Class 12 mathematics pre board sample paper 2023-24

65/C/3 JJJJ Page 1 P.T.O.
narjmWu àíZ-nÌ H$moS> >H$mo CÎma-nwpñVH$m Ho$
_wI-n¥ð >na Adí` {bIo§ &
Candidates must write the Q.P. Code on
the title page of the answer-book.
Series EF1GH/C SET~3
Q.P. Code
Roll No.
J{UV
MATHEMATICS
*
: 3 : 80
Time allowed : 3 hours Maximum Marks : 80
NOTE :
(i) - 23
Please check that this question paper contains 23 printed pages.
(ii) - - -
-
Q.P. Code given on the right hand side of the question paper should be written on the title
page of the answer-book by the candidate.
(iii) - 38
Please check that this question paper contains 38 questions.
(iv) -
Please write down the serial number of the question in the answer-book before
attempting it.
(v) - 15 -
10.15 10.15 10.30 -
-
15 minute time has been allotted to read this question paper. The question paper will be
distributed at 10.15 a.m. From 10.15 a.m. to 10.30 a.m., the students will read the
question paper only and will not write any answer on the answer-book during this period.
65/C/3
^

65/C/3 JJJJ Page 2
:
:
(i) 38
(ii)
(iii) 1 18 19 20
(iv) 21 25 (VSA)
(v) 26 31 (SA)
(vi) 32 35 (LA)
(vii) 36 38
(viii) 2 3
2 2
(ix)
IÊS> H$
1
1. AdH$b g_rH$aU x
dx
dy
2y = 4x2 H$m g_mH$bZ JwUH$ h¡ :
(a) x2 (b)
2
x
1
(c)
2
x
1
(d) x2
2. `h {X`m J`m h¡ {H$ X
1
1
2
3
=
3
2
1
4
h¡ & Vmo Amì`yh X h¡ :
(a)
1
0
0
1
(b)
1
1
1
0
(c)
1
1
1
1
(d)
1
1
1
1

General Instructions :
Read the following instructions very carefully and strictly follow them :
(i) This question paper contains 38 questions. All questions are compulsory.
(ii) This question paper is divided into five Sections A, B, C, D and E.
(iii) In Section A, Questions no. 1 to 18 are multiple choice questions (MCQs) and
questions number 19 and 20 are Assertion-Reason based questions of 1 mark
each.
(iv) In Section B, Questions no. 21 to 25 are very short answer (VSA) type
questions, carrying 2 marks each.
(v) In Section C, Questions no. 26 to 31 are short answer (SA) type questions,
carrying 3 marks each.
(vi) In Section D, Questions no. 32 to 35 are long answer (LA) type questions
carrying 5 marks each.
(vii) In Section E, Questions no. 36 to 38 are case study based questions carrying
4 marks each.
(viii) There is no overall choice. However, an internal choice has been provided in
2 questions in Section B, 3 questions in Section C, 2 questions in Section D and
2 questions in Section E.
(ix) Use of calculators is not allowed.
SECTION A
This section comprises multiple choice questions (MCQs) of 1 mark each.
1. Integrating factor of the differential equation x
dx
dy
2y = 4x2 is :
(a) x2 (b)
2
x
1
(c)
2
x
1
(d) x2
2. It is given that X
1
1
2
3
=
3
2
1
4
. Then matrix X is :
(a)
1
0
0
1
(b)
1
1
1
0
(c)
1
1
1
1
(d)
1
1
1
1

65/C/3 JJJJ Page 4
3. x Ho$ {H$g _mZ Ho$ {bE, gma{UH$
x
5
3
x
2
Am¡a
2
3
1
10
g_mZ h¢ ?
(a) 3 (b) 3 (c) 2 (d) 2
4. EH$ LPP Ho$ n[a~Õ gwg§JV joÌ Ho$ H$moUr` q~Xþ O(0, 0), A(250, 0), B(200, 50) Am¡a
C(0, 175) h¢ & `{X CÔoí` $bZ Z = 2ax + by H$m A{YH$V_ _mZ q~XþAm| A(250, 0)
Am¡a B(200, 50) na h¡, Vmo a Am¡a b Ho$ ~rM H$m g§~§Y hmoJm :
(a) 2a = b (b) 2a = 3b (c) a = b (d) a = 2b
5. aoImAm|
1
1
x
=
1
y
4
=
2
5
z
Am¡a
3
3
x
=
5
2
y
=
4
5
z
Ho$ ~rM H$m H$moU h¡ :
(a) cos 1
3
2
(b) cos 1
3
1
(c)
2
(d)
4
6. EH$ {Zînj nmgo H$mo CN>mbm J`m & KQ>ZmE± E Am¡a F H«$_e: E = {1, 3, 5} Am¡a F = {2, 3}
h¢ & P(E|F) H$m _mZ h¡ :
(a)
3
2
(b)
3
1
(c)
6
1
(d)
2
1

