4 SAMPLE PAPER

[Average Concept]
For KevskIn
Notes

|Marimum Marks: 80
Tine Allowed: 3Hours]
General Instructions:
Read the following instructions very carefully and strictly follow then:
() Thus Question paper contains 38 questions. All questions are compulsory.
(ü) This Question paper is divided into five Sections - A, B, C, D and E.
and Questions no. 19
(iü) In Section A. Questions no. I to 18 are multiple choice questions (MCQs)
and 20 are Assertion-Reason based questions of l mark each.
(VSA)-type questions, carryng
(iw) In Section B, Questions no. 21 to 25 are Very Short Answer
2 marks each.
questions, carrying 3 marks each.
() In Section C, Questions no. 26 to 31 are Short Answer (SA)-type
questions, carrying 5 marks each.
(vi) In Section D, Questions no. 32 to 35 are Long Answer (LA)-type
carrying 4 marks each.
(vii) In Section E, Questions no. 36 to 38 are Case study-based questions,
in 2 questions in Section B,
(vüi) There is nooverall choice. However, an internal choice has been provided Section E.
each in 2 questions of
3questions in SectionC, 2 questions in Section D and one subpart
(ix) Use of calculators is not allowed.

SECTION - A
of I mark each)
(This section comprises of multiple choice questions (MCQs)
Select the correct option (Question 1 -Question 18):
1. A matrixhaving n elements, where n isprime can be
(b) a row matrix
(a) a square matrix
(c) acolumn matrix (d) arow or a column matrix

2. IfA is asquare matrix of order 3and |4| =-5, then |3A| is

(b) 135 (c) -135 (d) -15
(a) 45
AB=r,then
3. If aand b are position vectors of points Aand B, when
(b) a -b=7 (c) la +b|=|7| (d) b= a+Y
(a) a+b=
sin 2x 0
4. For what value(s) of k, the function f(x) = tan 5x is continuous at x= 0?
3k, x=0
2 6
(6) (c) (4)
15

1
-dx is equal to
x+1
log| x + 1| 1 (d) tan+C
(a) log (x+ 1| + C (b) 2x
+C (c)
(e+ 1)'
+C

Sample Papers 101

 d'y
6. IImand nare degree and order of ditferential equation 5, then the
value of
An-nis
(a) 7 (b) 4 (c) 0 (d) 3

7. Thepoint which lies in the half planc ofx +2y>5 is

(a) (0,0) (b) (-1,) (c) (0, 3) (d) (4, -5)
8. ld andb are along oneside and diagonal of aparallelogram then area of parallelogram is
(a) ax6 (c) |2å x6|

9. The value of sin' du is

(a) (b) (c) - (d) 0
8 2

thenxis equal to
(a) 2 (b) 4 (c) -2 (d) 4
l1. The corner points of the feasible region of an LPP whose objective function is Z = 2r + 3y are
A(0, 6), B(1, 2), C(4, 3), D(6, 2). Then function attain maximum value at
(a) A (b) D
(c) C (d) anypoint on linesegment joining Aand D.
12. If points (2, 3). (3,k). (1,0) are collincar, then value of kis
(a) -6 (b) 6 (c) 2 (d) -2
2 -1|
13. The value of k, for which the matrix -2
S is skew symmetric is
1 -5 2k

(a) -l (b) 0 (c) 2 (d) 3

14. IfA and B are independent events then P(A/B) is equal to

(a) P(4) (b) P(B) (c) P(AnB) P(AnB)

(d) PA)
15. The general solution of the differentialequation cosec =1 is
d
(a) -cosec xcot x = (y (b) dy = sinx dr (c) y= cosx + C (d) y= -cosx + C
l6. Ify = logx, theny, is equal to
1
(a) (b) (c)
log x
17. If a and b are along adjacent sides of a rectangle then
(a) axb=0 (b) a -b=0 (c) ll=|BI
18. Direction cosines of z-axis are

(a) 1,0, 0 (6) 0, 1, 0 (c) 0, 0, 1 (d) 1, 1, 1

102 agether wtk EAD Mathematics--12

 ASSERTION-REASONBASED QUESTIONS
19 and 20 ar.
ASsertion-Reason based Reason(R).carrying I mark each. Two
Labelled Assertion (A) and the other labelledquestions
c)and() as given below.)
Select the correct answerstatements
from the
e and Risthe
adar correct explanation of A.
t and are true bu! Ris not the correct explanation ofA.
Ris false.
Ris true.
(sin )= sin
e Ausertion (4):
Rcason(}: sin Irepresents inverse of sine function, we can't use exponent laws.
Assertien():1Line = -j- k) +22+j+%) pases through afixed point(1, -1, -1).
Rcason (Ä}: In vector equation ofa line r=a+ib, a represents position vector of fixed point,
thrngh whih line pases

SECTION -B
Thissection comprisesof 5 very short answer (VSA) type questions of 2marks each.)

