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1
2
.
3 .
4
E ver
y
quadra
tic
eq
uat ion
ax
2
+
b
x
+
c
=
0
wh
ere a,
b, c e
R a *
0
has
(a)
ex
actly o
ne rea
l
r
oot.
(b)
at
leastone
real
root.
c)
at le
ast two
real root
s.
(d
)
a
t
m
ost tw
oreal
ro
ots.
The
relat io
n S is d
efined o
n
the set of int
egers
Z
as xSy
i f nteg
er
x d
ivides i
nteger
y. Then
(a)
S is
a
n
equiva
lence re
lation.
(b)
S
is only
reflexiv
e an
d
s
ymmetr
ic.
c)
S is o
nly refle
xive
and t rans
it ive.
(d)
Sis onl
ysymm
etric
and t ransi
t ive.
If
a
* b * c
are all
positive
,
then t
he
val
ue of
a
b c
the
d
etermi
nant b
c a
is
c
a b
(a)
non-
negat iv
e
(b)
n
on-posit
ive
c)
neg
at ive
(d)
positive
L e
t A
andB be tw
o mat r
ices suc
h
that AB = A
an
d BA =
B W hi
ch o
f
the follow
ing
stat
ements
are
corre
c t
?
1.
A
2
=
A
2.
B
2
=
B
3.
(
AB)
2
=AB
S
elect
the
correc
t answe
r using
the cod
e given
below
(a
) 1
and 2 onl
y
(b) 2
an
d
3
only
c
) 1 a
nd
3
only
(d)
1,
2
and
3
5
6
7
Wha
t
i
s (1001)
2
equal
t
o?
(a
)
5)10
(b)
9
)10
c)
(
17)10
(d)
(1 1
)10
.
. ..
6
--:
c
What
is
l
rqual
to, w he
re
i
=
?
(a)
1
(b)
1
/6
c)
6
(d)
2
Let
zbe a co
mplex
num
bers'uch
th a
t Iz I= 4
and
a
rg z
=
5
. Wh
atis
z
eq
ual to ?
6
a) 2. /3 +
i
(b)
2./3-
i
c)
- 2
. /3 + i
(d)
-
./3
+
i
wherei-
=F
i
6i -
i 1
8
If
4
3i
- 1
= x
+ iy wher
e
i
=
Fl
.
20
3
i
thenwhat is
x equa
l to ?
a)
3
b) 2
c)
d) 0
8 0E
BBN N
BS
2 -A
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1. ~ t1lii'h
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9
10
I
f a
f arethe
r
oots of
ax
2
bx c
= 0 and
12
+ ,
f +
a
re th
e roots
of px
2
+x
+ = 0,
If A
is an
inv
ert ible
m atr ix
, then
wha t
is
det
A
l
equ
a l
t
o ?
the
n what
i
s
h
equal
to ?
a)
E.
-
)
2
a
p
(b )
(_E.
+ )
2
a p
c)
(d)
I f
G
to?
a)
-4
1)
b)
:
c)
-1 4)
0
-1
d
)
. 1
4)
0
-1
A is such
th at
1
) ,
then w
ha tis
A
equal
-1
a)
(b )
c)
d
etA
1
detA
1
(d)
Non
e of
th
e above
13
From
the mat
rix eq u
a tion
AB = A
C, w
here
A, B
, Care
the squa
re m at
rices o f
sa m e o
rder,
we c
anconc
lu de B
=C pro
vided
(a
) A is
non-si
ngular .
b)
c)
d)
A
is sing
u lar .
A
is sym
m etric.
A is ske
w symm
etric.
14
I
f
A =
4
x
+
2
)
is symm
etric, the
n
2x-
3 x
+ l
w
ha t is
x e
q ual
to ?
a
)
2
b)
3
c)
- 1
d
)
5
a
b
0
a
b =
0
, then
w hich o
ne
of th
e
11
C onsid
er
the
f
ollowin
g st
atemen
ts
:
1
5 I f
0
b
0
a
1
. Determ
inant
i
s
a squ
are m
atrix .
2 Deter
minant
i
s a numb
er ass
ociated
witha s
quare m at
rix .
