NDA Mathematics Solved Paper 2024 | NDA (National Defence Academy) Past Year Papers PDF Download

 Page 1


Time Allowed : 2 hrs 30 min T otal Marks : 300
Instructions
1. This Test Booklet contains 120 items (questions). Each item is printed in English. Each item comprises four responses 
(answers). You will select the response which you want to mark on the Answer Sheet. In case you feel that there is more than 
one correct response, mark the response which you consider the best. In any case, choose ONLY ONE response for each item.
2. You have to mark all your responses ONLY on the separate Answer Sheet provided. See directions in the Answer Sheet.
3. All items carry equal marks.
4. Before you proceed to mark in the Answer Sheet the response to the various items in the Test Booklet, you have to fill in some 
particulars in the Answer Sheet as per instructions.
5. Penalty for wrong answers :
THERE WILL BE PENALTY FOR WRONG ANSWERS MARKED BY A CANDIDATE IN THE OBJECTIVE TYPE 
QUESTION PAPERS.
(i). There are four alternatives for the answer to every question. For each question for which a wrong answer has been given 
by the candidate, one-third of the marks assigned to that question will be deducted as penalty.
(ii). If a candidate gives more than one answer, it will be treated as a wrong answer even if one of the given answers happens 
to be correct and there will be same penalty as above to that question.
(iii). If a question is left blank, i.e., no answer is given by the candidate, there will be no penalty for that question.
NDA / NA
 National Defence Academy / 
Naval Academy I
QUESTION PAPER
2024
MATHEMATICS
1. Let A and B matrices of order 3 × 3. If |A| = 
1
22
 
and |B| = 
1
729
, then what is the value of
 
|2B[adj(3A)]|?
(a) 27 (b) 
27
22
 
(c) 
27
2
 (d) 1
2. If z is any complex number and iz
3
 + z
2
 – z + i 
= 0, where i = -1 , then what is the value of 
(|z| + 1)
2
?
(a) 1 (b) 4 (c) 81 (d) 121
3. What is the sum of all four digit numbers 
formed by using all digits 0, 1, 4, 5 without 
repetition of digits? 
(a) 44440 (b) 46460 (c) 46440 (d) 64440
4. If x , y and z are the cube roots of unity, then 
what is the value of xy + yz + zx?
(a) 0 (b) 1 (c) 2 (d) 3
5. A man has 7 relatives (4 women and 3 men). 
His wife also has 7 relatives (3 women and 4 
men). In how many ways can they invite 3 
women and 3 men so that 3 of them are man’s 
relatives and 3 of them are his wife’s relatives?
(a) 340 (b) 484 (c) 485 (d) 469
6. A triangle PQR is such that 3 points lie on the 
side PQ, 4 points on QR and 5 points on RP 
respectively. Triangles are constructed using 
these points as vertices. What is the number of 
triangles so formed?
(a) 205 (b) 206 (c) 215 (d) 220
7. If log
b
a = p, log
d
c = 2p and log
f
e = 3p, then 
what is () ace
p
1
 equal to?
(a) bd
2
f
3
 (b) bdf (c) b
3
d
2
f (d) b
2
d
2
f
2
8. If - 2
 
and 3
 
are roots of the equation a
0
 + 
a
1
x + a
2
x
2
 + a
3
x
3
 + x
4
 = 0 where a
0
, a
1
, a
2
, a
3
 
are integers, then which one of the following 
is correct?
(a) a
2
 = a
3
 = 0 (b) a
2
 = 0 and a
3
 = –5
(c) a
0
 = 6, a
3
 = 0 (d) a
1
 = 0 and a
2
 = 5
9. Let z
1
 and z
2
 be two complex numbers such 
that 
zz
zz
12
12


 
= 1, then what is Re
z
z
1
2






 
+ 1 
equal to?
(a) –1 (b) 0 (c) 1 (d) 5
10. If 26! = n8
k
, where k and n are positive integers, 
then what is the maximum value of k?
(a) 6 (b) 7 (c) 8 (d) 9
11. Consider the following statements in respect of 
two non-singular matrices A and B of the same 
order n:
1. adj(AB) = (adj A)(adj B)
2. adj(AB) = adj(BA)
3.  (AB)adj(AB) – |AB|I
n
 is a null matrix of 
order n
How many of the above statements are correct?
(a) None
(b) Only one statement
(c) Only two statements
(d) All three statements
Page 2


Time Allowed : 2 hrs 30 min T otal Marks : 300
Instructions
1. This Test Booklet contains 120 items (questions). Each item is printed in English. Each item comprises four responses 
(answers). You will select the response which you want to mark on the Answer Sheet. In case you feel that there is more than 
one correct response, mark the response which you consider the best. In any case, choose ONLY ONE response for each item.
2. You have to mark all your responses ONLY on the separate Answer Sheet provided. See directions in the Answer Sheet.
3. All items carry equal marks.
4. Before you proceed to mark in the Answer Sheet the response to the various items in the Test Booklet, you have to fill in some 
particulars in the Answer Sheet as per instructions.
5. Penalty for wrong answers :
THERE WILL BE PENALTY FOR WRONG ANSWERS MARKED BY A CANDIDATE IN THE OBJECTIVE TYPE 
QUESTION PAPERS.
(i). There are four alternatives for the answer to every question. For each question for which a wrong answer has been given 
by the candidate, one-third of the marks assigned to that question will be deducted as penalty.
(ii). If a candidate gives more than one answer, it will be treated as a wrong answer even if one of the given answers happens 
to be correct and there will be same penalty as above to that question.
(iii). If a question is left blank, i.e., no answer is given by the candidate, there will be no penalty for that question.
NDA / NA
 National Defence Academy / 
Naval Academy I
QUESTION PAPER
2024
MATHEMATICS
1. Let A and B matrices of order 3 × 3. If |A| = 
1
22
 
and |B| = 
1
729
, then what is the value of
 
|2B[adj(3A)]|?
(a) 27 (b) 
27
22
 
(c) 
27
2
 (d) 1
2. If z is any complex number and iz
3
 + z
2
 – z + i 
= 0, where i = -1 , then what is the value of 
(|z| + 1)
2
?
(a) 1 (b) 4 (c) 81 (d) 121
3. What is the sum of all four digit numbers 
formed by using all digits 0, 1, 4, 5 without 
repetition of digits? 
(a) 44440 (b) 46460 (c) 46440 (d) 64440
4. If x , y and z are the cube roots of unity, then 
what is the value of xy + yz + zx?
(a) 0 (b) 1 (c) 2 (d) 3
5. A man has 7 relatives (4 women and 3 men). 
His wife also has 7 relatives (3 women and 4 
men). In how many ways can they invite 3 
women and 3 men so that 3 of them are man’s 
relatives and 3 of them are his wife’s relatives?
(a) 340 (b) 484 (c) 485 (d) 469
6. A triangle PQR is such that 3 points lie on the 
side PQ, 4 points on QR and 5 points on RP 
respectively. Triangles are constructed using 
these points as vertices. What is the number of 
triangles so formed?
(a) 205 (b) 206 (c) 215 (d) 220
7. If log
b
a = p, log
d
c = 2p and log
f
e = 3p, then 
what is () ace
p
1
 equal to?
(a) bd
2
f
3
 (b) bdf (c) b
3
d
2
f (d) b
2
d
2
f
2
8. If - 2
 
and 3
 
are roots of the equation a
0
 + 
a
1
x + a
2
x
2
 + a
3
x
3
 + x
4
 = 0 where a
0
, a
1
, a
2
, a
3
 
are integers, then which one of the following 
is correct?
(a) a
2
 = a
3
 = 0 (b) a
2
 = 0 and a
3
 = –5
(c) a
0
 = 6, a
3
 = 0 (d) a
1
 = 0 and a
2
 = 5
9. Let z
1
 and z
2
 be two complex numbers such 
that 
zz
zz
12
12