3. For which value of x, are the determinants
x
5
3
x
2
and
2
3
1
10
equal ?
(a) 3 (b) 3 (c) 2 (d) 2
4. The corner points of the bounded feasible region of an LPP are O(0, 0),
A(250, 0), B(200, 50) and C(0, 175). If the maximum value of the objective
function Z = 2ax + by occurs at the points A(250, 0) and B(200, 50), then
the relation between a and b is :
(a) 2a = b (b) 2a = 3b (c) a = b (d) a = 2b
5. The angle between the lines
1
1
x
=
1
y
4
=
2
5
z
and
3
3
x
=
5
2
y
=
4
5
z
is :
(a) cos 1
3
2
(b) cos 1
3
1
(c)
2
(d)
4
6. A fair die is rolled. Events E and F are E = {1, 3, 5} and F = {2, 3}
respectively. Value of P(E|F) is :
(a)
3
2
(b)
3
1
(c)
6
1
(d)
2
1

65/C/3 JJJJ Page 6
7. `{X a , b Am¡a ( a + b ) g^r _mÌH$ g{Xe h¢ Am¡a a VWm b Ho$ ~rM H$m H$moU
h¡, Vmo H$m _mZ hmoJm :
(a)
3
2
(b)
6
5
(c)
3
(d)
6
8. `{X ABCD EH$ g_m§Va MVw^w©O h¡ Am¡a AC VWm BD BgHo$ {dH$U© h¢, Vmo AC + BD h¡ :
(a) 2DA (b) 2 AB (c) 2BC (d) 2BD
9. `{X x = a cos + b sin , y = a sin b cos h¡, Vmo {ZåZ{b{IV _| go H$m¡Z-gm
ghr h¡ ?
(a) y2
2
2
dx
y
d
x
dx
dy
+ y = 0 (b) y2
2
2
dx
y
d
+ x
dx
dy
+ y = 0
(c) y2
2
2
dx
y
d
+ x
dx
dy
y = 0 (d) y2
2
2
dx
y
d
x
dx
dy
y = 0
10. `{X Amì`yh A =
1
1
1
1
Am¡a A
2
= kA h¡, Vmo k H$m _mZ hmoJm :
(a) 1 (b) 2 (c) 2 (d) 1
11. AdH$b g_rH$aU
2
2
2
dx
y
d
+
3
dx
dy
+ x4 = 0 H$s H$mo{Q> Am¡a KmV _| A§Va h¡ :
(a) 1 (b) 2 (c) 1 (d) 0
12. dx
x
cos
x
sin
x
2
cos
2
2
.
~am~a h¡ :
(a) tan x cot x + C (b) cot x tan x + C
(c) cot x + tan x + C (d) tan x cot x C

7. If a , b and ( a + b ) are all unit vectors and is the angle between a
and b , then the value of is :
(a)
3
2
(b)
6
5
(c)
3
(d)
6
8. If ABCD is a parallelogram and AC and BD are its diagonals, then
AC + BD is :
(a) 2DA (b) 2 AB (c) 2BC (d) 2BD
9. If x = a cos + b sin , y = a sin b cos , then which one of the
following is true ?
(a) y2
2
2
dx
y
d
x
dx
dy
+ y = 0 (b) y2
2
2
dx
y
d
+ x
dx
dy
+ y = 0
(c) y2
2
2
dx
y
d
+ x
dx
dy
y = 0 (d) y2
2
2
dx
y
d
x
dx
dy
y = 0
10. If matrix A =
1
1
1
1
and A2
= kA, then the value of k is :
(a) 1 (b) 2 (c) 2 (d) 1
11. The difference of the order and the degree of the differential equation
2
2
2
dx
y
d
+
3
dx
dy
+ x4 = 0 is :
(a) 1 (b) 2 (c) 1 (d) 0
12. dx
x
cos
x
sin
x
2
cos
2
2
.
is equal to
(a) tan x cot x + C (b) cot x tan x + C
(c) cot x + tan x + C (d) tan x cot x C

