NIOS Class 12 Question Paper Apr 2019 - Mathematics

This Question Paper consists of 33 questions and 16 printed pages + Graph Sheet.
Bg àíZ-nÌ _| 33 àíZ VWm 16 ‘w{ ÐV n¥ð> + J« m’$$ erQ> h¢ &

Sl. No.
Roll No.
AZw H « $ _m§ H $
Code No.
H$mo S > Z§ .
58/OSS/1
MATHEMATICS
Set / go Q > A
(J{UV)
(311)
Day and Date of Examination :
(narjm H$m {XZ d {XZm§ H $)
Signature of Invigilators : 1.
({ZarjH$m| Ho $ hñVmja)
2.
General Instructions :
1. Candidate must write his/her Roll Number on the first page of the Question Paper.
2. Please check the Question Paper to verify that the total pages and total number of questions contained in
the Question Paper are the same as those printed on the top of the first page. Also check to see that the
questions are in sequential order.
3. Making any identification mark in the Answer-Book or writing Roll Number anywhere other than the
specified places will lead to disqualification of the candidate.
4. Write your Question Paper Code No. 58 / OSS /1, Set - A on the Answer-Book.
5. (a) The Question Paper is in English/Hindi medium only. However, if you wish, you can answer in any
one of the languages listed below :
English, Hindi, Urdu, Punjabi, Bengali, Tamil, Malayalam, Kannada, Telugu, Marathi, Oriya, Gujarati,
Konkani, Manipuri, Assamese, Nepali, Kashmiri, Sanskrit and Sindhi.
You are required to indicate the language you have chosen to answer in the box provided in the
Answer-Book.
(b) If you choose to write the answer in the language other than Hindi and English, the responsibility
for any errors/mistakes in understanding the question will be yours only.
gm_mÝ` AZw Xo e :
1. narjmWu àíZ-nÌ Ho$ nhbo n¥ð> na AnZm AZwH«$‘m§H$ Adí¶ {bI|&
2. H¥$n¶m àíZ-nÌ H$mo Om±M b| {H$ àíZ-nÌ Ho$ Hw$b n¥ð>m| VWm àíZm| H$s CVZr hr g§»¶m h¡ {OVZr àW‘ n¥ð> Ho$ g~go D$na N>nr h¡& Bg ~mV
H$s Om±M ^r H$a b| {H$ àíZ H«${‘H$ ê$n ‘| h¢&
3. CÎma-nwpñVH$m ‘| nhMmZ-{M• ~ZmZo AWdm {Z{X©ï> ñWmZm| Ho$ A{V[aº$ H$ht ^r AZwH«$‘m§H$ {bIZo na narjmWu H$mo A¶mo½¶ R>ham¶m Om¶oJm&
4. AnZr CÎma-nwpñVH$m na àíZ-nÌ H$r H$moS> g»¶m§. 58 / OSS /1, goQ> - A {bI|&
5. (H$) àíZ-nÌ Ho$db qhXr/A§J«oOr ‘mܶ‘ ‘| h¡& {’$a ^r, ¶{X Amn Mmh| Vmo ZrMo Xr JB© {H$gr EH$ ^mfm ‘| CÎma Xo gH$Vo h¢ :A§J«oOr,
qhXr, CXÿ©, n§Om~r, ~§Jbm, V{‘b, ‘b¶mb‘, H$ÝZ‹S>, VobwJy, ‘amR>r, C{‹S>¶m, JwOamVr, H$m|H$Ur, ‘{Unwar, Ag{‘¶m, Zonmbr,
H$í‘rar, g§ñH¥$V Am¡a qgYr&
H¥$n¶m CÎma-nwpñVH$m ‘| {XE JE ~m°³g ‘| {bI| {H$ Amn {H$g ^mfm ‘| CÎma {bI aho h¢&
(I) ¶{X Amn qhXr Ed§ A§J«oOr Ho$ A{V[a³V {H$gr Aݶ ^mfm ‘| CÎma {bIVo h¢, Vmo àíZm| H$mo g‘PZo ‘| hmoZo dmbr Ìw{Q>¶m|/Jb{V¶m|
H$s {Oå‘oXmar Ho$db AmnH$s hmoJr&

58/OSS/1/311-A] G-207 1  [ Contd......