Which
of th
e ab ove s
ta temen
tsis/ar
e correc
t ?
(a)
1 onl
y
b)
2 only
c)
Both
1and 2
(d)
Neithe
r 1
nor2
B OEB
B NNB
S
4
-A
followi
ng is co
rrect ?
(a
)
is one
of the cu
be root
s of unit
y .
(b)
is one
of
the
cube
roots
of -
1
c)
a i
s
o
ne o
f thecube
roots
of unity .
(d)
bis one
of the
r
u e
ro
ots o
f
un
ity .
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9.
~ ax
2
i -bx+c=0 ~ liR a f ~ ail\
12.
px
2
+ qx + r 0 ~ liR a h, f + h t m
~ < m a R t ?
~ A
~
ci : i'lli4Uft4
3lfOIIli t_
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16.
The
function f : N -- N, N being the set of 20. A
and
B are two
sets
having 3 elements in
17
natural
numbers, defined by f(x) =
2x
+ 3
is
common.
If
n(A) = 5, n(B) =4, then what is
(a) injective and sUijective
(b) injective
but
not
sUijective
c) not injective but surjective
(d)
neither
injective nor sUijective
What
is
(
1
+
i)4n + 5 .
- ---- -:
4
..---
3
equal to, where n IS a
(1-i) D+
natural
number
and
i
=
F1 ?
(a)
2
(b)
i
c)
2 i
(d)
i
n(A
x
B) equal
to
?
a)
0
b)
9
c) 15
d) 20
21.
I f f(x) =
ax
+ b and g(x) = ex + d such that
f(g(x)) = g(f(x)), then which one
of
the
following
is
correct ?
(a)
f c)
=g(a)
(b) f(a)
=
g(c)
c)
f c) = (d)
(d)
f(d) = (b)
22.
18.
What is the number of
ways in which one
can
post 5
letters in
7
letter
boxes ?
I f
A and
B are
square
matrices
of
second order
such that lA I = 1, IBl = 3, then what
is
I3AB I equal to ?
19
(a)
3
(b)
-9
c)
27
c)
5
7
(d)
None of the above
d) 2520
or the next three 03) items that follow :
What is the.
number
of ways
that
a cricket Consider the function f(x)
=
x -
1
.
x+1
team of
players can be made
out of
15
players?
a) 364
(b)
1001
c)
1365
d) 32760
23.
What
is
f(x)
+
1
al
to
?
+x equ
.
f(x) -1
(a)
0
(b)
1
c)
2x
(d)
4x
B-OEBB-N-NBS (
6 -A
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16.
f : N ~ N lit, ~ N ~ ilfqd l1t.f-m
(
1
+ i)4n +5
v:Tm
I;
f(x) -1
(a) 364
b)
1001
(c) 1365
(d) 32760
B-OEBB-N-NBS
(a) 0
b)
1
(c)
2x
(d) 4x
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2
4 . Wh
at
is f 2
x) eq
ual t
o ?
a)
b)
c)
d
)
f x)
+ 1
f x)
+
3
f x)
+
1
3f
x)
+ 1
3
f x
) + 1
f x)
+
3
f x)+
3
3
f x)+
1
25. W
hat
is f
f x))
equa
l to
?
a
)
x
b)
X
c)
1
X
d)
Non
e ofthe
ab
ove
or
th e
ne x
t fiv
e05
)
item
s t
hat
fo
llow
:
1
)15
C o
nsid
er
th
e
exp
ansio
n x
2
+;-
26 .
What
is
the
independent
term in the
giv en
e
xpan
sion
?
27.
a)
21 03
b)
30
03
c)
4
50 3
d)
N on
e ofth
e ab
ove
Wha
t
is the
ratio
o
f co e
fficie
nt o
f x
1
5
to
t
he
te
rm
ind
epen
dent
of
x
in
the
giv e
n
expansion
a)
1
b
)
1/2
c)
2 /3
d)
3/4
28.
C on
sid er
the
follow
ingstat
emen
ts
:
1
2.
Ther
e a
re 1
5
terms
n the
give
n
e
xpan
sio n.
T
he c
oef fi c
ie nt
of x
12
is e
qual
to t
hat o
f
x3.
