HBSE Class 12 Mathematics Question Paper 2017 Set B

CLASS : 12th (Sr. Secondary) Code No. 2031
Series : SS-M/2017
Roll No. SET : B

xf.kr GRAPH
MATHEMATICS
[ Hindi and English Medium ]
ACADEMIC/OPEN
(Only for Fresh Candidates)
(Evening Session)
Time allowed : 3 hours ] [ Maximum Marks : 80
• Ñi;k tk¡p dj ysa fd bl iz'u&i= esa eqfnzr iz'u 20 gSaA
Please make sure that the printed question paper are contains 20
questions.
• iz'u&i= esa nkfgus gkFk dh vksj fn;s x;s dksM uEcj rFkk lsV dks Nk= mÙkj&iqfLrdk ds
eq[;&i`"B ij fy[ksaA
The Code No. and Set on the right side of the question paper should be
written by the candidate on the front page of the answer-book.
• Ñi;k iz'u dk mÙkj fy[kuk 'kq: djus ls igys] iz'u dk Øekad vo'; fy[ksaA
Before beginning to answer a question, its Serial Number must be written.
• mÙkj&iqfLrdk ds chp esa [kkyh iUuk@iUus u NksMsa+A
Don’t leave blank page/pages in your answer-book.
• mÙkj&iqfLrdk ds vfrfjDr dksbZ vU; 'khV ugha feysxhA vr% vko';drkuqlkj gh fy[ksa vkSj fy[kk
mÙkj u dkVsaA
Except answer-book, no extra sheet will be given. Write to the point and do
not strike the written answer.
• ijh{kkFkhZ viuk jksy ua0 iz'u&i= ij vo'; fy[ksaA
Candidates must write their Roll Number on the question paper.
• d`i;k iz'uksa dk mÙkj nsus lss iwoZ ;g lqfuf'pr dj ysa fd iz'u&i= iw.kZ o lgh gS] ijh{kk ds
mijkUr bl lEcU/k esa dksbZ Hkh nkok Lohdkj ugha fd;k tk;sxkA
2031/ (Set : B) P. T. O.

(2) 2031/ (Set : B)
Before answering the question, ensure that you have been supplied the
correct and complete question paper, no claim in this regard, will be
entertained after examination.
lkekU; funsZ'k %
(i) bl iz'u-i= esa 20 iz'u gSa] tks fd pkj [k.Mksa % v] c] l vkSj n esa ck¡Vs x, gSa %
[k.M ^v* % bl [k.M esa ,d ç'u gS tks cgqfodYih; çdkj ds 16 (i-xvi) Hkkxksa esa gSA
izR;sd Hkkx 1 vad dk gSA
[k.M ^c* % bl [k.M esa 2 ls 11 rd dqy nl ç'u gSaA çR;sd ç'u 2 vadksa dk gSA
[k.M ^l* % bl [k.M esa 12 ls 16 rd dqy ik¡p ç'u gSaA çR;sd ç'u 4 vadksa dk
gSA
[k.M ^n* % bl [k.M esa 17 ls 20 rd dqy pkj ç'u gSAa çR;sd ç'u 6 vadksa dk gSA
(ii) lHkh ç'u vfuok;Z gSaA
(iii) [k.M ^n* ds dqN ç'uksa esa vkarfjd fodYi fn;s x;s gSa] muesa ls ,d gh iz'u dks pquuk
gSA
(iv) fn;s x;s xzkQ-isij dks viuh mÙkj-iqfLrdk ds lkFk vo'; uRFkh djsaA
(v) xzkQ-isij ij viuh mÙkj-iqfLrdk dk Øekad vo'; fy[ksaA
(vi) dSYD;qysVj ds ç;ksx dh vuqefr ugha gSA
General Instructions :
(i) This question paper consists of 20 questions which are divided into
four Sections : A, B, C and D :
Section 'A' : This Section consists of one question which is divided
into 16 (i-xvi) parts of multiple choice type. Each part
carry 1 mark.
Section 'B' : This Section consists of ten questions from 2 to 11. Each
question carries 2 marks.
Section 'C' : This Section consists of five questions from 12 to 16.
Each question carries 4 marks.
Section 'D' : This Section consists of four questions from 17 to 20.
Each question carries 6 marks.
(ii) All questions are compulsory.
(iii) Section 'D' contains some questions where internal choice have been
provided. Choose one of them.