 
= 1, then what is Re
z
z
1
2






 
+ 1 
equal to?
(a) –1 (b) 0 (c) 1 (d) 5
10. If 26! = n8
k
, where k and n are positive integers, 
then what is the maximum value of k?
(a) 6 (b) 7 (c) 8 (d) 9
11. Consider the following statements in respect of 
two non-singular matrices A and B of the same 
order n:
1. adj(AB) = (adj A)(adj B)
2. adj(AB) = adj(BA)
3.  (AB)adj(AB) – |AB|I
n
 is a null matrix of 
order n
How many of the above statements are correct?
(a) None
(b) Only one statement
(c) Only two statements
(d) All three statements
12. Consider the following statements in respect of 
non-singular matrix A of order n:
1. A(adj A
T
) = A(adj A)
T
2.  If A
2
 = A, then A is identity matrix of order n
3.  If A
3
 = A, then A is identity matrix of order n
Which of the statements given above are 
correct?
(a) 1 and 2 only (b) 2 and 3 only
(c) 1 and 3 only (d) 1, 2 and 3
13. How many four-digit natural numbers are 
there such that all of the digits are even?
(a) 625 (b) 500 (c) 400 (d) 256
14. If w ¹ 1 is a cube root of unity, then what are the 
solutions of (z – 100)
3
 + 1000 = 0?
(a) 10(1 – w), 10(10 – w
2
), 100
(b) 10(10 – w), 10(10 – w
2
), 90
(c) 10(1 – w), 10(10 – w
2
), 1000
(d) (1 + w), (10 + w
2
), –1
15. What is (1 + i)
4
 + (1 – i)
4
 equal to,
where i =
-1
?
(a) 4 (b) 0 (c) –4 (d) –8
16. Consider the following statements in respect of 
a skew-symmetric matrix A of order 3:
1. All diagonal elements are zero.
2.  The sum of all the diagonal elements of the 
matrix is zero.
3. A is orthogonal matrix.
Which of the statements given above are 
correct?
(a) 1 and 2 only (b) 2 and 3 only
(c) 1 and 3 only (d) 1, 2 and 3
17. Four digit numbers are formed by using the 
digits 1, 2, 3, 5 without repetition of digits. How 
many of them are divisible by 4?
(a) 120 (b) 24 (c) 12 (d) 6
18. What is the remainder when 2
120
 is divided by 
7?
(a) 1 (b) 3 (c) 5 (d) 6
19. For what value of n is the determinant
CC Cn
CC Cn
Cm Cm Cm
(, )( ,) (, )
(, )( ,) (, )
(, )( ,) (
94 93 10 2
11 6115 12
76

1 11
0
,) n

for every m > n?
(a) 4 (b) 5 (c) 6 (d) 7
20. If ABC is a triangle, then what is the value of 
the determinant
cossin
tansin
tan( )cos
?
CB
AB
BC C
0
0
0 

(a) –1 (b) 0 (c) 1 (d) 3
21. What is the number of different matrices, each 
having 4 entries that can be formed using 1, 2, 
3, 4 (repetition is allowed)?
(a) 72 (b) 216 (c) 254 (d) 768
22. Let A = {x Î R: –1 < x < 1}. Which of the 
following is/are bijective functions from A to 
itself?
1. f(x) = x|x|
2. g(x) = cos(px)
Select the correct answer using the code given 
below:
(a) 1 only (b) 2 only
(c) Both 1 and 2 (d) Neither 1 nor 2
23. Let R be a relation on the open interval (–1, 1) 
and is given by
R = {(x, y): |x + y| < 2}. Then which one of the 
following is correct?
(a) R is reflexive but neither symmetric nor 
transitive
(b)  R is reflexive and symmetric but not 
transitive
(c) R is reflexive and transitive but not 
symmetric
(d) R is an equivalence relation
24. For any three non-empty sets A, B, C, what is 
(A È B) – {(A – B) È (B – A) È (A Ç B)} equal to?
(a) Null set (b) A
(c) B (d) (A È B) – (A Ç B)
25. If a, b, c are the sides of triangle ABC, then what 
is 
ab Ac A
bA A
cA A
2
1
1
sinsin
sincos
sincos
 equal to?
(a) Zero
(b) Area of triangle
(c) Perimeter of triangle
(d) a
2
 + b
2
 + c
2
26. If a, b, c are in AP; b, c, d are in GP; c, d, e are in 
HP , then which of the following is/are correct?
1. a, c and e are in GP
2.
11 1
ac e
,, are in GP
Select the correct answer using the code given 
below:
(a) 1 only (b) 2 only
(c) Both 1 and 2 (d) Neither 1 nor 2
27. What is the number of solutions of
log
4
(x – 1) = log
2
(x – 3)?
(a) Zero (b) One (c) T wo (d) Three
Page 3


Time Allowed : 2 hrs 30 min T otal Marks : 300
Instructions
1. This Test Booklet contains 120 items (questions). Each item is printed in English. Each item comprises four responses 
(answers). You will select the response which you want to mark on the Answer Sheet. In case you feel that there is more than 
one correct response, mark the response which you consider the best. In any case, choose ONLY ONE response for each item.
2. You have to mark all your responses ONLY on the separate Answer Sheet provided. See directions in the Answer Sheet.
3. All items carry equal marks.
4. Before you proceed to mark in the Answer Sheet the response to the various items in the Test Booklet, you have to fill in some 
particulars in the Answer Sheet as per instructions.
5. Penalty for wrong answers :
THERE WILL BE PENALTY FOR WRONG ANSWERS MARKED BY A CANDIDATE IN THE OBJECTIVE TYPE 
QUESTION PAPERS.
(i). There are four alternatives for the answer to every question. For each question for which a wrong answer has been given 
by the candidate, one-third of the marks assigned to that question will be deducted as penalty.
(ii). If a candidate gives more than one answer, it will be treated as a wrong answer even if one of the given answers happens 
to be correct and there will be same penalty as above to that question.
(iii). If a question is left blank, i.e., no answer is given by the candidate, there will be no penalty for that question.
NDA / NA
 National Defence Academy / 
Naval Academy I
QUESTION PAPER
2024
MATHEMATICS
1. Let A and B matrices of order 3 × 3. If |A| = 
1
22
 
and |B| = 
1
729
, then what is the value of
 
|2B[adj(3A)]|?
(a) 27 (b) 
27
22
 
(c) 
27
2
 (d) 1
2. If z is any complex number and iz
3
 + z
2
 – z + i 
= 0, where i = -1 , then what is the value of 
(|z| + 1)
2
?
(a) 1 (b) 4 (c) 81 (d) 121
3. What is the sum of all four digit numbers 
formed by using all digits 0, 1, 4, 5 without 
repetition of digits? 
(a) 44440 (b) 46460 (c) 46440 (d) 64440
4. If x , y and z are the cube roots of unity, then 
what is the value of xy + yz + zx?
(a) 0 (b) 1 (c) 2 (d) 3
5. A man has 7 relatives (4 women and 3 men). 
His wife also has 7 relatives (3 women and 4 
men). In how many ways can they invite 3 
women and 3 men so that 3 of them are man’s 
relatives and 3 of them are his wife’s relatives?
(a) 340 (b) 484 (c) 485 (d) 469
6. A triangle PQR is such that 3 points lie on the 
side PQ, 4 points on QR and 5 points on RP 
respectively. Triangles are constructed using 
these points as vertices. What is the number of 
triangles so formed?
(a) 205 (b) 206 (c) 215 (d) 220
7. If log
b
a = p, log
d
c = 2p and log
f
e = 3p, then 
what is () ace
p
1
 equal to?
(a) bd
2
f
3
 (b) bdf (c) b
3
d
2
f (d) b
2
d
2
f
2
8. If - 2
 
