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Mathematics - solved paper of 2021 (I) Pathfinder NDA/NA National Defence Academy & Naval Academy Entrance Examination, ISBN: 9789325797963$8.09
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Mathematics - solved paper of 2021 (I) Pathfinder NDA/NA National Defence Academy & Naval Academy Entrance Examination, ISBN: 9789325797963
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Course
Mathematics
Institution
Book
Pathfinder NDA/NA National Defence Academy & Naval Academy Entrance Examination
Images are taken from book itself "Pathfinder NDA/NA National Defence Academy & Naval Academy Entrance Examination", ISBN: 7963
Its a pdf of solved paper of 2021 set (I) Nda exam _ Mathematics.
PAPERI: Mathematics
1. The smallest positive integer n for 3. If A is the value of the determinant and we know that
which Co+ C +" Cat...+ C, =2
"C,+ "C2 + .. + "C, =2" - C
= 1
=2" -1
where i =V-1, is then what is the value of the Hence, option (b) is correct.
(a) 2 (b) 4 ()6 (d) 8 following determinant? 5. Ifa + b+c =4 and
pa b ab + bc +ca = 0, then what is t
(a7 1
1, wherei=-1 pag ba qe value of the following
determinant?
pa bqe3|
(P #0 or 1,q 0 or 1) a b
(a) pA
(c) (p+9) A
(b) qA bca
(d) pqA c a b
a b C (a) 32 (b) - 64 (c) - 128 (
(d) Given, a2 b2 C2=A
(b) Let
(1-1-2 =1=1
1+1 ab c
pay biqc
(-iy = (-1 pa2 b2 gC2
A=b ab C a
n= 4 pag b3 qC
a + b +C b C
n = 2 =a+b +C C a
Hence, option (a) is correct. p: a2 b2 C2 | a + b +C a b
ag ba C3|
2. The value of x, satisfying the (by C C +C2
equation logcos x Sinx = 1, where pqA
Hence, option (d) is correct.
1b c
=
(a +b + c) 1 C a
0<x
4. If Co. C, C2,.., C, are the 1 ab
coefticients in the expansion of
12 ) ( (1+ x)', then what is the value of
(To take common a + b+Cfro
0b-C C-a
(c)log9cosx sinx =
1, where 0< x<
2 C +C2 +C3 t .+Cn? (a+b + C) 0 c-a a-b
b
( c o s X) = sin x c o S X = Sinx
(a) 2n
(c)2n-1
(b) 2" 1 1 a
tan x = 1 tanx = tan t / 4 (d)2-2 (byR R, -
Rz, Rg R
(b):(14+ X = ®Co+ C,x + "C2 ( a + b + c) [(b - c)(a - b) - C
x = T/4
Hence, option (c) is correct.
x + + 'Cx ( a + b + c)(ab - b - ca + b
- c2-a2+2
, NDA/NA Solved Paper 2021 () 31
=-
(a + b + c)(a2 + b +c2- ab - bc ca) to) +1.
(a+ b+c)[la log2n log3n log100
b+ c?-3(ab
-
+ bc
+ +ca)]
=log,2 log, 3 + log,4.+ log,100
= -
(4) [16- 0)= 64 -
= log,(2 3.4-5...100)
6. The number of integer values of k, for which the : n = 100!]
2sin x = 2k +1 has a solution, is
equation = lo9100 (100)
:log 1]
(a) zero (b) one
(C) two Hence, option (b) is correct.
(d) four
(c) Given, 10. Ifz =
1+i, wherei =
v-1, then what is the modulus of
2sinx =2k+ 1
z+
-
1s sinxs1 -2s2 sinxs 2
2 1s2sin x - 1 s 2 -1 (6) 2 (c) 3 (d) 4
(a) 1
3s 2ks 1 ( b ) z =1+ i, wherei = y-1
sks-1.5s ks 0:5
Integer values of k = - 1, 0
Hence, option (c) is correct. =(1i)+ 2 |1+i+1-il=|2] =2
7. Ifa, a2, ag, .., ag are in GP, then what is the value of the
Hence, option (b) is correct.
following determinant?
11. If A and B are two matrices such that AB is
of order n X n,
lna Ina2 lna | then which one of the following is correct?
Ina, Ina Ina (a) A andB should be square matrices of same
order.
Ina ln ag lna Either A B should be a square matrix.
(b) or
(c) Both A and B should be of same order
(a) 0 (b) 1
(d) Orders of A and B need not be the same.
c) 2 (d) 4
term and ratio of GP are a and r respectively. (d) Given that, order of matrix AB = n xn
( a ) Let first common
f we take Anxp and Bp xn then AB will be of order n xn.
loga, loga2 loga loga logar logar
So, orders of A and B need not be the same, is correct.
loga logas logas=|log ar logar logar Hence, option (d) is correct.
loga7 log ag logag log ar log ar logar
12. How many matrices of different orders are possible with
loga +logr loga+2 logr
loga elements comprising all prime numbers less than 30?
=|loga 3 logr + loga+ 4logr loga+ 5 logr (d) 6
(a) 2 (b) 3 (C)4
loga +6log r loga+ 7logr loga+8 logr (c):Prime numbers less than 30 {2,3, 5, 7, 11, 13, 17, 19,
:log mn= log m + log n]
23,29)
logr logr
loga Number of elements = 10
=| loga 3 logr logr logr
+ . Possible order of matrices with 10 elements
loga + 6logr log r logr 10x 1,1x 102 x 5,5 x2
(byC2C2-C, and CsCa-Ca :. Number of matrices of different order= 4
= 0
: C2Cal Hence, option (c) is correct.
equationx^ +2x + k =0 are
8. Ifthe roots of the quadratic 13. Let,A
real, then
(b) ks 0 where p, g, r and s are any four different prime numbers
(a) k< 0 (d) ks 1 less than 20. What is the maximum value of the
C)k<1
determinant?
( d ) Given quadraticequation, ..0 (a) 215 (b) 311
x + 2x + k= 0 (c) 317 (d) 323
Since, roots are real prime numbers less than 20
D2 0 b 4ac 2-
0 (o) A-
ks1 = {2,3,5,7., 11, 13, 17, 19)A = ps q
(2)-4(1)(K)2 0 4 24k
is correct.
For maximum values of A, p and s must be maximum and rand g
Hence, option (d) must be minimum.
is the value of the following?
9. Ifn = 100!, then what Then, p 17, S = 19, r =2.9 = 3
1 . A =17 x 19 -2 x3
log3
log4 n log100 323 6 317
log2 (b) 1
(a) 0 Hence, option (c) is correct.
(d) 3
(C)2
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