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DONBOSCOSCHOOL - ALAKNANDA - NEWDELHI-110019

CLASS: XII PERIODIC TEST- II 14-09-2019
TIME:3HRS MATHEMATICS {041) M.M.:80
GENERAL INSTRUCTIONS:
(i) All questions are compulsory
(ii) The question paper is divided in to two parts: PART I and PART II.
PART I is objective and divided into three sections A, Band C.
PART II is descriptive and divided into three sections D, E and F.
(iii) PART/:
SECTION A: QI to QI 6 are MCQ ofI mark each
SECTION B: Q/,7 and Q/8 are case-based questions. Each question has 4 parts of J mark each.
SECTION C: QI 9 to Q34 are very short answer of 1 mark each.
{iv) PART II:
SECTION D: Q35 to Q39 are of2 marks each
SECTION E: Q40 and Q44 are of 3 marks each.
SECTION F: Q45 to Q47 are of 5 marks each.

..
(v) All questions in section C are to be answered in one or two sentence(s) as per the exact requirement ofthe
question.
(vi) Use of calculators is not permitted.
(viij Two graph sheets required.

SECTION A (16 x 1= 16 marks)

1. Let f: N --+ N be defined as f(x) = 3x. then f(x) is:
(a) one-one and onto (b) many-one and onto (c) one-one and into (d) neither one-one nor onto

2. Let A= {l, 2, 3) and B = {4, 5, 6, 7}, then total number of functions possible from A to B is:
(a) 12 (bl 21 z (c) 81 _JdJ'64

3. The value of sin-1 ( cos (1~ ")) is:

(a) 3TC (b)~ (c) .!!.. (d)-2!..
s 5 10 10

4. Let y = sin- (2x), then 1

(a) x E [-1, 1] (b) x E [-2,2] (c) XE r-12 ,121 (d) XE JR(
/

5. The value of 2 sec- 1 2 + sin- 1 G) is:

(a) STC (b) 1rc - (c) 7r (d)~
6 6 3

6. If A is a square matrix su ch that Az I, t hen (/ = +A )3 - 4A =

(a) 4A L{b) 4/ (c) / - 3A (d) none of these

7. If A and Bare ·matrices of sa me order, then (A B' - BA') is a:

j!l skew-symmetric matrix (b) symmetric matrix (c) null matrix (d) unit matrix

8. Let A = [-1
1 3
2
] be a matrix such that Az + 51 = kA, t hen t he val ue of k is:
(a) 2 _(b) -3 • (c) 3 (d) none of these

9. Let A be a square matrix of order 3 x 3 such that A AT =I= AT A, then IA- 1 1=

(a ) ± 3 (b) 1 (c) ± 1 (d) not defined
 cosx -sinx
10. Let /1

(a)[-1, 1]
=
I
cos(x
sin x
+ y)
cos x
-sin(x + y)
(b) [-v'2, \/2)
I• where x, y E JR, then /1 lies in the Interval
(c) [0, 1] (d) [0, 2]

XZ X $ 1
11. Consider the function f (x) = ' . Then f (x)is :
2x, X > 1f
(a) continuous and differentiable everywhere in JR
(bl continuous everywhere in IR but not differentiable at x = 1
(c)differentiable at x = 1 as LHD = RHO but not continuous at x = 1 as LHL * RHL
(d) neither continuous nor differentiable at x = 1
12. The derivative of ,11 - x 2 with respect to sin- 1 vl - xz is
(a) x (b) -~ (c) -Yl - x 2 1-
(d)-
x

13. The maximum and minimum value of the function f (x) x 3 , x E (-2, 2) are: =
(a)Maximum value=8, minimum value= - 8 (b) Maximum value=8, minimum value= 0
(c) No maximum value, No minimum value (d) No maximum value, minimum value= O

14. If f'(c) =
0 and f'(x) changes its sign from negative to positive as x increasing through c, then x = c is:
(a) a point of discontinuity (b) a point of maxima (cl a point of minima (d) a point of inflexion