65/C/3 JJJJ Page 8
13. `{X A, H$mo{Q> 3 H$m EH$ dJ© Amì`yh h¡ Am¡a |A| = 6 h¡, Vmo |adj A| H$m _mZ h¡ :
(a) 6 (b) 36
(c) 27 (d) 216
14. Amì`yh g_rH$aU
z
2
y
z
x
z
y
x
=
1
2
3
, _| z H$m _mZ h¡ :
(a) 1 (b) 2
(c) 1 (d) 2
15. `{X$y = log
2
x
4
tan h¡, Vmo
dx
dy
h¡ :
(a) sec x (b) cosec x
(c) tan x (d) sec x tan x
16. AY©-Vb 2x + y 4 0 _| pñWV q~Xþ h¡ :
(a) (0, 8) (b) (1, 1)
(c) (5, 5) (d) (2, 2)
17. dx
x
3
sin
6
/
0
H$m _mZ h¡ :
(a)
2
3
(b)
3
1
(c)
2
3
(d)
3
1
18. g{Xe ^
i H$m g{Xe ^
i +
^
j + 2
^
k na àjon h¡ :
(a)
6
1
(b) 6 (c)
6
2
(d)
6
3

13. If A is a square matrix of order 3 and |A| = 6, then the value of |adj A|
is :
(a) 6 (b) 36
(c) 27 (d) 216
14. In the matrix equation
z
2
y
z
x
z
y
x
=
1
2
3
, the value of z is :
(a) 1 (b) 2
(c) 1 (d) 2
15. If y = log
2
x
4
tan , then
dx
dy
is :
(a) sec x (b) cosec x
(c) tan x (d) sec x tan x
16. The point which lies in the half-plane 2x + y 4 0 is :
(a) (0, 8) (b) (1, 1)
(c) (5, 5) (d) (2, 2)
17. The value of dx
x
3
sin
6
/
0
is :
(a)
2
3
(b)
3
1
(c)
2
3
(d)
3
1
18. The projection of vector
^
i on the vector
^
i +
^
j + 2
^
k is :
(a)
6
1
(b) 6 (c)
6
2
(d)
6
3

65/C/3 JJJJ Page 10
19 20 1
(A) (R)
(a), (b), (c) (d)
(a) A{^H$WZ (A) Am¡a VH©$ (R) XmoZm| ghr h¢ Am¡a VH©$ (R), A{^H$WZ (A) H$s ghr
ì¶m»¶m H$aVm h¡ &
(b) A{^H$WZ (A) Am¡a VH©$ (R) XmoZm| ghr h¢ Am¡a VH©$ (R), A{^H$WZ (A) H$s ghr
ì¶m»¶m H$aVm h¡ &
(c) A{^H$WZ (A) ghr h¡, naÝVw VH©$ (R) µJbV h¡ &
(d) A{^H$WZ (A) µJbV h¡, naÝVw VH©$ (R) ghr h¡ &
19. (A) : ímrfmªo A(0, 0, 0), B(3, 4, 5), C(8, 8, 8) Am¡a D(5, 4, 3) go ~Zm
MVw^w©O EH$ g_MVw^w©O h¡ &
(R) : ABCD EH$ g_MVw^©wO h¡, `{X AB = BC = CD = DA, AC BD h¡ &
20. (A) : cot 1 ( 3 ) H$m _w»` _mZ
6
h¡ &
(R) : cot 1 x H$m àm§V { 1, 1} h¡ &
IÊS> I
(VSA) 2
21. EH$ Eogm g{Xe kmV H$s{OE, {OgH$m n[a_mU 6 h¡ VWm dh àË`oH$ g{Xe a + b Am¡a
a b Ho$ bå~dV² h¡, Ohm±$ a =
^
i +
^
j +
^
k Am¡a b =
^
i + 2
^
j + 3
^
k h¢ &
22. (H$) `{X a , b Am¡a c VrZ g{Xe Bg àH$ma h¢ {H$ | a | = 7,
| b | = 24, | c | = 25 Am¡a a + b + c = 0 h¡, Vmo
a . b + b . c + c . a H$m _mZ kmV H$s{OE &
AWdm
(I) `{X EH$ aoIm x-Aj, y-Aj Am¡a z-Aj Ho$ gmW H«$_e: , Am¡a H$moU ~ZmVr
h¡, Vmo {gÕ H$s{OE {H$ sin2 + sin2 + sin2 = 2 h¡ &

