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Determinants — NDA Mathematics PYQs

130 previous year questions from Determinants with answers and solutions. Numbered list, year tags, and one-tap solutions — built for serious JEE / NEET practice.

130 questionsMathematicsSolutions on every page
1

Let p=(x+y+z) and q=xyz . If vmatrix x & 1 & 1 1 & y & 1 1 & 1 & z vmatrix is positive, then which one of the following is correct ?

NDA 2026 Solution View → 2

Let f(x)= vmatrix 3x^2 & x & - x 6 & -1 & 0 q & q^2 & q^3 vmatrix where q is any constant, then what is d^2 dx^2 (f(x) ) at x=0 equal to ?

NDA 2026 Solution View → 3

If M= bmatrix 2 & 0 & 0 0 & 2 & 0 0 & 0 & 2 bmatrix , then what is the value of |M| ,| adj M| ?

NDA 2026 Solution View → 4

If vmatrix a-b & p-q & x-y b-c & q-r & y-z c-a & r-p & z-x vmatrix =k vmatrix a & b & c p & q & r x & y & z vmatrix , then what is the value of k ?

NDA 2026 Solution View → 5

If A is a square matrix such that |A|=-2 , then |AA^ T | , where A^ T is the transpose of A , is equal to

NDA 2026 Solution View → 6

The system of equations 2x - 3y - 5 = 0 , 15y - 10x + 50 = 0

NDA 2025 Solution View → 7

The value of the determinant vmatrix a & b & c l & m & n p & q & r vmatrix is equal to

NDA 2025 Solution View → 8

If a , b , c are the sides of a triangle ABC and p is the perimeter of the triangle, then what is vmatrix p+c & a & b c & p+a & b c & a & p+b vmatrix equal to?

NDA 2025 Solution View → 9

Consider the following statements: Statement-I: If X is an n n matrix, then (mX) = m^n (X) , where m is a scalar. Statement-II: If Y is a matrix obtained from X by multiplying any

NDA 2025 Solution View → 10

If = vmatrix a & b & c d & e & f g & h & i vmatrix and A , B , C , D , G are the cofactors of the elements a , b , c , d , g respectively, then what is bB + cC - dD - gG equal to?

NDA 2025 Solution View → 11

If A^2 + B^2 + C^2 = 0 , then what is the value of vmatrix 1 & C & B C & 1 & A B & A & 1 vmatrix ?

NDA 2025 Solution View → 12

Consider the following statements in respect of the determinant = vmatrix k(k+2) & 2k+1 & 1 2k+1 & k+2 & 1 3 & 3 & 1 vmatrix : I. is positive if k > 0 . II. is negative if k III. i

NDA 2025 Solution View → 13

If vmatrix 2 & 3+i & -1 3-i & 0 & i-1 -1 & -1-i & 1 vmatrix = A + iB where i = -1 , then what is A + B equal to?

NDA 2025 Solution View → 14

Consider the following in respect of a non-singular matrix M : I. |M^2| = |M|^2 II. |M| = |M⁻¹| III. |M| = |M^T| How many of the above are correct?

NDA 2025 Solution View → 15

In obtaining the solution of the system of equations x + y + z = 7 , x + 2y + 3z = 16 and x + 3y + 4z = 22 by Cramer's rule, the value of y is obtained by dividing D by D₂ , where

NDA 2025 Solution View → 16

If Z = 1 3 | array ccc i & 2 i & 1 2 i & 3 i & 2 3 & 1 & 3 array |=x+i y ; i= -1 then what is modulus of Z equal to?

NDA 2024 Solution View → 17

For what value of n is determinant | array ccc C (9,4) & C (9,3) & C (10, n-2) C (11,6) & C (11,5) & C (12, n) C (m, 7) & C (m, 6) & C (m+1, n+1) array |=0 for every m n ?

NDA 2024 Solution View → 18

If D _n= | array lll n & 20 & 30 n^2 & 40 & 50 n^3 & 60 & 70 array | then what is the value of _ n=1 ^4 D _n ?