MATHEMATICS
(J{UV)
(311)
Time : 3 Hours] [Maximum Marks : 100
g_` : 3 KÊQ>o] [nyUmªH$ : 100
Note : (1) This question paper consists of four Sections A, B, C and D containing 33 questions.

(2) Question Number 1 to 10 in Section A are multiple choice questions (MCQ). Each question carries
one mark. In each question there are four choices (A), (B), (C) and (D) of which only one is correct.
You have to select the correct choice and indicate it in your answer book by writing (A), (B), (C) or (D)
as the case may be. No separate time is allotted for attempting MCQ.

(3) Question Number 11 to 16 in Section B are very short answer questions and carry 2 marks each.

(4) Question Number 17 to 28 in Section C are short answer questions and carry 4 marks each.

(5) Question Number 29 to 33 in Section D are long answer questions and carry 6 marks each.

(6) All questions are compulsory. There is no overall choice, however, alternative choices are given in
some questions. In such questions, you have to attempt only one choice.

{ZX} e : (1) Bg àíZ nÌ ‘| Hw$b 33 àíZ h¢, Omo Mma IÊS>m| A, ~, g VWm X ‘| {d^m{OV h¡&

(2) IÊS>-A ‘| àíZ g§»¶m 1 go 10 VH$ VWm ~hþ{dH$ënr¶ àíZ h¢, {OZ‘| à˶oH$ Ho$ {bE 1 A§H$ {ZYm©[aV h¡& à˶oH$ àíZ Ho$ CÎma
Ho$ ê$n ‘| (A), (B), (C) VWm (D) Mma {dH$ën {XE JE h¢ {OZ ‘| go H$moB© EH$ ghr h¡& AmnH$mo ghr {dH$ën MwZZm h¡ VWm AnZr
nwpñVH$m ‘| (A), (B), (C) VWm (D) ‘| Omo ghr hmo CÎma Ho$ ê$n ‘| {bIZm h¡& ~hþ{dH$ënr¶ àíZ hb H$aZo Ho$ {bE AbJ go g‘¶
Zht {X¶m J¶m h¡&

(3) IÊS> – ~ ‘| àíZ g§»¶m 11 go 16 VH$ A{V bKwCÎmar¶ àíZ h¡ VWm à˶oH$ Ho$ 2 A§H$ {ZYm©[aV h¢&

(4) IÊS >– g ‘| àíZ g§»¶m 17 go 28 VH$ bKwCÎmar¶ àíZ h¡ VWm à˶oH$ Ho$ 4 A§H$ {ZYm©[aV h¢&

(5) IÊS> – X ‘| àíZ g§»¶m 29 go 33 VH$ XrK© bKwCÎmar¶ àíZ h¡ VWm à˶oH$ Ho$ 6 A§H$ {ZYm©[aV h¢&

(6) g^r àíZ A{Zdm¶© h¢& nyU© àíZnÌ ‘| {dH$ën Zht h¢, {’$a ^r Hw$N> àíZmo§ ‘|, Am§V[aH$ {dH$ën h¢& Eogo g^r àíZm| ‘| go AmnH$mo
EH$ hr {dH$ën hb H$aZm h¢&

58/OSS/1/311-A] G-207 2  [ Contd......

SECTION-A
IÊS-A

1. If A is a square matrix of order 3×3 such that |adj A| = 324, then the possible value
of |A| is equal to [1]

(A) 24 (B) 72

(C) 18 (D) 27

¶{X A EH$ 3×3 H$mo{Q> H$m Amì¶wh h¡ Am¡a |adj A| = 324 hmo, Vmo |A| H$m g§^d ‘mZ hmoJm…

(A) 24 (B) 72

(C) 18 (D) 27

1 1 
2. If sin x  cos x  , then x is equal to [1]
6

1
(A)  (B)
2

1 3
(C) (D)
2 2

1 1 
¶{X sin x  cos x  h¡ Vmo x ~am~a hmoJm …
6

1
(A)  (B)
2

1 3
(C) (D)
2 2

58/OSS/1/311-A] G-207 3  [ Contd......

