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ЇЇЇuЇЇЇЇ╕╕╕D№д╕╕╕№╕╕╕дддддд MODEL PRACTICE TEST PAPER - II
MATHEMATICS
CLASS 12 - CBSE 2011
Time : 3 hrs Max. Marks: 100
General Instructions:
1. All questions are compulsory
2. The question paper consists of 29 questions divided into three sections A,B and C. Section A contains 10
questions of 1 mark each, Section B contains 12 questions of 4 marks each and section C contains 07
questions of 6 marks each.
3.
Section A
(Questions 1 10 carry one mark each)
1.
(
)
2. Write the principal value of cos
)
(
3. If #" #"= "3, #"
"! #"= 2 and
" . !=3,findtheanglebetweenand!"
4. Write down the equation of a line parallel to the line
2
3
=
453
2
=
657
and passing through the point
(1,2,3).
5. If matrix A = (1 2 3), write AAТ, where AТ is thr transpose of A
6. Evaluate 89:43
7. Write the order and degree of differential equation(
<▓4
<
▓
)▓ + ?
<4
>
@
3
+ 2A = 0
8. Find the value of p if (2CD+6ED+27 GF) x (CD+3ED+pGF) =
"0
9. Evaluate :[
2I "29:I
9:I I
[
10. Form the differential equation of the family of curves y = a cos(x+b), where a and b are arbitrary
constants.
Section B
(Questions 11 22 carry four marks each)
11. Using the properties of determinants, prove the following : [
1 1 + M 1 + M + N
2 3 + 2M 1 + 3M + 2N
3 6 + 3M 1 + 6M + 3N
] = 1
12. If y = (9:)
+ sin
", find
<4
>
13. Form a differential equation of the family of circles touching the x-axis at origin.
14. Solve the following for x : tan
+ 2cot
=
2
3
or, Prove that 2 tan
2
+ tan
= tan
3
15. Find the value of P so that the lines
3
=
4Q
2R
=
76S
and
3R
=
47
=
6
7
are perpendicular to
each other.
16. Find the equation of the tangent to the curve x▓+3y = 3, which is parallel to the line y-4x+5=0
Or
Find the intervals in which the function f given by f(x) = sinx + cosx, 0d" d" 2U, is strictly
increasing or strictly decreasing.
17. Using properties of definite integral, evaluate :
5VW▓
S
, or, Evaluate
>
"(7Q
2
X
(
18. Solve the differential equation :
<4
>
+ y = cosx-sinx
19. If f: NТ! Z be defined by f(n) =
5
2
, if n is odd
=
2
, if n is even
Find whether the function f is bijective.
20. The scalar product of the vector C + E +
"F with unit vector along sum of vectors 2C + 4E - 5"F and
[C + 2E + 3"F is equal to one. Find the value of [
21. Let * be the Binary operation on N given by a*b = LCM of a and b. Find the value of 20*16.
Is * (i) Commutative (ii) Associative.
22. Prove that the relation R in the set A = {1,2,3,4,5} given by R = {(a,b) : *▐4№·№UhI8.;Q < й
R
X
r
┬
╠
╨
т
ф
ь
Ї
<BLRV^Иж>?·····························gdI8▐4■?BEIMOSWYyГ╚■7
;
=
?
F
I
J
L
N
P
Y
Х
Щ
п
48·····························gdI88╬$░╘■(6Z\^jpДКМР"FISV\_ax{}·····························gdI8}ЕИКМФЧЪЭ╔╩═╧╙┘▄рцщяЁЇў∙¤ ·····························gdI8 -ТшР╪JLVXfhptx┤╢║╞╠╬╥╓▄*026Z·····························gdI8Zдм░╨@DHn─WНэ.*\*А+╘+И,&-4-@-L-P-n-p-v-x-Ж-·····························gdI8#"a-b#" is even}, is an
equivalence relation.
Email:contactvcs@rediffmail.com V.C.Srikanthan Phone : 9831071644 vcs/cbse/MPTII/2011 Section C
(Questions 23 29 carry Six marks each)
23. Find the point on the curve y▓=2x which is at a minimum distance from the point (1,4)
24. Obtain inverse of the following matrix using elementary operations[
3 0 "1
2 3 0
0 4 1
[
25. Evaluate
>
▓VW▓
5\▓ ▓
S
26. Find the area of the region enclosed between the two circles x▓+y▓=9 and (x-3)▓ + y▓=9
27. Evaluate
▓
]5
▓5
dx or, Evaluate (2^_9: " ^_9:2)
`
X
S
28. Find the equation of the plane through the point (-1,3,2) and perpendicular to each of the planes
x + 2y + 3z = 5 and 3x + 3y + 3z = 0
29. A diet is to contain atleast 80 units of vitamin A and 100 units of minerals. Two foods X and Y are
available. Food X costs Rs.4 per unit and Food Y costs Rs.6 per unit. One unit of Food X contains 3
units of vitamin A and 4 units of minerals. One unit of Food Y contains 6 units of vitamin A and 3
units of minerals. Formulate this as a Linear Programming Problem and find graphically the
minimum cost for diet that consists of mixture of these two foods and also meets the minimal
nutritional requirements.
V.C.Srikanthan
Chartered Engineer, MIE
Cell : 9831071644
Email : contactvcs@rediffmail.com
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