NDA 2024 Solution View → 19

Let A be a skew-symmetric matrix of order 3 . What is the value of det (4 A^4 )- det (3 A^3 )+ det (2 A^2 )- det (A)+ det (-I) Where I is the identity matrix of order 3 ?

NDA 2023 Solution View → 20

If all elements of a third order determinant are equal to 1 or –1, then the value of the determinant is:

NDA 2023 Solution View → 21

The value of the determinant of a matrix A of order 3 is 3 . If C is the matrix of cofactors of the matrix A , then what is the value of determinant of C ^2 ?

NDA 2023 Solution View → 22

The system of linear equations x+2 y+z=4,2 x+4 y+2 z=8 and 3 x+6 y+3 z=10 has

NDA 2023 Solution View → 23

Let A X=B be a system of 3 liner equations with 3 -unknowns. Let X₁ and X₂ be its two distinct solutions. If the combination a X₁+b X₂ is a solution of A X=B ; where a, b , are rea

NDA 2023 Solution View → 24

What is the sum of the roots of the equation

NDA 2023 Solution View → 25

[QUESTION] Paragraph: If (7,4,2, )=0 , then is a root of which one of the following equation?

NDA 2023 Solution View → 26

Paragraph: What is f(0) equal to?

NDA 2023 Solution View → 27

Paragraph: What is _ x 0 f(x) x equal to?

NDA 2023 Solution View → 28

Paragraph: What is _ x 0 f(x) x^2 equal to?

NDA 2023 Solution View → 29

If a, b, c are in AP , then what is | array lll x+1 & x+2 & x+3 x+2 & x+3 & x+4 x+a & x+b & x+3 array | equal to?

NDA 2023 Solution View → 30

Paragraph: If (a, b, c, )=0 for every 0 , then which one of the following is correct?

NDA 2023 Solution View → 31

If | array ccc x^2-3 x & x-1 & x+3 x+1 & -2 x & x-4 x-3 & x+4 & 3 x array |=a x^4+b x^3+c x^2+d x+e , then what is the value of e ?

NDA 2023 Solution View → 32

If A_k= [ array cc k-1 & k k-2 & k+1 array ] , then what is det (A₁ )+ det (A₂ )+ det (A₃ )+ + det (A₁₀₀ ) equal to?

NDA 2023 Solution View → 33

Consider the determinant = | array lll a₁₁ & a₁₂ & a₁₃ a₂₁ & a₂₂ & a₂₃ a₃₁ & a₃₂ & a₃₃ array | If a₁₃=y z, a₂₃=z x, a₃₃=x y and the minors of a₁₃, a₂₃ , a₃₃ are respectively (z-y),

NDA 2022 Solution View → 34

Paragraph: Let be the determinant of a matrix A , where A= ( array lll a₁₁ & a₁₂ & a₁₃ a₂₁ & a₂₂ & a₂₃ a₃₁ & a₃₂ & a₃₃ array ) and C₁₁, C₁₂, C₁₃ be the cofactors of a₁₁, a₁₂, a₁₃ r

NDA 2022 Solution View → 35

Paragraph: Let be the determinant of a matrix A , where A= ( array lll a₁₁ & a₁₂ & a₁₃ a₂₁ & a₂₂ & a₂₃ a₃₁ & a₃₂ & a₃₃ array ) and C₁₁, C₁₂, C₁₃ be the cofactors of a₁₁, a₁₂, a₁₃ r

NDA 2022 Solution View → 36

Paragraph: Let be the determinant of a matrix A , where A= ( array lll a₁₁ & a₁₂ & a₁₃ a₂₁ & a₂₂ & a₂₃ a₃₁ & a₃₂ & a₃₃ array ) and C₁₁, C₁₂, C₁₃ be the cofactors of a₁₁, a₁₂, a₁₃ r