1 1
3. Let f(x)= and g(x)= 3 . The value of fog(–2) is equal to [1]
x x
1 1
(A) (B) 
8 8

(C) 8 (D) –8

1 1
‘mZm f(x)= VWm g(x)= 3 . fog(–2) H$m ‘mZ hmoJm…
x x

1 1
(A) (B) 
8 8

(C) 8 (D) –8

 sin x dx is equal to
o
4. [1]

 
(A) cos x o  c (B)  cos x o  c
180 180

180 180
(C) cos x o  c (D)  cos x o  c
 

 sin x dx ~am~a hmoJm …
o

 
(A) cos x o  c (B)  cos x o  c
180 180

180 180
(C) cos x o  c (D)  cos x o  c
 
58/OSS/1/311-A] G-207 4  [ Contd......

5. Area of a triangle whose vertices are (1,2,1), (2,2,1) and (2,1,1) is equal to [1]
1
(A) (B) 1
2
1
(C) (D) 2
3
EH$ {Ì^wO {OgHo$ erf© (1,2,1), (2,2,1) VWm (2,1,1) h¢, H$m joÌ’$b hmoJm,
1
(A) (B) 1
2
1
(C) (D) 2
3

dy
6. If y  x x , then is equal to [1]
dx
(A) x x (1  log x) (B) x(1  log x)
(C) y (1  log x ) (D) x x (1  log x)
dy
¶{X y  x x , Vmo ~am~a hmoJm …
dx
(A) x x (1  log x) (B) x(1  log x)
(C) y (1  log x ) (D) x x (1  log x)

1
d2y  dy  3
7. Degree of the differential equation 2  x    0 is equal to [1]
dx  dx 
(A) 3 (B) 2
(C) 5 (D) 1
1
d2y  dy  3
AdH$b g‘rH$aU 2
 x    0 H$s KmV hmoJr …
dx  dx 
(A) 3 (B) 2
(C) 5 (D) 1

58/OSS/1/311-A] G-207 5  [ Contd......

8. The negation of the statement 'all real numbers are rational or irrational' is [1]

(A) all real numbers are rational but not irrational

(B) all real numbers are rational and irrational

(C) all real numbers are not rational but irrational

(D) all real numbers are not rational or irrational

H$WZ ""g^r dmñV{dH$ g§»¶mE| n[a‘o¶ ¶m An[a‘o¶ g§»¶mE| hmoVr h¡§'' H$m {ZfoYZ h¡ …

(A) g^r dmñV{dH$ g§»¶mE| n[a‘o¶ hmoVr h¢ na§Vw An[a‘o¶ Zht

(B) g^r dmñV{dH$ g§»¶mE| n[a‘o¶ Am¡a An[a‘o¶ hmoVr h¢

(C) g^r dmñV{dH$ g§»¶mE| An[a‘o¶ hmoVr h¢ naÝVw n[a‘o¶ Zht

(D) g^r dmñV{dH$ g§»¶mE| Z Vmo n[a‘o¶ Am¡a Z hr An[a‘o¶ hmoVr h¡

9. Slope of the normal to the curve xy – 3x + 2y = 1 at (3, 2) is [1]
1 1
(A) (B) 
5 5

(C) –5 (D) 5

dH«$ xy – 3x + 2y = 1 Ho$ {~ÝXw (3, 2) na A{^b§~ H$s àdUVm h¡…

1 1
(A) (B) 
5 5

(C) –5 (D) 5

58/OSS/1/311-A] G-207 6  [ Contd......

