HBSE Class 12 Mathematics Question Paper 2017 Set C

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

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 : C) P. T. O.

(2) 2031/ (Set : C)
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 : C)

(3) 2031/ (Set : C)
(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) ;fn f}vk/kkj lafØ;k ∗ N ij a ∗ b = a3 + b3 }kjk ifjHkkf"kr gks] rks ∗ gS % 1
(A) nksuksa Øefofues; rFkk lkgp;Z
(B) lkgp;Z fdUrq Øefofues; ugha
(C) Øefofues; fdUrq lkgp;Z ugha
(D) u lkgp;Z vkSj u gh Øefofues;
If the binary operation ∗ on N defined as a ∗ b = a3 + b3, then ∗
is :
(A) Both associative and commutative
(B) Commutative but not associative
(C) Associative but not commutative
(D) Neither commutative nor associative
(ii) tan −1( 3 ) − cot −1(− 3 ) dk eku gS % 1
π
(A) − (B) 2 3 (C) 0 (D) π
2
tan −1( 3 ) − cot −1(− 3 ) is equal to :
π
(A) − (B) 2 3 (C) 0 (D) π
2
;fn vkO;wg   7 6
2x + 3 6
(iii) = cjkcj gks]a rks x vkSj y ds eku gSa %
 15 2y − 4 15 14
1
(A) x = 4, y = 5 (B) x = −2, y = 1
(C) x = 3, y = 9 (D) x = 2, y = 9

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

(4) 2031/ (Set : C)
2x + 3 6  7 6
If the matrices  = are equal, then the
 15 2y − 4 15 14
values of x and y are :
(A) x = 4, y = 5 (B) x = −2, y = 1
(C) x = 3, y = 9 (D) x = 2, y = 9
2 3 
(iv) vkO;wg A =   dk lg[k.Mt gS % 1
1 − 4
 − 4 3 − 4 − 1
(A)   (B)  
 1 2 − 3 2 
 − 4 − 3  − 4 − 1
(C) −1 2  (D)  3 − 2
   
2 3 
Ad joint of the matrix A =   is :
1 − 4
 − 4 3 − 4 − 1
(A)   (B)  
 1 2 − 3 2 
 − 4 − 3  − 4 − 1
(C)   (D)  
−1 2   3 − 2
 kx + 1, ;fn x ≤ 5
(v) ;fn Qyu f(x), f (x ) =  }kjk ifjHkkf"kr x = 5 ij larr gks]
3x − 5, ;fn x > 5
rks k dk eku gS % 1
9 4 5
(A) (B) (C) (D) 0
5 5 9
 kx + 1, if x ≤5
If the function f(x) defined f (x ) = 
by is
3x − 5, if x >5
continuous at x = 5, then the value of k is :
9 4 5
(A) (B) (C) (D) 0
5 5 9
(vi) ,d o`Ùk dh f=T;k 0.7 lseh/ls0 dh nj ls c<+ jgh gSA bldh ifjf/k dh o`fn~/k dh nj
gS % 1
(A) 3.3 π lseh/ls0 (B) 1.4 π lseh/ls0

2031/ (Set : C)

(5) 2031/ (Set : C)
(C) 2.2 π lseh/ls0 (D) 4.4 π lseh/ls0
The radius of a circle is increasing at the rate of 0.7 cm/sec. The
rate of increase of its circumference is :
(A) 3.3 π cm/sec. (B) 1.4 π cm/sec.
(C) 2.2 π cm/sec. (D) 4.4 π cm/sec.
x 2 y2
(vii) oØ + = 1 ij og fcUnq] ftl ij Li'kZ js[kk y-v{k ds lekUrj gS] gS %
9 16
1
(A) (±4, 0) (B) (±3, 0)
(C) (0, ±2) (D) (0, ±4)
x 2 y2
The point on the curve + = 1 at which the tangents are
9 16
parallel to y-axis, is :

(A) (±4, 0) (B) (±3, 0)