2031/ (Set : B)

(3) 2031/ (Set : B)
(iv) You must attach the given graph-paper along with your answer-book.
(v) You must write your Answer-book Serial No. on the graph-paper.
(vi) Use of Calculator is not permitted.
[k.M – v
SECTION – A
1. (i) eku yhft, fd f(x) = 3x }kjk ifjHkkf"kr Qyu f : R → R gS] lgh
mÙkj dk p;u dhft, % 1
(A) f cgq,d vkPNknd gS
(B) f ,dSdh vkPNknd gS
(C) f u rks ,dSdh gS vkSj u gh vkPNknd gS
(D) f ,dSdh gS] fdUrq vkPNknd ugha gS
Let f : R → R be defined as f(x) = 3x, choose the correct answer :
(A) f is many-one onto
(B) f is one-one onto
(C) f is neither one-one nor onto
(D) f is one-one but not onto
1 1
(ii) cos −1  + 2 sin−1  dk eku gS % 1
2 2
2π π 3π
(A) (B) (C) (D) π
3 2 4
1 1
cos −1  + 2 sin−1  is equal to :
2 2
2π π 3π
(A) (B) (C) (D) π
3 2 4
2 − 1 10
(iii) ;fn x   + y   =   gks] rks x vkSj y ds eku gSa % 1
3 1  5
(A) x = 2, y = −3 (B) x = 10, y = 0
(C) x = 3, y = −4 (D) x = 5, y = −1
2 − 1 10
If x   + y   =   , the values of x and y are :
3 1  5
2031/ (Set : B) P. T. O.

(4) 2031/ (Set : B)
(A) x = 2, y = −3 (B) x = 10, y = 0
(C) x = 3, y = −4 (D) x = 5, y = −1
(iv) ;fn A dksfV 2 dk O;qRØe.kh; vkO;wg gS] rks det(A–1) cjkcj gS % 1
(A) 0 (B) det A
1
(C) 1 (D)
det A
–1
If A is an invertible matrix of order 2, then det(A ) is equal to :
(A) 0 (B) det A
1
(C) 1 (D)
det A
 2
(v) ;fn Qyu f (x ) = kx , ;fn x ≤2
}kjk ifjHkkf"kr x = 2 ij larr gks] rks k
 3, ;fn x >2
dk eku gS % 1
4 3 2
(A) (B) (C) (D) 3
5 4 3
kx 2 , if x ≤2
If the function f(x) defined by f (x ) =  is continuous
 3, if x >2
at x = 2, then the value of k is :
4 3 2
(A) (B) (C) (D) 3
5 4 3
(vi) o`Ùk ds {ks=Qy ds ifjorZu dh nj] bldh f=T;k r ds lkis{k r = 4 lseh ij gS % 1
11 π lseh /ls0 (B) 10 π lseh /ls0
2 2
(A)

8 π lseh /ls0 12 π lseh /ls0
2 2
(C) (D)
The rate of change of the area of a circle w.r.t. its radius r at r = 4
cm is :
(A) 11 π cm2/sec. (B) 10 π cm2/sec.
(C) 8 π cm2/sec. (D) 12 π cm2/sec.