and 3
 
are roots of the equation a
0
 + 
a
1
x + a
2
x
2
 + a
3
x
3
 + x
4
 = 0 where a
0
, a
1
, a
2
, a
3
 
are integers, then which one of the following 
is correct?
(a) a
2
 = a
3
 = 0 (b) a
2
 = 0 and a
3
 = –5
(c) a
0
 = 6, a
3
 = 0 (d) a
1
 = 0 and a
2
 = 5
9. Let z
1
 and z
2
 be two complex numbers such 
that 
zz
zz
12
12


 
= 1, then what is Re
z
z
1
2






 
+ 1 
equal to?
(a) –1 (b) 0 (c) 1 (d) 5
10. If 26! = n8
k
, where k and n are positive integers, 
then what is the maximum value of k?
(a) 6 (b) 7 (c) 8 (d) 9
11. Consider the following statements in respect of 
two non-singular matrices A and B of the same 
order n:
1. adj(AB) = (adj A)(adj B)
2. adj(AB) = adj(BA)
3.  (AB)adj(AB) – |AB|I
n
 is a null matrix of 
order n
How many of the above statements are correct?
(a) None
(b) Only one statement
(c) Only two statements
(d) All three statements
12. Consider the following statements in respect of 
non-singular matrix A of order n:
1. A(adj A
T
) = A(adj A)
T
2.  If A
2
 = A, then A is identity matrix of order n
3.  If A
3
 = A, then A is identity matrix of order n
Which of the statements given above are 
correct?
(a) 1 and 2 only (b) 2 and 3 only
(c) 1 and 3 only (d) 1, 2 and 3
13. How many four-digit natural numbers are 
there such that all of the digits are even?
(a) 625 (b) 500 (c) 400 (d) 256
14. If w ¹ 1 is a cube root of unity, then what are the 
solutions of (z – 100)
3
 + 1000 = 0?
(a) 10(1 – w), 10(10 – w
2
), 100
(b) 10(10 – w), 10(10 – w
2
), 90
(c) 10(1 – w), 10(10 – w
2
), 1000
(d) (1 + w), (10 + w
2
), –1
15. What is (1 + i)
4
 + (1 – i)
4
 equal to,
where i =
-1
?
(a) 4 (b) 0 (c) –4 (d) –8
16. Consider the following statements in respect of 
a skew-symmetric matrix A of order 3:
1. All diagonal elements are zero.
2.  The sum of all the diagonal elements of the 
matrix is zero.
3. A is orthogonal matrix.
Which of the statements given above are 
correct?
(a) 1 and 2 only (b) 2 and 3 only
(c) 1 and 3 only (d) 1, 2 and 3
17. Four digit numbers are formed by using the 
digits 1, 2, 3, 5 without repetition of digits. How 
many of them are divisible by 4?
(a) 120 (b) 24 (c) 12 (d) 6
18. What is the remainder when 2
120
 is divided by 
7?
(a) 1 (b) 3 (c) 5 (d) 6
19. For what value of n is the determinant
CC Cn
CC Cn
Cm Cm Cm
(, )( ,) (, )
(, )( ,) (, )
(, )( ,) (
94 93 10 2
11 6115 12
76

1 11
0
,) n

for every m > n?
(a) 4 (b) 5 (c) 6 (d) 7
20. If ABC is a triangle, then what is the value of 
the determinant
cossin
tansin
tan( )cos
?
CB
AB
BC C
0
0
0 

(a) –1 (b) 0 (c) 1 (d) 3
21. What is the number of different matrices, each 
having 4 entries that can be formed using 1, 2, 
3, 4 (repetition is allowed)?
(a) 72 (b) 216 (c) 254 (d) 768
22. Let A = {x Î R: –1 < x < 1}. Which of the 
following is/are bijective functions from A to 
itself?
1. f(x) = x|x|
2. g(x) = cos(px)
Select the correct answer using the code given 
below:
(a) 1 only (b) 2 only
(c) Both 1 and 2 (d) Neither 1 nor 2
23. Let R be a relation on the open interval (–1, 1) 
and is given by
R = {(x, y): |x + y| < 2}. Then which one of the 
following is correct?
(a) R is reflexive but neither symmetric nor 
transitive
(b)  R is reflexive and symmetric but not 
transitive
(c) R is reflexive and transitive but not 
symmetric
(d) R is an equivalence relation
24. For any three non-empty sets A, B, C, what is 
(A È B) – {(A – B) È (B – A) È (A Ç B)} equal to?
(a) Null set (b) A
(c) B (d) (A È B) – (A Ç B)
25. If a, b, c are the sides of triangle ABC, then what 
is 
ab Ac A
bA A
cA A
2
1
1
sinsin
sincos
sincos
 equal to?
(a) Zero
(b) Area of triangle
(c) Perimeter of triangle
(d) a
2
 + b
2
 + c
2
26. If a, b, c are in AP; b, c, d are in GP; c, d, e are in 
HP , then which of the following is/are correct?
1. a, c and e are in GP
2.
11 1
ac e
,, are in GP
Select the correct answer using the code given 
below:
(a) 1 only (b) 2 only
(c) Both 1 and 2 (d) Neither 1 nor 2
27. What is the number of solutions of
log
4
(x – 1) = log
2
(x – 3)?
(a) Zero (b) One (c) T wo (d) Three
SOLVED PAPER – 2024-I
28. For x ³ y > 1, let loglog
xy
x
y
y
x













= k, then 
the value of k can never be equal to
(a) –1 (b) -
1
2
(c) 0 (d) 1
29. If A = 
sinsin
cossin cos
22 10
22 0
00 1
2

 











, then which of 
the following statements is/are correct?
1. A
–1
 = adj A
2. A is skew-symmetric matrix
3. A
–1
 = A
T
Select the correct answer using the code given 
below:
(a) 1 only (b) 1 and 2
(c) 1 and 3 (d) 2 and 3
30. What is the coefficient of x
10
 in the expansion 
of () 12
1
2202
2
5
 







xx
x
?
(a) –1
(b) 1
(c) 10
(d) Coefficient of x
10
 does not exist
31. If the 4
th
 term in the expansion of mx
x
n







1
is 
5
2
, then what is the value of mn?
(a) –3 (b) 3 (c) 6 (d) 12
32. If a, b and c (a > 0, c > 0) are in GP , then consider 
the following in respect of the equation
ax
2
 + bx + c = 0;
1. The equation has imaginary roots.
2.  The ratio of the roots of the equation is 1: w 
where w is a cube root of unity.
3.  The product of roots of the equation is 
b
a
2
2






.
Which of the statements given above are 
correct?
(a) 1 and 2 only (b) 2 and 3 only
(c) 1 and 3 only (d) 1, 2 and 3
33. If x
2
 + mx + n is an integer form all integral 
values of x, then which of the following is/are 
correct?
1. m must be an integer
2. n must be an integer
Select the correct answer using the code given 
below:
(a) 1 only (b) 2 only
(c) Both 1 and 2 (d) Neither 1 nor 2
34. In a binomial expansion of (x + y)
2n + 1
(x – y)
2n + 1
, 
the sum of middle terms is zero. What is the
value of 
x
y
2
2








