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4 Sample Paper
CBSE Term-l & 2, 2021-22@
Thora
MATHEMATICS _ iittiaiinennamaa
TERM-1
Time Allowed : 1.5 hours Maximum Marks : 40
General Instructions: «
‘Same instructions as given in the Sample Paper 1.
0
@
This questions paper comprise 50 questions out of which 40 questions are to be attempted as per
-instructions. All. questions carry equal marks.
The question paper consists of three Sections - Section A. 8 and
Gi) Sections ~ A contains 20 questions. Attempt any 16 questions from Q. No. 1 to 20.
(iy) Section - B also contains 20 questions. Attmept any 16 questions from Q. No. 21 to-40.
”
Section “- C contains. 10 questions including one ‘Case Study Attmept any 8. from
Q.Na. 41 t0 50.
(Theres only one correct option fer avery Multiple Choice Question (MCQ). Morks will not be awarded for
‘answering more than one option.
(vi) There is no negative marking
SECTION - A 16 Marks
Iin this section, attempt any 16 questions out of Questions 1-20 Each question is of one mark]
1. Differential of log flog (log x3] wart. x is 3. Afunction f:R Ris defined as ffx) =x°+ 1.
5 Then the function has:
© Fiog()legtogx) (© no minimum value
5 (0) no maximum value
Oath (©) both maximum and minimum values
(@ neither maximum value nor minimum
(ee value 1
log(x*)log(logx*)
a
o— 1 4. Iesiny = xeos (a+ y).then Sis:
log x' log(log x") a
cos =
2. The number of all possible matrices of © Sern © Sern
order 2 x 3 with each entry 1 or 2 is: cost(aty) emai
@16 6 cosa cosa
64 @ 24 1 © ant y @ sin’y “5.
10.
11.
xe
The points on the curve +
tangent is parallel to x-axis are:
(@ G5, 0) ©) (0, 5)
© ©, +3) @ (3, 0) 1
Three points P(2x, x + 3), Q(0, x) and
R(x + 3, x + 6) are collinear, then x is equal
to:
@o
©3
2
@1 1
The principal value of cos* (3) sin
[se
x
os x
x x
© 3 @) 6
dy
WOE + YP? = xy, then Fist
Y+4xO? +4?)
Ay (x? +u7)— x
yr4xGP+y?)
Ay? +y)—x
yn 4x0e+y")
G) x440e+y')
(b)
Ay(x? +y?)-x
© y-4x(x7+y2)
If a matrix A is both symmetric and skew
symmetric, then A is necessarily a:
(@) Diagonal matrix
(b) Zero square matrix
(© Square matrix
(@) Identify matrix 1
Let set X = (1, 2, 3) and a relation R is
defined in X as ; R= [(1, 3), (2, 2. 3, 2h,
then minimum ordered pairs which should
be added in relation R to make it reflexive
and symmetric are:
@ {, ), 2, 3), , 2)}
(©) {G, 3), G, 1), (, 2)}
© 2, 2, B, 3) B 0, (2 3}
@ {, D, G, 3), B, 1), 4, 25 1
A Linear Programming Problem is as
follows:
Minimise z= 2x +y
Subject to the constraints x2 3,x<9,y20
x-y20,x+ys14
The feasible region has
(@ 5S.comer points including (0, 0) and (9, 5)
(b) 5 comer points including (7, 7) and (3, 3)
* Not examinable for 2024 exam.
12.
13.
16.
“17.
18.
19.
© 5 comer points including (14, 0) and
(9,0)
@ 5 corer points including (3, 5) and
@,5) 1
if,x#0
The function fx) = |x
k ifx=0
is continuous at x = 0 for the value of k, as:
@3 ms
@2 @s 1
If, denotes the cofactor of element P, of
1-1 2
the matrixP=|0 2 -3
sie2e
then the value of C,..C,, is:
@s (&) 24
(©-24 @-5 1
The function y = x'e* is decreasing in the
interval
(@) (0,2) (b) (2, ==)
O,2 @) (=, 0) U (2,1
IFR= {&%, Wx, ye Z,x? + y?< 4} is a relation
in set Z. then domain of R is:
@) {0, 1, 2} 1,0, 1, 2}
(©) {0, -1, -25 0, 1} rt
The system of linear equations,
will be consistent i
(@k+-3 k=
kes @kses Eb
‘The equation of the tangent to the curve
y(t +2) = 2 ~ x, where it crosses the x-axis
ist
@x-5y=2
(© x+Sy=2
w[2t8 o-d] [12 2
at+d 2-3b]=|-8 —4]o7e equal,
then value of ab ~ ed is:
@4 16
o-4 @-16 1
(b) 5x-y=2
@5x+y=2 1
‘The principal value of tan (sande
@% =
eo @-% fi 34
20. For two matrices P = E | and Qt =
o1
-1 2 1) p_ois
[22 thr -e%
SECTION - B
[in this section attempt any 16 questions out of the Questions 24 - 40. Each questions is of one mark]
24. The function fix) = 2x? ~ 15x? + 36x + 6 is
increasing in the interval,
@2U3,~) — (b) (-~, 2)
© AUB) @3, =) 1
22. Ifx = 2 cos® - cos 26 and y = 2 sind - sin 26,
6080+¢0520 cos@—cos20
(© “Gino -sin20 ©) “in20—sin@
23. What is the domain of the function
cos (2x - 3)?