(C) (0, ±2) (D) (0, ±4)
e x (1 + x )
(viii) ∫ dx cjkcj gS % 1
cos2 (e x .x )

(A) − cot (e x .x x ) + c (B) tan (e x ) + c

(C) cot (e x ) + c (D) tan (xe x ) + c
e x (1 + x )
∫ cos2(e x .x ) dx is equal to :

(A) − cot (e x .x x ) + c (B) tan (e x ) + c

(C) cot (e x ) + c (D) tan (xe x ) + c

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

(6) 2031/ (Set : C)
2
3
dx
(ix) ∫ 4 + 9x 2 cjkcj gS % 1
0

π π
(A) (B)
24 4
π π
(C) (D)
12 6
2
3
dx
∫ 4 + 9x 2 is equal to :
0

π π
(A) (B)
24 4
π π
(C) (D)
12 6

(x) oØksa y + c sin x = 0 ds dqy ds fy, vody lehdj.k gS % 1
dy dy
(A) − y cot x = 0 (B) + cos x = 0
dx dx
dy
(C) − y = cos x (D) buesa ls dksbZ ugha
dx
The differential equation for the family of curves y + c sin x = 0 is :
dy dy
(A) − y cot x = 0 (B) + cos x = 0
dx dx
dy
(C) − y = cos x (D) None of these
dx
dy
(xi) ;fn x 2. = 2 gks] rks bl vody lehdj.k dk gy gS % 1
dx

(A) y = 2x + c (B) y = x2 + c

2031/ (Set : C)

(7) 2031/ (Set : C)
2 −2
(C) y = +c (D) y= +c
x x
dy
If x 2. = 2 , then the solution of this differential equation is :
dx

(A) y = 2x + c (B) y = x2 + c

2 −2
(C) y= +c (D) y= +c
x x

(xii) ;fn lfn'k 5iˆ + 2 ˆj − kˆ rFkk λiˆ − ˆj + 5kˆ ijLij yEcor~ lfn'k gks] rks λ dk eku
gS % 1
3 5
(A) (B)
5 7
7 2
(C) (D)
5 5

If the vectors 5iˆ + 2 ˆj − kˆ and λiˆ − ˆj + 5kˆ are orthogonal vectors,
then the value of λ is :
3 5
(A) (B)
5 7
7 2
(C) (D)
5 5

(xiii) js[kkvksa
x y z
= = rFkk x − 5 = y − 2 = z − 3 ds ;qXe ds chp dk dks.k gS %
2 2 1 4 1 8
1
π 2
(A) (B) cos −1 
2 3
 1  1
(C) cos −1  (D) cos −1 
 65  3

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

(8) 2031/ (Set : C)
x y z
The angle between the pair of lines = = and
2 2 1
x −5 y −2 z −3
= = is :
4 1 8
π 2
(A) (B) cos −1 
2 3
 1  1
(C) cos −1  (D) cos −1 
 65  3
(xiv) ;fn A vkSj B nks ,slh ?kVuk,¡ gSa fd P(A) ≠ 0 vkSj P(B/A) = 1, rc % 1
(A) A⊂B (B) B⊆A
(C) B ≠ ϕ (D) A = ϕ
If A and B are two events such that P(A) ≠ 0 and P(B/A) = 1, then :
(A) A⊂B (B) B⊆A
(C) B≠ϕ (D) A=ϕ
(xv) ;fn iklksa dk ,d tksM+k mNkyk tkrk gS rks izR;sd ikls ij fo"ke vHkkT; la[;k izkIr
djus dh izkf;drk gS % 1

1
(A) (B) 0
3
1 1
(C) (D)
9 36
The probability of obtaining an odd prime number on each die,
when a pair of die is rolled, is :

1
(A) (B) 0
3
1 1
(C) (D)
9 36

1
(xvi) ;fn P(A) = rFkk P(B) = 0 gks] rks P(A/B) gS % 1
4
2031/ (Set : C)