NDA 2022 Solution View → 37

Paragraph: Let A= ( array ccc 0 & ^2 & ^2 ^2 & 0 & ^2 ^2 & ^2 & 0 array ) and A=P+Q where P is symmetric matrix and Q is skew-symmetric matrix. Question: What is the minimum value

NDA 2022 Solution View → 38

If ₁= | array lll 1 & p & q 1 & q & r 1 & r & p array | and ₂= | array lll 1 & 1 & 1 q & r & p r & p & q array | where p q r , then ₁+ ₂ is

NDA 2022 Solution View → 39

If (a-b)(b-c)(c-a)=2 and a b c=6 , then what is the value of | array ccc a & b & c a^2 & b^2 & c^2 a^3 & b^3 & c^3 array | ?

NDA 2022 Solution View → 40

Under which of the following conditions does the determinant | array lll a & b & c b & c & a c & a & b array | vanish? 1. a+b+c=0 2. a^3+b^3+c^3=3 a b c 3. a^2+b^2+c^2-a b-b c-c a=

NDA 2022 Solution View → 41

For what value of k is the matrix [ array ccc 2 2 & 2 2 & 6 1-2 ^2 & 2 ^2 -1 & 3 k & 2 k & 1 array ] singular?

NDA 2022 Solution View → 42

What is the value of the following determinant? | array ccc C & A & 0 B & 0 & - A 0 & B & C array |

NDA 2021 Solution View → 43

If = | array lll a & b & c d & e & f g & h & i array | then what is | array ccc 3 d+5 g & 4 a+7 g & 6 g 3 e+5 h & 4 b+7 h & 6 h 3 f+5 i & 4 c+7 i & 6 i array | equal to?

NDA 2021 Solution View → 44

If is the value of the determinant | array lll a₁ & b₁ & c₁ a₂ & b₂ & c₂ a₃ & b₃ & c₃ array | then what is the value of the following determinant? | array lll p a₁ & b₁ & q c₁ a₂ &

NDA 2021 Solution View → 45

If | array ccc a & -b & a-b-c -a & b & -a+b-c -a & -b & -a-b+c array |-k a b c=0 (a 0, b 0, c 0) then what is the value of k ?

NDA 2021 Solution View → 46

Let A= | array lr p & q r & s array | where p, q, r and s are any four different prime numbers less than 20 . What is the maximum value of the determinant?

NDA 2021 Solution View → 47

If the determinant | array lll x & 1 & 3 0 & 0 & 1 1 & x & 4 array |=0 then what is x equal to?

NDA 2021 Solution View → 48

If f(x)= | array ccc 1 & x & x+1 2 x & x(x-1) & x(x+1) 3 x(x-1) & 2(x-1)(x-2) & x(x+1)(x-1) array | then what is f(-1)+f(0)+f(1) equal to?

NDA 2021 Solution View → 49

The element in the i^ th row and the j^ th column of a determinant of third order is equal to 2(i+j) . What is the value of the determinant?

NDA 2021 Solution View → 50

With the numbers 2,4,6,8 , all the possible determinants with these four different elements are constructed. What is the sum of the values of all such determinants?

NDA 2021 Solution View → 51

If a+b+c=4 and a b+b c+c a=0 , then what is the value of the following determinant? | array lll a & b & c b & c & a c & a & b array |

NDA 2021 Solution View → 52

If x= a b-c , y= b c-a , z= c a-b then what is the value of the following? | array ccc 1 & -x & x 1 & 1 & -y 1 & z & 1 array |

NDA 2021 Solution View → 53

For what values of k is the system of equations 2 k^2 x+3 y-1=0,7 x-2 y+3=0,6 k x+y+1=0 consistent?

NDA 2021 Solution View → 54

If a₁, a₂, a₃, , a ₉ are in GP, then what is the value of the following determinant? | array lll a₁ & a₂ & a₃ a₄ & a₅ & a₆ a₇ & a₈ & a₉ array |

NDA 2021 Solution View → 55

Let p, q and r be three distinct positive real numbers. If D= | array lll p & q & r q & r & p r & p & q array | , then which one of the following is correct?