W hic
h of the
ab
ove stat
emen
ts
is/a
re co
rrect
?
{a)
1
only
{b)
2 o
nly
c)
B
oth1
and
2
d)
Neit
her1
nor 2
29
. C on
sid er
thef
ollow
ing
state
ment
s :
1
2.
T h
e te r
m co
nta in
ing x
2
do e
s not
ex is t
i
n
th
e
give
n exp
ansi
on.
T he
sum
of t
he c
oe ffi
cients
of
all th
e
t
. t
h .
. .
2
1
5
erm
s m
eg
iVen
ex pan
siOn
IS
.
W
hic h
o f th
e abo
ve state
men
ts
is/are
corr
ect ?
a )
1
only
{b)
2 on ly
{c)
B
oth1
an
d
2
{d)
Ne
ither
1 no
r 2
30. W
hat
is the
sum
ofth
e c
oe ffi c
ie nts
ofthe
m
idd
le
t
erms
in the
giv en
e xp
ansio
n ?
a)
C
15, 9
)
{
b)
C 16
, 9 )
c
)
C
16, 8
)
{d
)
N o
ne o
f
the
above
8-0E
BB-N
-NBS
8
-A
3
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24. R2x) a m < ~ \ ~
?
(a)
f x) + 1
f x) + 3
b)
f x) + 1
3 f x) + 1
I
3f x)+1
c)
f x) + 3
d)
f x) + 3
3f x)+1
25.
f{f{x))
~ a m < ~ \ ~
?
(a)
X
(b)
X
c)
1
d)
i i ~ ~ - : r @
m : m : l ~ r i k l ( o s ) J J R i i i l ~ f F n :
(x
2
+ ~ r i i > ~ R h ~ 1
26.
~ '11 ;
ii
tfG
1fliT ~
?
a)
2103
b)
3003
c)
4503
d)
i i ~ ~ : r @
a) 1
b) 1/2
c) 2/3
(d) 3/4
B-OEBB-N-NBS
28.
f . i ' l ~ f u l d R fcRm: :
1.
~
'11 ;
if
15
tfG
~ I
2. x
12
Iii
x
3
il il a m < ~ \ ~
I
ii
cm mffi l M?
a)
~
(b) ~ 2
c) 1afu: ~
d) ~ o) 1 f t \ ~ ~ 2
29. f . i ' l ~ f u l d R
fcRm:
~ :
9 -A )
1. ~
'11 ;
ii x
2
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31. What is
.}1
+ sin 29 equal to ?
(a)
cos
9 - sin 9
(b)
cos
9
+
sin 9
c)
2 cos 9 + sin 9
(d) cos 9 + 2
sin
9
'
32. A lamp post stands on a horizontal plane.
From a point situated
at
a distance 150 m
from
its
foot, the angle of elevation of the top
Wh
.
1 3 14.
al
?
35. at1s
SID
- SID
-
equ to.
5 5
' '
(a) tr/2
(b) tr/3
(c) tr/4
' (d) tr/6
is 30 .
What
is
the
height of the lamp post ? 36.
What 18
cos
7x - oos 3x al to ?
. . . equ .
SID
7x
2siD5x
+siD3x
a)
50m
b)
(c)
d)
lOOm
33.
If
cot A = 2 and cot B = 3, then
what
is
the
value of
A
+
B
?
34.
(a) tr/6
b) 1I
(c) tr/2
(d) tr/4
What is
sin
2
66. .
2
(a) sin 47
b) cos
47
c)
2 sin 47
(d) 2 cos 47
D
D
equal to?
2 231
S I D
-
2
37.
(a)
tanx
(b)
cotx
(c)
tan2x
d)
cot2x
In a triangle ABC, c =2, A =45 , a =2 J2
then what is
C
equal to
?
a)
o
(b) 15
(c)
45
(d) None of the above
38. In a triangle ABC, sin A cos B
=
cos
C, then
what
is B equal to. ?
