VBQs Determinants Class 12 Mathematics with solutions has been provided below for standard students. We have provided chapter wise VBQ for Class 12 Mathematics with solutions. The following Determinants Class 12 Mathematics value based questions with answers will come in your exams. Students should understand the concepts and learn the solved cased based VBQs provided below. This will help you to get better marks in class 12 examinations.

**Determinants VBQs Class 12 Mathematics**

**Question. (image 36) then K is equal to:**

(a) 1

(b) –1

(c) αβ

(d) 1/αβ

**Answer**

A

**Question. (image 41) then α is equal to**

(a) α + b + c

(b) αbc

(c) 4

(d) 1

**Answer**

C

**Question. The area of triangle formed by the lines oining the vertex of the parabola, x ^{2} = 8y, to the extremities of its latus rectum is**

(a) 2

(b) 8

(c) 1

(d) 4

**Answer**

B

**Question. Let ω be a complex number such that 2ω + 1 = where = -3. If (image 34) then k is equal to :**

(a) 1

(b) –z

(c) z

(d) –1

**Answer**

B

**Question. Let α, b, c be such that b(α + c) ≠ 0 if (image 44) then the value of n is :**

(a) any even integer

(b) any odd integer

(c) any integer

(d) zero

**Answer**

B

**Question. If 2 1,ω,ω ^{2} are the cube roots of unity, then (image 2) is equal to**

(a) ω

^{2}

(b) 0

(c) 1

(d) ω

**Answer**

C

**Question. If (image 45) for x ≠ 0, y ≠ 0 , then D is**

(a) divisible by x but not y

(b) divisible by y but not x

(c) divisible by neither x nor y

(d) divisible by both x and y

**Answer**

D

**Question. If a > 0 and discriminant of ax ^{2}+2bx+c is –ve, then (image 49) is equal to**

(a) +ve

(b) (ac-b

^{2})(ax

^{2}+2bx+c)

(c) –ve

(d) 0

**Answer**

C

**Question. If (image 56) then the matrix A ^{–50} when θ = π/12, is equal to:**

(image 56)

**Answer**

C

**Question. (image 37)**

(a) depends only on a

(b) depends only on n

(c) depends both on a and n

(d) is independent of both a and n

**Answer**

D

**Question. If (image 38) then k is equal to:**

(a) 4λabc

(b) – 4λabc

(c) 4λ^{2}

(d) – 4λ^{2}

**Answer**

C

**Question. l, m, n are the p ^{th}, q^{th} and r^{th} term of a G. P. all positive, then (image 50) equals**

(a) –1

(b) 2

(c) 1

(d) 0

**Answer**

D

**Question. Let A = {X = (x, y, z) ^{T} : PX = 0 and x^{2} + y^{2} + z^{2} = 1}, where (image 86), then the set A :**

(a) is a singleton

(b) is an empty set

(c) contains more than two elements

**(d) contains exactly two elements**

**Answer**

D

**Question. If the matrices (image 52) B = adj A and C = 3A, then | adj B | / |C| is equal to :**

(a) 8

(b) 16

(c) 72

(d) 2

**Answer**

A

**Question. The system of linear equations**

λx + 2y + 2z = 5

2λx + 3y + 5z = 8

4x + λy + 6z = 10 has:

(a) no solution when λ = 8

(b) a unique solution when λ = –8

(c) no solution when λ = 2

(d) infinitely many solutions when λ = 2

**Answer**

C

**Question. (image 54)**

(image 54)

**Answer**

B

**Question. Let A be a 3 × 3 matrix such that (image 65) Then A ^{–1} is:**

(image 65)

**Answer**

A

**Question. Let A be a matrix such that A. (image 57) is a scalar matrix and |3A| = 108. Then A ^{2} equals**

(image 57)

**Answer**

D

**Question. If (image 66) is the adjoint of a 3 × 3 matrix A and |A| = 4, then α is equal to :**

(a) 4

(b) 11

(c) 5

(d) 0

**Answer**

B

**Question. Suppose A is any 3 × 3 non-singular matrix and (A – 3I) (A – 5I) = O, where I = I _{3} and O = O_{3}. If αA + βA^{–1} = 4I, then α + β is equal to**