Questions number 19 and 20 are Assertion and Reason based questions carrying
1 mark each. Two statements are given, one labelled Assertion (A) and the other
labelled Reason (R). Select the correct answer from the codes (a), (b), (c) and (d)
as given below.
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the
correct explanation of the Assertion (A).
(b) Both Assertion (A) and Reason (R) are true and Reason (R) is not
the correct explanation of the Assertion (A).
(c) Assertion (A) is true, but Reason (R) is false.
(d) Assertion (A) is false, but Reason (R) is true.
19. Assertion (A) : Quadrilateral formed by vertices A(0, 0, 0), B(3, 4, 5),
C(8, 8, 8) and D(5, 4, 3) is a rhombus.
Reason (R) : ABCD is a rhombus if AB = BC = CD = DA, AC BD.
20. Assertion (A) : The principal value of cot 1 ( 3 ) is
6
.
Reason (R) : Domain of cot 1 x is { 1, 1}.
SECTION B
This section comprises very short answer (VSA) type questions of 2 marks each.
21. Find a vector of magnitude 6, which is perpendicular to each
of the vectors a + b and a b , where a =
^
i +
^
j +
^
k and
b =
^
i + 2
^
j + 3
^
k .
22. (a) If a , b and c are three vectors such that | a | = 7, | b | = 24,
| c | = 25 and a + b + c = 0 , then find the value of
a . b + b . c + c . a .
OR
(b) If a line makes angles , and with x-axis, y-axis and z-axis
respectively, then prove that sin2 + sin2 + sin2 = 2.

65/C/3 JJJJ Page 12
23. $bZ
f(x) =
1
x
,
b
2
ax
5
1
x
,
11
1
x
,
b
ax
3
`{X
`{X
`{X
x = 1 na g§VV h¡ & a Am¡a b Ho$ _mZ kmV H$s{OE &
24. dh A§Vamb kmV H$s{OE {Og_| $bZ f(x) = x3 +
3
x
1
, x 0 õmg_mZ h¡ &
25. (H$) gab H$s{OE :
tan 1
x
sin
1
x
cos
AWdm
(I) {gÕ H$s{OE {H$ f(x) = [x] Ûmam àXÎm _hÎm_ nyUmªH$ $bZ f : Z Vmo
EH¡$H$s h¡ Am¡a Z hr AmÀN>mXH$ h¡ &
IÊS> J
(SA) 3
26. (H$) Xmo W¡bm| _| go W¡bo A _| 2 gµo$X Am¡a 3 bmb J|X| h¢ Am¡a W¡bo B _| 4 gµo$X Am¡a
5 bmb J|X| h¢ & `mÑÀN>`m EH$ J§oX H$mo EH$ W¡bo _| go {ZH$mbm J`m Am¡a nm`m J`m
{H$ `h bmb h¡ & àm{`H$Vm kmV H$s{OE {H$ Bgo W¡bo B _| go {ZH$mbm J`m Wm &
AWdm
(I) 50 ì`{º$`m| Ho$ g_yh _| go 20 gX¡d gM ~mobVo h¢ & Bg g_yh _| go `mÑÀN>`m
2 ì`{º$`m| H$mo MwZm J`m ({~Zm à{VñWmnZm Ho$) & MwZo JE CZ ì`{º$`m| H$s g§»`m
H$m àm{`H$Vm ~§Q>Z kmV H$s{OE Omo gX¡d gM ~mobVo h¢ &
27. kmV H$s{OE :
2
cos
sin
3
3
cos
d