(a) "
(b) tr/3
(c)
tr/2
(d) tr/4
B OEBB N NBS
( 10-A)
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39
If
-
m ( -x_+ ....::y...:..) a + b th h t . tan x
=-- enw a ts
sin
x -
y)
a - b tan y
equal to
a)
b
a
b)
a
b
c)
ab
d) 1
40
I f
sin A sin 60 - A) sin 60 + A) = k sin 3A,
then what is k equal to ?
a)
1/4
b) 1/2
c)
42 Which one of the following
is
one of the
solutions of the equation tan 28 tan 8 = 1 ?
a) n/12
b)
n 6
c)
n/4
d) n/3
For
the
n xt three
03)
items
that
follow
Given
that
16
sin
5
x
=
p sin 5x
+
q sin 3x
+
r
sin
x.
43 Wh:at is the value ofp
a) 1
b)
2
c) 1
d) 2
B OEBB N NBS
2
A
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sin x+y) =a+ b
39. '
sin
(x - y) a - b
GIU'R ~
a)
b
a
b)
a
b
c) ab
d)
1
tan x
tany
42.
f . l ~ < t W l f < S i d it it
m
1.% fl Jil'fiiiT ~
~ TJ
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orth enext tw
o 02 )items thot
follow :
50. T he
two circles tou
ch each oth r if
Let Sn de
note
th
e sumo
f first n term
s
o
f an AP and
3Sn =
S2n
4 6
What i s S
3
n :
Sn equal
to
?
a) 4
:
1
b) 6
: 1
c)
8:
1
d
) 10: 1
47. W
hat is
S
an : S
2
n eq
ual
to?
a) 2
:. 1
b) :
1
c) 4: 1
d) 5
:1
48.
What is thele
ngth of the
latus rec tum of the
ellipse2 5
x
2
+
16fl
= 40
0 ?
51.
a) 25/
2
52.
b) 25
/4
c)
16/5
d) 32/5
o
r the
n
ext two
02)
i
tems thot
fo
llow
:
a
)
b)
c
)
d)
c= +b
2
1
1 1
=
c a2 b2
1
1
c=
-
a
2 b2
1
c = -- -
::---,-
a2 +
b2
A 3, 4)
and B 5, - 2)
are two points
and P is a
p
oint such tha
t PA = PB.
If
the
area o
f
tr
iangle PAB is
10 square u n i
t,
wha
t are the
coordinates ofP?
a) 1,0)
only
b) 7,2
)only
c)
1, 0) o r
7,
2)
d)
Neither
1
, 0)
no r 7, 2)
W h a t is
the product of
the perpend
iculars
drawn from the
points
~ a - b
2
, 0) up
on
the
l ine bx
cos a
+
ay sin
a
= ab
?
a) a
2
d
) a+ b
Co
nsider the ci rc
les +
f
+2
ax + c = 0 a
nd
x
2
+ y
2
+ 2by + c
= 0
53
. Which one
of
t
he follo
wing is corre
ct in
49. W
hat is the dis
tance between
the centres o
f
the
two circ
les ?
a)
+ b
2
b)
a
2
+ b
2
c)
a+ b
d)
2 a+ b)
8-0EB
B-N-NBS
4 A)
f
h
t
.
x 1
y 2
a
nd
respec
t o t e equ a
Ions =
2
3
2x+3y=5
?
a)
They rep
resent two li n
es
whi ch
are
parallel .
b)
They rep
resent
tw
o lines whic
h
are
perp
endicular.
c) The
y represent t
wo lines whi
ch are
neither paral
lel nor perpend
icular.
d)
The f i rs t e qu
ation does no t
represent a
line.
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m ; m ; ~ m ;;7 02)
J RiiiT
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F
or
th
e ne
xt three
03
) items t
hatfo
llow :
o
r the
next
t
wo
02) items
that
follow
:
Conside
r a sp h
ere p a
ssin g t
hrough
the orig
in a
nd . The l
ine join
ing
the p o ints
2,
1
, 3
) an d
4, _ 2,
5)
the poin
ts 2,
1 - 1), 1
,
5
- 4),
- 2, 4 -
6).
54
.
What is
the rad iu s
of
the
s
p h ere ?
a)
Ji2
b)
/14
c) 12
d) 14
5
5 .
What is
the
centr
e of
the sp h e re
?
