HBSE Class 12 Mathematics Question Paper 2017 Set A

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

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 : A) P. T. O.

(2) 2031/ (Set : A)
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 : A)

(3) 2031/ (Set : A)
(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 : R → R, f(x) = x4 }kjk ifjHkkf"kr gS] lgh mÙkj dk p;u dhft,
% 1

(A) f ,dSdh vkPNknd gS

(B) f cgq,d vkPNknd gS

(C) f ,dSdh gS] fdUrq vkPNknd ugha gS

(D) f u rks ,dSdh gS] vkSj u gh vkPNknd gS

Let f : R → R be defined as f(x) = x4, choose the correct answer :

(A) f is one-one onto

(B) f is many-one onto

(C) f is one-one but not onto

(D) f is neither one-one nor onto

(ii) tan−1( 3 ) − sec −1(−2) dk eku gS % 1

π π 2π
(A) (B) − (C) (D) π
3 3 3

tan−1( 3 ) − sec −1(−2) is equal to :

π π 2π
(A) (B) − (C) (D) π
3 3 3

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

(4) 2031/ (Set : A)
3x + 7 5  5 y − 2
(iii) ;fn vkO;wg  = gks] rks x vkSj y ds eku gSa % 1
 y + 1 2 − 3x  8 4 

1 1 2
(A) x =− , y = 7 (B) x =− ,y = −
3 3 3

2 2
(C) x =− , y = 7 (D) x = 5, y = −
3 3
3x + 7 5  5 y − 2
If the matrices  = 4 
, then the values of x
 y + 1 2 − 3x  8
and y are :
1 1 2
(A) x = − , y = 7 (B) x = − , y = −
3 3 3
2 2
(C) x = − , y = 7 (D) x = 5, y = −
3 3
(iv) ;fn A, 3 × 3 dksfV dk O;qRØe.kh; oxZ vkO;wg gS] rks |adj A| dk eku gS % 1
(A) |A|3 (B) |A|
(C) 3|A| (D) |A|2
Let A be a non-singular square matrix of order 3 × 3. Then |adj A|
is :
(A) |A|3 (B) |A|
(C) 3|A| (D) |A|2
kx + 1, ;fn x≤π
(v) ;fn Qyu f (x ) =  , x = π ij larr gks] rks k dk eku gS %
 cos x , ;fn x >π
1
2
(A) –1 (B) −
π
(C) –2 (D) buesa ls dksbZ ugha
kx + 1, if x≤π
If the function f (x ) =  is continuous at x = π,
 cos x , if x>π
then the value of k is :

2031/ (Set : A)

(5) 2031/ (Set : A)
2
(A) –1 (B) −
π
(C) –2 (D) None of these
(vi) o`Ùk ds {ks=Qy ds ifjorZu dh nj] bldh f=T;k r ds lkis{k r = 5 ij gS % 1
(A) 10 π (B) 8π
(C) 12 π (D) 13 π
The rate of change of the area of a circle with respect to its radius
r at r = 5 is :
(A) 10 π (B) 8 π
(C) 12 π (D) 13 π
(vii) oØ y = x 3 − 11x + 5 ij og fcUnq gS] ftl ij Li'kZ js[kk y = x − 11 gS % 1
(A) (–2, 0) (B) (3, 7)
(C) (0, 2) (D) (2, –9)
The point on the curve y = x 3 − 11x + 5 at which the tangent is
y = x − 11 , is :
(A) (–2, 0) (B) (3, 7)
(C) (0, 2) (D) (2, –9)
x2
(viii) ∫ 6 dx dk eku gS % 1
x +1
1
(A) tan −1 x 3 + c (B) tan−1 x + c
3
(C) sin−1 x 3 + c (D) buesa ls dksbZ ugha
x2
∫ x 6 + 1 dx is equal to :
1
(A) tan −1 x 3 + c (B) tan−1 x + c
3
(C) sin−1 x 3 + c (D) None of these
1

∫ sin x dx dk eku gS %
−1
(ix) 1
0

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

(6) 2031/ (Set : A)
π
(A) 2 (B) −1
2
(C) –1 (D) 1
1

∫ sin x dx is equal to :
−1

0

π
(A) 2 (B) −1
2
(C) –1 (D) 1

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

d 2y d 2y
(A) 2
+1 = 0 (B) −1 = 0
dx dx 2
d 2y d 2y
(C) 2
−y = 0 (D) +y = 0
dx dx 2
Which of the following differential equation has y = c1e x + c 2e − x as
the general solution ?
d 2y d 2y
(A) 2
+1 = 0 (B) −1 = 0
dx dx 2
d 2y d 2y
(C) 2
−y = 0 (D) +y = 0
dx dx 2
(xi) vody lehdj.k e x dy + (ye x + 2x ) dx = 0 dk O;kid gy gS % 1

(A) ye x + x 2 = c (B) xe x + x 2 = c

(C) xe y + y 2 = c (D) ye y + x 2 = c

2031/ (Set : A)