?
(a) 1 (b) 2 (c) 4 (d) 8
35. Let A = {1, 2, 3, 4, 5} and B = {6, 7}. What is the 
number of onto functions from A to B?
(a) 10 (b) 20 (c) 30 (d) 32
36. What is 
31010
25 25
cossin
sincos
 

equal to?
(a) 1 (b)
3 (c) 2 (d) 4
37. What is (sin 9° – cos 9°) equal to?
(a) -
- 55
2
(b) -
- 53
2
(c) 
55
2
-
(d)
55
4
-
38. If in a triangle ABC, sin
3
A + sin
3
B + sin
3
C = 
3sinA sinB sinC, then what is the value of the 
determinant 
ab c
bc a
ca b
; where a, b, c are sides of 
the triangle?
(a) a + b + c
(b) ab + bc + ca
(c) (a + b)(b + c)(c + a)
(d) 0
39. If cos
–1
x = sin
–1
x, then which one of the 
following is correct?
(a) x = 1 (b) x = 
1
2
(c) x = 
1
2
(d) x = 
1
3
40. What is the number of solutions of
(sinq – cosq)
2 
= 2 where –p < q < p?
(a) Only one (b) Only two
(c) Four (d) No solution
41. ABC is a triangle such that angle C = 60°, then 
what is 
coscos
cos
AB
AB

 





2
equal to?
Page 4


Time Allowed : 2 hrs 30 min T otal Marks : 300
Instructions
1. This Test Booklet contains 120 items (questions). Each item is printed in English. Each item comprises four responses 
(answers). You will select the response which you want to mark on the Answer Sheet. In case you feel that there is more than 
one correct response, mark the response which you consider the best. In any case, choose ONLY ONE response for each item.
2. You have to mark all your responses ONLY on the separate Answer Sheet provided. See directions in the Answer Sheet.
3. All items carry equal marks.
4. Before you proceed to mark in the Answer Sheet the response to the various items in the Test Booklet, you have to fill in some 
particulars in the Answer Sheet as per instructions.
5. Penalty for wrong answers :
THERE WILL BE PENALTY FOR WRONG ANSWERS MARKED BY A CANDIDATE IN THE OBJECTIVE TYPE 
QUESTION PAPERS.
(i). There are four alternatives for the answer to every question. For each question for which a wrong answer has been given 
by the candidate, one-third of the marks assigned to that question will be deducted as penalty.
(ii). If a candidate gives more than one answer, it will be treated as a wrong answer even if one of the given answers happens 
to be correct and there will be same penalty as above to that question.
(iii). If a question is left blank, i.e., no answer is given by the candidate, there will be no penalty for that question.
NDA / NA
 National Defence Academy / 
Naval Academy I
QUESTION PAPER
2024
MATHEMATICS
1. Let A and B matrices of order 3 × 3. If |A| = 
1
22
 
and |B| = 
1
729
, then what is the value of
 
|2B[adj(3A)]|?
(a) 27 (b) 
27
22
 
(c) 
27
2
 (d) 1
2. If z is any complex number and iz
3
 + z
2
 – z + i 
= 0, where i = -1 , then what is the value of 
(|z| + 1)
2
?
(a) 1 (b) 4 (c) 81 (d) 121
3. What is the sum of all four digit numbers 
formed by using all digits 0, 1, 4, 5 without 
repetition of digits? 
(a) 44440 (b) 46460 (c) 46440 (d) 64440
4. If x , y and z are the cube roots of unity, then 
what is the value of xy + yz + zx?
(a) 0 (b) 1 (c) 2 (d) 3
5. A man has 7 relatives (4 women and 3 men). 
His wife also has 7 relatives (3 women and 4 
men). In how many ways can they invite 3 
women and 3 men so that 3 of them are man’s 
relatives and 3 of them are his wife’s relatives?
(a) 340 (b) 484 (c) 485 (d) 469
6. A triangle PQR is such that 3 points lie on the 
side PQ, 4 points on QR and 5 points on RP 
respectively. Triangles are constructed using 
these points as vertices. What is the number of 
triangles so formed?
(a) 205 (b) 206 (c) 215 (d) 220
7. If log
b
a = p, log
d
c = 2p and log
f
e = 3p, then 
what is () ace
p
1
 equal to?
(a) bd
2
f
3
 (b) bdf (c) b
3
d
2
f (d) b
2
d
2
f
2
8. If - 2
 
and 3
 
are roots of the equation a
0
 + 
a
1
x + a
2
x
2
 + a
3
x
3
 + x
4
 = 0 where a
0
, a
1
, a
2
, a
3
 
are integers, then which one of the following 
is correct?
(a) a
2
 = a
3
 = 0 (b) a
2
 = 0 and a
3
 = –5
(c) a
0
 = 6, a
3
 = 0 (d) a
1
 = 0 and a
2
 = 5
9. Let z
1
 and z
2
 be two complex numbers such 
that 
zz
zz
12
12


 
= 1, then what is Re
z
z
1
2






 
+ 1 
equal to?
(a) –1 (b) 0 (c) 1 (d) 5
10. If 26! = n8
k
, where k and n are positive integers, 
then what is the maximum value of k?
(a) 6 (b) 7 (c) 8 (d) 9
11. Consider the following statements in respect of 
two non-singular matrices A and B of the same 
order n:
1. adj(AB) = (adj A)(adj B)
2. adj(AB) = adj(BA)
3.  (AB)adj(AB) – |AB|I
n
 is a null matrix of 
order n
How many of the above statements are correct?
(a) None
(b) Only one statement
(c) Only two statements
(d) All three statements
12. Consider the following statements in respect of 
non-singular matrix A of order n:
1. A(adj A
T
) = A(adj A)
T
2.  If A
2
 = A, then A is identity matrix of order n
3.  If A
3
 = A, then A is identity matrix of order n
Which of the statements given above are 
correct?
(a) 1 and 2 only (b) 2 and 3 only
(c) 1 and 3 only (d) 1, 2 and 3
13. How many four-digit natural numbers are 
there such that all of the digits are even?
(a) 625 (b) 500 (c) 400 (d) 256
14. If w ¹ 1 is a cube root of unity, then what are the 
solutions of (z – 100)
3
 + 1000 = 0?
(a) 10(1 – w), 10(10 – w
2
), 100
(b) 10(10 – w), 10(10 – w
2
), 90
(c) 10(1 – w), 10(10 – w
2
), 1000
(d) (1 + w), (10 + w
2
), –1
15. What is (1 + i)
4
 + (1 – i)
4
 equal to,
where i =
-1
?
(a) 4 (b) 0 (c) –4 (d) –8
16. Consider the following statements in respect of 
a skew-symmetric matrix A of order 3:
1. All diagonal elements are zero.
2.  The sum of all the diagonal elements of the 
matrix is zero.
3. A is orthogonal matrix.
Which of the statements given above are 
correct?
(a) 1 and 2 only (b) 2 and 3 only
(c) 1 and 3 only (d) 1, 2 and 3
17. Four digit numbers are formed by using the 
digits 1, 2, 3, 5 without repetition of digits. How 
many of them are divisible by 4?
(a) 120 (b) 24 (c) 12 (d) 6
18. What is the remainder when 2
120
 is divided by 
7?
(a) 1 (b) 3 (c) 5 (d) 6
19. For what value of n is the determinant
CC Cn
CC Cn
Cm Cm Cm
(, )( ,) (, )
(, )( ,) (, )
(, )( ,) (
94 93 10 2
11 6115 12
76