2031/ (Set : B)

(5) 2031/ (Set : B)
2 2
x y
(vii) oØ + = 1 ij og fcUnq] ftl ij Li'kZ js[kk x-v{k ds lekUrj gS % 1
9 16

(A) (–1, ±4) (B) (0, ±4)

(C) (–1, ±8) (D) (±3, 0)
x 2 y2
The point on the curve + = 1 at which the tangent is parallel
9 16
to x-axis, is :

(A) (–1, ±4) (B) (0, ±4)

(C) (–1, ±8) (D) (±3, 0)
dx
(viii) ∫ 2
dk eku gS % 1
x + 2x + 2

(A) tan−1(x + 1) + c (B) (x + 1) tan−1 x + c

(C) x tan−1(x + 1) + c (D) tan−1 x + c

dx
∫ x 2 + 2x + 2 is equal to :

(A) tan−1(x + 1) + c (B) (x + 1) tan−1 x + c

(C) x tan−1(x + 1) + c (D) tan−1 x + c
π
2
∫ (x + x cos x + tan x + 1)dx dk eku gS %
3 5
(ix) 1
−π
2

(A) π (B) 1

(C) 2 (D) 0

2031/ (Set : B) P. T. O.

(6) 2031/ (Set : B)
π
2
The value of ∫ (x 3 + x cos x + tan5 x + 1) dx is :
−π
2

(A) π (B) 1
(C) 2 (D) 0

(x) vody lehdj.k dy = e x +y dk O;kid gy gS % 1
dx
(A) e − x + e −y = c (B) e x + e −y = c

(C) e x + ey = c (D) e −x + e y = c
dy
The general solution of the differential equation = e x +y is :
dx
(A) e − x + e −y = c (B) e x + e −y = c

(C) e x + ey = c (D) e −x + e y = c
(xi) fuEufyf[kr vody lehdj.kksa esa ls fdl lehdj.k dk O;kid gy
x
y = c1e + c 2e −x
gS % 1

d 2y d 2y
(A) 2
+1 = 0 (B) +y = 0
dx dx 2

d 2y d 2y
(C) −1 = 0 (D) −y = 0
dx 2 dx 2
Which of the following differential equation has y = c1e x + c 2e − x as
the general solution :

d 2y d 2y
(A) +1 = 0 (B) +y = 0
dx 2 dx 2

d 2y d 2y
(C) 2
−1 = 0 (D) −y = 0
dx dx 2

2031/ (Set : B)

(7) 2031/ (Set : B)
(xii) lfn'k i − j dk lfn'k i + j ij iz{ksi gS %
ˆ ˆ ˆ ˆ 1

(A) − 2 (B) 2

(C) 0 (D) buesa ls dksbZ ugha
The projection of the vector iˆ − ˆj on the vector iˆ + ˆj is :

(A) − 2 (B) 2

(C) 0 (D) None of these
x −3 y −2 z +4
(xiii) js[kkvksa = = rFkk x − 5 = y + 2 =
z
ds ;qXe ds chp dk
1 2 2 3 2 6
dks.k gS % 1
 19   19 
(A) cos −1  (B) cos −1 
 29   21 
 36   29 
(C) cos −1  (D) cos −1 

 7   19 
x −3 y −2 z +4
The angle between the pair of lines = = and
1 2 2
x −5 y +2 z
= = is :
3 2 6
 19   19 
(A) cos −1  (B) cos −1 
 29   21 
 36   29 
(C) cos −1  (D) cos −1 

 7   19 
(xiv) ,sls ikls] ftlds rhu Qydksa ij ,d] vU; nks ij 2 vkSj ,d Qyd ij 5 fy[kk x;k
gS] dks mNkyus ij izkIr la[;kvksa dk ek/; gS % 1
8
(A) 5 (B)
3
(C) 1 (D) 2

2031/ (Set : B) P. T. O.

(8) 2031/ (Set : B)
The mean of the numbers obtained on throwing a die having
written one on three faces, 2 on two faces and 5 on one face, is :
8
(A) 5 (B)
3
(C) 1 (D) 2

(xv) ;fn P(A) = 0 rFkk P(B) = 1 gks] rks P(B/A) dk eku gS % 1
3
1
(A) 1 (B)
3
(C) 0 (D) ifjHkkf"kr ugha
1
If P(A) = 0 and P(B) = then P(B/A) is :
3
1
(A) 1 (B)
3
(C) 0 (D) Not defined

