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7/29/2019 Mathematics set 3, Model papers of Madhya Pradesh Board of Secondary Education, XIIth Class
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(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
7/29/2019 Mathematics set 3, Model papers of Madhya Pradesh Board of Secondary Education, XIIth Class
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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
7/29/2019 Mathematics set 3, Model papers of Madhya Pradesh Board of Secondary Education, XIIth Class
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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.
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(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.
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(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.
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(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)=
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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.
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'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
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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
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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.
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'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
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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)
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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
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= 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)
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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
7/29/2019 Mathematics set 3, Model papers of Madhya Pradesh Board of Secondary Education, XIIth Class
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
7/29/2019 Mathematics set 3, Model papers of Madhya Pradesh Board of Secondary Education, XIIth Class
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
7/29/2019 Mathematics set 3, Model papers of Madhya Pradesh Board of Secondary Education, XIIth Class
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
7/29/2019 Mathematics set 3, Model papers of Madhya Pradesh Board of Secondary Education, XIIth Class
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
7/29/2019 Mathematics set 3, Model papers of Madhya Pradesh Board of Secondary Education, XIIth Class
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
7/29/2019 Mathematics set 3, Model papers of Madhya Pradesh Board of Secondary Education, XIIth Class
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
7/29/2019 Mathematics set 3, Model papers of Madhya Pradesh Board of Secondary Education, XIIth Class
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
7/29/2019 Mathematics set 3, Model papers of Madhya Pradesh Board of Secondary Education, XIIth Class
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
7/29/2019 Mathematics set 3, Model papers of Madhya Pradesh Board of Secondary Education, XIIth Class
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
7/29/2019 Mathematics set 3, Model papers of Madhya Pradesh Board of Secondary Education, XIIth Class
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
7/29/2019 Mathematics set 3, Model papers of Madhya Pradesh Board of Secondary Education, XIIth Class
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
7/29/2019 Mathematics set 3, Model papers of Madhya Pradesh Board of Secondary Education, XIIth Class
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
7/29/2019 Mathematics set 3, Model papers of Madhya Pradesh Board of Secondary Education, XIIth Class
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
7/29/2019 Mathematics set 3, Model papers of Madhya Pradesh Board of Secondary Education, XIIth Class
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
7/29/2019 Mathematics set 3, Model papers of Madhya Pradesh Board of Secondary Education, XIIth Class
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
7/29/2019 Mathematics set 3, Model papers of Madhya Pradesh Board of Secondary Education, XIIth Class
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
+
7/29/2019 Mathematics set 3, Model papers of Madhya Pradesh Board of Secondary Education, XIIth Class
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
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