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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
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1. What is the eligibility criteria for appearing in the NDA exam?
Ans. To appear in the NDA exam, candidates must be unmarried male candidates between the ages of 16.5 to 19.5 years and must have completed their 10+2 education from a recognized board.
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Ans. The selection process for the NDA exam consists of a written exam conducted by the UPSC followed by an interview conducted by the Services Selection Board (SSB).
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Ans. The NDA written exam includes subjects such as Mathematics and General Ability Test which covers topics like English, General Knowledge, Physics, Chemistry, Biology, History, Geography, etc.
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Ans. To prepare for the NDA exam effectively, candidates should create a study schedule, practice previous year question papers, take mock tests, and focus on improving their time management skills.
5. What are the career opportunities available after clearing the NDA exam?
Ans. After clearing the NDA exam, candidates have the opportunity to join the Indian Army, Indian Navy, or Indian Air Force as officers and pursue a rewarding career in the defense services.
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