(7) 2031/ (Set : A)
The general solution of the differential equation
x x
e dy + (ye + 2x ) dx = 0 is :

(A) ye x + x 2 = c (B) xe x + x 2 = c

(C) xe y + y 2 = c (D) ye y + x 2 = c
→ →
(xii) lfn'k a = 2iˆ + 3 ˆj + 2kˆ dk lfn'k b = iˆ + 2 ˆj + kˆ ij iz{ksi gS % 1

5 2
(A) (B) 6
6 3
3 5
(C) (D) 6
2 3
→ →
The projection of vector a = 2iˆ + 3 ˆj + 2kˆ on b = iˆ + 2 ˆj + kˆ is :

5 2
(A) (B) 6
6 3
3 5
(C) (D) 6
2 3

(xiii) js[kkvksa
x + 3 y −1 z + 3
= = rFkk x + 1 = y − 4 = z − 5 ds ;qXe ds chp dk
3 5 4 1 1 2
dks.k gS % 1
8 3  5 7 
(A) cos −1 
 (B) cos −1 

 15   15 
 15  3 8 
(C) cos −1  (D) cos −1 

8 3   15 
x + 3 y −1 z + 3
Angle between the pair of lines = = and
3 5 4
x +1 y − 4 z − 5
= = is :
1 1 2

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

(8) 2031/ (Set : A)
8 3  5 7 
(A) cos −1 
 (B) cos −1 

 15   15 
 15  3 8 
(C) cos −1  (D) cos −1 

8 3   15 
(xiv) ;fn iklksa dk ,d tksM+k mNkyk tkrk gS] rks izR;sd ikls ij le vHkkT; la[;k izkIr
djus dh izkf;drk gS % 1
1 1
(A) (B)
3 36
11
(C) 0 (D)
12
The probability of obtaining an even prime number on each die,
when a pair of dice is rolled is :
1 1
(A) (B)
3 36
11
(C) 0 (D)
12
(xv) ;fn ,d U;k¸; flDds dks 10 ckj mNkyk x;k gks] rks Bhd N% fpr izkIr djus dh
izkf;drk gS % 1

193 290
(A) (B)
512 512
105
(C) (D) buesa ls dksbZ ugha
512
If a fair coin is tossed ten times, the probability of getting exactly
six heads is :

193 290
(A) (B)
512 512
105
(C) (D) None of these
512

2031/ (Set : A)

(9) 2031/ (Set : A)
1
(xvi) ;fn P(A) = , P(B) = 0 gks] rks P(A/B) gS % 1
2

(A) 0 (B) 1

1
(C) (D) ifjHkkf"kr ugha
2

1
If P(A) = , P(B) = 0, then P(A/B) is :
2

(A) 0 (B) 1

1
(C) (D) Not defined
2
[k.M – c
SECTION – B
2. ;fn Qyu f : R → R, f(x) = 4x + 3 }kjk iznÙk gks] rks fn[kkb, fd f O;qRØe.kh; gS vkSj f
dk izfrykse Kkr dhft,A 2
Let f : R → R, given by f(x) = 4x + 3. Show that f is invertible and find
the inverse of f.
3. fln~/k dhft, % 2
3 24
2 sin−1 = tan−1
5 7
Prove that :
3 24
2 sin−1 = tan−1
5 7

vkO;wg A = 
1 5
4.  ds fy, lR;kfir dhft, fd A + A' ,d lefer vkO;wg gSA 2
6 7

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

( 10 ) 2031/ (Set : A)
1 5 
For the matrix A =   , verify that A + A' is a symmetric matrix.
6 7
0 sin α − cos α
5. − sin α 0 sin β dk eku Kkr dhft,A 2
cos α − sin β 0

0 sin α − cos α
Evaluate : − sin α 0 sin β
cos α − sin β 0

∫ x + 2x + 5 dx dk eku fudkfy,A
2
6. 2

Find ∫ x 2 + 2x + 5 dx

1
7. eku Kkr dhft, ∫ sin5 x cos4 x dx 2
−1

1
Evaluate : ∫ sin5 x cos4 x dx
−1

8. y-v{k dks ewy fcUnq ij Li'kZ djus okys o`Ùkksa ds dqy dk vody lehdj.k Kkr dhft,A 2

Find the differential equation of the family of circles touching the y-axis
at the origin.

d 2y
9. ;fn y = A sin x + B cos x gks] rks fln~/k dhft, fd + y = 0 gSA 2
dx 2

d2y
If y = A sin x + B cos x , then prove that + y = 0.
dx2

10. sin2 x dk e cos x ds lkis{k vodyu Kkr dhft,A 2

Differentiate sin2 x w.r.t. e cos x .

