NDA Solved Paper 2018 - I | NDA (National Defence Academy) Past Year Papers PDF Download

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1. If n N ? , then
121 25 190 4
n n n n
- + - - ( ) is divisible
by which one of the following?
(a) 1904 (b) 2000 (c) 2002 (d) 2006
Ê
(b) We have,
121 25 1900 4
n n n n
- + - - ( )
On puttingn = 1 , we get
( ) ( ) ( ) ( ) 121 25 1900 4
1 1 1 1
- + - -
= - + + 121 25 1900 4
= 2000
Which is divisible by 2000.
2. If n = ( )! 2017 , then what is
1 1 1
2 3 4
log log log n n n
+ +
+ … +
1
2017
log n
equal to ?
(a) 0 (b) 1 (c)
n
2
(d)n
Ê
(b) We have,
1 1 1 1
2 3 4 2017
log log log
....
log n n n n
+ + + +
= + + + + log log log .... log
n n n n
2 3 4 2017
Q log
log
b
a
a
b
=
?
?
?
?
?
?
1
= · · log ( ...... )
n
2 3 4 2017
[Qlog log log a b ab + = ]
= · · · log ( ...... )
n
1 2 3 4 2017
= log ( )!
n
2017
[Qn n n n ( )( ) ! - - · · = 1 2 3 2 1 K ]
= log ( )!
( )! 2017
2017 [ !] Qn = 2017
= 1 [ log ] Q
a
a = 1
3. In the expansion of ( ) 1
43
+ x , if the
coefficients of ( ) 2 1 r + th and ( ) r + 2
th terms are equal, then what is the
values of r r ( ) ? 1 ?
(a) 5 (b) 14 (c) 21 (d) 22
Ê
(b) We have, ( ) 1
43
+ x
? General term,T C x
r r
r
+
=
1
43
Now,T C x
r r
r
2 1
43
2
2
+
=
and T C x
r r
r
+ +
+
=
2
43
1
1
Now, according to the question
Coefficients of ( ) 2 1 r + th and ( ) r + 2 th
terms are equal
?
43
2
43
1
C C
r r
=
+
? 2 1 43 r r + + =
[ ] Q if
n
x
n
y
C C x y n = ? + =
? 3 42 r =
? r = 14
4. What is the principal argument of
( ) - - 1 i , where i = -1?
(a)
p
4
(b) -
p
4
(c) -
3
4
p
(d)
3
4
p
Ê
(c) Let z i - - 1
Now, tan
| |
| |
a =
b
a
=
-
-
| |
| |
1
1
[Qa b = - = - 1 1 , ]
? a
p
= =
-
tan ( )
1
1
4
Sincea b , both are negative,
? arg( ) z = - a p
= -
p
p
4
=
-3
4
p
5. Let a and ß be real number and z be a
complex number. If z z
2
0 + + = a ß
has two distinct non-real roots with
Re ( ) z = 1, then it is necessary that
(a) ß ? - ( , ) 1 0 (b)| | ß = 1
(c) ß ? 8 ( , ) 1 (d) ß ?( , ) 0 1
Ê
(c) Let z x iy = +
Now, we have
z z
2
0 + + = a ß
?( ) ( ) x iy x iy + + + + =
2
0 a ß
?x y ixy x i y
2 2
2 0 - + + + + = a a ß
[Qi
2
1 = - ]
?( ) ( ) x y x xy y i
2 2
2 0 - + + + + = a ß a
On comparing,
x y x
2 2
0 - + + = a ß and 2 0 xy y + = a
? x y x
2 2
0 - + + = a ß and ( ) 2 0 x y + = a
? x y x
2 2
0 - + + = a ß and 2 0 x + = a
[Qy ? 0]
? x y x
2 2
0 - + + = a ß and a = - 2x
? x y x
2 2
0 - + + = a ß and a = - 2
[QRe( ) z x = = 1 ]
? 1 2 0
2
- - + = y ß [Qx = = - 1 2 , a ]
? ß = + y
2
1
? ß ? 8 ( , ) 1
[Qy
2
0 = ? y
2
1 1 + = ]
6. Let A and B be subsets of X and
C A B A B = n ' ? ' n ( ) ( ), where A'
and B' are complements of A and B
respectively in X . What isC equal to ?
(a) ( ) ( ) A B A B ? ' - n '
(b) ( ) ( ) A B A B ' ? - ' n
(c) ( ) ( ) A B A B ? - n
(d) ( ) ( ) A B A B ' ? ' - ' n '
PAPER : I Mathematics