@-411 (b) (1,2)
y @ [1,21 1
24, Amatrix A= [aj] ,, ,is defined by.
+3), ij
The number of elements in A which is more
than 5, is:
@3
@5
b) 4
@6 1
25. Ifa function f defined by.
keosx inp
m-2x! 2
3, ifx=®
2
fx) =
is continuous at x = , then the value of k
@2 3
@é6 (d) -6 1
oid
26. Forthe matrixX=|1 0 1), 0@-xX)is:
110
* Not examinable for 2024 exam.
27.
28.
*29.
30.
31.
23 43
@|-3 0 &|-3 0
0-3 -1 -2
43 23
lo -3 @]o -3 1
12 0 -
16 Marks
(@) 21 (®) 31
ol @51 1
Let X = {2 : x © N} and the function F : N
— X is defined by f(x) = x2, x € N. Then this
function is:
(©) injective only
(©) surjective only
(&) not bijetive
@) bijective 1
The corner points of the feasible region for
a Linear Programming problem are P(O, 5),
Q(4, 5), R(4, 2) and §(12, 0). The minimum
value of the objective function Z = 2x + 5y
is at the point:
@P
OR
©)Q
@s 1
The equation of the normal to the curve
ay? = x at the point (am?, am?) is:
(@) 2y - 3mx + am? =0
(b) 2x + 3my ~ 3am‘ — am?
(© 2x + 3my + 3am4 ~ 2am’
© 2x + 3my + 3am* - 2am’
=0
1
IFAis a square metrix of order 3 and |A|=-5,
then [adjA] i
(125 (b)-25
©2s (#225 1
The simplestform oftan* [
is:
nx nx
O42 O42
Bloggs BM oot
© j-jente (@) Ft jeostx 1
If for the matrix A ic ZI. 1A} = 125,
then the value of a is:
@s3 &)-3
@©s1 @1 1 34.
35.
36.
44.
If y = sin (m sin »), then which one of the
following equations is true?
1-2) Shi Bem'y =0
Fy WY mye
wa-S4-xYemy=0
dy dy.
©a+x Box my=0
YW nieeo 4
dy
(14x Sax
The principal value of
{tan V3 - cot (-V3) J is:
@n
x
© ->
@ 23 1
©o
The maximum vue (2) i
@e«
of)
Let matrix X = [x] is given by
bye
@e 1
1-4 2
3 4-5], Then the matrix Y = [m,],
2-13
where m, = Minor of x, ist
7-5 -3 7 -19 -11
@}10 1 -11) @ ls -1 -1
“11167 341 7
37.
39.
‘A function f: R-> R defined by fx) = 2 +x?
is:
(@) not one-one
(©) one-one
(©) not onto
(@) neither one-one nor onto 1
Programming Problem is as
objective function
Za 2-ys5
Subject to the constraints.
3x4 4ys 60
x +3y230
x20,y20.
If the corner points of the feasible region
are A (0, 10), B(12, 6), C(20, 0) and 0(0, 0),
then which of the following is true?
(©) Maximum value of Z is 40
(®) Minimum vatue of Z is - 5
(©) Difference of maximum and
values of Zis 35
(@) At two comer points, value of Z are
equal. a
Ix 2 3)
Ifx=-4isarootof|1 x 1| =0,thenthe
3 2 x|
sum of the other two roots is:
@4 &)-3
2 @s a
The absolute maximum value of the
function fix) = 4x - 2x? in the interval
ye
7 19 41 7 19 -11
@|-3 117 (@j-1 -1 4
<5 -1 -1 2 3-11 7 @s @)9
1 oeé 10 1
SECTION - C 8 Marks
[Attempt any 8 questions out of the Questions 41-50 Each questions is of one mark]
In a sphere of radius r, a right circular 42+ The corner of the feasible
cone of height h having maximum curved region determined by a set of
constraints (linear inequalities) are
surface area is inscribed. The expression for
the square of curved surface of cone is:
(©) 2x°rh (2rh +h?)
(©) whr(2rh +h?)
(©) 2n’r (2rh?- h’)
@ 2x*F(2rh +h’) 1
* Not examinable for 2024 exam.
PO, 5), Q(3, 5), R(5, 0) and S(4, 1)
and the objective function is Z = ax + 2by
where a, b > 0, the condition on a and b
such that the maximum Z occurs at Q and
Sis:
(a)a-5b=0
(a-2b=0
()a-3b=0
@a-8=0 1 *43.
44,
45.
If curves y? = 4x and xy = ¢ cut at right
angles, then the value of c is
(@) 4V2 ms
(© 22 @ -4v2 1
2
The inverse of the matrix X = | 0
0
1/2 0 Oo
(a)24) 0 1/3 0
o Oo 1/4
oo
10
o1
00
30
o4
o o
1/3 0 1
o 41/4
For an L.P.P. the objective function is
Z = 4x + 3y, and the feasible region
determined by a set of constraints (linear
inequations) is shown in the graph.