56.
a
) -1
, 2 - 3)
b)
1,- 2, 3 )
c)
1, 2
- 3)
. d
)
-1 -
2 - 3)
C onsi
der
the followin
g stat
ements
:
1.
T h e sp
here p a
sse s t
hrough
t
he
p o
int
0
, 4, 0).
2. T h
e p o in
t 1,
1, 1)
is at a dis
ta nce o
f
5u
nit from
thecen
t re of
the sph er
e.
W
hichof
th
e
abov
e s
tatemen
ts
i
s /are co
rrect ?
a)
1o
nly
b)
2on
ly
c)
B
oth 1
and 2
d)
N
ei ther
1 no
r
2
cuts
th
e
plan
e 2x
+ y - z
=
3
.
57.
W h ere
do es
the l in e c
u t
the
p
lan e ?
a
) 0, - 4,
- 1)
b)
0,-
4 ,1)
c)
1, 4, 0)
d)
0, 4,
1)
58. W
hat is
the ratio
in
w hi
ch
the plan e d
ivides
the
line?
a)
1 :
1
b)
2:3
c)
3 :4
d)
N
oneof
the abo v
e
or
the
ne
xt two 02
)
ite
ms
that f
ollow
Co
nsider
th
e
plan
e p ass i
ng t h ro
u g h
the points
A 2, 2,
1), B
3, 4 2)
an d C
7, 0, 6
).
59.
. 60 .
W h
ich on e
of
the followin
g point
s li es o
n
the
plane?
a)
1,
0, 0)
b)
1, 0,
1)
c)
0, 0
, 1)
d)
N o
ne of th
e above
W
hat are
th
e
dire
ction r
atio s of
t
he
nor
m al
to
th
e
pl
ane?
a)
b)
< 0,
1, 0
>
c)
< 1, 0 -
1
>
d
)
N
one of
the ab o
ve
8-0
EBB-N-
NBS
6A)
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1@ f.r.s. 3fu:
f.r.s.m
c2. 1 ,- 1), o.
5 ,-
4),
~ ~ ~ ~ ~ 1 \ f < r . m : ~ I
54.
~ c l i ' t ~ < ~ m ~ ?
a)
- 12
b)
J14
c)
12
d)
14
55.
~ ' P T ~ < i m ~ ?
a)
-1, 2 ,- 3)
b)
1,- 2, 3)
c)
1, 2 ,- 3)
d) -
1 , - 2 , -
3)
56.
f.l' lfeif&d
'11: f
7/25/2019 NDA Paper Mathematics 2014
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For the
n
ex t
thr
ee 03
) it e
ms th
at
follo
w
{
x2
5
Con
sider
the f
uncti
on f(
x)
=
~
X 13
,
x:>3
X>3
61
Wha
t
i
s
lim
fx)
equ
al to
?
x->
3
(a)
2
(
b)
4
c
)
5
(d)
1
3
62
C on
side
r th
e follow
ing
sta
teme
nt s:
1. T
he fu
nctio
n is
disco
ntinu
ous
at x =
3.
For the
n
ext
th
ree 0
3) it
ems th
at
follo
w
The
line
2y
=
3
x +
12
cuts the
para
bola
4y
=3x
2
64
W
here
does
th
e
l
ine cu
t
thep ar
abola
?
(a)
A
t - 2, 3
)only
b)
A t
4, 1
2) on
ly
c) A
tboth
-
2, 3) a
nd (4
, 12)
(d
)
Neit
her
a t 2,
3) no
r a
t (4, 12
)
65
W
hat
is
the
are
a
e
nclo
sed b
y th
e para
bola
and
the
li ne
?
(a)
2
7 s
quare
uni
t
b)
36
s
quare
un
it
c
)
48
squa
re
unit
(d)
54
squa
re
unit
2
. The
func
tion
is
no
t
d
iffer
entiab
le
at 6
6
Wh
at is th
eare
aenc
losed
b y
the pa
rabo
la, th
e
x
=
0.
li n
e an
dthe
yax
is in
the
fi
rst
qu a
dran
t?
W h
ich
ofth
e above
state
men
ts is
/are c
orrec
t ?
