CBSE Class 12 Mathematics Determinants MCQs Set A

Question: For 3 × 3 matrices M and N, which of the following statement(s) is (are) not correct ?
(a) T N M N is symmetric or skew-symmetric, according as M is symmetric or skew-symmetric
(b) MN – NM is is symmetric for all symmetric matrices M and N
(c) M N is symmetric for all symmetric matrices M and N
(d) (adj M) (adj N) = adj (MN) for all invertible matrices M and N

Answer: c,d

Question: If ω is a cube root of unity and △ (Image 80) then △2 is equal to:
(a) −ω
(b) ω
(c) 1
(d) ω2

Answer: b

Question: If the system of linear equations x + 2ay + az = 0 x + 3by + bz = 0, x +3cy + cz = 0 has a non-zero solution, then a,b,c ?
(a) Are in AP.
(b) Are in G.P.
(c) Are in H.P.
(d) Satisfy

Answer: c

Question: If the value of a third order determinant is 11, then the value of the square of the determinant formed by the cofactors will be:
(a) 11
(b) 121
(c) 1331
(d) 14641

Answer: d

Question: Let M be a 2 × 2 symmetric matrix with integer entries. Then, M is invertible, if:
(a) the first column of M is the transpose of the second row of M
(b) the second row of M is the transpose of the first column of M
(c) M is a diagonal matrix with non-zero entries in the main diagonal
(d) the product of entries in the main diagonal of M is not the square of an integer.

Answer: c,d

Question: If the system of equations x+ ay= 0, az+y = 0 and ax+ z + = 0 has infinite solutions, then the value of a is
(a) 0
(b) – 1
(c) 1
(d) no real values

Answer: c

Question: The equations x+ y + z = 6, x + 2y +3z =10, x+2y +mz = n give infinite number of values of the triplet (x, y, z) if:
(a) m = 3, n ∈ R
(b) m = 3, n ≠ 10
(c) m = 3, n = 10
(d) None of these

Answer: c

Question: The three lines ax + by + c = 0, bx + cy + a = 0, cx + ay + b = 0 are concurrent only when:
(a) a + b + c = 0
(b) a2 + b2 + c2 = ab + bc + ca
(c) a3 + b3 + c3 = ab + bc + ca
(d) None of these

Answer: a,b

Question: If the system of equations x + 2y − 3z =1, (k + 3)z = 3, (2k +1)x + z = 0 is inconsistent, then the value of k is:
(a) –3
(b) 1/2
(c) 0
(d) 2

Answer: a

Question: Let M and N be two 3×3 non-singular skew-symmetric matrices such that MN = NM. If PT denotes the transpose of P, then M2 N2 (M^T N)^-1 (MN⁻¹⁾ is equal to: 
(a) M2
(b) –N²
(c) –M²
(d) MN

Answer: C

Question: Let ω be a complex cube root of unity with ω ≠ 1 and P = [pij] be a n × n matrix with . Pe =ω^i+j Then, p2 ≠ 0 when n is equal to:   
(a) 57
(b) 55
(c) 58
(d) 56

Answer: B,C,D

Question: Let M and N be two 3×3 matrices such that MN = NM.        
Further, if M ≠ N² and M² = N4 , then:
(a) determinant of (M² + MN2) is 0
(b) there is a 3×3 non-zero matrix U such that (M² + MN²) U is zero matrix
(c) determinant of (M² + MN²)≥1
(d) for a 3×3 matrix U, if (M² + MN²) U equals the zero matrix, then U is the zero matrix.

Answer: A,B

Question: If the system of equations x + λy + 2 = 0, λx + y – 2 = 0, λx + λy + 3 = 0 is consistent, then
(a) λ = ±1
(b) λ = ± 2
(c) λ = 1, – 2
(d) λ = –1, 2

Answer: A

Question: The system of simultaneous linear equations kx + 2y – z = 1, (k – 1) y – 2z = 2 and (k + 2) z = 3 have a unique solution if k equals:
(a) – 1
(b) – 2
(c) 0
(d) 1

Answer: A

Question: If any two rows (or columns) of a determinant are identical (all corresponding elements are same), then the value of determinant is
(a) 1
(b) –1
(c) 0
(d) 2

Answer: C

Question: If A and B are two square matrices such that B = – A^–1 BA, then (A + B)² =
(a) 0
(b) A² + B²
(c) A² + 2 AB + B²
(d) A + B

Answer: B

Question: If a square matrix satisfies the relation A² + A – I = 0 then A–1:
(a) exists and equals I + A
(b) exists and equals I – A
(c) exists and equals A²
(d) None of these

Answer: A

Question: If I₃ is the identity matrix of order 3, then I₃^–1 is
(a) 0
(b) 3I₃
(c) I₃
(d) Does not exist

Answer: C

Question: The equations 2x + 3y + 4 = 0; 3x + 4y + 6 = 0 and 4x + 5y + 8 = 0 are
(a) consistent with unique solution
(b) inconsistent
(c) consistent with infinitely many solutions
(d) None of the above

Answer: A

Question: If B is a non-singular matrix and A is a square matrix, then det (B^–1 AB) is equal to
(a) det (A^–1)
(b) det (B^–1)
(c) det (A)
(d) det (B)

Answer: C

STATEMENT TYPE QUESTIONS

Question: Consider the following statements
I. Matrix cannot be reduced to a number.
II. Determinant can be reduced to a number.
(a) Only I is true
(b) Only II is true
(c) Both I and II are true
(d) Neither I nor II is true

Answer: C

Question: Consider the following statements
I. | A | is also called modulus of square matrix A.
II. Every matrix has determinant.
(a) Only I is true
(b) Only II is true
(c) Both I and II are true
(d) Neither I nor II is true

Answer: D

Question: Consider the following statements
I. If any two rows (or columns) of a determinant are interchanged, then sign of determinant changes.
II. If any two rows (or columns) of a determinant are interchanged, then the value of the determinant
remains same.
(a) Only I is true
(b) Only II is true
(c) Both I and II are true
(d) Neither I nor II is true

Answer: A

Question: Consider the following statements
I. To every rectangular matrix A = [a_ij] of order n, we can associate a number (real or complex) called determinant of A.
II. Determinant is a function which associates each square matrix with a unique number (real or complex).
(a) Only I is true
(b) Only II is true
(c) Both I and II are true
(d) Neither I nor II is true

Answer: B

Assertion and Reason

Note: Read the Assertion (A) and Reason (R) carefully to mark the correct option out of the options given below:
(a) If both assertion and reason are true and the reason is the correct explanation of the assertion.
(b) If both assertion and reason are true but reason is not the correct explanation of the assertion.
(c) If assertion is true but reason is false.
(d) If the assertion and reason both are false.
e. If assertion is false but reason is true.

Question: Let A be a 2×2 matrix with non-zero entries and let A2 =I
where I is 2×2 identity matrix. Define Tr (A) = sum of diagonal elements of A and |A| = determinant of matrix A:
Assertion: Tr (A) =0
Reason: |A|=1.

Answer: b

Question: Let A be a 2×2matrix
Assertion: adj(adjA) = A
Reason: |adjA| = |A|

Answer: b

Question: Assertion: If A∈Mn(R), A ≠ O with det (A) = 0, then det
(Adj A) = 0
Reason: For (R), A∈Mn det (Adj A) = (det 1 )n An-1
Answer: a

Question: Consider the system of equations x − 2y + 3z = −1,
x − 3y + 4z =1 and −x + y − 2z = k

Answer: a