(xvi) ,d ikls dks 6 ckj mNkyk tkrk gSA ;fn ikls ij ^fo"ke la[;k izkIr gksuk* ,d lQyrk

gS rks 5 lQyrkvksa dh izkf;drk gS % 1

3 23
(A) (B)
32 5
32 15
(C) (D)
5 32

A die is thrown 6 times. If 'getting an odd number' is a success,
then the probability of 5 successes is :
3 23
(A) (B)
32 5
32 15
(C) (D)
5 32

2031/ (Set : B)

(9) 2031/ (Set : B)
[k.M – c
SECTION – B

2. ;fn f : R+ → [4, ∞), f(x) = x2 + 4 }kjk iznÙk Qyu gS] rks fn[kkb, fd f O;qRØe.kh; gS
vkSj f –1 fudkfy,A 2
+
Let f : R → [4, ∞) is given by f(x) = x2 + 4. Show that f is invertible and
–1
find of f .
3. fln~/k dhft, % 2
8 3 77
sin−1 + sin−1 = tan−1
17 5 36
Prove that :
8 3 77
sin−1 + sin−1 = tan−1
17 5 36

vkO;wg A = 
3 5
4.  ds fy, lR;kfir dhft, fd A – A' fo"ke lefer vkO;wg gSA 2
1 − 1
3 5 
For the matrix A =   , verify that A – A' is a skew-symmetric
1 − 1
matrix.
1 1 −2
5. eku Kkr dhft, % 2 1 −3 2
5 4 −9

1 1 −2
Evaluate : 2 1 − 3
5 4 −9

dx
6. ∫ 6 − x − x 2 dk eku Kkr dhft,A 2

2031/ (Set : B) P. T. O.

( 10 ) 2031/ (Set : B)
dx
Find ∫
6 − x − x2
π
2
dx
7. eku Kkr dhft, % ∫ 2
0
1 + cos x

π
2
dx
Evaluate : ∫ 1 + cos x
0

8. lHkh o`Ùkksa dk vody lehdj.k Kkr dhft, tks fd ewy-fcUnq ls xqtjrs gksa vkSj ftldk dsUnz
y-v{k ij gksA 2
Find the differential equation of all circles which passes through the
origin and whose centre lies on y-axis.
2
9. fn[kkb, fd y = e −x + ax + b vody lehdj.k e x d y2 = 1 dk gy gSA 2
dx
Show that y = e −x + ax + b is a solution of the differential equation
d 2y
ex = 1.
dx 2

10. ;fn x = cos θ − cos 2θ vkSj y = sin θ − sin 2θ gks] rks dy fudkfy;sA 2
dx
dy
Find , if x = cos θ − cos 2θ and y = sinθ − sin2θ .
dx

11. ,d ikls dks nks ckj mNkyk x;k vkSj izdV gqbZ la[;kvksa dk ;ksx 6 ik;k x;kA la[;k 4 ds
U;wure ,d ckj izdV gksus dh lizfrcU/k izkf;drk Kkr dhft,A 2
A dies is thrown twice and the sum of numbers appearing is observed
to be 6. What is the conditional probability that the number 4 has
appeared at least once ?
[k.M – l
SECTION – C

2031/ (Set : B)

( 11 ) 2031/ (Set : B)
1 1 1 1 π
12. fln~/k dhft, % tan−1 + tan−1 + tan−1 + tan−1 = 4
5 7 3 8 4
1 1 1 1 π
Prove that : tan−1 + tan−1 + tan−1 + tan−1 = .
5 7 3 8 4

13. (sin x )x + sin−1( x ) dk x ds lkis{k vodyu dhft,A 4

Differentiate (sin x )x + sin−1( x ) w.r.t. x.

14. vUrjky [1, 5] esa f (x ) = 2x 3 − 15x 2 + 36x + 1 }kjk iznÙk Qyu ds fujis{k mPpre
rFkk fujis{k fuEure ekuksa dks Kkr dhft,A 4
Find the absolute maximum and absolute minimum values of a
function f given by f (x ) = 2x 3 − 15x 2 + 36x + 1 on the interval [1, 5].