15. The corner points of the feasible region determined by the constraints 2x + y ::; 10, x + 3y::;; 15, x, y Oare
A(O, OJ, 8(5, 0), C(3,4) and D(0,5). Let the objective function be z =ax+ by, where a, b >0. Then the relation between a
and b so that maximum of Z occurs at 8(5, O) and C(3, 4), is:
(a) 2a - b = O (b) a - 2b = O (c) 3a - b = o (d) a - 3b = o

16. The linear progra mmi ng problem is to Maximise z = Bx +12y subject to 3x + 4y 6,x + 3y::;; 3,x O, y 0. The
feasi ble region has:
(a} infinitely many points and is bounded (b) exactly one point
(c} infinitely many points and is unbounded (d) no points

SECTION B (2 x 4 = 8 marks)
(2 case-based questions with 4 parts each)
17. Shyam, RaVi and Zi',\aan goes to grocery.store to buy ~ugar, whea~ and ricefor their respective houses. Th~y decided tp
purchase grocery itJms of t he same brand. After buying, at tAe bill coun!er, Shyam pays a total of ~ k g sugar,
3kg wheat and 2kg rice. Ravi pays 1'460 for 2kg sugar, 4kg wheat and 6kg rice. Ziyaan pays 1'330 for buying 6kg
sugar, 2kg wheat and 3kg rice. ii;;:~~ ,,.•tW,.
. ·-, 'l ....., ,

Based on the information given above, answer t he following questions:

(i) Let x, y ,z be cost per kg of sugar, wheat and rice respective ly. Then the matrix equation AX= 8 for the above is:

(a) 3
[
4 42 6]
2 [x y z] = [270]
460 [4
(bl 3
61
42 2 [X [2701
Y ] = 460

2 J j y]x = [~460-
70 ] 2( 4 6 3 3 2]z[x] 3[~~
= 33001
330
(c)
[H-H[ "
z 3 z 330
(d) 2 4 6 Y
6 2 3 z 460
 (ii) When sol ving

1 [ 0
(a) 10 30 5
· t he matrix

0 10]
20
20 10 10
· equation AX= B, the inv~rse A, A-1 =

(b)
1 [
10
6O -1
-4 2
2
o -4
2
l [ /
0
0
30
-5 10
0 -20
-20 10 10
l (d) none of these

(iii) The cost per kg of wheat, rice and sugar are respectively
(a) ~30, ~50 and ~20 (b) ~40, ~30 and ~20 (c) ~20, ~30 and ~50 (d) ~20, ~30 and ~40

(iv) The total amount received by the grocery owner for _selling rice to Shyam, Ravi and Ziyaan is:
(a) ~1060 (b) ~550 (c) ~270 (d) ~240

l8. A water ta ~ located l!!_ a park has the shape of an inverted cone as (shown in the image) with its semi-vertical angle
a _= tan- 1 G)·The tank is filled everyday for daily usage of water inthe neighbouring societies.

Based on the above information, answer the following questions.

(i) Let 'h' be the height of the tank and 'd' be the diameter of the top of the tank. The relation between h and Pis:
(a)h= 2r----u;~ d- (c)h=~ - - (d) h =-./d . ' • •
2

(ii) Let r be the radius of the top of the water level at a particular moment, then volume of the water in the tank is
given by:
(a) V = !. nr 2
3
(b) V = .!.m-
3
3 (c) V = ~m-
3
3
(d)
.
V = .!.nr 3
6

(iii) Jf water is pumped into the tank at a constant rate of 5 cm 3 /min, the rate of increase of the radius of water
surface, when r = 5 m, is :
1
1
(a)-m/min ( ) 1
b -m / ·,
mm (c)--m/min (d) 5 rr m/min
lOrr 5 rr 20 rr

(iv) If maximum capacity of the water ta nk is 486 m3, the rate at which the height of the water level is increasing, when
water level is Sm below from the top of the tank, is:
1
5
(a)-m/min (b) -5m / mm ' (
c) -m 1 /mm
. (d)--m/min
16 rr 32 rr 5 rr 10 rr

SECTION C (16 x 1= 16 marks)

19. Simplify: tan ( cos- 1 (-b))

20. Write cos - 1 ( ..,
1
:x 2
1
) in simplest form.
21. Eva luate: tan(tan- (-2))
22. Let A= [ -1, 1] and f(x) = x Ix I be a function defined on A. State whet her f(x) is bijective or not.
23. For a set, A= {a, b, c}, define a relation Ron A as R= { (a, a), (a, b), (b, b), (c, cl}. Write the ordered pair(s) to be added in
Rto make it the smallest equivalence relation.

