vad

SD =[1(15 16) 2(10 12) 2(8 9)]

6

+ 1 vad

SD =

1 4 2

6

+

SD =1

62 vad

iz'u&16 dk dk vkok dk mkj&ekuk xY dk lehdj.k&

x2 + y2 + z2 + 2ux + 2vy +2wz + d = 0 --------1fcUnq (1, 3, 4) ] (1, 5, 2) vj (1, 3, 0) bl ij fLkr gSAvr% 1+9+16+2u6v+8w+d = 0

;k 26+2u6v+8w+d = 0 --------21+25+4+2u10v+4w+d = 0

;k 30 + 2u 10v + 4w + d = 0 ......(3)1+9+0+2u6v+0+d = 0

;k 10 + 2u 6v + d = 0 ......(4) 1 vadlehdj.k 2 esa ls lehdj.k 3 d ?kVkus ij&

4v + 4w 4 = 0;k v + w 1 = 0 ------5 1 vad

lehdj.k 2 esa ls lehdj.k 4 ?kVkus ij8w + 16 = 0

;k w = 2 6 1 vadlehdj.k 5 esa j[kus ij&

v 2 1 = 0;k v = 3 ------7 1 vad

xY dk dsU (u, v, w) lerYk x + y + z = 0 ij fLkr gS] blfYk,u v w = 0

u + v + w = 0 --------8v = 3 vj w = 2 j[kus ij


31/32

(31)

I

u + 3 2 = 0

u = 1 1 vadlehdj.k 4 esa u o v ds eku j[kus ij

2 18 + d + 10 = 0

;k d = 10

lehdj.k 1 esa u, v, w, d ds eku j[kus ij xY dk lehdj.k&x2 + y2 + z2 + 2x + 6y 4z + 10 = 0 1 vad

mkj& 17

L.H.S. = a b , b c , c a + + + H H H H H H

= ( )a b . b c c a + + + H H H H H H

= ( )a b . b c b a c c c a + + + + H H H H H H H H H H

= ( )a b . b c b a c a + + + H H H H H H H H

1 vad

= ( )a b . b c a b c a + + H H H H H H H H

= ( ) ( ) ( )a. b c a a b a c a + H H H H H H H H H

1 vad

= a b c a . a b a. c . a b b c + + H H H H H H H H H H H H

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

1 vad

= a b c H H H

0 + 0 + 0 0 + b c a H H H

1 vad

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

1 vad

= 2 a b c H H H

vkok dk mkj&

lfn'k i 2j 2k + dh fn'kk esa ekud lfn'k

=i 2j 2k

1 4 4

++ +

=i 2 j 2k

3

+ 1 vad

fn;k gqvk lfn'k cYk

FH

= 15

i 2j 2k

3

+


32/32

= 5 ( )i 2 j 2k + 1 vadekuk lfn'k i j k+ + rkk 2i 2 j 2k + e'k% fcUnq O o P d fu:fir djrs

gSaA rc&

OP

KKKH

= r

H

=

( )

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

1 vad= i 3j k +

cYk FH

dk fcUnq O ds ifjr% vk?kw.kZ

= r FH H

1 vad

= ( ) ( )i 3j k i 2j 2k + +

= 5

i j k

1 3 11 2 2

= 4 ( )4i j k + 2 vad

Documents

Mathematics set 3, Model papers of Madhya Pradesh Board of Secondary Education, XIIth Class