DELHI PUBLICSCHOOL BANGALORE EAST

PREBOARD EXAMINATION-I
CLASS XII 2024-25

Mathematics (Code-041)
Time Allowed: 3 hours
Date: 11/t1/2024 Maximum Marks: 80

General Instructions:
1. This Question paper contains -five sections A. B, C, Dand E. Each section is compulsory.
2. Section Ahas 18 MCQ's and 02 Assertion-Reason based questions of I mark cach.
3. Section Bhas 5 Very Short Answer (VSA-type questions of 2 marks each.
4. Scction Chas 6 Short Answer (SA)-type questions of 3 marks cach.
5. Section D has 4 Long Answer (LA)-type questions of 5 marks each.
6. Section Ehas 3 source based/case based/passage bascd/integrated units of assessment of 4
marks cach with sub-parts
SECTION-A
XArea bounded by the curve y= sinx and the x- axis betweenx = 0and x = 2n is
(a) 2 sq units (b) 0 sq units
(c) 3 sq units ( 4 sq units

2: The Feasible region for a LPP is shown in the adjacent figure. Let Z = 3x 4y be the
objective function. Then the minimum of Z occurs at

(G.8)

(GS) D

0.0)

(a) (0, 0) (b) (0, 8) (c) (5, 0) (dy (4, 10})

3: a'and bare two unit vectors and a be the angle between them. Then å+ b is a unit vector,
if a is

(a) (b) (c) (d) 2

dy
4. y= sin'(6x19x).-<x< then dy

(c) S-9x (d) V1-9x2

Ifa matrix A andthenfind the vahue of (1+ A+A' +A* +4+414).

(a) 2|
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 1 Sinp 1
-sino 1 sino|. where 0 < s2n. Then
-1 -sin) 1
(d) Det A(2.4)
(a) DetA=0 (b) Det A (2,2) (yDet Ae [2,4)
2.x
f«) = is increasing in
logx (d) (-o,e)
fa) (0, I) (b) (1, e) (c) (e, co )
equation(2+32= 5xis
8. The sum ofthe order and degree of the differential dx.

(a) 5 (b) 6 (c) 7 (d) 8

[2 -1 31
9 The matrix p 0 7 is not invertible for
1 4

(a) p = -1 (b) p=3 (c)p =0 (a) peR -{1)

19 f() dx =
(a) f(x- )dx (b5 Sfa+c) dx (c) f() dx
. IFA is a non-singular matrix of order 3, then which of the following is not true?
(a) |A| * |4'| (b) |A-|* |A|-1 (c) |A4'| # J4| (d) |A| |A'|0

j2. Domain of the function f(x) = cos() is equal to

(a) [-6,6} (6) (-o, 2) u(2,3) (c) (-6,2) u (2,3) (d) (2,3)
13.(i+ j )x (j +k). (k+ ) is equal to
(a) 0 (b) 1 (c) 2 (d) 3
4. Jfa line makes angles a, b and c with the co-ordinate axes then find the value of
cos 2a + cos 2b + cos 2c

(a) 3 (b)1 (c) 2 (d) -1 12

8. Corner points of the feasible region for an LPP are (0, 2),(3, 0), (6, 0). (6,8) and (0,5). Let
F= 4x + 6ýbe the objective function. Find the difference between maximum and minimumvalue
of Z.

(a) 72 (b) 60 (ef 48 (d) 0

16. f xVz4 1 dx is
(a) xVr-4 -loglxvr' -4|+C
(6) V-1 - log|r +Vrt-1|]+ c
(e) x*Vz-4 +loglx +Vr?-4] +c
(d) x-1-log|x' +Vr- 1|+C

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17. ifA and B be the events such that P(A) = .P(B) 1
and P(A 0B) = hen find P(A/B)
23 37
() 30 (b) 60 (c) 40
(d)
dy 1+y
I8.The integrating factor of thegiven differential equation dx + y= is
(a) x+ logx (b) e* (c) x- logx (d) xe
ASSERTION-REASON BASED QUESTIONS
In the following questions, a statement of assertion (A) is followed by a statement of
Reason (R). Choose the correct answer out of the following choices.
(a) Both A and R are true and R is the correct expBanation of A.
(b) Both A and R are true but R is not the correct expBanation of A.
(c) A is true but R is false.
(d) Ais false but R is true.
19. Assertion (A): Inverse of sine function exists in interval [0, n).
Reason (R): sin-function becomes bijective if we restrict its domain to -1,1}.
20. Assertion (A) : f(x) = |x}sinx is diflerentiable at x 0.

Reason (R): If f) is not differentiable arnd g(x)is differentiabBe at x = a, then f(x). g(r)will be
differentiable at x = .

SECTION-B
21. Prove that cos'x + cos ( + =
2 Find the local maximum value for the function f(x) = sin2x - x,in -s<;, also find the
point of local maxima.

