2nd PU Maths 2014.pdf
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)
(-+
-\/-
-+\
-
l-l
2l .lf
a
is
a
unit
vector such
that
[r-
,
Jlr*
o
)=
f,
U*
l'l
22.\f
theposition
vectors
of
the
points
A
and
B
are
?+
Z
i-li
&
i-lc
respectively,
then
find
the
direction
10X3-30
(o,b)*(c,d)=(a+c,b+d),
prove that
*
is
both
'
cosines
of.
AB .
23.
Find
the
equation
the
plane
passing througlr
the
intersection
of the
planes x+
y + z
=
6
and
2x+3y+42-5=0 andthepoint(1,1,1).
+:..
24.
A
dieis
thrown.
If E is the
event
'the
number
appea{rng
is
a
multiple
of
3'
and
F is the
event
'
-,,qr1.p1 aOpeanne
is
even'
then
find
whether
E
and
F
are
independent?
i
PART.
C
Answer
any
TEN
questions:
25.\f
x
is abinary
operation
defined
on
A=N XN'
by
commutative
and
associative.
Find
the
identity
if
it
exists'
26. Prove
that
2tan-r(;).,*-,[})
=,*-,(lJ
lr
21
27. Exeress
L;
Ol
*
*.
sum
of
a symmetric
and
skey
symmetric
matrices'
28.rf ,=,;-,ri-"\
1 I
''dv
'
(1
'=T?)*({E
'nndfi'
zg
.
tf
x
=
a(0+
sin d)
ar.rd
y
=
a(l
-cos
d)
,
prove
*"
*=
,*(;)
30' Find
the
intervals
in which
the
function
'f
given
by
f
(x)
=
4xt
-
6x2
-72x
+ 30
is
(i)
strictly
increasing
(ii)
strictly
decreasing.
31. Find l---dtt-
J
x
* xlogx
32.
Find
I,*-'
x.dx.
33' Find
the
are
bounded by
the
parabola
y2
=
5x
and
the
line
y
-
r'
34.
Form
the
differential equation
representing
the
family
of curves
/ =
c sin(-x
+
b)
,
where
a,
b
are
arbitrary
constants.
'+
'+ -'
-t
-+ -')
-)
'
-)
'-t
-)
-+
-+
.+
35.If
i,bordc
arethreeunitvectorssuchthat
a+b+c=0,findthevalueof
a-b+b.c+c'a'
36.
Find
a
unit
vector
perpendicular to
each
of the
vectors
(i+i)ana(;-b,
where
+ A-+ A
A
a
=
i+
j+k,b
=
i+2
j+3k
.
37. Find
the shortest
distance
between
thelines
/,
and
/,
whose
vector
equations
are;
1
^
\ '+
A A
A
A
-A
-
T
i
=
i+
j+
1(2i-7+k)
and
r
=2i+
j-k+
p(3i-5
j+Tk).
38.
A bag
contains
4 red
and
4 black balls,
another
bag contains
2
red
and
6
black
balls.
One
of
the
two
I
bags is
setected at
random
and
a ballis
Oru*n
tori
ttre bag
which
is
found
to
be
red'
Find
the
probability that the ball
is
drawn
from
the
firstbag'
2
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8/20/2019 2nd PU Maths 2014.pdf
3/3
PART
_
D
Answer
any
Six questions:_
6X5=30
"
::::,.t1*"-Y:':::,
-{
'
^*.:
[4,
""]
defined
bv
/(x)
=
)c' +4
,
where
R+ being
the
set
of
al
non
-
egative
realfl*b:T
is
invertible. Also
write
the inverse
of
(x).
oo
o
n=1f,
?],'=[?
:)*,,=l:
l],;;;;;;;;@+c)
Veri&
that
A(B
#-C)
=
AB
+
rrI
tll.Sclvethefeilowtngsystgili..rf
equar.-.rtrudlru*_-lX:rrlthoo,
x+-,.,+7=u
i
;,*sz.=ii
airr.,
x--2y*
z
=
0.
42.If
y
=
(tan-rx)2
then
show
that
(*,
+gt +2x(x2
+1.)g
43.lstone
is dropped
into
a
quiet
lake
*;f;"r.";,";*"r;;
x2
instant,*r,.niri"u
jil;;il"ffi
ffiXffi
tr;:ffiTltHffi1,::,.'I;1il
j,".Il,lrro'*.
44'
Find
the
integral
of
+
with
respect
to
x
and
hence
evaluate
[g
,tx-
-d'
I
i*z
q5
-r*
w,
45'
Find
the
area
of
the
region
in
the
first
quadrant
encrosed
by
the
x_axis,
the
rine
y:
x and
the
circle
x2 +
yz
_
32.
46.
Solve
the
differential
equ4io
n,
*9*2y
=
xr.logx.
-'
U;Hjle
eeuatlon
of
a
plane
purfne
through
three
non
-
colrinear
points
both
in
the
vector
and
-t
*i::J,::T?,i'"f,:ff1,ll.::$li
an
odd
number"
is
a success,
what
is
the
probability
or(i)5
"i
Answer
any
One
question:
PART
_
E
a
trXI0:trO
2a
ag.
@)Prove
that
-i
a'
.,.1^-
t,n
n
1f
@)dr
=
z
lf
G)dx,
when
[tzr
-
x)
=
f
(x)
and
hence
evaruate
jl*.rFr.
ogo
(b)
Determine
the
value
of
K,
ir
f(x)
=ffi,*
**r,*=
*i
is continuou,
o
,
=+
50' (a)
solve
the
following
linear
programming
problem
graphically;
H,]|*andMaximiie
z=x+2"y,
subjr.tio.or,rtilr,ir";;';;>100,
Z*_ys0,
Zx+y