10. The unit vectors which is perpendicular to both vectors 2iˆ  3 ˆj  6 kˆ and 3 ˆj  4 kˆ is
[1]
1 1
(A) (3iˆ  4 ˆj  3kˆ) (B) (3iˆ  4 ˆj  3kˆ)
34 34

1 1
(C) (3iˆ  4 ˆj  3kˆ ) (D) (3iˆ  4 ˆj  3kˆ)
34 34

g{Xem| 2iˆ  3 ˆj  6kˆ VWm 3 ˆj  4kˆ Ho$ bå~dV² EH$ ‘mÌH$ g{Xe h¡ …

1 1
(A) (3iˆ  4 ˆj  3kˆ) (B) (3iˆ  4 ˆj  3kˆ)
34 34

1 1
(C) (3iˆ  4 ˆj  3kˆ) (D) (3iˆ  4 ˆj  3kˆ)
34 34

SECTION-B

IÊS - ~

 2  1  10 
11. Solve for x and y if x    y      . [2]
 3 1 5

 2  1  10 
x Am¡a y Ho$ {bE hb H$s{OE, ¶{X x    y      .
 3 1 5

12. Prove that the function f :    given by f(x) = |x| is neither one-one nor onto.[2]

{gÕ H$s{OE {H$ ’$bZ … f :    Omo f(x) = |x| Ûmam n[a^m{fV h¡, Z Vmo EH¡$H$s Am¡a Z hr AmÀN>mXH$
’$bZ h¡&

58/OSS/1/311-A] G-207 7  [ Contd......

dy  x 
at x = 0 if y  sin  e  .
2

13. Find [2]
dx 6 

 x  dy
¶{X y  sin  e  hmo, Vmo x= 0 na
2

kmV H$s{OE&
6   dx

OR/AWdm

dy y 1
Prove that   if xy = 5.
dx x log x

dy y 1
¶{X x = 5 hmo, Vmo {gÕ H$s{OE {H$
y   .
dx x log x

 1  cos 2 x 
14. Evaluate lim
x 0  . [2]
 3tan x 
2

 1  cos 2 x 
‘mZ kmV H$s{OE … lim
x 0  
 3tan x 
2

x 3 y  2 z 5 x  8 y  3 z  10
15. Find the angle between the lines   and   .
2 2 6 9 3 4
[2]

x 3 y  2 z 5 x  8 y  3 z  10
aoImAm|   VWm   Ho$ ~rM H$m H$moU kmV H$s{OE&
2 2 6 9 3 4

58/OSS/1/311-A] G-207 8  [ Contd......

16. If p and q are two statements given by

p: x+y is an even integer, x, y  

q: x+y is an odd integer, x, y   .

Write the compound statement connecting these two statements with 'OR' and
check its validity. [2]

¶{X H$WZ p Am¡a q {ZåZñdê$n go n[a^m{fV hmo …

p: x+y EH$ g‘ g§»¶m h¡; x, y  

q: x+y EH$ {df‘ g§»¶m h¡; x, y   .

Vmo BZ H$WZm| H$mo ""AWdm'' g§¶moOH$ Ûmam OmoµS>H$a EH$ {‘l H$WZ {b{IE& BgH$s d¡YVm H$s Om±M ^r H$s{OE&

SECTION - C

IÊS - g

 22 20 11 
8 6 23 
17. Express   , as the sum of a symmetric and a skew symmetric
 15 20 9 
 
matrices. [4]

 22 20 11 
8 6 23 
Amì¶yh   H$mo EH$ g‘{‘V Amì¶h d EH$ {df‘ g‘{‘V Amì¶wh Ho$ ¶moJ Ho$ ê$n ‘| 춳V
 15 20 9 
 
H$s{OE&

58/OSS/1/311-A] G-207 9  [ Contd......

x2 y2 z2
18. Prove that yz zx xy  (x–y) (y–z) (z–x) (xy+yz+zx). [4]
x y z

x2 y2 z2
{gÕ H$s{OE H$s yz zx xy  (x–y) (y–z) (z–x) (xy+yz+zx).
x y z

3 8 84
19. Prove that sin
1
 sin 1  cos 1 . [4]
5 17 85

3 8 84
{gÕ H$s{OE … sin
1
 sin 1  cos 1 .
5 17 85

20. A binary operation "*' is defined on  by a*b =a+b+ab for all a, b  . Prove
that the operation * is commutative and associative. [4]

 na EH ~mBZar g§{H«$¶m * {ZåZ ê$n ‘| Xem©B© JB© h¡…$ a*b =a+b+ab à˶oH$ a, b  .

