Seri es : Z6Y WX -· • SET -1
·-·-·-· -·-·-·- ·-·-·-· -·-·-·- ·~
i ~-~q ifs 65/ 6/1 ;;
! Q.P. Code
L--·-· -·-·-·- ·-·-·-· -·-·-·- ·-·
~;t. i-·-·-· -·-·-·- ·-·-·-· -·-·-·- ·-·-·-· -·1
Roll No. i ~ ~-'CJ'f cf;ls q;)- ~-4~c fil i *
!~-~ ~~f ffi«I " !
~ Candidates must write the Q.P. Code;
; on the title page of the answer-book.
---- ---- ---- ---- - ---- ---- ---- ---- ---- ---- -
. .
SI
~'f/ip ;[:3'f lUZ
a1fu1a
MATHEMATICS 111111111111111111111~ ll11111~ 111111·111111111111111
~3:r cn:8 0
Time allowed : 3 hours Maxim um Marks : 80
NOTE
(I) cf>441 ~ ~ ~ fcJ>- ~ ~-'CJ' f -q- ~ (I) Please check that this questio n paper
• contain s 23 printed pages. •
~23 ll
hand side
(II) ~-~ -q- ~ ~ ~ 3W ~ ~ (II) Q.P. Code given on the right
~-~ ~ q;1" ~ ~-3~ cfil * of the questio n paper shQiµd be written
on the· title page of the answer-book by
~-~'4\~ I the candidate. •
paper
(III) cfi441 'ffl" ~ ~ Ff; ~ ~-'CJ' f -q- (III) Please check that this questio n
38 ~ l I . contains 38 questions.
(IV) cfiqq I ~ c6T '3ffl' ~-O;ll..f I ~ ~ ~ (IV) Please write
down the Serial
~ ' 3ffl-~ fla¥1 i ~ ~ ~ m- Numb er of the questi on in the
cnT a,qicfi ~c:1~4 fmi I answe r-book at the given place
before attem pting it.
(V)
15 minute time has been allotted to read
this questio n paper. The questio n paper
will be distrib uted at 10.15 a.m. From
10.15 a.m. to 10.30 a.m., the candid ates
will read the questio n paper only and·
will not write any answer on the
answer-book during this period. #
Page 1 of23 P.T.O.
65/6/1
Gen era l Inst ruc tion s :
strictly follow them :
Rea d the following instructions very carefully and
tions are .com puls ory.
(i)
This ques tion pap er contains 88 questions. All ques
B, C, D and E.
(ii) This ques tion pap er is divi ded into five Sections -A,
ce questions (MCQs) and
(iii) In Sec tion A, Questions no. 1 to 18 ar,e mul tiple choi
base d questions of 1 mar k
·questions numb'er 19 and 20 are Assertion-Reason
each.
very shor t answ er (VSA) type
(iv) In Sec tion B, Questions no. 21 to 25 are
questions, carr ying 2 mar ks each. •
answ er (SA) type questions,
(v) In Sec tion C, Que stion s no. 26 to 81 are shor t
carr ying 8 mar ks each.
long answ er (LA) type ques tion s
(vi) In Sec tion D, Questions no. 82 to 85 are
carr ying 5 mar ks each.
stud y base d questions carr ying
(vii) In Sec tion E, Questions no. 86 to 88 are case
4 mar ks each.
rnal choice has been prov ided in
(viii) There is no overall choice. However, an inte
2 ques tion s in Sect ion D and
2 ques tion s in Section B, 3 questions in Section C,
2 ques tion s in Sect ion E.
(ix) Use of calc ulat or is not allowed.
SEC TIO N A
of 1 mar k each.
Thi s sect ion comprises mul tiple choice ques tion s (MCQs)
and B. If orde r of A.is
1. Let both AB' and B'A' be defined for mat rice s A
n x m, then the orde r of B is :
n xn (B) n x m
(A)
(C) mxm _a) ) m xn
2.
-l 0
IfA = 0 3
[
0 0
~J then A is a/an :
(A) scal ar mat rix (B) iden tity mat rix
(C) sym met ric mat rix (D) skew -sym met ric mat rix
,- P.T.O.