1 11
0
,) n

for every m > n?
(a) 4 (b) 5 (c) 6 (d) 7
20. If ABC is a triangle, then what is the value of 
the determinant
cossin
tansin
tan( )cos
?
CB
AB
BC C
0
0
0 

(a) –1 (b) 0 (c) 1 (d) 3
21. What is the number of different matrices, each 
having 4 entries that can be formed using 1, 2, 
3, 4 (repetition is allowed)?
(a) 72 (b) 216 (c) 254 (d) 768
22. Let A = {x Î R: –1 < x < 1}. Which of the 
following is/are bijective functions from A to 
itself?
1. f(x) = x|x|
2. g(x) = cos(px)
Select the correct answer using the code given 
below:
(a) 1 only (b) 2 only
(c) Both 1 and 2 (d) Neither 1 nor 2
23. Let R be a relation on the open interval (–1, 1) 
and is given by
R = {(x, y): |x + y| < 2}. Then which one of the 
following is correct?
(a) R is reflexive but neither symmetric nor 
transitive
(b)  R is reflexive and symmetric but not 
transitive
(c) R is reflexive and transitive but not 
symmetric
(d) R is an equivalence relation
24. For any three non-empty sets A, B, C, what is 
(A È B) – {(A – B) È (B – A) È (A Ç B)} equal to?
(a) Null set (b) A
(c) B (d) (A È B) – (A Ç B)
25. If a, b, c are the sides of triangle ABC, then what 
is 
ab Ac A
bA A
cA A
2
1
1
sinsin
sincos
sincos
 equal to?
(a) Zero
(b) Area of triangle
(c) Perimeter of triangle
(d) a
2
 + b
2
 + c
2
26. If a, b, c are in AP; b, c, d are in GP; c, d, e are in 
HP , then which of the following is/are correct?
1. a, c and e are in GP
2.
11 1
ac e
,, are in GP
Select the correct answer using the code given 
below:
(a) 1 only (b) 2 only
(c) Both 1 and 2 (d) Neither 1 nor 2
27. What is the number of solutions of
log
4
(x – 1) = log
2
(x – 3)?
(a) Zero (b) One (c) T wo (d) Three
SOLVED PAPER – 2024-I
28. For x ³ y > 1, let loglog
xy
x
y
y
x













= k, then 
the value of k can never be equal to
(a) –1 (b) -
1
2
(c) 0 (d) 1
29. If A = 
sinsin
cossin cos
22 10
22 0
00 1
2

 











, then which of 
the following statements is/are correct?
1. A
–1
 = adj A
2. A is skew-symmetric matrix
3. A
–1
 = A
T
Select the correct answer using the code given 
below:
(a) 1 only (b) 1 and 2
(c) 1 and 3 (d) 2 and 3
30. What is the coefficient of x
10
 in the expansion 
of () 12
1
2202
2
5
 







xx
x
?
(a) –1
(b) 1
(c) 10
(d) Coefficient of x
10
 does not exist
31. If the 4
th
 term in the expansion of mx
x
n







1
is 
5
2
, then what is the value of mn?
(a) –3 (b) 3 (c) 6 (d) 12
32. If a, b and c (a > 0, c > 0) are in GP , then consider 
the following in respect of the equation
ax
2
 + bx + c = 0;
1. The equation has imaginary roots.
2.  The ratio of the roots of the equation is 1: w 
where w is a cube root of unity.
3.  The product of roots of the equation is 
b
a
2
2






.
Which of the statements given above are 
correct?
(a) 1 and 2 only (b) 2 and 3 only
(c) 1 and 3 only (d) 1, 2 and 3
33. If x
2
 + mx + n is an integer form all integral 
values of x, then which of the following is/are 
correct?
1. m must be an integer
2. n must be an integer
Select the correct answer using the code given 
below:
(a) 1 only (b) 2 only
(c) Both 1 and 2 (d) Neither 1 nor 2
34. In a binomial expansion of (x + y)
2n + 1
(x – y)
2n + 1
, 
the sum of middle terms is zero. What is the
value of 
x
y
2
2








?
(a) 1 (b) 2 (c) 4 (d) 8
35. Let A = {1, 2, 3, 4, 5} and B = {6, 7}. What is the 
number of onto functions from A to B?
(a) 10 (b) 20 (c) 30 (d) 32
36. What is 
31010
25 25
cossin
sincos
 

equal to?
(a) 1 (b)
3 (c) 2 (d) 4
37. What is (sin 9° – cos 9°) equal to?
(a) -
- 55
2
(b) -
- 53
2
(c) 
55
2
-
(d)
55
4
-
38. If in a triangle ABC, sin
3
A + sin
3
B + sin
3
C = 
3sinA sinB sinC, then what is the value of the 
determinant 
ab c
bc a
ca b
; where a, b, c are sides of 
the triangle?
(a) a + b + c
(b) ab + bc + ca
(c) (a + b)(b + c)(c + a)
(d) 0
39. If cos
–1
x = sin
–1
x, then which one of the 
following is correct?
(a) x = 1 (b) x = 
1
2
(c) x = 
1
2
(d) x = 
1
3
40. What is the number of solutions of
(sinq – cosq)
2 
= 2 where –p < q < p?
(a) Only one (b) Only two
(c) Four (d) No solution
41. ABC is a triangle such that angle C = 60°, then 
what is 
coscos
cos
AB
AB

 





2
equal to?
(a) 2 (b) 2 (c) 1 (d)
1
2
42. What is 15
4
23
21








cotcot

equal to?
(a) 1 (b) 7 (c) 8 (d) 16
43. What is the value of sin 10° . sin 50° + sin 50° . 
sin 250° + sin 250° . sin 10° equal to?
(a) -
1
4
(b) -
3
4
(c)
° 3sin10
4
(d) 
 310
4
cos
44. What is tantan
















11
a
b
ab
ab
equal to?
(a) 