(a)
7 sq
uare
un
it
(a
)
1only
(b)
14
sq
uare
un it
b
) 2
only
c
) 20
sq
uare
un it
(d
)
21
sq
uare
un
it
c)
Bot
h a
nd 2
(d
)
Neit
her 1
nor
2
67
Cons
ider
the
fu
nctio
n
63
W
hat is
the
d
iffere
ntial
coef
ficien
t of
f x)
at
X 1
2?
a) 5
/2
b)
5
c)
1/5
(d)
1/10
B
OEB
B NN
BS
18
-A )
l
an
kx
f x)
=
x
3x
+2k
2
,
x
7/25/2019 NDA Paper Mathematics 2014
19/40
{
x2 -5
q
)ff.f fl:x) =
~ +13,
x,;3
x>
3
61
lim fl
:x) 3
(a) 2
(b) 4
(c) 5
(d) 13
62.
~ < t B f R
:
1.
q)ff.f X =
3
R 3Rimr I
2. q) ff.f X=
0 R 3jq
7/25/2019 NDA Paper Mathematics 2014
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68.
C o
nside
r th
e fo llo
wing
state
men
ts :
1
.
The
fu nc
ti on
f x)
= [x]
, w h e
re
[,] is t
he
g
reate
stin
teger
fun
ction
defi n
ed o
n
,
is con
ti nuo
us a
t a ll
po i
n ts
exc
ept at
x=O
.
2
.
T he f
uncti
onf
x) =
sin Ix
I i s
conti
nuou
s
fora
llxE
R.
W
hic h of
t
he ab ov
e sta
teme
ntsi
s /a re c
orrec
t?
a)
1o
nly
b)
2
only
c)
B
oth 1
and
2
d)
Neith
er 1
n
or 2
or tM
n x
t
two
02
)
i
tem
that
fo
llow
:
Co
nsid
er t
he curv
e x =a
c
os
e + e
si
n e
and
y
= a s
in
a - e
co
s 0).
69.
Wh
ati s
dy
equa
l t
o
?
dx
a)
tan
a
b )
cote
c
)
sin 2
0
{
d)
cos 2
0
d2
70
.
W
hat is
--f
eq u a
l
to
?
d
x
a) se c
2
e
b )
-cos
ec
2
e
c)
sec
3
e
e
d)
N
oneof
th
e ab ov
e
71.
Wh
at is
th
e area
of
the
p
arab
o la
l =
4bx
boun
ded b
y
its lat
us rect
um?
a)
2b
2
/3
squ
are u
nit
b)
4b
2
/3 squa
re
unit
c)
b
2
squ
are
un
it
d)
8b
2
/3
sq
uare
unit
72
. 1f
y=
x ln
x +
xex,
then
w
hat is
the va
lueof
dy
atx
= 1
?
dx
7
3.
74.
7
5.
a)
1+
e
b)
1
e
c)
1+ 2e
d
)
N
one o
f th
e abo
ve
l
ogs
l +x)
W
hat
is
lim
eq u
a l
to ?
x
-- 0
a)
b)
c)
d)
1
l
og s e
log
5
e
5
W
hat
is
lim
X-->0
a)
log 5
e
b)
log
s e
c)
5
d
)
1
W
ha ti
s lim
a) 5
b)
2
c)
1
d) 0
5x
1
eq
ua
l o?
X
-
.;1:.. .c..::::2... _ 3_ -=-
._.
. c . . : : : : n ~
a
?
eq
u to
.
12
+2
2+3
2
+
... n2
B-OE
BB-N
-NBS
2
0- A)
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6
8.
f
.tq
f
ft
fl
io
~
14
tI
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9
1
9
If
I a
I
= 2
, I
b I
=
5
and
I a
x b
I =
8,
t
hen
W
h
ich
on
e
o
the
fo l
low
ing
is
th
e
uni
tve
cto
r
wh
at
is a
b
e
qu
al t
o ?
a
)
6
b)
7
c
)
8
(
d)
9
-+
92
I
f I a
+
b
I =
I
a
-
b
I th
en
w h
ic h
on
e
o
the
foll
ow
ing
i s
cor
rec
t ?
(a)
b
)
I
a I =
I b
-+
a
is
p a
ra l
lel
to
b .