(a) 8

(b) 12

(c) 13

(d) 7

**Answer**

A

**Question. If (image 61) and A adj A = A A ^{T}, then 5a + b is equal to:**

(a) 4

(b) 13

(c) –1

(d) 5

**Answer**

D

**Question. If (image 53) is the inverse of a 3 × 3 matrix A, then the sum of all values of α for which det (A) + 1 = 0, is :**

(a) 0

(b) –1

(c) 1

(d) 2

**Answer**

C

**Question. If A is a 3 × 3 matrix such that |5.adj A| = 5, then |A| is equal to :**

(a) ± 1/5

(b) ± 1/25

(c) ±1

(d) ±5

**Answer**

A

**Question. If the system of linear equations**

x + ky + 3 = 0

3x + ky – 2 = 0

2x + 4y – 3 = 0

has a non-zero solution (x, y, ), then xZ/y^{2} is equal to :

(a) 10

(b) – 30

(c) 30

(d) – 10

**Answer**

A

**Question. If A ^{T} denotes the transpose of the matrix (image 69) where a, b, c, d, e and f are integers such that abd ≠ 0, then the number of such matrices for which A^{–1} = A^{T} is**

(a) 2(3!)

(b) 3(2!)

(c) 2

^{3}

(d) 3

^{2}

**Answer**

C

**Question. Let A be a 3 × 3 matrix such that adj (image 51) and B = adj(adj A). If | A|=λ and |(B ^{-1})^{T} | = μ then the ordered pair, (| λ |, μ) is equal to :**

(a) (3, 1/81)

(b) (9, 1/9)

(c) (3, 81)

(d) (9, 1/81)

**Answer**

A

**Question. If A is an 3 × 3 non-singular matrix such that AA’ = A’A and B = A ^{–1}A’, then BB’ equals:**

(a) B

^{–1}

(b) ( B

^{-1})’

(c) I + B

(d) I

**Answer**

D

**Question. Let P and Q be 3 × 3 matrices P ≠ Q. If P ^{3}= Q^{3} and P^{2}Q = Q^{2}P then determinant of (P^{2} + Q^{2}) is equal to :**

(a) – 2

(b) 1

(c) 0

(d) – 1

**Answer**

C

**Question. The number of real values of λ for which the system of linear equations**

2x + 4y – λ = 0

4x + λy + 2 = 0

λx + 2y + 2 = 0

has infinitely many solutions, is :

(a) 0

(b) 1

(c) 2

(d) 3

**Answer**

B

**Question. If the system of linear equations**

x + y + = 5

x + 2y + 2z = 6

x + 3y + λz = μ, (λ, μ ∈ R), has infinitely many solutions, then the value of λ + μ is :

(a) 12

(b) 9

(c) 7

(d) 10

**Answer**

D

**Question. If (image 59) then adj (3A ^{2} + 12A) is equal to :**

(image 59)

**Answer**

C

**Question. Let A be any 3 × 3 invertible matrix. Then which one of the following is not always true ?**

(a) adj (A)= |A| . A^{–1}

(b) adj (adj (A)) = |A|.A

(c) adj (adj (A)) = |A|^{2} .(adj (A))^{–1}

(d) adj (adj (A)) = |A|.(adj (A))^{–1}

**Answer**

B

**Question. Let (image 68) If u _{1} and u_{2} are column matrices such that (image 68) then u_{1} + u_{2} is equal to :**

(image 68)

**Answer**

D

**Question. Consider the following relation R on the set of real square matrices of order 3.**

R = { (A,B)A = P^{–1} BP for some invertible matrix P}**Statement-1 :** R is equivalence relation.**Statement-2 :** For any two invertible 3 × 3 matrices M and N, (MN)^{-1} = N^{-1}M^{-1}.

(a) Statement-1 is true, statement-2 is true and statement-2 is a correct explanation for statement-1.

(b) Statement-1 is true, statement-2 is true; statement-2 is not a correct explanation for statement-1.

(c) Statement-1 is true, stement-2 is false.