NDA 2020 Solution View → 56

What is the value of the determinant | array ccc i & i^2 & i^3 i^4 & i^6 & i^8 i^9 & i¹² & i¹⁵ array | where i= -1 ?

NDA 2020 Solution View → 57

For how many values of k , is the matrix [ array ccc 0 & k & 4 -k & 0 & -5 -k & k & -1 array ] singular?

NDA 2020 Solution View → 58

What are the values of x that satisfy the equation | array lll x & 0 & 2 2 x & 2 & 1 1 & 1 & 1 array |+ | array lll 3 x & 0 & 2 x^2 & 2 & 1 0 & 1 & 1 array |=0 ?

NDA 2019 Solution View → 59

If x+a+b+c=0 , then what is the value of | array ccc x+a & b & c a & x+b & c a & b & x+c array | ?

NDA 2019 Solution View → 60

If [ array ccc x & -3 i & 1 y & 1 & i 0 & 2 i & -i array ]=6+11 i , then what are the values of x and y respectively?

NDA 2019 Solution View → 61

A is a square matrix of order 3 such that its determinant is 4. What is the determinant of its transpose?

NDA 2019 Solution View → 62

If A is a square matrix of order n 1 , then which one of the following is correct?

NDA 2019 Solution View → 63

What is the value of the determinant | array lll 1! & 2! & 3! 2! & 3! & 4! 3! & 4! & 5! array | ?

NDA 2019 Solution View → 64

Paragraph: Let A and B be (3 3) matrices with det A=4 and det B=3 . Question: What is det (2 AB ) equal to ?

NDA 2019 Solution View → 65

Which one of the following factors does the expansions of the determinant ( array ccc x & y & 3 x^2 & 5 y^3 & 9 x^3 & 10 y^3 & 27 array ) contain?

NDA 2018 Solution View → 66

If a+b+c=0 , then one of the solution of | array ccc a-x & c & b c & b-x & a b & a & c-x array |=0 is

NDA 2018 Solution View → 67

The system of equation aligned & 2 x+y-3 z=5 & 3 x-2 y+2 z=5 and & 5 x-3 y-z=16 aligned

NDA 2018 Solution View → 68

What is the determinant of the matrix ( array ccc x & y & y+z z & z & z+x y & z & x+y array ) ?

NDA 2018 Solution View → 69

A train covers the first 5 km of its journey at a speed of 30 km / hr and the next 15 km at a speed of 45 ~km / hr . What is the average speed of the train?

NDA 2018 Solution View → 70

If A, B and C are the angles of a triangle and | array ccc 1 & 1 & 1 1+ A & 1+ B & 1+ C ~A + ^2 ~A & ~B + ^2 ~B & C + ^2 C array |=0 then which one of the following is correct?

NDA 2018 Solution View → 71

The value of the determinant | array cc ^2 2 & ^2 2 ^2 2 & ^2 2 array | for all values of , is

NDA 2017 Solution View → 72

The value of the determinant | array ccc 1- & - ^2 & ^2 1- & - ^2 & ^2 1- & - ^2 & ^2 array | is equal to

NDA 2017 Solution View → 73

If p+q+r=a+b+c=0 , then the determinant | array lll p a & q b & r c q c & ra & p b r b & p c & q a array | equals

NDA 2017 Solution View → 74

What is the value of the determinant? | array ccc 1 & 1 & 1 1 & 1+ xyz & 1 1 & 1 & 1+ xyz array | ?

NDA 2017 Solution View → 75

If a b c , then one value of x which satisfies the equation | array ccc 0 & x-a & x-b x+a & 0 & x-c x+b & x+c & 0 array |=0 is given by

NDA 2017 Solution View → 76

If | array ccc x & y & 0 0 & x & y y & 0 & x array |=0 , then which one of the following is correct?