L. Evaluate 2 tan (3)-cosec

OR

Find the domain ofy = sin (-4).

2 The revemue function for acertain commadity is given by R(r) = ar-St +9,rrepresents number
of units sold. Find the marginal revenue whenx = 5.
A Find the angle berween the vectors a =i - and b=j+3k using cross product of vectors.
OR
Find a vector whose magnitude is 5 units and is along the vector 2-j+2k.
* Find the integrating factor for the differential equation (1 +r-(tany -xhy=0.
X ix=hxe+o. Show that a+c= tb, where tis ascalar.
SECTION -C
of 3marks each.)
(Ihis section comprises of 6short answer (SA) type questions
4 Evaluate 2r-4r+ 3dr.
afamily has two children, what
27. Assume that each born child is equally likely to be a boy or a girl. If
child is a girl.
Se conditional probabilitv that both are girls given that youngest
OR
transferred
Bag l contains 3red and 4 black balls and bag lI containss4 red and 5 black balls. Aballis
probability that the ball drawn is
bag I to bag I and then a ballis drawn from bag II. Find the
red in colour.
3 Evaluate -dr.
1+ sin x OR

Evaluatesin xkdr.
Sample Papers 103
 29. Solve the differential equation (1 +e*dy +(1 +y'e dr = 0. given that when x=
r 0,y =1
OR

Solve the differential equation (x-y) dy =I+ 2y.

dr
30. Solve the following LPP graphically:
Maximise Z = 40r + 300y
Subject to constraints
x 20,y> 0,x + ys 200, y>4r
31. Evaluate -dr.
x+x+x+1
SECTION -D
(This section comprises of 4 long answer (LA) type questions of 5 marks each)
32. Draw sketch of the following region and find its
area
{(r,y): x+yslsxt y}.
33. Consider f: R ’[-9, ) given by fx) = 5x + 6x9.Show that function is one-one
and onto.
OR
Show that the relation R in the set of real numbers defined as R= {(a, b) :
nor symmetric nor transitive.
asb} is neither reflexive
34. The Cartesian equation of a line is 6r -2 = 3y + 1 = 2z-2.
Find
Write Cartesian and vector equations of a line passing through pointthe(2,direction
-1,
cosines of the line.
given line. 5)which are parallel to
OR
Show that thelines =(i +j-k) + 2(3i - ) and Y=(4i -k) +n(2i+3Á) intersect. Also,
point of intersection. find their
-1 0 2 4
35. Given A=|2 3 4 andB=4 2 4 verify that BA = 6/ and hence solve the system of
1 2 2 -1 5

equationsx=y+ 3, 2r + 3y + 4z = 17,y + 2z =7.

SECTION -E
(This section comprises of 3 case-study/passage-based questions of 4 marks each with subparts. The first
twocase study questions have three subparts (i), (i), (üü) of marks 1, 1,2 respectively. The third case study
question has two subparts of 2 marks each)
Case Study - 1
36. Read the following passage and answer the questions given below.
A company received an order tomake closed cylindrical boxes, which are to be made in such a way
that capacity should be maximum for agiven surface area. If V, S, rand h represent volume, surtace
area, radius of base and height of cylinder respectively, then
()) write the expression for total surface area.
(i) write the expression for volume in terms of r.
(iüi) find critical value of r, for volume to be maximum.
OR
(ii) write the relation between r and h for maximum volume.

104 agether witk EAD Mathematics-12

 2
CaseStudy-
37. Read the following passage and answer the questions given below.
cfudent is preparing for examinations and he came across a
function
Be=-12r + 36x + 17. He consider xas number of hours he is able to
studyand concentrate on
studies at a stretch and its depends upon when function decreases.
() Find fr).
(ii) Findthe critical points of f.
(ii) In which limits of hours he is able to concentrate?
OR
(ii) In which limits of hours he is not able to concentrate?

Case Study -3
38. Read the following passage and answer the questions given below.
Three friendsA, B, Care playing with acoin. They are tossing acoin in turn and waiting for the head
to turn up and as soon as head turns up, game is stopped. But somehow it is showing tail every time
coin is tossed, coin is assumed to be unbiased. So they said let's find probability.
() IfA,B, Cthrow a coin in turn starting with Afollowed by B and C, what is the probability that
C gets head in sixth throw?
(ii) What is the probability of Bgetting head first?