(d) Statement-1 is false, statement-2 is true.

**Answer**

B

**Question. Let A and B be real matrices of the form (image 70) respectively.**

Statement 1: AB – BA is always an invertible matrix.

Statement 2: AB – BA is never an identity matrix.

(a) Statement 1 is true, Statement 2 is false.

(b) Statement 1 is false, Statement 2 is true.

(c) Statement 1 is true, Statement 2 is true; Statement 2 is a correct explanation of Statement 1.

(d) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation of Statement 1.

**Answer**

A

**Question. Let A be a 2 × 2 matrix****Statement -1 :** adj (adj A) = A**Statement -2 :** |adj A |= |A|

(a) Statement-1 is true, Statement-2 is true. Statement-2 is not a correct explanation for Statement-1.

(b) Statement-1 is true, Statement-2 is false.

(c) Statement -1 is false, Statement-2 is true.

(d) Statement-1 is true, Statement -2 is true. Statement-2 is a correct explanation for Statement-1.

**Answer**

A

**Question. Let λ ∈ R . The system of linear equations**

2x_{1} – 4x_{2} + λx_{3} =1

x_{1} – 6x_{2} + x_{3} = 2

λx_{1} -10x_{2} + 4x_{3} = 3**(a) exactly one negative value of λ**

(b) exactly one positive value of λ

(c) every value of λ

(d) exactly two value of λ

**Answer**

A

**Question. Let A be a square matrix all of whose entries are integers. Then which one of the following is true?**

(a) If det A = ± 1, then A^{–1} exists but all its entries are not necessarily integers

(b) If det A ≠ ± 1, then A^{–1} exists and all its entries are non integers

(c) If det A = ± 1, then A^{–1} exists but all its entries are integers

(d) If det A = ± 1, then A^{–1} need not exists

**Answer**

C

**Question. Suppose the vectors x _{1}, x_{2} and x_{3} are the solutions of the system of linear equations, Ax = b when the vector b on the right side is equal to b_{1}, b_{2} and b_{3} respectively. If (image 82) then the determinant of A is equal to :**

(a) 4

(b) 2

(c) 1/2

(d) 3/2

**Answer**

B

**Question. Let A and B be two invertible matrices of order 3 × 3. If det (ABA ^{T}) = 8 and det (AB^{–1}) = 8, then det (BA^{–1} B^{T}) is equal to :**

(a) 1/4

(b) 1

(c) 1/16

(d) 16

**Answer**

C

**Question. If A ^{2} – A+ I = 0 , then the inverse of A is**

(a) A + I

(b) A

(c) A – I

(d) I – A

**Answer**

D

**Question. Let S be the set of all λ ∈ R for which the system of linear equations**

2x – y + 2z = 2

x – 2y + λz = -4

x + λy + z = 4

has no solution. Then the set S

(a) contains more than two elements.

(b) is an empty set.

(c) is a singleton.

(d) contains exactly two elements.

**Answer**

D

**Question. Let (image 75) If B is the inverse of matrix A, then α is**

(a) 5

(b) –1

(c) 2

(d) –2

**Answer**

A

**Question. If the system of linear equations**

x + y + 3z = 0

2 x + 3y + k^{2}z = 0

3x + y + 3z = 0

has a non- ero solution (x, y, z) for some k ∈ R, then x + (y/z) is equal to :

(a) – 3

(b) 9

(c) 3

(d) – 9

**Answer**

A

**Question. If the system of equations x – 2y +3z = 9 , 2x + y + z = b x -7y + az = 24, has infinitely many solutions, then a – b is equal to __________.**

**Answer**

5000

**Question. The sum of distinct values of λ for whcih the system of equations**

(λ -1)x + (3λ +1) y + 2λz = 0

(λ -1)x + (4λ – 2) y + (λ + 3)z = 0

2x + (3λ + 1) y + 3(λ – 1) z = 0,

has non- ero solutions, is ______.

**Answer**

300

**Question. Let S be the set of all integer solutions, (x, y, z), of the system of equations**

x – 2y + 5z = 0

-2x + 4y + z = 0

-7x +14y + 9z = 0

such that 15 ≤ x^{2} + y^{2} + z^{2} ≤ 150. Then, the number of elements in the set S is equal to ____________.