)
~
di
u
c
a
ISpe
rpe
n
c
a
rto
b .
d)
a
is
a
u
ni t
ve
cto
r.
93 .
Wh
at
is t
he
are
a of t
he
tri a
ng
le
OA
B
w
he
re
0
A
A
A
is
the
ori
gin
, O
A
= 3
i
- j
+
k
a
nd
-
.
A
A
A
O
B
= 2
i
j -
3 k
?
(a
)
b)
c)
5
../6
sq
ua
re
u
ni
t
5.
./6
t
-
-
sq
ua
reu
m
2 .
J
sq
ua r
e
u
ni t
d)
J3
ri
squ
are
un
it
7
li
pe
rpe
ndi
cu
la r
to
bot
h a
= -
i
+
j +
k
an
d
4
/1
A
A
b
=
i
-j
+
k
?
A
A
a)
i j
J
2
A
b)
k
A
A
c)
j
k
J
2
A
A
d)
i -
j
J2
9
5 .
W
ha
t
is
th
e i
n te
rio
r
acu
te
an
gle
o
f
t
he
p
ara
ll e
lo g
ra m
wh
ose
si
de
s
are
re
pre
sen
ted
1
A
1
A
A
by
th
e ve
cto
rs
J2
i +
J2
j +
k
and
}
A
A
A
- i
-
-
j
k
?
J
2
J2
a)
60
c
)
30
d)
1
5
96
Fo
r wh
at
v
alu
e
o
Aar
e th
e v
ect
ors
A
A
A
Ai +
1+
A
j
+
1 +
2
A k
an
d
A
A
A
(
1-
A)
i
+
A
+ 2
k
pe
rpe
nd
icu
la r
?
(a
)
-
1/
3
(b
).
1/
3
c)
2/3
(d
) 1
B-
OE
BB
-N -
NB
S
2
6 -
A )
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~
- -
\
II.
A
-
\
\
11
a
=
-
i
j
k
3f
u:
b
=
i
-
j
k
G
A
T
R ~
t
?
a
)
6
A
A
b
)
7
a
)
i
j
J
c)
8
(
b )
A
k
(
d)
9
c)
A
A
j
k
J
A
A
d
)
i
-
j
J
a
)
Ia
I=
I
b
1
~
~
t
l
b
)
a
,
b
R
~
t
I
c
)
a
,
b
(d)
a
l f;
l< l
i
t
I
(a
)
6
0
(b
)
4
5
c
)
3
0
9
3.
O
AB
fiT
f
ci;
o;n
t,
0
~
-
~
O
A
3
i
-
j
+
k
(l l
fl
---
- >
O
= 2
i
+
j
3
k
t
?
A
A
A
A
i
(1
A
.
j
(1
2
1..
)
k 3f
u:
(
a)
A
,
1
-A
.)
i
+
A
.j
+
2
k '
ff i
'R
7/25/2019 NDA Paper Mathematics 2014
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F
or t
hene
xt
fou
r 04
)
ite
ms tha
t f
o llo
w :
101
.
- - 1
- ~ -
- 1
- - -
- -
4
a +
b
+
c
=0
su
ch
tha
t
I
a I
=
3,
I
b I=
5 a
nd
rr
/2
W
ha
ti s
0
--
I
c I =
7.
a
)
2ab
b
)
27t
ab
--
97
.
--
hat i
st
hean
gle
bet
w ee
n a
a
nd
b
?
c
)
t
2
ab
a
)
n
6
b)
rr./
4
c)
rr./
3
d
)
rr./2
--+
--+
--+-
-+
--
--
98.
hat
is
a
b
+
b
c
+
c
. a
to ?
a)
-
83
b)
-8
3/2
c)
7
5
d
)
-
75/
2
99.
Whatis co sine of
the
angle betw een
--
c
?
a)
1
1/1
2
b)
13
/14
c)
-1
1/1
2
d)
-13
/14
--+
--
100. Whatis
I
a
+
b
I
equal
to
?
a)
7
b)
8
c
)
10
d)
11
8
-0E
BB
-N -N
BS
d)
t
ab
F
or th
e nex
t
two
02
)
ite
ms
tha
t
f
o ll o
w:
A cy
li nd
er i s
in s
cri b
ed in
a s
phe
re o
f ra
diu
sr.
equ
al
1
02.
hat
is
th
e h
eig
ht o
f t
he
cy
lin d
er
of
m
ax i
m um
vo
lum
e?