{gÕ H$s{OE H$s ¶h g§{H«$¶m * H«$‘{d{Z‘¶ d ghMmar h¡&

58/OSS/1/311-A] G-207 10  [ Contd......

21. Determine the constants a and b so that the function f(x) is continuous every
where. [4]

 2, x2

f ( x)  ax  b, 2  x  8
 15, x  8.


AMa a Am¡a b Ho$ dmo ‘mZ kmV H$s{OE, {OZHo$ {bE, {ZåZ ’$bZ f(x) g~ OJh gVV² h¡…

 2, x2

f ( x)  ax  b, 2  x  8
 15, x  8.


dy cos 2 (l  y )
22. Given cos y = x cos(l+y). Prove that  . [4]
dx sin l

dy cos 2 (l  y )
{X¶m J¶m h¡ : cos y = x cos(l+y), {gÕ H$s{OE H$s  .
dx sin l

1  2x 
23. Evaluate :  sin  2 
dx. [4]
 1  x 

1  2x 
‘mZ kmV H$s{OE …  sin  2 
dx.
1 x 

OR/AWdm

58/OSS/1/311-A] G-207 11  [ Contd......

x2 1
Evaluate  4 dx.
x  3x2 1

x2 1
‘mZ kmV H$s{OE …  4 dx.
x  3x2 1

 1
24. Evaluate 0
2
dx. [4]
5  4cos x
 1
‘mZ kmV H$s{OE … 0
2
dx.
5  4cos x
OR/AWdm

4
Verify Rolle's theorem for f ( x)  x  when x  [1,4] .
x

4
’$bZ f ( x)  x  H$mo AÝVamb [1,4] na amobo à‘o¶ H$mo g˶m{nV H$s{OE&
x

25. Solve the following differential equation : [4]

dy y 
x  y  x  x tan , given y = when x = 1.
dx x 2

{ZåZ AdH$b g‘rH$aU hb H$s{OE…
dy y 
x  y  x  x tan {X¶m J¶m h¡ y = O~ x = 1.
dx x 2

58/OSS/1/311-A] G-207 12  [ Contd......

26. Show that the three points with position vectors

       
a  2b  3c ,  2 a  3b  2 c ,  8 a  13b are collinear.. [4]

  
{XImB©E {H$ VrZ {~ÝXþ {OZHo$ pñW{V g{Xe H«$‘e… a  2b  3c ,  2 a  3b  2 c ,  8 a  13b
g§ a o I h¡&

27. Find the equations of tangent to the curve y=x3+3x2–5 which is perpendicular to
the line 2x–6y+1 = 0. [4]

dH«$ y = x3 +3x2–5 na ñne© aoIm, Omo aoIm 2x–6y+1 = 0 Ho$ bå~dV² h¡, H$m$ g‘rH$aU kmV H$s{OE&

28. Using vector method, prove that the diagonals of a rhombus are perpendicularly
bisect each other. [4]

g{Xem| H$m à¶moJ H$aHo$, [gÕ H$s{OE H$s EH$ g‘ MVw^©wO Ho$ {dH$U© nañna bå~dV² hmoVo h¢&

OR/AWdm

   
Find a unit vector perpendicular to both the vectors (a  b ) and (a  b ) where
 
a  iˆ  ˆj  kˆ and b  iˆ  2 ˆj  3kˆ .

 
g{Xe (a  b ) Am¡a (a  b ) ‘| go à˶oH$ Ho$ bå~dV² ‘mÌH$ g{Xe kmV H$s{OE, Ohm± a  iˆ  ˆj  kˆ

VWm b  iˆ  2 ˆj  3kˆ h¢&

58/OSS/1/311-A] G-207 13  [ Contd......