**Answer**

8

**Question. Let (image 76) The only correct statement about the matrix A is**

(a) A^{2} = I

(b) A = (–1) I, where I is a unit matrix

(c) A^{-1} does not exist

(d) A is a zero matrix

**Answer**

A

**Question. The set of all values of λ for which the system of linear equations**

x – 2y – 2 = λx

x + 2y + = λy

–x – y = λ2

has a non-trivial solution :

(a) is a singleton

(b) contains exactly two elements

(c) is an empty set

(d) contains more than two elements

**Answer**

A

**Question. If the system of equations**

x + y + z = 2

2x + 4y – z = 6

3x + 2y + λz = μ

has infinitely many solutions, then :

(a) λ + 2μ = 14

(b) 2λ – μ = 5

(c) λ – 2μ = -5

(d) 2λ + μ = 14

**Answer**

D

**Question. If the system of linear equations**

2x + 2y + 3 = a

3x – y + 5 = b

x – 3y + 2 = c

where, a, b, c are non- ero real numbers, has more than one solution, then :

(a) b – c + a = 0

(b) b – c – a = 0

(c) a + b + c = 0

(d) b + c – a = 0

**Answer**

B

**Question. If a, b, c are non- ero real numbers and if the system of equations**

(a – 1)x = y + z,

(b – 1)y = z + x,

(c – 1)z = x + y,

has a non-trivial solution, then ab + bc + ca equals:

(a) a + b + c

(b) abc

(c) 1

(d) – 1

**Answer**

B

**Question. An ordered pair (α, β) for which the system of linear equations**

(1 + α) x + β+ = 2

αx + (1 + β)y + = 3

αx + βy + 2 = 2

has a unique solution, is :

(a) (2, 4)

(b) (–3, 1)

(c) (–4, 2)

(d) (1, – 3)

**Answer**

A

**Question. The greatest value of c ∈ R for which the system of linear equations x – cy – cz = 0; cx – y + cz = 0; cx + cy – z = 0 has a non-trivial solution, is :**

(a) –1

(b) 1/2

(c) 2

(d) 0

**Answer**

B

**Question. The set of all values of λ for which the system of linear equations :**

2x_{1} – 2x_{2} + x_{3} = λx_{1}

2x_{1} – 3x_{2}+ 2x_{3} = λx_{2}

–x_{1} + 2x_{2} = λx_{3}

has a non-trivial solution,

(a) contains two elements.

(b) contains more than two elements

(c) is an empty set.

(d) is a singleton

**Answer**

A

**Question. For which of the following ordered pairs (μ, δ), the system of linear equations**

x + 2y + 3z = 1

3x + 4y + 5z = μ

4x + 4y + 4z = δ

is inconsistent?

(a) (4, 3)

(b) (4, 6)

(c) (1, 0)

(d) (3, 4)

**Answer**

A

**Question. If the system of linear equations**

x – 2y + kz = 1

2x + y + z = 2

3x – y – kz = 3

has a solution (x, y, z), z ≠ 0, then (x, y) lies on the straight line whose equation is :

(a) 3x – 4y – 1 = 0

(b) 4x – 3y – 4 = 0

(c) 4x – 3y – 1 = 0

(d) 3x – 4y – 4 = 0

**Answer**

B

**Question. The values of λ and μ for which the system of linear equations**

x + y + z = 2

x + 2y + 3z = 5

x + 3y + λz = μ

has infinitely many solutions are, respectively :

(a) 6 and 8

(b) 5 and 7

(c) 5 and 8

(d) 4 and 9

**Answer**

C

**Question. If the system of linear equations**

2x + 2ay + az = 0

2x + 3by + bz = 0

2x + 4cy + cz = 0,

where a, b, c ∈ R are non-zero and distinct; has a non-zero solution, then:

(a) 1/a, 1/b, 1/c are in A.P.

(b) a, b, c are in G.P.

(c) a + b + c = 0

(d) a, b, c are in A.P.