(9) 2031/ (Set : C)
1
(A) 1 (B)
2
(C) 0 (D) ifjHkkf"kr ugha
1
If P(A) = and P(B) = 0, then P(A/B) is :
4
1
(A) 1 (B)
2
(C) 0 (D) Not defined

[k.M – c
SECTION – B
x
2. fn[kkb, fd f : [–1, 1] → R, f(x) = }kjk iznÙk ,dSdh Qyu gSA Qyu f : [–1, 1]
x +2
→ R dk O;qRØe Hkh Kkr dhft,A 2
x
Show that f : [–1, 1] → R, given by f(x) = is one-one. Find the
x +2
inverse of the function f : [–1, 1] → R.
3. fln~/k dhft, % 2
1 1 31
2 tan−1 + tan−1 = tan−1
2 7 17
Prove that :
1 1 31
2 tan−1 + tan−1 = tan−1
2 7 17

izkjfEHkd lafØ;kvksa dk iz;ksx djds vkO;wg 
2 − 6
4.  dk O;qRØe Kkr dhft,A 2
1 − 2
Using elementary transformations, find the inverse of the matrix
2 − 6
1 − 2 .
 

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

( 10 ) 2031/ (Set : C)
5. ;fn A vkSj B nks lefer vkO;wg gks] rks fln~/k dhft, fd AB – BA ,d fo"ke lefer
vkO;wg gSA 2
If A and B are symmetric matrices, prove that AB – BA is a skew-
symmetric matrix.

∫ 8 + 2x − x dx dk eku Kkr dhft,A
2
6. 2

Evaluate : ∫ 8 + 2x − x 2 dx
π
4
7. eku Kkr dhft, % ∫ tan3 x dx 2
0
π
4
∫ tan x dx
3
Evaluate :
0

8. ,sls lHkh o`Ùkksa dh] tks mn~xe fcUnq ls xqtjrs gksa vkSj ftudk dsUnz x-v{k ij fLFkr gks] dh
vody lehdj.k Kkr dhft,A 2
Find the differential equation of all circles passing through the origin
and having their centres on x-axis.
9. vody lehdj.k ydx − xdy = xy dx dks gy dhft,A 2
Solve the differential equation ydx − xdy = xy dx .

10. Qyu f (x ) = x 2 + 2x − 8 , x ∈ [–4, 2] ds fy, jksys ds izes; dks lR;kfir dhft,A 2

Verify Rolle's theorem for the function f (x ) = x 2 + 2x − 8 for x ∈ [–4, 2].

11. ,d cDls esa nl dkMZ 1 ls 10 rd iw.kkZad fy[kdj j[ks x;s vkSj mUgsa vPNh rjg feyk;k
x;kA bl cDls ls ,d dkMZ ;kn`PN;k fudkyk x;kA ;fn ;g Kkr gks fd fudkys x;s dkMZ ij
la[;k 3 ls vf/kd gS] rks bl la[;k ds le gksus dh D;k izkf;drk gS \ 2
Ten cards numbered one to ten are placed in a box, mixed up
thoroughly and then one card is drawn randomly. If it is known that
the number on the drawn card is more then 3, what is the probability
that it is an even number ?

2031/ (Set : C)

( 11 ) 2031/ (Set : C)
[k.M – l
SECTION – C

12. ;fn sin  sin−1 1 + cos−1 x  = 1 gks] rks x dk eku Kkr dhft,A 4
 5 
 1 
If sin  sin−1 + cos−1 x  = 1 , then find the value of x.
 5 

13. ;fn y x + x x + x y = a b gks] rks dy fudkfy;sA 4
dx
dy
Find , if y x + x x + x y = a b .
dx
14. og vUrjky Kkr dhft, ftlesa Qyu f (x ) = 2x 3 − 3x 2 − 36x + 7 fujUrj o/kZeku vkSj
fujUrj Ðkleku gksA 4