23. The function
f(x) =
1
x
if
,
b
2
ax
5
1
x
if
,
11
1
x
if
,
b
ax
3
is continuous at x = 1. Find the values of a and b.
24. Find the interval in which the function f(x) = x3 +
3
x
1
, x 0 is decreasing.
25. (a) Simplify :
tan 1
x
sin
1
x
cos
OR
(b) Prove that the greatest integer function f : , given by
f(x) = [x], is neither one-one nor onto.
SECTION C
This section comprises short answer (SA) type questions of 3 marks each.
26. (a) Out of two bags, bag A contains 2 white and 3 red balls and bag B
contains 4 white and 5 red balls. One ball is drawn at random from
one of the bags and is found to be red. Find the probability that it
was drawn from bag B.
OR
(b) Out of a group of 50 people, 20 always speak the truth. Two
persons are selected at random from the group (without
replacement). Find the probability distribution of number of
selected persons who always speak the truth.
27. Find :
2
cos
sin
3
3
cos
d

65/C/3 JJJJ Page 14
28. (H$) _mZ kmV H$s{OE :
x
cos
x
sin
x
cos
x
sin
x
4
4
2
/
0
dx
AWdm
(I) _mZ kmV H$s{OE :
3
1
(|x 1| +|x 2|) dx
29. kmV H$s{OE :
dx
)
1
x
(
)
1
x
(
x
2
30. {ZåZ{b{IV a¡{IH$ àmoJ«m_Z g_ñ`m H$mo AmboIr` {d{Y go hb H$s{OE :
{ZåZ{b{IV ì`damoYm| Ho$ A§VJ©V,
z = 6x + 7y H$m Ý`yZV_rH$aU H$s{OE :
2x + y 8
x + 2y 10
x, y 0
31. (H$) AdH$b g_rH$aU
dx
dy
= 2
2
y
x
xy
H$m {d{eîQ> hb kmV H$s{OE, {X`m J`m h¡
{H$ O~ x = 0 h¡, Vmo y = 1 h¡ &
AWdm
(I) AdH$b g_rH$aU (1 + x2)
dx
dy
+ 2xy = 2
x
1
1
H$m {d{eîQ> hb kmV H$s{OE,
{X`m J`m h¡ {H$ O~ x = 1 h¡, Vmo y = 0 h¡ &

28. (a) Evaluate :
x
cos
x
sin
x
cos
x
sin
x
4
4
2
/
0
dx
OR
(b) Evaluate :
3
1
(|x 1| +|x 2|) dx
29. Find :
dx
)
1
x
(
)
1
x
(
x
2
30. Solve the following Linear Programming Problem graphically:
Minimise z = 6x + 7y
subject to the constraints
2x + y 8
x + 2y 10
x, y 0
31. (a) Find the particular solution of the differential equation
dx
dy
= 2
2
y
x
xy
, given that y = 1 when x = 0.
OR
(b) Find the particular solution of the differential equation
(1 + x2)
dx
dy
+ 2xy = 2
x
1
1
, given that y = 0 when x = 1.

65/C/3 JJJJ Page 16
IÊS> K
(LA) 5
32. g_mH$bZ H$m Cn`moJ H$aVo hþE, EH$ Eogo {Ì^wO go {Kao joÌ H$m joÌ$b kmV H$s{OE {OgHo$
erf© ( 1, 2), (1, 5) Am¡a (3, 4) h¢ &
33. `{X A =
1
1
3
2
0
1
1
1
1
h¡, Vmo A 1 kmV H$s{OE, AV: {ZåZ{b{IV a¡{IH$ g_rH$aU
{ZH$m` H$mo hb H$s{OE :
x + y + z = 6
x + 2z = 7
3x + y + z = 12
34. (H$) _|
S = {(a, b) : a b3, a , b }
Ûmam n[a^m{fV g§~§Y S Z Vmo ñdVwë` h¡, Z g_{_V h¡ Am¡a Z hr g§H«$m_H$ h¡ &
AWdm
(I) A = {1, 2, 3, 4, 5, 6, 7} _| g§~§Y R Bg àH$ma n[a^m{fV h¡
R = {(a, b) : a Am¡a b XmoZm| `m Vmo {df_ h¢ `m g_ h¢} Xem©BE {H$ R EH$
Vwë`Vm g§~§Y h¡ & AV:, Vwë`Vm dJ© [1] Ho$ Ad`d kmV H$s{OE &
35. (H$) Xem©BE {H$ aoImE±
3
1
x
=
5
3
y
=
7
5
z
Am¡a
1
2
x
=
3
4
y
=
5
6
z
à{VÀN>oXr aoImE± h¢ & BZH$m à{VÀN>oXZ {~ÝXþ ^r kmV H$s{OE &
AWdm
(I) aoIm `w½_m|
2
1
x
=
3
1
y
= z Am¡a
5
1
x
=
1
2
y
; z = 2 Ho$ ~rM H$s Ý`yZV_
Xÿar kmV H$s{OE &