15. A vkSj B ckjh-ckjh ls ,d ikls dks mNkyrs gSa tc rd fd muesa ls dksbZ ,d ikls ij N%
izkIr dj [ksy dks thr ugha ysrkA ;fn A [ksy dks 'kq: djs rks muds thrus dh Øe'k%
izkf;drk Kkr dhft,A 4
A and B throw a die alternatively till one of them gets a 6 and wins the
games. Find their respective probability of winning, if A starts first.

16. ;fn a = 2iˆ − ˆj + kˆ rFkk b = 3iˆ + 4 ˆj − kˆ gks] rks a vkSj b nksuksa ij yfEcr ,d ek=d
lfn'k Kkr dhft, rFkk bu nksuksa lfn'kksa ds chp dk lkbu dk dks.k Hkh Kkr dhft,A 4

If a = 2iˆ − ˆj + kˆ and b = 3iˆ + 4 ˆj − kˆ , then find a unit vector perpendicular
to both a and b . Also calculate the sine of angle between these two
vectors.
[k.M – n
SECTION – D

17. fuEu jSf[kd lehdj.kksa dks vkO;wg fof/k }kjk gy dhft, % 6
2x + 3y + 3z = 5
x − 2y + z = −4
3x − y − 2z = 3

2031/ (Set : B) P. T. O.

( 12 ) 2031/ (Set : B)
Solve the system of linear equations by matrix method :
2x + 3y + 3z = 5
x − 2y + z = −4
3x − y − 2z = 3

18. oØ y = x 2 + 5 rFkk y = x 3 vkSj js[kkvksa x = 1 rFkk x = 2 ds chp
f?kjs gq, Hkkx dk {ks=Qy Kkr dhft,A 6
Find the area between the curves y = x 2 + 5 and y = x 3 and the lines x
= 1 and x = 2.
vFkok
OR

oØ y = x 2 − 4 rFkk js[kkvksa y = 0 vkSj y = 5 ds chp f?kjs gq, Hkkx dk {ks=Qy Kkr
dhft,A
Find the area bounded by the curve y = x 2 − 4 and the lines y = 0 and
y = 5.

19. leryksa r .(iˆ + 3 ˆj − kˆ ) = 5 rFkk r .(2iˆ − ˆj + kˆ ) = 3 ds izfrPNsnu rFkk fcUnq (2, 1, –2) ls
xqtjrs gq, lery dk lehdj.k Kkr dhft,A 6
Find the equation of the plane through the intersection of the planes
r .(iˆ + 3 ˆj − kˆ ) = 5 and r .(2iˆ − ˆj + kˆ ) = 3 and passing through the point
(2, 1, –2).

vFkok
OR

lekUrj js[kkvksa r = 2iˆ + 3 ˆj + kˆ + λ(2iˆ + 5 ˆj + 3kˆ ) rFkk
r = 3iˆ + 4 ˆj + 5kˆ + µ(4iˆ + 10 ˆj + 6kˆ ) ds chp U;wure nwjh Kkr dhft,A

2031/ (Set : B)

( 13 ) 2031/ (Set : B)
Find the shortest distance between the parallel lines :
r = 2iˆ + 3 ˆj + kˆ + λ(2iˆ + 5 ˆj + 3kˆ ) and r = 3iˆ + 4 ˆj + 5kˆ + µ(4iˆ + 10 ˆj + 6kˆ ) .

20. fuEu jSf[kd izksxzkeu leL;k dks xzkQh; gy dhft, % 6
O;ojks/kksa 2x + y ≤ 104 ; x + 2y ≤ 76 rFkk x ≥ 0, y ≥ 0 ds vUrxZr z = 6x + 11y
dk vf/kdrehdj.k dhft,A
Solve the following linear programming problem graphically :
Maximize z = 6x + 11y subject to the constraints 2x + y ≤ 104 ;
x + 2y ≤ 76 and x ≥ 0, y ≥ 0 .

s

2031/ (Set : B) P. T. O.