SECTION-D

IÊS -X

29. Solve the following system of equations, using matrix inversion method [6]

x + 2y + 2z = 4

3x – 3y + 5z = 11

x + 12y – 5z = –3

Amì¶yh {d{Y go, {ZåZ g‘rH$aU {ZH$m¶ H$mo hb kmV H$s{OE …

x + 2y + 2z = 4

3x – 3y + 5z = 11

x + 12y – 5z = –3

OR/AWdm

2 2 1
 
Find inverse of a matrix A   1 0 2  .
2 1 2
 

2 2 1
 
Amì¶wh A   1 0 2  H$m ì¶wËH«$‘ kmV H$r{OE&
2 1 2
 

58/OSS/1/311-A] G-207 14  [ Contd......

30. Find the area of the region bounded by the curves x2 =16y, y =1, y = 4 and the
y-axis in the first quadrant, using integration. [6]

àW‘ MVwWmªe ‘| dH«$ x2 =16y, y =1, y = 4 VWm y-Aj go {Kao joÌ H$m joÌ’$b, g‘mH$bZ {d{Y go kmV
H$s{OE&

OR/AWdm

4 36
If x+y= 2, show that the maximum value of  is less than its minimum value.
x y

4 36
¶{X x+y= 2 h¡, Vmo {XImB©E {H$  H$m A{YH$V‘ ‘mZ ݶyZV‘ ‘mZ go H$‘ h¡&
x y

dy
31. For the differential equation xy  ( x  2)( y  2) , find the solution curve passing
dx
through the point (1, –1). [6]

dy
AdH$b g‘rH$aU xy  ( x  2)( y  2) Ûmam àX{e©V dH«$ H$m g‘rH$aU kmV H$s{OE Omo {~ÝXþ
dx
(1, –1) go hmoH$a OmVm h¡&

32. Find the equation of the plane through the point (2,3,5) and perpendicular to the
planes 2x–3y+z = 2, 4x+y–3z+1= 0 [6]

{~ÝXþ (2,3,5) go JwOaZo dmbo g‘Vb, Omo g‘Vbm| 2x–3y+z = 2, 4x+y–3z+1= 0 Hoo$ bå~dV² h¢, H$m
g‘rH$aU kmV H$s{OE&

58/OSS/1/311-A] G-207 15  [ Contd......

33. A producer has 30 and 17 units of labour and capital respectively which he can use
to produce two types of goods X and Y. To produce one unit of X, 2 units of
labour and 3 units of capital are required . Similarly, 3 units of labour and 1 unit of
capital is required to produce one unit of Y. If X and Y are priced at 100 and
120 per unit respectively, how should the producer use his resources to maximise
the total revenue? Formulate the above problem as a LPP and solve it graphically.
[6]

EH$ CËnmXH$ Ho$ nmg 30 BH$mB© l‘ H$s d 17 BH$mB© nw±Or h¡, {OZH$m à¶moJ dh Xmo àH$ma X VWm Y H$s
dñVyAm| H$m CËnmXZ H$aZo Ho$ {bE H$a gH$Vm h¡& X H$s EH$ BH$mB© Ho$ CËnmXZ Ho$ {bE 2 BH$mB© l‘ VWm 3
BH$mB© n±wOt H$s Amdí¶H$Vm hmoVr h¡& Y H$s EH$ BH$mB© Ho$ CËnmXZ Ho$ {bE 3 BH$mB© l‘ VWm 1 BH$mB© n±wOt H$s
Amdí¶H$Vm hmoVr h¡& CËnmXm| X VWm Y H$s à{V BH$mB© H$m ‘wë¶ H«$‘e… ê$. 100 VWm ê$. 120 h¡& A{YH$V‘
‘wë¶ A{O©V H$aZo Ho$ {bE CËnmXH$ H$mo AnZr CnbãY l‘ d nw±Or BH$mB©¶m| H$m {H$g àH$ma Cn¶moJ H$aZm
Mm{hE& Cnamoº$ H$o {bE EH$ a¡{IH$ àmoJ«m‘Z g‘ñ¶m ‘| n[ad{V©V H$a, AmboIr¶ {d{Y go hb H$s{OE&



58/OSS/1/311-A] G-207 16 