**Answer**

A

**Question. The following system of linear equations**

7x + 6y – 2z = 0

3x + 4y + 2z = 0

x – 2y – 6z = 0, has

(a) infinitely many solutions, (x, y, z) satisfying y = 2z.

(b) no solution.

(c) infinitely many solutions, (x, y, z) satisfying x = 2z.

(d) only the trivial solution.

**Answer**

C

**Question. The number of values of q ∈ (0, π) for which the system of linear equations**

x + 3y + 7z = 0

– x + 4y + 7z = 0

(sin 3θ)x + (cos 2θ)y + 2z = 0

has a non-trivial solution, is

(a) three

(b) two

(c) four

(d) one

**Answer**

B

**Question. If the system of linear equations :**

x_{1} + 2x_{2} + 3x_{3} = 6

x_{1}+ 3x_{2} + 5x_{3} = 9

2x_{1} + 5x_{2}+ ax_{3} = b

is consistent and has infinite number of solutions, then :

(a) a = 8, b can be any real number

(b) b = 15, a can be any real number

(c) a ∈ R -{8} and b ∈ R -{15}

(d) a = 8, b = 15

**Answer**

D

**Question. If the system of linear equations**

x – 4y + 7z = g

3y – 5z = h

– 2x + 5y – 9z = k

is consistent, then :

(a) g + 2h + k = 0

(b) g + h + 2k = 0

(c) 2g + h + k = 0

(d) g + h + k = 0

**Answer**

C

**Question. The number of values of k for which the linear equations 4x + ky + 2z = 0 , kx + 4y + z = 0 and 2x + 2y + z = 0 possess a non-zero solution is**

(a) 2

(b) 1

(c) zero

(d) 3

**Answer**

A

**Question. The system of equations**

ax + y + z = a – 1

x + ay + z = a – 1

x + y + az = a – 1

has infinite solutions, if a is

(a) – 2

(b) either – 2 or 1

(c) not – 2

(d) 1

**Answer**

A

**Question. Let S be the set of all real values of k for which the system of linear equations**

x + y + z = 2

2x + y – z = 3

3x + 2y + kz = 4

has a unique solution. Then S is

(a) an empty set

(b) equal to R – {0}

(c) equal to {0}

(d) equal to R

**Answer**

B

**Question. Consider the system of linear equations;**

x_{1} + 2x_{2} + x_{3} = 3

2x_{1}+ 3x_{2} + x_{3} = 3

3x_{1} + 5x_{2} + 2x_{3}= 1

The system has

(a) exactly 3 solutions

(b) a unique solution

(c) no solution

(d) infinite number of solutions

**Answer**

C

**Question. Let a, b, c be any real numbers. Suppose that there are real numbers x, y, z not all ero such that x = cy + bz, y = az + cx, and z = bx + ay. Then a ^{2} + b^{2} + c^{2} + 2abc is equal to**

(a) 2

(b) –1

(c) 0

(d) 1

**Answer**

D

**Question. If the system of equations 2x + 3y – z =0, x + ky – 2z = 0 and 2x – y + z = 0 has a non-trivial solution (x, y, z), then x/y + y/z + z/x + k is equal to :**

(a) 3/4

(b) 1/2

(c) – 1/4

(d) –4

**Answer**

B

**Question. If the system of linear equations**

x + ay + z = 3

x + 2y + 2z = 6

x + 5y + 3z = b

has no solution, then

(a) a = 1, b ≠ 9

(b) a ≠ – 1, b = 9

(c) a = – 1, b = 9

(d) a = – 1, b ≠ 9

**Answer**

D

**Question. The system of linear equations**

x + λy – z = 0

λx – y – z = 0

x + y – λz = 0

has a non-trivial solution for:

(a) exactly two values of λ.

(b) exactly three values of λ.

(c) infinitely many values of λ.

(d) exactly one value of λ.