Find the intervals in which the function f (x ) = 2x 3 − 3x 2 − 36x + 7 is
strictly increasing and strictly decreasing.
15. ,d O;fDr ds ckjs esa Kkr gS fd og 4 esa ls 3 ckj lR; cksyrk gSA og ,d ikls dks
mNkyrk gS vkSj crykrk gS fd ml ij vkus okyh la[;k 6 gSA bldh izkf;drk Kkr dhft,
fd ikls ij vkus okyh la[;k okLro esa 6 gSA 4
A man is known to speak truth 3 out of 4 times. He throws a die and
reports that it is a 6. Find the probability that it is actually a six.
16. ;fn a = iˆ + 2 ˆj + 3kˆ rFkk b = 2iˆ + 4 ˆj − 5kˆ ,d lekUrj prqHkZqt dh layXu Hkqtk,¡
iznf'kZr djrh gks] rks lekUrj prqHkZqt ds fod.kZ ds lekUrj ,d ek=d lfn'k Kkr dhft,A
4
If a = iˆ + 2 ˆj + 3kˆ and b = 2iˆ + 4 ˆj − 5kˆ represent two adjacent sides of a
parallelogram, find unit vector parallel to the diagonals of the
parallelogram.
[k.M – n
SECTION – D

17. fuEu jSf[kd lehdj.kksa dks vkO;wg fof/k }kjk gy dhft, % 6

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

( 12 ) 2031/ (Set : C)
2x + y + z = 1
3
x − 2y − z =
2
3y − 5z = 9
Solve the system of linear equations by matrix method :
2x + y + z = 1
3
x − 2y − z =
2
3y − 5z = 9

18. ijoy; y 2 = 4ax rFkk blds ukfHkyEc ls f?kjs {ks= dk {ks=Qy Kkr dhft,A 6

Find the area of the region bounded by the parabola y 2 = 4ax and its
latus-rectum.
vFkok
OR

o`Ùk x 2 + y 2 = 16 ] js[kk y = x rFkk x-v{k ls f?kjs gq, Hkkx dk izFke prqFkkZ'a k esa {ks=Qy
Kkr dhft,A
Find the area bounded by the circle x 2 + y 2 = 16 , the line y = x and x-
axis in the first quadrant.

19. leryksa r .(2iˆ − 7 ˆj + 4kˆ) = 3 rFkk r .(3iˆ − 5 ˆj + 4kˆ) + 11 = 0 ds izfrPNsnu ls xqtjrs gq,
vkSj fcUnq (–2, 1, 3) ls xqtjrs gq, lery dk lfn'k lehdj.k Kkr dhft,A 6
Find the vector equation of the plane passing through the intersection
of the planes r .(2iˆ − 7 ˆj + 4kˆ ) = 3 and r .(3iˆ − 5 ˆj + 4kˆ ) + 11 = 0 and passing
through the point (–2, 1, 3).

vFkok
OR

2031/ (Set : C)

( 13 ) 2031/ (Set : C)
lekUrj js[kkvksa r = i + 2 j − 4k + λ(2i + 3 ˆj + 6kˆ )
ˆ ˆ ˆ ˆ rFkk
r = 3iˆ + 3 ˆj − 5kˆ + µ(2iˆ + 3 ˆj + 6kˆ ) ds chp U;wure nwjh Kkr dhft,A

Find the shortest distance between the parallel lines :
r = i + 2 j − 4k + λ(2i + 3 j + 6k ) and r = 3i + 3 j − 5k + µ(2i + 3 ˆj + 6kˆ ) .
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ

20. fuEu jSf[kd izksxzkeu leL;k dks xzkQh; fof/k ls gy dhft, % 6
O;ojks/kksa 2x − y + 1 ≥ 0 ; x + y ≤ 3 ; x ≤ 2, x, y ≥ 0 ds vUrxZr z = x + y dk
vf/kdrehdj.k dhft,A
Solve the following linear programming problem graphically :
Maximize z = x + y subject to the constraints 2x − y + 1 ≥ 0 ; x + y ≤ 3 ;
x ≤ 2, x, y ≥ 0.

s

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