4
(b)
p
4
(c) tan









1
22
22
ab
ab
(d)
tan








1
22
2ab
ab
45. Under which one of the following conditions 
does the equation (cos b – 1)x
2
 + (cos b)x + sin b
= 0 in x have a real root for b Î [0, p]?
(a) 1 – cos b < 0 (b) 1 – cos b £ 0
(c) 1 – cos b > 0 (d) 1 – cos b ³ 0
46. In a triangle ABC, AB = 16 cm, BC = 63 cm and 
AC = 65 cm. What is the value of cos 2A + cos 2B
+ cos 2C?
(a) –1 (b) 0 (c) 1 (d)
76
65
47. If f(q) = 
1
1tan
and a + b = 
p 5
4
, then what 
is the value of f(a) f(b)?
(a) -
1
2
(b)
1
2
(c) 1 (d) 2
48. If tan a and tan b are the roots of the equation 
x
2
 – 6x + 8 = 0, then what is the value of
cos(2a + 2b)?
(a) 
13
75
(b)
13
85
(c)
17
85
(d)
19
85
49. What is the value of tan 65° + 2tan 45° – 2tan 40°
– tan 25°?
(a) 0 (b) 1 (c) 2 (d) 4
50. Consider the following statements:
1.  In a triangle ABC, if cot A . cot B . cot C > 0, 
then the triangle is an acute angled triangle.
2.  In a triangle ABC, if tan A . tan B . tan C 
> 0, then the triangle is an obtuse angled 
triangle.
Which of the statements given above is/are 
correct?
(a) 1 only (b) 2 only
(c) Both 1 and 2 (d) Neither 1 nor 2
51. If (a, b) is the centre and c is the radius of the 
circle x
2
 + y
2
 + 2x + 6y + 1 = 0, then what is 
the value of a
2
 + b
2
 + c
2
?
(a) 19 (b) 18 (c) 17 (d) 11
52. If (1, –1, 2) and (2, 1, –1) are the end points of a 
diameter of a sphere x
2
 + y
2
 + z
2
 + 2ux + 2vy
+ 2wz – 1 = 0, then what is u + v + w equal to?
(a) -2 (b) –1 (c) 1 (d) 2
53. The number of points represented by the 
equation x = 5 on the xy-plane is
(a) Zero (b) One
(c) T wo (d) Infinitely many
54. If < l, m, n > are the direction cosines of a 
normal to the plane 2x – 3y + 6z + 4 = 0, then 
what is the value of 49(7l
2
 + m
2
 – n
2
)?
(a) 0 (b) 1 (c) 3 (d) 71
55. A line through (1, –1, 2) with direction ratios 
< 3, 2, 2 > meets the plane x + 2y + 3z = 18. 
What is the point of intersection of line and 
plane?
(a) (4, 4, 1) (b) (2, 4, 1)
(c) (4, 1, 4) (d) (3, 4, 7)
56. If p is the perpendicular distance from origin to 
the plane passing through (1, 0, 0), (0, 1, 0) and 
(0, 0, 1), then what is 3p
2
 equal to?
(a) 4 (b) 3 (c) 2 (d) 1
57. If the direction cosines < l, m, n > of a line are 
connected by relation l + 2m + n = 0, 2l – 2m + 
3n = 0, then what is the value of l
2
 + m
2
 – n
2
?
(a) 
1
101
(b)
29
101
(c)
41
101
(d)
92
101
58. If a variable line passes through the point of 
intersection of the lines x + 2y – 1 = 0 and 2x
– y – 1 = 0 and meets the coordinate axis in A 
and B, then what is the locus of the mid-point 
of AB?
(a) 3x + y = 10xy (b) x + 3y = 10xy
(c) 3x + y = 10 (d) x + 3y = 10
59. What is the equation to the straight line 
passing through the point (–sinq, cosq) and 
perpendicular to the line xcos q + ysin q = 9? 
(a) xsin q – ycos q – 1 = 0
(b) xsin q – ycos q + 1 = 0
(c) xsin q – ycos q = 0
(d) xcos q – ysin q + 1 = 0
Page 5


Time Allowed : 2 hrs 30 min T otal Marks : 300
Instructions
1. This Test Booklet contains 120 items (questions). Each item is printed in English. Each item comprises four responses 
(answers). You will select the response which you want to mark on the Answer Sheet. In case you feel that there is more than 
one correct response, mark the response which you consider the best. In any case, choose ONLY ONE response for each item.
2. You have to mark all your responses ONLY on the separate Answer Sheet provided. See directions in the Answer Sheet.
3. All items carry equal marks.
4. Before you proceed to mark in the Answer Sheet the response to the various items in the Test Booklet, you have to fill in some 
particulars in the Answer Sheet as per instructions.
5. Penalty for wrong answers :
THERE WILL BE PENALTY FOR WRONG ANSWERS MARKED BY A CANDIDATE IN THE OBJECTIVE TYPE 
QUESTION PAPERS.
(i). There are four alternatives for the answer to every question. For each question for which a wrong answer has been given 
by the candidate, one-third of the marks assigned to that question will be deducted as penalty.
(ii). If a candidate gives more than one answer, it will be treated as a wrong answer even if one of the given answers happens 
to be correct and there will be same penalty as above to that question.
(iii). If a question is left blank, i.e., no answer is given by the candidate, there will be no penalty for that question.
NDA / NA
 National Defence Academy / 
Naval Academy I
QUESTION PAPER
2024
MATHEMATICS
1. Let A and B matrices of order 3 × 3. If |A| = 
1
22
 
and |B| = 
1
729
, then what is the value of
 
|2B[adj(3A)]|?
(a) 27 (b) 
27
22
 
(c) 
27
2
 (d) 1
2. If z is any complex number and iz
3
 + z
2
 – z + i 
= 0, where i = -1 , then what is the value of 
(|z| + 1)
2
?
(a) 1 (b) 4 (c) 81 (d) 121
3. What is the sum of all four digit numbers 
formed by using all digits 0, 1, 4, 5 without 
repetition of digits? 
(a) 44440 (b) 46460 (c) 46440 (d) 64440
4. If x , y and z are the cube roots of unity, then 
what is the value of xy + yz + zx?
(a) 0 (b) 1 (c) 2 (d) 3
5. A man has 7 relatives (4 women and 3 men). 
His wife also has 7 relatives (3 women and 4 
men). In how many ways can they invite 3 
women and 3 men so that 3 of them are man’s 
relatives and 3 of them are his wife’s relatives?
(a) 340 (b) 484 (c) 485 (d) 469
6. A triangle PQR is such that 3 points lie on the 
side PQ, 4 points on QR and 5 points on RP 
respectively. Triangles are constructed using 
these points as vertices. What is the number of 
triangles so formed?
(a) 205 (b) 206 (c) 215 (d) 220
7. If log
b
a = p, log
d
c = 2p and log
f
e = 3p, then 
what is () ace
p
1
 equal to?
(a) bd
2
f
3
 (b) bdf (c) b
3
d
2
f (d) b
2
d
2
f
2
8. If - 2
 
and 3
 
are roots of the equation a
0
 + 
a
1
x + a
2
x
2
 + a
3
x
3
 + x
4
 = 0 where a
0
, a
1
, a
2
, a
3
 
are integers, then which one of the following 
is correct?
(a) a
2
 = a
3
 = 0 (b) a
2
 = 0 and a
3
 = –5
(c) a
0
 = 6, a
3
 = 0 (d) a
1
 = 0 and a
2
 = 5
9. Let z
1
 and z
2
 be two complex numbers such 
that 
zz
zz
12
12