**Answer**

B

**Question. The number of values of k, for which the system of equations:**

(k + 1) x + 8y = 4k

kx + (k + 3)y = 3k – 1

has no solution, is

(a) infinite

(b) 1

(c) 2

(d) 3

**Answer**

B

**Question. If S is the set of distinct values of ‘b’ for which the following system of linear equations**

x + y + z = 1

x + ay + z = 1

ax + by + z = 0

has no solution, then S is :

(a) a singleton

(b) an empty set

(c) an infinite set

(d) a finite set containing two or more elements

**Answer**

A

**Question. Consider the system of equations :**

x + ay = 0, y + az = 0 and z + ax = 0. Then the set of all real values of ‘a’ for which the system has a unique solution is:

(a) R – {1}

(b) R – { – 1}

(c) {1, – 1}

(d) {1, 0, –1}

**Answer**

B

**Question**. **Statement-1:** The system of linear equations

x + (sin α) y + (cos α) z = 0

x + (cos α) y + (sin α) z = 0

x – (sin α) y – (cos α) z = 0

has a non-trivial solution for only one value of α lying in the interval (0, π/2).**Statement-2:** The equation in α (image 114) has only one solution lying in the interval (0, π/2).

(a) Statement-1 is true, Statement-2 is true, Statement-2 is not correct explantion for Statement-1.

(b) Statement-1 is true, Statement-2 is true, Statement-2 is a correct explantion for Statement-1.

(c) Statement-1 is true, Statement-2 is false.

(d) Statememt-1 is false, Statement-2 is true.

**Answer**

C

**Question. If the system of equations**

x + y + z = 5

x + 2y + 3z = 9

x + 3y + αz = β

has infinitely many solutions, then β – α equals:

(a) 21

(b) 8

(c) 18

(d) 5

**Answer**

B

116. **Statement 1:** If the system of equations x + ky + 3z = 0, 3x + ky – 2z = 0, 2x + 3y – 4z = 0 has a non-trivial solution, then the value of k is 31/2.**Statement 2:** A system of three homogeneous equations in three variables has a non trivial solution if the determinant of the coefficient matrix is zero.

(a) StatemenBt 1 is false, Statement 2 is true.

(b) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.

(c) Statement 1 is true, Statement 2 is true, , Statement 2 is not a correct explanation for Statement 1.

(d) Statement 1 is true, Statement 2 is false.

**Answer**

A

**Question. If the system of equations**

x + y + z = 6

x + 2y + 3z = 10

x + 2y + λz = 0

has a unique solution, then λ is not equal to

(a) 1

(b) 0

(c) 2

(d) 3

**Answer**

D

**Question. If the trivial solution is the only solution of the system of equations**

x – ky + z = 0

kx + 3y – kz = 0

3x + y – z = 0

then the set of all values of k is :

(a) R -{2,-3}

(b) R -{2}

(c) R -{-3}

(d) {2,-3}

**Answer**

A

**Question. Let λ be a real number for which the system of linear equations:**

x + y + z = 6

4x + λy – λz = λ –2

3x + 2y – 4z = –5

has infinitely many solutions. Then l is a root of the quadratic equation :

(a) λ^{2} + 3λ– 4 = 0

(b) λ^{2} – 3λ – 4 = 0

(c) λ^{2} +λ – 6 = 0

(d) λ^{2} – λ – 6 = 0

**Answer**

D

**Question. If the system of linear equations**

x + 2ay + az = 0 ; x + 3by + bz = 0 ;

x + 4cy + cz = 0 has a non -zero solution, then a, b, c.

(a) satisfy a + 2b + 3c = 0

(b) are in A.P

(c) are in G..P

(d) are in H.P.

**Answer**

B

**Question. The number of values of k for which the system of linear equations, (k + 2) x + 10y = k, kx + (k + 3) y = k – 1 has no solution, is**

(a) Infinitely many

(b) 3

(c) 1

(d) 2

**Answer**

C

**Question. Let A be a 3 × 3 matrix such that A ^{2} – 5A + 7I = 0.**

**Statement–I :**A

^{-1}= 1/7 (5I – A)

**Statement II :**the polynomial A

^{3}– 2A

^{2}– 3A + α can be reduced to 5 (A – 4I).

Then :

(a) Both the statements are true.

(b) Both the statements are false.

(c) Statement–I is true, but Statement-II is false.

(d) Statement I is false, but Statement-II is true.

**Answer**

D