SECTION D
This section comprises long answer type questions (LA) of 5 marks each.
32. Using integration, find the area of the region bounded by the triangle
whose vertices are ( 1, 2), (1, 5) and (3, 4).
33. If A =
1
1
3
2
0
1
1
1
1
, find A 1 and hence solve the following system of
linear equations :
x + y + z = 6
x + 2z = 7
3x + y + z = 12
34. (a) Show that the relation S in set of real numbers defined by
S = {(a, b) : a b3, a , b }
is neither reflexive, nor symmetric, nor transitive.
OR
(b) Let R be the relation defined in the set A = {1, 2, 3, 4, 5, 6, 7} by
R = {(a, b) : both a and b are either odd or even}. Show that
R is an equivalence relation. Hence, find the elements of
equivalence class [1].
35. (a) Show that the lines
3
1
x
=
5
3
y
=
7
5
z
and
1
2
x
=
3
4
y
=
5
6
z
intersect. Also find their point of
intersection.
OR
(b) Find the shortest distance between the pair of lines
2
1
x
=
3
1
y
= z and
5
1
x
=
1
2
y
; z = 2.

65/C/3 JJJJ Page 18
IÊS> L>
3 4
àH$aU AÜ``Z 1
36. EH$ hmCqgJ gmogmBQ>r AnZo {Zdm{g`m| Ho$ {bE V¡amH$s hoVw EH$ nyb (Vmbm~) ~ZmZm MmhVr
h¡ & BgHo$ {bE CÝh| EH$ dJm©H$ma ^y{_ IarXZr h¡ Am¡a Bg JhamB© VH$ ImoXZm h¡ {H$ Bg nyb
H$s j_Vm 250 KZ _rQ>a hmo OmE & ^y{_ H$s H$s_V < 500 à{V dJ© _rQ>a h¡ &
ImoXZo H$s H$s_V _| JhamB© H$s A{YH$Vm Ho$ AZwgma d¥{Õ hmoVr OmVr h¡ VWm nyao nyb H$s bmJV
< 4000 (JhamB©)2 h¡ &
_mZ br{OE {H$ dJm©H$ma ßbm°Q> H$s ^wOm x _rQ>a Am¡a JhamB© h _rQ>a h¡ &
Cn`w©º$ gyMZm Ho$ AmYma na, {ZåZ àíZm| Ho$ CÎma Xr{OE :
(i) bmJV $bZ C(h) H$mo h Ho$ nXm| _| {b{IE & 1
(ii) H«$m§{VH$ q~Xþ kmV H$s{OE & 1

SECTION E
This section comprises 3 case study based questions of 4 marks each.
Case Study 1
36. A housing society wants to commission a swimming pool for its residents.
For this, they have to purchase a square piece of land and dig this to such
a depth that its capacity is 250 cubic metres. Cost of land is < 500 per
square metre. The cost of digging increases with the depth and cost for
the whole pool is < 4000 (depth)2.
Suppose the side of the square plot is x metres and depth is h metres.
On the basis of the above information, answer the following questions :
(i) Write cost C(h) as a function in terms of h. 1
(ii) Find critical point. 1