 
= 1, then what is Re
z
z
1
2






 
+ 1 
equal to?
(a) –1 (b) 0 (c) 1 (d) 5
10. If 26! = n8
k
, where k and n are positive integers, 
then what is the maximum value of k?
(a) 6 (b) 7 (c) 8 (d) 9
11. Consider the following statements in respect of 
two non-singular matrices A and B of the same 
order n:
1. adj(AB) = (adj A)(adj B)
2. adj(AB) = adj(BA)
3.  (AB)adj(AB) – |AB|I
n
 is a null matrix of 
order n
How many of the above statements are correct?
(a) None
(b) Only one statement
(c) Only two statements
(d) All three statements
12. Consider the following statements in respect of 
non-singular matrix A of order n:
1. A(adj A
T
) = A(adj A)
T
2.  If A
2
 = A, then A is identity matrix of order n
3.  If A
3
 = A, then A is identity matrix of order n
Which of the statements given above are 
correct?
(a) 1 and 2 only (b) 2 and 3 only
(c) 1 and 3 only (d) 1, 2 and 3
13. How many four-digit natural numbers are 
there such that all of the digits are even?
(a) 625 (b) 500 (c) 400 (d) 256
14. If w ¹ 1 is a cube root of unity, then what are the 
solutions of (z – 100)
3
 + 1000 = 0?
(a) 10(1 – w), 10(10 – w
2
), 100
(b) 10(10 – w), 10(10 – w
2
), 90
(c) 10(1 – w), 10(10 – w
2
), 1000
(d) (1 + w), (10 + w
2
), –1
15. What is (1 + i)
4
 + (1 – i)
4
 equal to,
where i =
-1
?
(a) 4 (b) 0 (c) –4 (d) –8
16. Consider the following statements in respect of 
a skew-symmetric matrix A of order 3:
1. All diagonal elements are zero.
2.  The sum of all the diagonal elements of the 
matrix is zero.
3. A is orthogonal matrix.
Which of the statements given above are 
correct?
(a) 1 and 2 only (b) 2 and 3 only
(c) 1 and 3 only (d) 1, 2 and 3
17. Four digit numbers are formed by using the 
digits 1, 2, 3, 5 without repetition of digits. How 
many of them are divisible by 4?
(a) 120 (b) 24 (c) 12 (d) 6
18. What is the remainder when 2
120
 is divided by 
7?
(a) 1 (b) 3 (c) 5 (d) 6
19. For what value of n is the determinant
CC Cn
CC Cn
Cm Cm Cm
(, )( ,) (, )
(, )( ,) (, )
(, )( ,) (
94 93 10 2
11 6115 12
76

1 11
0
,) n

for every m > n?
(a) 4 (b) 5 (c) 6 (d) 7
20. If ABC is a triangle, then what is the value of 
the determinant
cossin
tansin
tan( )cos
?
CB
AB
BC C
0
0
0 

(a) –1 (b) 0 (c) 1 (d) 3
21. What is the number of different matrices, each 
having 4 entries that can be formed using 1, 2, 
3, 4 (repetition is allowed)?
(a) 72 (b) 216 (c) 254 (d) 768
22. Let A = {x Î R: –1 < x < 1}. Which of the 
following is/are bijective functions from A to 
itself?
1. f(x) = x|x|
2. g(x) = cos(px)
Select the correct answer using the code given 
below:
(a) 1 only (b) 2 only
(c) Both 1 and 2 (d) Neither 1 nor 2
23. Let R be a relation on the open interval (–1, 1) 
and is given by
R = {(x, y): |x + y| < 2}. Then which one of the 
following is correct?
(a) R is reflexive but neither symmetric nor 
transitive
(b)  R is reflexive and symmetric but not 
transitive
(c) R is reflexive and transitive but not 
symmetric
(d) R is an equivalence relation
24. For any three non-empty sets A, B, C, what is 
(A È B) – {(A – B) È (B – A) È (A Ç B)} equal to?
(a) Null set (b) A
(c) B (d) (A È B) – (A Ç B)
25. If a, b, c are the sides of triangle ABC, then what 
is 
ab Ac A
bA A
cA A
2
1
1
sinsin
sincos
sincos
 equal to?
(a) Zero
(b) Area of triangle
(c) Perimeter of triangle
(d) a
2
 + b
2
 + c
2
26. If a, b, c are in AP; b, c, d are in GP; c, d, e are in 
HP , then which of the following is/are correct?
1. a, c and e are in GP
2.
11 1
ac e
,, are in GP
Select the correct answer using the code given 
below:
(a) 1 only (b) 2 only
(c) Both 1 and 2 (d) Neither 1 nor 2
27. What is the number of solutions of
log
4
(x – 1) = log
2
(x – 3)?
(a) Zero (b) One (c) T wo (d) Three
SOLVED PAPER – 2024-I
28. For x ³ y > 1, let loglog
xy
x
y
y
x













= k, then 
the value of k can never be equal to
(a) –1 (b) -
1
2
(c) 0 (d) 1
29. If A = 
sinsin
cossin cos
22 10
22 0
00 1
2

 











, then which of 
the following statements is/are correct?
1. A
–1
 = adj A
2. A is skew-symmetric matrix
3. A
–1
 = A
T
Select the correct answer using the code given 
below:
(a) 1 only (b) 1 and 2
(c) 1 and 3 (d) 2 and 3
30. What is the coefficient of x
10
 in the expansion 
of () 12
1
2202
2
5
 







xx
x
?
(a) –1
(b) 1
(c) 10
(d) Coefficient of x
10
 does not exist
31. If the 4
th
 term in the expansion of mx
x
n







1
is 
5
2
, then what is the value of mn?
(a) –3 (b) 3 (c) 6 (d) 12
32. If a, b and c (a > 0, c > 0) are in GP , then consider 
the following in respect of the equation
ax
2
 + bx + c = 0;
1. The equation has imaginary roots.
2.  The ratio of the roots of the equation is 1: w 
where w is a cube root of unity.
3.  The product of roots of the equation is 
b
a
2
2






.
Which of the statements given above are 
correct?
(a) 1 and 2 only (b) 2 and 3 only
(c) 1 and 3 only (d) 1, 2 and 3
33. If x
2
 + mx + n is an integer form all integral 
values of x, then which of the following is/are 
correct?
1. m must be an integer
2. n must be an integer
Select the correct answer using the code given 
below:
(a) 1 only (b) 2 only
(c) Both 1 and 2 (d) Neither 1 nor 2
34. In a binomial expansion of (x + y)
2n + 1
(x – y)
2n + 1
, 
the sum of middle terms is zero. What is the
value of 
x
y
2
2








?
(a) 1 (b) 2 (c) 4 (d) 8
35. Let A = {1, 2, 3, 4, 5} and B = {6, 7}. What is the 
number of onto functions from A to B?
(a) 10 (b) 20 (c) 30 (d) 32
36. What is 
31010
25 25
cossin
sincos
 

equal to?
(a) 1 (b)
3 (c) 2 (d) 4
37. What is (sin 9° – cos 9°) equal to?
(a) -
- 55
2
(b) -
- 53
2
(c) 
55
2
-
(d)
55
4
-
38. If in a triangle ABC, sin
3
A + sin
3
B + sin
3
C = 
3sinA sinB sinC, then what is the value of the 
determinant 
ab c
bc a
ca b
; where a, b, c are sides of 
the triangle?
(a) a + b + c
(b) ab + bc + ca
(c) (a + b)(b + c)(c + a)
(d) 0
39. If cos
–1
x = sin
–1
x, then which one of the 
following is correct?
(a) x = 1 (b) x = 
1
2
(c) x = 
1
2
(d) x = 
1
3
40. What is the number of solutions of
(sinq – cosq)
2 
= 2 where –p < q < p?
(a) Only one (b) Only two
(c) Four (d) No solution
41. ABC is a triangle such that angle C = 60°, then 
what is 
coscos
cos
AB
AB

 





2
equal to?
(a) 2 (b) 2 (c) 1 (d)
1
2
42. What is 15
4
23
21








cotcot

equal to?
(a) 1 (b) 7 (c) 8 (d) 16
43. What is the value of sin 10° . sin 50° + sin 50° . 
sin 250° + sin 250° . sin 10° equal to?
(a) -
1
4
(b) -
3
4
(c)
° 3sin10
4
(d) 
 310
4
cos
44. What is tantan
















11
a
b
ab
ab
equal to?
(a) 