23.If @and b are two vectors such that a' +b|= |aI, then prove that vector 2å + b is
perpendicular to vector b.

24. Check the continuity and find the points of discontinuity for the function f(x) = [x}, -3 <x<3
where [x} represents greatest integer function.
OR

If sin y = XCos(a + y), then show that 2= cos a, if x= 0.

1

25. Findthe acute angle which the line with the directioncosines n makes with the positive
direction of z -axis.
OR

Find the vecior equation of a line passing through the point A(1,2, -1) and parallel to the ine
5x-25 = 14- 7y= 35z
 SECTION -C

26. Maximize Z = 8x + 9y. subject to the constraints:

Zx + 3y s6,3x - 2y < 6,y s 1,x > 0,y >0
3 Three critics review a book. Odds in favour of the book are 5:2,4:3 and 3:4 respectively lor three
critics. Find the probability that majority are in favour of thc book.
OR

Let X denotes the number of times 'a total of 9' appears in two throws of a pair of dice. Find the
probability distribution of X. Also, find the mean of X.

28.The slope of thetangent to acurve at any point (x,y) on it is given by (cot) (cos), where
x> 0.y > 0. If thecurve passes through the point (1,), find the cquation ofthecurve.
29. Evaluate (Vtanx +Vcotr)dx
OR

sin x+Osdx
Evaluate Vsin 2x

0. The volume of a cube is increasing at a constant rate. Prove that increase its surface area varies
inversely as the length of edge of the cube.

Ifa, band care unit vectors satisfying lå- b|+|B-+ |- äj? = 9, then find |lä +b+d|
and (2ä +S× +5¿|.
OR

Find the angle between the pair of lines=-*0 and 1

- y* - 3Z-18
1
3 -5 8 6

SECTION-D

2. The sum of three numbers is 6. If we multiply third number by 3 and add the second number to it,
we get 11. By adding first and third numbers, we get double ofthe second number. Represent it
algebraically and find the numbers using matrix method.
33.Draw the rough sketch and find the area of the region bounded by the curve y = x and y= x.
(Using the method of Intergration)
OR

Draw the rough sketch and find the area bounded by s,):x+y's4 and x+y>2
(Using the method of Intergration)
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 341x -(2cos0 -cos20) and y - (2sin0 -sin20). find dyz al 0=

j5. Find he cquation oftheline which intersects the lines *2 y-3-1

2 4 and -
2 23_ 4
and
passes through the point (I, 1, ).
OR
Find the shortest distance between Y= (1+ k)i + (2 k)j +(k + 1)k and
= (2i-j - R) + p(20 + + 2k).
SECTION E
(This section comprises of3 case-study/ passage-based questions of 4 marks. First two case
study questions have three sub-parts (i), (i), (ü) of marks 1, 1, 2 respectively. The third case
study question has two sub-parts of 2 marks each.)
36. In answering a question on a multiple choice test for class XII, a student either knows the
answer or guesses. Letbe the probabiity that he knows the answer and be the probability that
he guesses. Assume that a student who guesses at the answer willbe correct with probability 3 Let
El, E2,E be the events that the student knows the answer, guesses the answer and answers correctBy
respectively.

(i Find P(E/EI).
öiy Find the value of SP(E )
(iji) What is the probability that the student knows the answer given that he answered it
correctly?
OR

Find the value of KE P(E/E)P(E)

37.In order to set up rain water harvesting system, a tank to colBect rain water is to be dug. The tank
should have a square base and a capacity of 250m².The cost of land is 25000 per square metre and the
cost of digging increases with depth and for the whole tank, it is 240,000 , where his the depth of
the tank in metres. x is the sicde of the square base of the tank in metres.

ELEMENTS OF ATYPICAL RAIN WATER HARVESTING SYSTEM

-CATCHVMENT

CONDUIT

TANK

RRECHARGE,
FACIIATEy
STORAGEO
FACILITY

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Baeln the VCn
\Vhee C()lotesnlouuton,;aswe
the ollowmy jucstionS:
cost lmcion
() Fd the otal cost of
diygug tlhe ank in tems ol x.
() Find
(im) Find the value ofx or which (is
minimunm.
OR

Check vhether the cost function C()

expressed in lerms ofx is inereasing or not, where x > 0.
38. Students of Grade XIl, planned to
plant saplings along straight lines, parallel to cach
one side of the playground ensuring that they had other to
enough play arca. Let us assumc that they planted
one of the rows of the saplings
along the line y 4. Let L be the set ofall lines which are
parallel on the ground and R be a relation on ..

Answer the following using the above information.

(i) Let relation R be defined by R {(LI, L2}: Li| L2 where L1, L2 E L} then prove that R 0s an
cquivalence relation.
(ii) If the function f:R ’ R defined by f (x) -x-4 then check whether the
function is bijective
or not. Also, find the range of f(x).