Pag e3 of23
65/6/1 #
•
3. The following graph is a combination of :
y
-57t·
2
.....
x'~~ ...... ..,.'- ....:. ...~- ~-1e -~~- ---.- -~~~ x
57t
2
57t
·····-2-
Y'
/4) y =sin-1 x and y = cos-1 x
(B) y := cos-1 x and y =cos x
(C) y =sin-1 x and y =sin x
(D) y =cos-1 x and y =sin x
4.· Sum of two skew-symmetric matrices of sam~ order is always a/an : .,.
(A) skew-symmetric matri x
(B) symmetric matri x
(C) null matri x
(BJ identity matri x
5.
1
[ sec-' (--./2 )- tan- (}a)] is equal to :
ll1t 51t
(A) - (B) -
12 12
51t 71t
(C) -- (D) -
12 12
~
Page S of23 P.T.O.
65/6/1 #
•
'
log (1 + ~) + log (1. - bx) for x :t 0
6. If ftx) = x
k , for x =0
is continuous at x_= 0, then the value ofk is:
(A) a (B) a+ b
)Q} a-b (D) b
7. If tan-1 (x2 -y2) =a, where 'a' is a constant, then dy is:
. dx
X X
(A) JB)
y y
(C)
a
- ( -. (D) -a
X • " 1.
y
8. Ify = a cos (log x) + b sin (log x), then x2y2 + xy1 is:
(A) cot (log x) (B) y
~ - y (D) tan (log x)
9. Let ftx) = Ix I, x e R. Then, which of the following statements is
incorrect?
(A) fhas a minimum value at x =0.
( ~ f has no maximum value in R.'
(C) fis continuous at x = 0.
(D) fis differentiable at x = 0.
10. Let f'(x) = 3 (x2 + 2x)- ~ + 5, ftl) = 0. Then, ftx) fs:
·x
2 ' 2 '
(A) x3 + 3x2 +. 2 + 5x + 11 (B) x3 + 3x2 + 2 + 5x - 11
X X
2
x3 + 3x2 - - + 5x - 11 (D) x3 - 3x2 - -2 + 5x - 11
% x2 x2
x + 5 ex dx is equal to :
11.
J +6)2
log (x + 6) + C (B) ex+ C
~
ex -1
(C) +C +C
~ x+6 ~) (x+6)2
65/6/1 Page 7 of23 P.T.O.
#
•
12. The order and degree of •the following differential equation are,
respectively :
d4
- _x + 2edy / dx + y2 = 0
dx4
(A) - 4, 1 . (B) 4, not defined
~ 1, 1 (D) 4, 1
13. The solution for the differential equation log ( : ) =3x + 4y is :
/4 3e4Y + 4e-3x + C = 0 (B) e3x+4y + C = 0 .
(C) 3e-3Y + 4e4x + 12C = 0 (D) 3e- 4Y+ 4e3x + 12C = 0
function is
14. . Programming Problem (LPP), the .given objective
For a Linear .
Z = x + 2y. The feasible region PQRS determined by the set of constrai nts
is shown as a shaded region in the graph.
II y
..
!)
(Note : The figure is not to scale)
p= (1- 24J Q = (3 15J R == (7 3J S = (18 2J
- 13 ' 13 ' - 2 ' 4 ' - 2 ' 4 ' - 7 ' 7
Which of the following stateme nts is correct ?
Z is minimum at S ( : , ~
1
(A) J
)B1 Z is maximum atRG, !J
~ (Value ofZ at P) > (Value ofZ at Q)
(D) (Value of Z at Q) < (Value of Z at R)
65/6/1 ,Page 9 of23 P.T.O.
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15. ,_hi . a .. Linear Progr11mming Problem (LPP), · the obje~ive functio
. Z = 2x + 5y is to be maximised under the following constraints.:
X + y ~ 4, 3x + 3y ~ 18, X, y ~ 0
Study the graph an~ select the corr~ct option.
y
l-
p
Y'
x+y= 4
(Note: The figure is not to scale)
The solution of the given LPP :
( ~ ~es ~ t?e shaded unbounded· region. .,
.)fl) lies m !),, AOB. • •
(C) does not exist. ·~ 1
(D) lies in the combined regio~ of Ii AOB and unbounded shaded
region.• I
.• r ,
' ➔
16. Let I ➔
a I = 5 and - 2 ~ A ~ 1. Then, the range of p. a I is :
(A) [5, 10] (B) [- 2, 5]
~ [- 2, 1) (D) [-10, 5) -~
2
17. The area of the region bounded by. the curve y = x between x = 0 and
X = 1 is
:J
2 .