4
(b)
p
4
(c) tan









1
22
22
ab
ab
(d)
tan








1
22
2ab
ab
45. Under which one of the following conditions 
does the equation (cos b – 1)x
2
 + (cos b)x + sin b
= 0 in x have a real root for b Î [0, p]?
(a) 1 – cos b < 0 (b) 1 – cos b £ 0
(c) 1 – cos b > 0 (d) 1 – cos b ³ 0
46. In a triangle ABC, AB = 16 cm, BC = 63 cm and 
AC = 65 cm. What is the value of cos 2A + cos 2B
+ cos 2C?
(a) –1 (b) 0 (c) 1 (d)
76
65
47. If f(q) = 
1
1tan
and a + b = 
p 5
4
, then what 
is the value of f(a) f(b)?
(a) -
1
2
(b)
1
2
(c) 1 (d) 2
48. If tan a and tan b are the roots of the equation 
x
2
 – 6x + 8 = 0, then what is the value of
cos(2a + 2b)?
(a) 
13
75
(b)
13
85
(c)
17
85
(d)
19
85
49. What is the value of tan 65° + 2tan 45° – 2tan 40°
– tan 25°?
(a) 0 (b) 1 (c) 2 (d) 4
50. Consider the following statements:
1.  In a triangle ABC, if cot A . cot B . cot C > 0, 
then the triangle is an acute angled triangle.
2.  In a triangle ABC, if tan A . tan B . tan C 
> 0, then the triangle is an obtuse angled 
triangle.
Which of the statements given above is/are 
correct?
(a) 1 only (b) 2 only
(c) Both 1 and 2 (d) Neither 1 nor 2
51. If (a, b) is the centre and c is the radius of the 
circle x
2
 + y
2
 + 2x + 6y + 1 = 0, then what is 
the value of a
2
 + b
2
 + c
2
?
(a) 19 (b) 18 (c) 17 (d) 11
52. If (1, –1, 2) and (2, 1, –1) are the end points of a 
diameter of a sphere x
2
 + y
2
 + z
2
 + 2ux + 2vy
+ 2wz – 1 = 0, then what is u + v + w equal to?
(a) -2 (b) –1 (c) 1 (d) 2
53. The number of points represented by the 
equation x = 5 on the xy-plane is
(a) Zero (b) One
(c) T wo (d) Infinitely many
54. If < l, m, n > are the direction cosines of a 
normal to the plane 2x – 3y + 6z + 4 = 0, then 
what is the value of 49(7l
2
 + m
2
 – n
2
)?
(a) 0 (b) 1 (c) 3 (d) 71
55. A line through (1, –1, 2) with direction ratios 
< 3, 2, 2 > meets the plane x + 2y + 3z = 18. 
What is the point of intersection of line and 
plane?
(a) (4, 4, 1) (b) (2, 4, 1)
(c) (4, 1, 4) (d) (3, 4, 7)
56. If p is the perpendicular distance from origin to 
the plane passing through (1, 0, 0), (0, 1, 0) and 
(0, 0, 1), then what is 3p
2
 equal to?
(a) 4 (b) 3 (c) 2 (d) 1
57. If the direction cosines < l, m, n > of a line are 
connected by relation l + 2m + n = 0, 2l – 2m + 
3n = 0, then what is the value of l
2
 + m
2
 – n
2
?
(a) 
1
101
(b)
29
101
(c)
41
101
(d)
92
101
58. If a variable line passes through the point of 
intersection of the lines x + 2y – 1 = 0 and 2x
– y – 1 = 0 and meets the coordinate axis in A 
and B, then what is the locus of the mid-point 
of AB?
(a) 3x + y = 10xy (b) x + 3y = 10xy
(c) 3x + y = 10 (d) x + 3y = 10
59. What is the equation to the straight line 
passing through the point (–sinq, cosq) and 
perpendicular to the line xcos q + ysin q = 9? 
(a) xsin q – ycos q – 1 = 0
(b) xsin q – ycos q + 1 = 0
(c) xsin q – ycos q = 0
(d) xcos q – ysin q + 1 = 0
SOLVED PAPER – 2024-I
60. Two points P and Q lie on line y = 2x + 3. 
These two points P and Q are at a distance 2 
units from another point R(1, 5). What are the 
coordinates of the points P and Q?
(a) 1
2
5
5
4
5
1
2
5
5
4
5














,, ,
(b) 3
2
5
5
4
5
1
2
5
5
4
5







 






,, ,
(c) 
1
2
5
5
4
5
1
2
5
5
4
5














,, ,
(d) 3
2
5
5
4
5
1
2
5
5
4
5







 






,, ,
61. If two sides of a square lie on the lines 2x + y – 
3 = 0 and 4x + 2y + 5 = 0, then what is the area 
of the square in square units?
(a) 6.05 (b) 6.15 (c) 6.25 (d) 6.35
62. ABC is a triangle with A(3, 5). The mid-points 
of sides AB, AC are at (–1, 2), (6, 4) respectively. 
What are the coordinates of centroid of the 
triangle ABC?
(a) 
8
3
11
3
,





 (b)
7
3
7
3
,






(c) 2
8
3
,





 (d)
8
3
2 ,






63. ABC is an acute angled isosceles triangle. T wo 
equal sides AB and AC lie on the lines 7x – y – 3 
= 0 and x + y – 5 = 0. If q is one of the equal 
angles, then what is cot q equal to?
(a) 
1
3
(b)
1
2
(c)
2
3
(d) 2
64. In the parabola y
2
 = 8x, the focal distance of 
a point P lying on it is 8 units. Which of the 
following statements is/are correct?
1. The coordinates of P can be (, ) 64 3 .
2.  The perpendicular distance of P from the 
directrix of parabola is 8 units.
Select the correct answer using the code given 
below:
(a) 1 only (b) 2 only
(c) Both 1 and 2 (d) Neither 1 nor 2
65. What is the eccentricity of the ellipse if the 
angle between the straight lines joining the 
foci to an extremity of the minor axis is 90°?
(a) 
1
3
(b)
1
2
(c)
1
3
(d)
1
2
66. Let a
?
= ij k
?? ?
 and b
?
= ij k
?? ?
 2 . If ab a
?? ?
×× ()
=   ij k
?? ?
 , then what is the value of a + b
+ g?
(a) 8 (b) 7 (c) 6 (d) 0
67. If a vector of magnitude 2 units makes an angle 
p
3
with 2i
?
,
p
4
with 3 j
?
and an acute angle q
with 4k
?
, then what are the components of the
vector?
(a) (, ,) 12 1 (b) (, ,) 12 1 -
(c) (, ,) 12 1 -- (d) (, ,) 12 1 -
68. Consider the following in respect of moment 
of a force:
1.  The moment of force about a point is 
independent of point of application of 
force.
2.  The moment of a force about a line is a 
vector quantity.
Which of the statements given above is/are 
correct?
(a) 1 only (b) 2 only
(c) Both 1 and 2 (d) Neither 1 nor 2
69. For any vector r
?
, what is
(. )( )( .)() (. )( ) ri ri rj rj rk rk












   equal to?
(a) 0
?
(b) r
?
(c) 2r
?
(d) 3r
?
70. Let a
?
and b
?
are two vectors of magnitude 4 
inclined at an angle 
p
3
, then what is the angle 
between a
?
and ab
??
- ?
(a)
p
2
(b)
p
3
(c)
p
4
(d)
p
6
71. Let y
1
(x) and y
2
(x) be two solutions of the 
differential equation 
dy
dx
 = x. If y
1
(0) = 0 and 
y
2
(0) = 4, then what is the number of points of 
intersection of the curves y
1
(x) and y
2
(x)?
(a) No point
(b) One point
(c) T wo points
(d) More than two points
72. The differential equation, representing the 
curve y = e
x
(a cosx + b sinx) where a and b are 
arbitrary constants, is