65/C/3 JJJJ Page 20
(iii) (H$) {ÛVr` AdH$bO narjU Ûmam h H$m dh _mZ kmV H$s{OE, {OgHo$ {bE nyb
? 2
AWdm
(iii) (I) àW_ AdH$bO narjU go nyb H$s Eogr JhamB© kmV H$s{OE {H$ nyb ~ZmZo
H$s bmJV Ý`yZV_ hmo & Ý`yZV_ bmJV Ho$ {bE x Am¡a h Ho$ ~rM H$m g§~§Y
^r kmV H$s{OE & 2
àH$aU AÜ``Z 2
37. EH$ H¥${f g§ñWmZ _|, d¡km{ZH$ ~rOm| H$s {H$ñ_m§o H$mo AbJ-AbJ dmVmdaUm| _| CJmZo H$m
à`moJ H$aVo h¢ {Oggo {H$ ñdñW nm¡Yo CJ| Am¡a A{YH$ CnO àmá hmo &
EH$ d¡km{ZH$ Zo AdbmoH$Z {H$`m {H$ EH$ {deof ~rO A§Hw$[aV hmoZo Ho$ ~mX ~hþV VoµOr go
CgZo H$hm {H$ Bg d¥{Õ H$mo $bZ f(x) =
3
1
x3 4x2 + 15x + 2, 0 x 10
go n[a^m{fV {H$`m Om gH$Vm h¡, Ohm± x {XZm| H$s dh g§»`m h¡ {OZ_| nm¡Ym gy`© Ho$ àH$me
go COmJa Wm &
(i) Bg $bZ f(x) Ho$ H«$m§{VH$ q~Xþ H$m¡Z-go h¢ ? 2
(ii) {ÛVr` AdH$bO narjU H$m à`moJ H$aHo$, $bZ H$m Ý`yZV_ _mZ kmV H$s{OE & 2

(iii) (a) Use second derivative test to find the value of h for which cost
of constructing the pool is minimum. What is the minimum
cost of construction of the pool ? 2
OR
(iii) (b) Use first derivative test to find the depth of the pool so that
cost of construction is minimum. Also, find relation between x
and h for minimum cost. 2
Case Study 2
37. In an agricultural institute, scientists do experiments with varieties of
seeds to grow them in different environments to produce healthy plants
and get more yield.
A scientist observed that a particular seed grew very fast after
germination. He had recorded growth of plant since germination and he
said that its growth can be defined by the function
f(x) =
3
1
x3 4x2 + 15x + 2, 0 x 10
where x is the number of days the plant is exposed to sunlight.
(i) What are the critical points of the function f(x) ? 2
(ii) Using second derivative test, find the minimum value of the
function. 2

65/C/3 JJJJ Page 22
àH$aU AÜ``Z 3
38. EH$ g_yh {H«$`mH$bmn H$s H$jm _| 10 {dÚmWu h¢ {OZH$s Am`w 16, 17, 15, 14, 19,
17, 16, 19, 16 Am¡a 15 df© h¢ & EH$ {dÚmWu H$mo `mÑÀN>`m Bg àH$ma MwZm J`m {H$
àË`oH$ {dÚmWu Ho$ MwZo OmZo H$s g§^mdZm g_mZ h¡ Am¡a MwZo JE {dÚmWu H$s Am`w H$mo {bIm
J`m &
(i) àm{`H$Vm kmV H$s{OE {H$ MwZo JE {dÚmWu H$s Am`w EH$ ^mÁ` g§»`m h¡ & 1
(ii) _mZm X MwZo hþE {dÚmWu H$s Am`w h¡, Vmo X ? 1
(iii) (H$) `mÑÀN>`m Ma X H$m àm{`H$Vm ~§Q>Z kmV H$s{OE VWm _mÜ` Am`w kmV
H$s{OE & 2
AWdm
(iii) (I) EH$ `mÑÀN>`m MwZo JE {dÚmWu H$s Am`w 15 df© go A{YH$ nmB© JB© &
àm{`H$Vm kmV H$s{OE {H$ CgH$s Am`w EH$ A^mÁ` g§»`m h¡ & 2

Case Study 3
38. In a group activity class, there are 10 students whose ages are 16, 17, 15,
14, 19, 17, 16, 19, 16 and 15 years. One student is selected at random
such that each has equal chance of being chosen and age of the student is
recorded.
(i) Find the probability that the age of the selected student is a
composite number. 1
(ii) Let X be the age of the selected student. What can be the value
of X ? 1
(iii) (a) Find the probability distribution of random variable X and
hence find the mean age. 2
OR
(iii) (b) A student was selected at random and his age was found to be
greater than 15 years. Find the probability that his age is a
prime number. 2

Class 12 mathematics pre board sample paper 2023-24

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