KENDRIYA VIDYALAYA DANAPUR CANTT, (FS)
SUB:- MATHEMATICS
CLASS:- XII
Time:- 3hrs
MM:-100
General Instructions:1. All questions are compulsory
2. The question paper consists of 26 questions divided into three sections A,B &
C. Section A comprises of 6 questions of 1 Marks each. Sections B comprises of
13 questions of 4 Marks each and Section C comprises of 7 questions of 6
marks each.
3. All question 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. Uses of calculator are not permitted. You may ask for logarithmic table if
required.
Section – A
3
1. Find the value of tan(sin-1 )
5
1 𝑑𝑥
2. ∫0
1+𝑥2
3. The degree of differential eqn .
𝑑2𝑦
𝑑𝑦
+ 5x (𝑑𝑥 ) 2 – 6y = 10 g x is?
𝑑𝑥 2
4. Find the projection of the vector 𝑎⃗ = 2i + 3j +2k on the vector ⃗𝒃⃗ = ^𝒊+2 ^𝒋+ 𝒌^
5. Show that the vectors 2 ^𝒊 - 3 ^𝒋 + 4 𝒌^ and -4 ^𝒊 +6 ^𝒋 - 8 𝒌^ are collinear.
6. Write the vector eqn of the following line
𝑥−5
3
=
𝑦+4
7
=
6−𝑧
2
Section – B
7.
Prove that the relation R in the set A= {1,2,3,4,5} given by R=} (a-b):|a-b| is even is
an equivalence relation.
OR
gf f:RR and g: RR are given by f(x) = sin 𝑥 and g(x) = 5x 2 find gof (x) and fog (x).
8. Solve it
1−𝑥 1
tan-1
= tan−1 𝑥 (x > 0)
1+𝑥 2
9. Obtain the inverse of the following matrix using elementary operations.
012
A= [1 2 3]
311
10. By using properties of determinants show that
1 x x2
x2 1 x =(1-x3 )2
x x2 1
OR
a2 +1 ab
ac
ab
b2+1 bc
ca
cb
= 1+a2+b2+c2
c2 +1
11. Find the value of K so that the function f is continuous at the indicated point.
𝑲𝒄𝒐𝒔 𝒙
𝝅−𝟐𝒙
if x ≠
𝝅
if x =
𝝅
𝟐
f(x) =
3
12.
𝟐
Differentiate the following function w.r.t. x.
cos x. cos2x. cos3x
𝝅
at x = 𝟐
or
(log x )x + x logx
13.
A balloon, which always remains spherical on inflation, is being inflated by pumping
in 900 cubic centimeters of gas per second. Find the rate at which the radius of the
balloon increases when the radius is 15cm.
14.
Integrate the following function –
e2x - 1
e2x +1
or
Find
x2+1
x2
𝜋/2
-5x+b
dx
15.
Evaluate ∫0
16.
Find the general solution of the differential equation ydx – (x+2y2) d y = 0
17.
If a, b ,c are unit vectors such that
log sin 𝑥 𝑑𝑥
a + b + c = O find the value of
a. b + b . c + c . a
18.
Find the shortest distance between the lines whose vector equations are
and
19.
^
^
^
^
^
r = ( 𝒊 + 2 𝒋 + 3𝒌^) + ⋏ ( 𝒊 - 3 𝒋 + 2𝒌 )
^
^
^
^
^ ^
r = 4 + 5 + 6 +𝜇 (2 + 3 + )
𝒊
𝒋
𝒌
𝒊
𝒋
𝒌
Maximise
z = 3x + 4y
subject to the constraints
x+y≤4
x≥O
y≥O
SECTION – C
20.
21.
A binary operation * on set R-} -1} is defined as a*b = a + b + ab ∀ a,b ∈ R-}-1}. Prove
that * is commutative and associative. Also find the identity element for a ∈ R -} -1} if
exists.
Prove that the volume of the largest cone that can be inscribed in a sphere of radius
R is
22.
23.
8
27
or
Show that the semi vertical angle of the cone of the maximum volume and of given
slant height is tan-1 √2.
Find the area of the the region in the first quadrant enclosed by the x – ais, the
Line y = x and the circle x2 = y2 = 32.
or
Find the area of the region bounded by the cures y= x2 + 2, y = x, x = 0 and x = 3.
Show that the family of curves for which the slope of the tangent at any point (x ,y)
on it is
24.
of the volume of the sphere.
𝑥2+𝑦2
2𝑥𝑦
is given by x2 – y2 = cx.
Find the equation of the plane through the intersection of the planes 3x – y + 2z – 4
= 0 and x + y + z -2 = 0 and the point (2,2,1)
or
Find the angle between the line
25.
26.
𝑥+1
2
𝑦
=3=
𝑧−3
6
and the plane 10x + 2y – 11z = 3.
A company manufactures two types of stickers A: “ SAVE EVVIRONMENT” and B: “
BE COURTEOUS” Type A requires 5 minutes each for cutting and 10 minutes each
for assembling. Type B requires 8 minutes each for cutting and 8 minutes each for
assembling. There are 3 hours and 20 minutes available for cutting and 4 hours
available for assembling in a day. He earns a profit of Rs 50 on each type A and Rs 60
on each type B. How many stickers of each type should be company manufacture in
a day to maximize profit? Give your views about “SAVE ENVIRONMENT” and “BE
COURT EOUS”.
There are three coins. One is a two headed coin (heaving head on both
faces)another is a biased coin that comes up heads 75% of the time and third is an
unbiased coin. One of the three coins is chosen at random and tossed, it shows
heads, what is the probability that it was the two heads lion?
-------------------------------------------------end------------------------------------------Sh. Bhola Singh
PGT, (Maths)
K.V Danapur Cantt (FS).
Kendriay Vidyalaya Danapur Cantt, FS
Mathematics
Class – XII
Blue print
SI. NO
Name of Chapter
1
Relation
functions
2
Algebra
3
V. Short
Q. 1 Marks
Long Q.
4Marks
Long Q.
6 Marks
Total
1 (4)
1 (6)
2 (10)
1 (1)
3 (4)
-
4 (13)
Calculus
2 (1)
6 (4)
3 (6)
11(44)
4
Vectors and 3D
3 (1)
2 (4)
1 (6)
6 (17)
5
Linear
programming
-
-
1 (6)
1 (6)
6
Probability
-
1 (4)
1 (6)
2 (10)
7
Total
6 (1)
13 (4)
7 (6)
26 (100)
and -