2031/ (Set : A)

( 11 ) 2031/ (Set : A)
11. ,d vufHkur (unbiased) ikls dks nks ckj mNkyk x;kA eku ysa A ?kVuk ^igyh mNky ij
fo"ke la[;k izkIr gksuk* vkSj B ?kVuk ^f}rh; mNky ij fo"ke la[;k izkIr gksuk* n'kkZrs gSaA
?kVukvksa A vkSj B ds Lokra×; dk ijh{k.k dhft,A 2

An unbiased die is thrown twice. Let the event A be 'odd number on the
first throw' and B be the event 'odd number on the second throw'.
Check the independence of the events A and B.

[k.M – l
SECTION – C

12. fn[kkb, fd sin−1 12 + cos −1 4 + tan−1 63 = π gSA 4
13 5 16

12 4 63
Show that sin−1 + cos −1 + tan −1 = π.
13 5 16

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

Differentiate (log x )x + x log x w.r.t. x.

14. og vUrjky Kkr dhft, ftlesa f (x ) = sin x + cos x , 0 ≤ x ≤ 2π }kjk iznÙk Qyu f
fujUrj o/kZeku ;k fujUrj Ðkleku gSA 4

Find the interval in which the function f given by
f (x ) = sin x + cos x, 0 ≤ x ≤ 2π is strictly increasing or strictly decreasing.

15. ,d fu'kkusckt ds y{;-Hksnu dh izkf;drk 3 gSA og de ls de fdruh ckj xksyh pyk, fd
4
y{; dks de ls de ,d ckj Hksnus dh izkf;drk 0.99 ls vf/kd gks \ 4
3
The probability of a shooter hitting a target is
. How many minimum
4
number of times must he/she fire so that the probability of hitting the
target at least once is more than 0.99 ?

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

( 12 ) 2031/ (Set : A)
16. lfn'k iˆ + ˆj + kˆdk lfn'kksa 2i + 4 j − 5k vkSj λi + 2 j + 3k ds ;ksxQy dh fn'kk esa]
ˆ ˆ ˆ ˆ ˆ ˆ
ek=d lfn'k ds lkFk vfn'k xq.kuQy ,d ds cjkcj gS] rks λ dk eku Kkr dhft,A 4

The scalar product of the vector iˆ + ˆj + kˆ with a unit vector along the
sum of vectors 2iˆ + 4 ˆj − 5kˆ and λiˆ + 2 ˆj + 3kˆ is equal to one, find the
value of λ.

[k.M – n
SECTION – D

17. fuEu lehdj.k fudk;ksa dks vkO;wg fof/k }kjk gy dhft, % 6

x − y + 2z = 7
3x + 4y − 5z = −5
2x − y + 3z = 12

Solve the system of equations by matrix method :

x − y + 2z = 7
3x + 4y − 5z = −5
2x − y + 3z = 12

18. x-v{k ds Åij rFkk o`Ùk x 2 + y 2 = 8x ,oe~ ijoy; y 2 = 4x ds e/;orhZ {ks= dk
{ks=Qy Kkr dhft,A 6

Find the area lying above x-axis and included between the circle
x 2 + y 2 = 8x and the parabola y 2 = 4x .

vFkok
OR

js[kk y = 3x + 2 , x-v{k ,oe~ dksfV;ksa x = –1 rFkk x = 1 ls f?kjs {ks= dk {ks=Qy Kkr
dhft,A
2031/ (Set : A)

( 13 ) 2031/ (Set : A)
Find the area of the region bounded by the line y = 3x + 2 and the
ordinates x = –1 and x = 1.

19. leryksa r .(iˆ + ˆj + kˆ ) = 6 vkSj r .(2iˆ + 3 ˆj + 4kˆ ) = −5 ds izfrPNsnu rFkk fcUnq (1, 1, 1)
ls tkus okys 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 .(iˆ + ˆj + kˆ ) = 6 and r .(2iˆ + 3 ˆj + 4kˆ ) = −5 and the point (1,
1, 1).

vFkok
OR

js[kkvksa r = (iˆ + 2 ˆj + kˆ ) + λ(iˆ − ˆj + kˆ) rFkk r = (2iˆ − ˆj − kˆ ) + µ(2iˆ + ˆj + 2kˆ ) ds chp
U;wure nwjh Kkr dhft,A
Find the shortest distance between the lines :
r = (iˆ + 2 ˆj + kˆ ) + λ(iˆ − ˆj + kˆ ) and r = (2iˆ − ˆj − kˆ ) + µ(2iˆ + ˆj + 2kˆ ) .

20. vkys[kh; fof/k }kjk fuEu jSf[kd-lehdj.kksa dks gy dhft, % 6

fuEu O;ojks/kksa ds vUrxZr
x + 3y ≤ 60

x + y ≥ 10

x ≤y

x ≥ 0, y ≥ 0

z = 3x + 9y dk U;wure vkSj vf/kdre eku Kkr dhft,A
Solve the following linear programming graphically :

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

( 14 ) 2031/ (Set : A)
Minimize and maximize z = 3x + 9y subject to the constraints
x + 3y ≤ 60

x + y ≥ 10

x ≤y

x ≥ 0, y ≥ 0

s

2031/ (Set : A)