!A) : sq units (B) sq umts
3
/. (C) 3 sq units
4
(D) ·a sq uruts
.
. ... . --
18. A box has 4 green, ~ blue and 3 ~ed pens. A student picks up a pen at
• .
random,. •checks its· colour and ~eplaces it in the box. He repeats tp:is
1
process 3 times. The probability that at least one pen picked was red is· :
(A) 124 JB-) 1
125 125
61 64
(C) .. (D)
125 125
65/6/1 Page 11 of 23 P.T.O.
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I) ~ v"9
-· y
Questipns number, 19 and 20 are Assertion and Reason based questions. Two
stateriients are given, one labelled Assertion (A) and the other labelled Reason
(R). Select the correct answer from the codes (A), (BJ, (CJ and (D) as given below.
~A) Both
.
Assertion (A) Iand Reason (R) are
~ •
true and Reason (R) is the
correct explanation of the Assertion (A). •
I
(B) Both Assertion (A) and ·Reason (R) are true, but 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 Reas<?n ,<R) is true.
f
➔ ➔ 2 ➔
➔ ➔
19. Assertion (AJ : If I a x b 12 + I a .b
I
1 = 256 and I b I = 8, then
➔
I a I =2. ,
Reason (RJ: sin2 0 + cos2 0 =1·and c_
➔➔ ➔ ➔
I
➔
a x b
➔
I = I ➔a If ➔-.
b •.I sm 0 and a . b =. I a I I b I cos e.
20. Assertion (AJ : Let f(x) ~· ex ahd g(x) =log x. Then (f + g) x = ex + log x
where domain of(f + g) is R. (,,
Reason (R): Do~(f + g) =Dom(f) n Dom(g).-
SECTIONB
This section comprises 5 Very Short Answer (VSA) type questions of 2 marks each.
0 Find the domain of •ftx) = _s:hll-1 (- x2). .1
22. (a) Differentiate ..Je"2i with respect to e./2i for x > 0.
OR
(b) If (x)Y =(y)", then find : . ,,
6S/6/1 Page 13 of23 P.T.O.
•------------- ---------=-._ .:;.::.....----- ---- #
..
I
23. Determ~e the values of x for which f(x) = x - : , x * -1 is an increasing
x+
or a decreasing function.
➔ ➔
24. (a) If a and b are position vectors of point A and •point B
respectively, find the position vector of point C on BA produced
such that BC =3BA.
OR
(b) Vector 1 is inclined at equal angles to the three axes x, y and z. If
magnitude of 1 is 5Jg units, then find 1.
25. Determine if the lines 1 =(i + j - k) + A(3·i - j) and
➔ /\ A /\ A
r =(4 i - k) + µ (2 i + 3 k) intersect with each other.
SECTIONC.
This section comprises 6 Short Answer (SA) type questions of 3 marks each.
1 3 4 2
26. Let A= 4 and C = 12 16 8 be two matrices. Then, find the
✓ -2 ~6 -s· -4
matrix B if AB =C.
27✓ fa) Differentiate y =sin-1 (3x- 4x3) w.r.t. x, ifx e [- !,!J
OR
1
(b)- Differentiate y = cos-1 ( - x: J with respect to x, when x e (0, 1).
l+x
65/6/1 Page 15 of 23 . P.T.O.
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A student wants to pair up natural numbers in such a way that
they satisfy the equation 2x + y = 4i, x, y EN. Find the domain
and range of the relation. Check. if the relation thus formed is
·reflexive, symmetric and transitive. Hence, state whether it is an
equivalence relation or not.
OR
(b) Show that the function f : N ➔ N, where N is a set of natural
. {n - 1, it'h is even
numbers, given by fl:n) = if . dd- is a bijection•
. n+l, n1so
Consider the Linear Programming Problem, where the objective function .
Z = (x + 4y) needs to be minimized subject to constraints
2x+y~ 1000 . 11; •
x+ 2y~800
x, y ~ 0.
Draw a neat graph of the feasible region and find the minimum value
ofZ. •
·30. rla) }i Find the-r9istance of, th~ _poµit P(2, 4, -1) from the line·
x+5 y+3 z-6
'1 = 4 =.---:-g. )- .