Ea Patipey
Eieat
~
EEN z
LENE Tey
Lek ® |
| p. 1
a {
a
a Sheto 1
ON bt
—t Lars
i i
Which one of the following statement is
true?
(@) Maximum value of Z is at R.
(b) Maximum value of Z is at Q.
(©) Value of Z at Ris less than the value at
P.
(d) Value of Z at Q is less than the value at
R 1
9 eer ceetieaiile tr BR
47.
In a residential society comprising of 100
houses, there were 60 children between the
ages of 10-15 years, they were inspired by
their teachers to start composting to ensure
that biodergradable waste is recycled. For
this purpose, instead of each child doing it
for only his/her house, children convinced
the Residents welfare association to do it as
@ society initiative, For this they identified a
square area in the local park. Local authorities
charged amount of & 50 per square metre
for space so that there is no misues of the
space and Resident welfare association takes
it seriously, Association hired a labourer
for digging out 250 m° and he charged
% 400 x (depth)?. Association will like to have
minimum cost.
Based on this information, answer any 4 of
the following questions,
Let side of square plot is x m and its depth
is h metres, then cost c for the pi
© 20 + 400h?
@ 52. soon?
250
‘“ 250 .
oath @ A+ 400n* 1
ch HE
Value of h (in m) for which
@15 2
©25 @3 1
ane ie given by:
@ aoe +800 ®) a +800
h h
100 500
© 4p +800 @ a2 1
Value of x (in m) for minimum cost is:
@s ©) 10,
© 5 @ 10 1
Total minimum cost of digging the pit (in 2)
is:
(@) 4,100
(©) 7,820
() 7,500
@) 3,220 a TERM-2
Time Allowed : 2 hours
General Instructions:
@. This questions paper contains three Sections-A, Band G
(@ Each section is compulsory.
Gii) Section-A has 6 short answer type-I questions of 2 marks each.
Gv Section-B has 4 short-answer type-Il questions of 3 marks each.
(Y Section-C has 4 long-answer type questions of 4 marks each.
(W) There is an internal choice in some questions.
(vi) Question 14 Is a case study based questions with two sub-parts of 2 marks each,
1
3.
7
SECTION - A
Maximum Marks : 40
12 Marks
[Question Numbers 1 to 6 carry 2 marks each]
Find: [2% — 2
V4x—x?
Find the general solution of the following
differential equation:
Baertexter 2)
Let X be a random variable which assumes
values x,, x,y Xy x, such that 2P(K = x,) = 3P
(=x) =P () = SP K=x,).
Find the probability distribution of X. 2
5. Ifa line makes an angle a, 8, y with the
coordinate axes, then find the value of
cos 20+ cos 28 + cos 2y. 2
6. Events A and B are such that P(A) = 4,
7 1
= and P(AUB)=>
P@) = 75 and (AUB) 4
Find whether the events A and B are
Independent or not.
OR
AboxB, contains 1 white ball. and 3 red balls.
+ gee rd Another box B, contains 2 white balls and
a=i+ j+ka.b=1and axb=j-k, then 3 red balls. If one balll is drawn at random
Fe from each of the boxes B, and B,, then find
find |b1. 2 the probability that the two balls drawn
are of the same colour. 2
SECTION - B 12 Marks
[Questions Number from 7 to 10 carry 3 marks each]
Evaluate: *9. Find the equation of the plane passing
dx through the line of intersection of the
J istane 3
If @and & are two vectors such that
|a+b|=|B], then prove that (a+2b) is
perpendicular to a.
oR
Hf Gand bare unit vectors and @ is the
angle between them, then prove that
* Not examinable for 2024 exam.
planes r(i+j+k) = 10 and
r(2is3j-h)+4 = 0 and passing
through the point (-2, 3, 1). 3
10. Find:
Jet-sin2x dx
OR
Find:
2x
x
oP + 1)(x? +2) ii.
12.
13.
SECTION - C
16 Marks
[Questions Number from 11 to 14 carry 4 marks each]
‘Three persons A, B and C apply for a job
of manager in a private company.
Chances of their selection are in the ratio
1:2: 4. The probability that A, B and C
can introduce changes to increase the
profits of a company are 0.8, 0.5 and 0.3
respectively. If increase in the profit does
not take place, find the probability that it
is due to the appointment of A. 4
Find the area bounded by the curves
y= |x-1] and y = 4, using integration. 4
Solve the following differential equation:
U- sin? x)dx + tan x dy = 0
OR
Find the general solution of the differential
equation:
Oe + y") dy =27y dx 7 4
14. Two motorcycles A and B are running at the
speed more than the allowed speed on the
roads represented by the lines r= Mi+2j-k)
andr =(31+3})+u(2i+ j+b) respectively.
Based on the above information, answer
the following questions:
(A) Find the shortest distance between the
given lines. 2
() Find the point at which the motorcycles
may collide. 2
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