2r
J3
a)
-
-
b
)
r
b and
c)
d)
J3
2r
J
r
103
. W
hat
i
s th
e
radi
us
of
th
e
cyli
nder
o
f
m a
xim
um
volu
me
?
a)
2r
J
3
b)
f
ir
J
3
c)
r
d
)
J
r
2
8 -
A
)
J
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-
-
-
-
-
-
a +
b +
c
= o
~
i
f
7/25/2019 NDA Paper Mathematics 2014
30/40
F
orthe n x
t t
wo 0
2)
i
tems
that
follow
:
106
. Supp
ose A
an
d
B are
tw oe
vents.
vent B ha
s
occ
urred a
nd it is
known
that P(
B) < 1.
What
C
onsider
the
fun
ction f
(x)
=
se
c
4
x +4
withf(O) =
0
i
s P(A B
e) equa
l
to?.
J
and
f'(O) =
0.
10
4. W
hat
is
f'(x
) equal
to
?
a)
tan
x
ta n
3
x
+
4x
3
;Jc
(b)
tanx
+
ta
n
3
x
+
4x
3
(c)
sec
3
x
tan
x+
+4x
3
(d)
t a n x
tan
3
x
3
+
4x
105.
What is :f(x)
equ
al to ?
(a)
l
nsecx
ta n
2
x
+ 2x
2
+
3
6
(b)
3
/nsecx
oot
2
x
2x 2
--
2
6
(c)
4ln
secx
sec
2
x
.
2
2
--
X
3
6
(d)
l
n secx
+
t
an
4
x
12
+ 2x
2
B OEB
B N NB
S
a )
(b)
(c)
(d
)
P
A)-P
B)
P(B)
P
A)-P A
B)
P(B
)
P(A) +P
(Bc)
P(B )
N
one
o
f the
ab ov
e
F
orthe n x
t fo
ur
04)
ite
ms t
hat
fol low
:
C onsid
er even
ts A B
, C, D,
E
of
the
sam p
le spac
e
S =
ln : n i
s
an
in t
eger su
ch th at
10 s; n
s;
2
0} gi
ven
by
:
1
07.
30
-A )
A i sthe se
tof
al l eve
n num b
ers.
Bi
s the set of
all p
r ime nu
m bers .
c
=
151.
Dis
the set of
all in
tegers
s; 16.
E
s
the
se
t of
all d
ouble
digit n
u m b ers
ex
pressib
le
a
s a
powerof
2.
A,B
andDa
re
(a)
M u
tually
ex
clusive
even t
s
but
not
e
xhaustiv
e even
ts
(b)
E xh
aust ive
events
but not
m utu
al ly
exclusi
ve even
ts
(c)
M
utual ly
e
xclusiv
e
and
exhaus
t ive
even
ts
(
d)
lem
entary
ev
ents
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3
f T
3
IR
m
o
2 >
W
til
iT
f> f
Rrt
f
(x )
=se
c
4
x
+4
R
,
f(
O)
=0
alR
f
'(O)
= 0
t,
fcr
.m
I
10
4.
f'(x
)
i ?
ta
n
3
x
a
)
ta
n x-
+4
x
3
(b
)
tan
x
+
tan
3
x +
4x
3
c)
s
t
an
x+
+4
x
3
ta
n
3
x
(d
)
- t
an
x
+ 4
x
3
10
5.
f x
)
i
?
(a
)
2
/n s
ec
x
tan
2
x
2
2
+
+ X
3
6
(
b)
3/
ns
ecx
r
ot
2
x
2
2
-
-
X
2
6
.
c)
4/n
se
cx
se
c
2
x
2
2
_ : : :
+
+ X
3
6
d
)
n
se
c x
+
ta
n4
x +
2
x2
1
2
10
6.
l iH
f. ;
A
alR
B
l
~
I
f
Rr
t
S =
{
n
:
n
~
~ ~
~
i
f
.
i
;
10
s n
s
20}