NDA 2017 Solution View → 77

The system of equations k x+y+z=1, x+k y+z=k and x+y+k z=k^2 has no solution if k equals

NDA 2017 Solution View → 78

The equations aligned & x+2 y+3 z=1 & 2 x+y+3 z=2 & 5 x+5 y+9 z=4 aligned

NDA 2017 Solution View → 79

Paragraph: Let a x^3+b x^2+c x+d= | array lcc x+1 & 2 x & 3 x 2 x+3 & x+1 & x 2-x & 3 x+4 & 5 x-1 array | then Question: What is the value of c ?

NDA 2016 Solution View → 80

Paragraph: Let a x^3+b x^2+c x+d= | array lcc x+1 & 2 x & 3 x 2 x+3 & x+1 & x 2-x & 3 x+4 & 5 x-1 array | then Question: What is the value of a+b+c+d ?

NDA 2016 Solution View → 81

Paragraph: For the system of linear equations 2 x+3 y+5 z=9,7 x+3 y-2 z=8 and 2 x+3 y+ z= Question: Under what condition does the above system of equations have unique solutions?

NDA 2016 Solution View → 82

Paragraph: For the system of linear equations 2 x+3 y+5 z=9,7 x+3 y-2 z=8 and 2 x+3 y+ z= Question: Under what condition does the above system of equations have infinitely many sol

NDA 2016 Solution View → 83

Consider the function f(x)= | array ccc x^3 & x & x 6 & -1 & 0 p & p^2 & p^3 array | , where p is a constant. What is the value of f ^ (0) ?

NDA 2016 Solution View → 84

Consider the function f(x)= | array ccc x^3 & x & x 6 & -1 & 0 p & p^2 & p^3 array | , where p is a constant. What is the value of p for which f^ (0)=0 ?

NDA 2016 Solution View → 85

Which of the following determinants have value 'zero'? 1. | array lll 41 & 1 & 5 79 & 7 & 9 29 & 5 & 3 array | 2. | array lll 1 & a & b + c 1 & ~b & c + a 1 & c & a + b array | 3.

NDA 2016 Solution View → 86

The system of linear equations kx + y + z =1, x + ky + z =1 and x+y+k z=1 has a unique solution under which one of the following conditions?

NDA 2016 Solution View → 87

Which of the following are correct in respect of the system of equations x+y+z=8, x-y+2 z=6 and 3 x-y+5 z=k ? 1. They have no solution. if k =15 . 2. They have infinitely many solu

NDA 2016 Solution View → 88

The matrix A = [ array ccc 1 & 3 & 2 1 & x-1 & 1 2 & 7 & x-3 array ] will have inverse for every real number x except for

NDA 2015 Solution View → 89

If the value of the determinant | array lll a & 1 & 1 1 & ~b & 1 1 & 1 & c array | is positive, where a b c , then the value of a b c

NDA 2015 Solution View → 90

Consider the following statements in respect of the determinant | array cc ^2 2 & ^2 2 ^2 2 & ^2 2 array | where , are complementary angles 1. The value of the determinant is 1 2 (

NDA 2015 Solution View → 91

If a, b, c are real numbers, then the value of the determinant | array lll 1-a & a-b-c & b+c 1-b & b-c-a & c+a 1-c & c-a-b & a+b array |

NDA 2015 Solution View → 92

The value of | array ccc 1 & 1 & 1 1 & 1+x & 1 1 & 1 & 1+y array | is

NDA 2015 Solution View → 93

If A and B are square matrices of second order such that | A |=-1,| ~B |=3 , then what is |3 AB | equal to?

NDA 2014 Solution View → 94

If a b c are all positive, then the value of the determinant | array lll a & b & c b & c & a c & a & b array | is

NDA 2014 Solution View → 95

If | array ccc 6 i & -3 i & 1 4 & 3 i & -1 20 & 3 & i array |=x+i y , where i= -1 , then what is x equal to?