24. The value of x + y + z if A = x

o y1 zo]is a skew-symmetric matrix.
[0 -8 0
2 3

25. Let A = [! 3 ~] , the give an example of a non-zero matrix B of order 2 x 2 such that AB is a null matrix.
 ,._,,,/1
26. Simplify: cos x [ co~ x sin x] + sin x [sin x - ~os x]
/ 4 .Find the value of a~ :n:
/
[!:s
a
x~
3

1
!j
-2
i::ss;ngul::n:atrix.

v-f
z . ss· determinants, find the value(s) of 'k' if area of the triangle with vertices A(-1, 1), 8(3, 5) and C(5, k) is 4 sq. units.
kx + 1, x S rr
he value of 'k' if f(x)
[ = is continuous at x TC. =
, /. cos XI X > 1C
/4. :nddy ify = xY
31. The !'st
function for producing x units is given by C(x) = 10 - 18x + 3x 2 • Find the marginal cost when 25 units are
produced.
A It is given that at x = 1, f(x) = x 4 2
82x +ax+ 8 attains its maximum value on the interval [O, 2]. Find the value of a.
-
The corner points of the feasible region are A(3, 5),8(5, 3) and C(4, 1). Find the minimum value of the objective function
L,,<, Z=3x_-y.
34. Determine whether the point P(2, 3) lies in the feasible region of a LPP with constraints
2x + y 5, 2x -y S 0, x,y 0

pllfq
/ SECTION D (5 JC 2= 10 marks)
--is. Solve the linear programming problem graphically:
/ Maximise: Z = x + y subject to x + y s 4,x 0,y o.
ii'6,t5implify and find the value of sin ( 2 cot- 1 (¾))
.Ii Determine whether/: Ill - {-1} lR - {-1} defined by f(x) =~is one-one or onto. Justify.
y Prove that adjoint of a singular matrix is a singular matrix. l+x

Y ,Find the intervals on which f(x) = (x - 1) 3 (x + 1) is increasing.

SECTION E (5 JC 3 = 1s marks)
tr.,. (,ff+x--ff=i)
4 - 2cos x, -
-1 rr _1 1 1
'WI. Prove that tan ..ff+x +-.rr=x = ..fl :5 x :5 1

y.1'how that the matrix A= [~ !1]

satisfies the equation x 3 - 2 - 4x 3x + 11 '== 0. Hence find A-
1.
1 2 3
Y.;;J a2y
= ea cos- 1 x, -1 s x :s; 1, prove that (1 - x 2 ) axz ay
- x ax - a2 Y = 0

~ ! ,et x = a sin t , y = a (cost + log ( tan find: when x = . _ D) ,

Yoetermine graphically the minimum (if any) of the objective function Z = -Sox+ 20y
subject to the constraints x + 2y 5, 3x + y 3, 2x - 3y $ 12, x,y 0.

/ SECTION F (3 JC s =15 marks)

v45. Let N denote the set of all natural numbers and R be a relation on N x N defined by (a, b) R(c, d) if and only if
ad(b+c)=bc(a+d). Determine whether Ris an equivalence relation or not. ff yes, find [(S, S)J, the equivalence class
of the element (5,5)

4~ Determine the product -7 1

-4 4 -1 1 41 [1 1
3 1 -2 -2 and using the product solve the system of equations
f
5 -3 -1 2 1 . 3
x + y + 2z ::: 1, -x - 2y + z = 4, x - 2y + 3z = 0
4, . Prove that the volume of the largest cone that can be inscribed in a sphere of radius Ris-;; of the volume of the sphere.