OR
I •
I\ I\ I\
(b) Let .the position vectors of the points A, B and C be 3 i - j - 2 k,
I\ I\ I\ I\ I\ I\ •
i + 2 j - k and i + 5 j + 3 k respectively. Find tlie vector and
cartesian equations of the line passing through A ~d parallel to
line BC.
A person is Head of two independent selection committees I and II. If the
probability of making ,a wrong selection in committee I is 0·03 and that in
colllllrittee II is 0·01, then find the probability that the person makes the
correct decision of selec~on :
(i) in both committees
r
(ii) in only one committee
65/6/1 Page 17 of'23 P.T.O.
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J....__ __ !·
(I • • (I'
SECTIOND
. . -
This section comprises 4 Long Answer (LA) type ques~ion,~ of 5 marks each.·
♦, ••
.,..' ... .,, .. : OR·
.,
(b) -#
Evaluate: '.
•. I
/ '4 I
' '
''
,
~ Draw a rough sketch for the curve y = 2 + Ix +t I, Using integration, find
V the area of the region bounded .by the curve y = 2 + Ix + 11, x = - 4, x = 3 , .,
•I.;
andy = 0.
(9 (a) Solve the differential equation : x2y dx """. (x3 ·+ y3) dy = O.
OR
(b) Solve the differential equation (1 + x2) : ,+ ;xy - 4x2 = 0 subject
to initial condition y(O) = 0.
_.
. .l : )'" '
!; i ~ • t 1' I,' i : .:' ,.
2
Let the polished side of the mi';°r be, along th~ ~~ ~ 1
= _c..; _ z; ;
I
. ' -
'A p~int P(l; 6, :3), some distance away·'rro~--the mirror, has its imag~
•,· ·•,,r1-,,•.~• r•~ • • . I-:'", ,: - •·-
I ' ' • ~ V' • , ..
·, formed behind the mirror. Find the coordinates of the image point and
• (•' • •
. • • •
. ,# ' • • • • ' , 4
, the distance between the point P and its image.
65/6/1 Page 19 of23 P.T.O.
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-~
-~- "'9,
y
f rt SECTIONE
.
This section comprises 3 case study based questions of 4 marks each.
CaseStu dy-1 . - •
r
1 •
Three students, Neha, Rani and Sam go to a m_arket to purchase
( stationery items. Neha buys 4 pens, 3 notepads and 2 erasers and pays
, 60. Rani buys 2·pens, 4 notepads and 6 erasers f~r, 90. Sam,pay s, 70
for 6 pens, 2 notepads and 3 erasers.
·Based upon the above information, answer the following questions :
~ Form the equations required to solve the problem of finding the
price of each item, and express it in the matrix form AX = B. 1
~ Find IAI and confirm.if
I , •I
it is possible
.
to' findA- 1. 1
A
(iii) (a) Find A-1, if possible, and write the formula to find X. 2
r (b)
OR ,
~
Find A2 - 81, where I is an identity matrix.
Case Study - 2
2
sy
.
A ladder of fixed length 'h' is to be placed along the wall such that it is
free to move along the height of the wall.
Based upon the above information, answer the following questions :
Express the distance (y) between the wall and foot of the ladder in
terms of 'lf and height (x) on the wall at a certain instant. Also,
write an expression in terms of h and x for the area (A) of the right
triangle, as seen from the side by an observer. 1
65/6/1 Page 21 of23
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1
Find the derivative of the area (A) with respect to the height on the
wall (x), and find its critical point. 1
(iii) (a) Show that the area (A) of the right triangle is maximum at
the critical point. 2
OR
If the foot· of the ladder whose length is 5 m, is being pulled
towards the wall such .that the rate ..of decrease of distance
(y) is 2 mis, then at what rate is the height on the wall
(x) increasing, when the foot of the ladder is 3 m away from
the wall? 2
Case Study- 3
u.
• shop selling electronic items sells smartphones of only three reputed
ompanies A, B and C because chances of their manufacturing a defective
smartphone are only 5%, 4% and 2% respectively. In his inventory he has
25% smartphones from company A, 35% smartphones from company
Band 40% smartphones from company C.
A person buys a smartphone from this shop.
(i) Find'the probability that.it was defective. 2
(ii) What is the probability that this defective smartphone was
manufactured by company B ? 2
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