NDA 2014 Solution View → 96

Consider the following statements: 1. Determinant is a square matrix. 2. Determinant is a number associated with a square matrix. Which of the above statements is/are correct?

NDA 2014 Solution View → 97

If | array lll a & b & 0 0 & a & b b & 0 & a array |=0 , then which one of the following is correct?

NDA 2014 Solution View → 98

One of the roots of | array ccc x+a & b & c a & x+b & c a & b & x+c array |=0 is

NDA 2014 Solution View → 99

The determinant of an odd order skew symmetric matrix is always:

NDA 2014 Solution View → 100

If any two adjacent rows or columns of a determinant are interchanged in position, the value of the determinant:

NDA 2014 Solution View → 101

What is the value of the determinant | array lll 1 & b c & a(b+c) 1 & c a & b(c+a) 1 & a b & c(a+b) array | ?

NDA 2013 Solution View → 102

If 8 x-9 y=20 and 7 x-10 y=9 , then what is 2 x-y equal to?

NDA 2013 Solution View → 103

Consider the following statements: 1. The matrix ( array ccc 1 & 2 & 1 a & 2 a & 1 b & 2 b & 1 array ) is singular. 2. The matrix ( array ccc c & 2 c & 1 a & 2 a & 1 b & 2 b & 1 ar

NDA 2013 Solution View → 104

For what value of k , the equations 3 x-y=8 and 9 x-k y=24 will have infinitely many solutions?

NDA 2013 Solution View → 105

What is the value of the minor of the element 9 in the determinant | array ccc 10 & 19 & 2 0 & 13 & 1 9 & 24 & 2 array | ?

NDA 2013 Solution View → 106

The value of the determinant | array ccc m & n & p p & m & n n & p & m array |

NDA 2013 Solution View → 107

If D is a determinant of order 3 and D^ is the determinant obtained by replacing the elements of D by their cofactors, then which one of the following is correct?

NDA 2013 Solution View → 108

The cofactor of the element 4 in the determinant | array lll 1 & 2 & 3 4 & 5 & 6 5 & 8 & 9 array | is

NDA 2013 Solution View → 109

The roots of the equation | array ccc 1 & t-1 & 1 t-1 & 1 & 1 1 & 1 & t-1 array |=0 are

NDA 2013 Solution View → 110

If each element in a row of a determinant is multiplied by the same factor r , then the value of the determinant?

NDA 2012 Solution View → 111

The value of the determinant | array lll x^2 & 1 & y^2+z^2 y^2 & 1 & z^2+x^2 z^2 & 1 & x^2+y^2 array | is :

NDA 2012 Solution View → 112

The solution of the simultaneous linear equations 2 x+y=6 and 3 y=8+4 x will also be satisfied by which one of the following linear equations?

NDA 2012 Solution View → 113

What is | array ccc -a^2 & a b & a c a b & -b^2 & b c a c & b c & -c^2 array | equal to?

NDA 2012 Solution View → 114

If | array ccc 8 & -5 & 1 5 & x & 1 6 & 3 & 1 array |=2 , then what is the value of x ?

NDA 2012 Solution View → 115

If two rows of a determinant are identical, then what is the value of the determinant?

NDA 2012 Solution View → 116

If the matrix [ array ccc & 2 & 2 -3 & 0 & 4 1 & -1 & 1 array ] is not invertible, then :

NDA 2012 Solution View → 117

The simultaneous equations 3 x+5 y=7 and 6 x+10 y=18 have

NDA 2011 Solution View → 118

The roots of the equation | array lll x & & 1 & x & 1 & & 1 array |=0 are independent of

NDA 2011 Solution View → 119

What is the value of the determinant | array lll a-b & b+c & a b-c & c+a & b c-a & a+b & c array | ?

NDA 2011 Solution View → 120

| array ccc p & -q & 0 0 & p & q q & 0 & p array |=0 , then which one of the following is correct?

NDA 2011 Solution View →
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