CLASS 12 Maths by nehalwan

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```									                                      Class – XII
Subject: Maths

Time allowed: 3 hrs.
M.M. 100
General Instructions:
1. All questions are compulsory.
2. The question paper consists of 29 questions divided into three sections A,
B & C. Section A comprises of 10 questions of 1 mark each. Section B
comprises of 12 questions of 4 marks each and section C Comprises of 7
Questions of 6 marks each.
3. All questions in Section A are to be answered in one word, one sentence or
as per the exact requirement of the question.
4. There is no overall choice. However, internal choice has been provided in
4 questions of 4 marks each and 2 questions of 6 mark each. You have to
attempt only one of the alternatives in all such questions.
5. Use of calculator is not permitted.

SECTION – A
1. What is the number of binary operations on set {a, b}?
2. What is the principal value of

2                     2
Cos-1 (Cos            )  Sin-1 (Sin      )
3                     3
x 2         6 2
3. If             =                Find x.
18 x         18 6
4.If the points (2, -3) (  , -1) and (0,4) are collinear, find the value of  .

1 1 
5. Evaluate      ex   2  dx
x x 
6 Evaluate    Logx dx
7. What is the angle between the vectors a and b with magnitude               3 and 2
respectively? Given a . b = 3.

8. Find a unit vector in the direction of a = 3 i - 2 j + 6 k
2x  1     4 y   z 1
9. If the Cartesian equation of a line AB is          =       =
2          7      2
Write the direction ratios of a line parallel to AB.
1 5 
10. For the matrix A =   show that ( A + A’) is a symmetric matrix.
6 7 

SECTION – B
11. Show that the relation R on the set
A = {x  z: 0  x  12}, given by
R = {(a,b): |a – b| is a multiple of 4}
is an equivalence relation.
 1 x  1 x         1
12. Prove tan-1                   =    -    cos-1x
 1 x  1 x        4 2

OR

Solve for x:
x 1         x 1 
If tan-1      + tan-1     =   , then find the value of ‘x’.
x2          x2   4
13 Prove that
(b  c )  a        a
b         (c  a ) b       = 2abc (a+b+c)3
c         c        (a  b)

14. Find ‘a’ and ‘b’ such that the function defined by
5       if
x  2

ax+b        if 2<x<10

21
if x  10

OR

Verify mean value theorem for the function: f(x) = x2 – 4x -3       in [1, 4]

15. If x 1  y + y 1  x =0
dy      1
Prove that    =-
dx (1  x) 2

OR

If x=a sec3  and y=atan3  ,
dy           
Find      at  =
dx           3


2
16. Show that     (
0
tan x  cot x) dx =     2

17.Prove that the curves x=y2 and xy =k intersect at right angles if 8k2 = 1

18. Solve the following differential equation

dy
x             + 2y = x2logx
dx
OR
Solve the differential equation
sec2x tany dx + sec2y tanx dy =0

19. The scalar product of the vector I + j + k with a unit vector along the sum of
vectors 2i+4j-5k and i+2j+3k is equal to one. Find the value of .

1 x   7 y  14   z 3
20. Find the values of p so that the lines            =          =      and
3        2p        2
7  7x   y 5   6 z
=      =      are at right angles.
3p       1     5
1
21. Probability of solving specific problem independently by A and B are               and
2
1
respectively. If both try to solve the problem independently, find the
3
probability that
a. The problem is solved.
b. Exactly one of them solves the problem.

22. Form the differential equation not containing the arbitrary constants and
satisfied by the equation y = aebx, a and b are arbitrary constants.

Section –C
6 Marks Questions
23. Using elementary transformations, find the inverse of A,Where the matrix.

2        -3      3
A=       2        2       3
3        -2      2

OR
Solve by matrix method.
x  y  5 z  26
x  2 y  z  4
x  3 y  6 z  29

24. Show that the volume of the largest cone that can be inscribed in a sphere of
8
radius R is    of the volume of the sphere.
27

25. Using integration find the area of region bounded by the triangle whose
vertices are(1,0) ,(2,2) and (3,1).

26. Evaluate :
 /4

 log(1  tan x)dx
0

OR
Evaluate

   (2x 2 + x+7) dx, as a limit of sum .

27.  Find the distance of the point (1, -2, 3) from the plane x – y + z =5
measured along a line
x      y      z
parallel to    =      =
2      3     6

28 A Company manufacturer’s two types of toys A and B. Toy A requires 4
minutes for cutting and 8 minutes for assembling and 8 minutes for cutting. There
are 3 hrs. and 20 minutes available in a day for cutting and 4 hrs. for assembling.
The Profit on a piece of toy A is Rs. 50 and that on toy B is Rs. 60. How many toys
of each type should be made daily to have maximum profit? Solve the problem
graphically

29 . A man is known to speak truth 3 out of 4 times. He throws a die and reports
that it is a six. Find the probability that it is actually a six.

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