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Determinants Class 12 MCQs Questions with Answers
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Question 1.
If \(\left|\begin{array}{rr}
x & 2 \\
18 & x
\end{array}\right|\) = \(\left|\begin{array}{rr}
6 & 2 \\
18 & 6
\end{array}\right|\), then x is equal to
(a) 6
(b) ±6
(c) -6
(d) 6, 6
Answer
Answer: (a) 6
Question 2.
Let A be a square matrix of order 3 × 3. Then |kA| is equal to
(a) k|A|
(b) k²|A|
(c) k³|A|
(d) 3k|A|
Answer
Answer: (c) k³|A|
Question 3.
Which of the following is correct?
(a) Determinant is a square matrix
(b) Determinant is a number associated to a matrix
(c) Determinant is a number associated to a square matrix
(d) None of these.
Answer
Answer: (c) Determinant is a number associated to a square matrix
Question 4.
If area of triangle is 35 sq. units with vertices (2, -6), (5, 4) and (k, 4). Then k is
(a) 12
(b) -2
(c) -12, -2
(d) 12, -2.
Answer
Answer: (d) 12, -2.
Question 5.
If A = \(\left[\begin{array}{lll}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{array}\right]\) and Aij is co-factors of aij, then A is given by
(a) a11A31 +a12A32 + a13A33
(b) a11A11 + a12A21 + a13A33
(c) a21A11 + a22A12 + a23A13
(d) a11A11 + a21A21 + a31A31
Answer
Answer: (d) a11A11 + a21A21 + a31A31
Question 6.
Let A be a non-singular matrix of order 3 × 3. Then |adj. A| is equal to
(a) |A|
(b) |A|²
(c) |A|³
(d) 3|A|
Answer
Answer: (b) |A|²
Question 7.
If A is any square matrix of order 3 x 3 such that |a| = 3, then the value of |adj. A| is?
(a) 3
(b) \(\frac { 1 }{3}\)
(c) 9
(d) 27
Answer
Answer: (c) 9
Question 8.
If A is an invertible matrix of order 2, then det (A-1) is equal to
(a) det (A)
(b) \(\frac { 1 }{det(A)}\)
(c) 1
(d) 0
Answer
Answer: (b) \(\frac { 1 }{det(A)}\)
Question 9.
If a, b, c are in A.P., then determinant
(a) 0
(b) 1
(c) x
(d) 2x
Answer
Answer: (a) 0
Question 10.
If x, y, z are non-zero real numbers, then the inverse of matrix A = \(\left[\begin{array}{lll}
x & 0 & 0 \\
0 & y & 0 \\
0 & 0 & z
\end{array}\right]\) is
Answer
Answer:
Question 11.
Let A = 
where 0 ≤ θ ≤ 2π then
(a) Det (A) = 0
(b) Det (A) ∈ (2, ∞)
(c) Det (A) ∈ (2, 4)
(d) Det (A) ∈ [2, 4]
Answer
Answer: (d) Det (A) ∈ [2, 4]
Question 12.
If \(\left|\begin{array}{rr}
2x & 5 \\
8 & x
\end{array}\right|\) = \(\left|\begin{array}{rr}
6 & -2 \\
7 & 3
\end{array}\right|\), then value of ‘x’ is
(a) 3
(b) ±3
(c) ±6
(d) 6
Answer
Answer: (c) ±6
Question 13.
Let Δ = \(\left|\begin{array}{lll}
\mathbf{A} x & x^{2} & 1 \\
B y & y^{2} & 1 \\
C z & z^{2} & 1
\end{array}\right|\) and Δ1 = \(\left|\begin{array}{rrr}
\mathbf{A} & \mathbf{B} & \mathbf{C} \\
\boldsymbol{x} & \boldsymbol{y} & z \\
z \boldsymbol{y} & z x & x y
\end{array}\right|\), then
(a) Δ1 = -Δ
(b) Δ ≠ Δ1
(c) Δ – Δ1 = 0
(d) None of these
Answer
Answer: (c) Δ – Δ1 = 0
Question 14.
If x, y ∈R, then the determinant:
lies in the interval
(a) [-√2, √2]
(b) [-1, 1]
(c) [√2, 1]
(d) [-1, √2]
Answer
Answer: (a) [-√2, √2]
Question 15.
The area of a triangle with vertices (-3, 0), (3, 0) and (0, k) is 9 sq. units. The value of ‘k’ will be:
(a) 9
(b) 3
(c) -9
(d) 6.
Answer
Answer: (b) 3
Question 16.
If A, B and C are angles of a triangle, then the determinant:
\(\left|\begin{array}{ccc}
-1 & \cos C & \cos B \\
\cos C & -1 & \cos A \\
\cos B & \cos A & -1
\end{array}\right|\) is equal to
(a) 0
(b) -1
(c) 1
(d) None of these.
Answer
Answer: (a) 0
Question 17.
Let A be a square matrix all of whose entries are integers. Then which of the following is true?
(a) If det A = ± 1, then A-1 need not exist
(b) If det A = ± 1, then A-1 exists but all entries are not necessarily integers.
(c) If det A ≠ ± 1, then A-1 exists and all its entries are non-integers
(d) If det A = ± 1, then A-1 exists and all its entries are integers.
Answer
Answer: (d) If det A = ± 1, then A-1 exists and all its entries are integers.
Hint:
Since each entry of A is an integer,
∴ co-factor of each entry is also an integer.
Hence, each entry of the adjoint is an integer.
Also det A = ± 1 and A-1 = \(\frac { 1 }{det(A)}\) (adj A).
Hence, all entries of A-1 are integers.
Question 18.
The number of values of ‘k’ for which the linear equations:
4x + ky + 2z = 0
kx + 4y + z = 0
2x + 2y + z = 0
possesses a non-zero solution is
(a) 3
(b) 2
(c) 1
(d) zero.
Answer
Answer: (b) 2
Hint:
The system possesses non-zero solution
If \(\left|\begin{array}{lll}
4 & k & 2 \\
k & 4 & 1 \\
2 & 2 & 1
\end{array}\right|\) = 0
If 4(4 – 2) + k (k – 2) + 2(2k – 8) = 0
if = 8 – k² + 2k + 4k – 16 = 0
if k² – 6k + 8 = 0
if (k – 2)(k – 4) = 0
if k = 2 or 4
k = 2.
Question 19.
If A = \(\left[\begin{array}{lll}
1 & \alpha & 3 \\
1 & 3 & 3 \\
2 & 4 & 4
\end{array}\right]\) is the adjoint of a 3 × 3 matrix A and |A| = 4 then α is equal to
(a) 11
(b) 5
(c) 0
(d) 4
Answer
Answer: (a) 11
Hint:
Here |adj A| = |A|3-1
= |A|² = 4²
= 16
⇒ 1. (12 – 12) -α (4 – 6)+ 3(4 – 6) = 16
⇒ 2α – 6 = 16
⇒ 2α = 22.
Hence, α = 11.
Question 20.
If α, ß ≠ 0 and f(x) = α” + ß” and
= k (1 – α)²(1 – ß)²(α – ß)², then ‘4k’ is equal to:
(a) \(\frac { 1 }{αß}\)
(b) 1
(c) -1
(d) αß
Answer
Answer: (b) 1
Hint:
= [(α – 1) (ß² – 1) – (α² – 1) (ß – 1)]²
= (α – 1)²(ß – 1)²(α – ß)2.
Hence, k = 1
Question 21.
The system of linear equations:
x + λy – z = 0
λr – y – z = 0
x + y – λz = 0
has a non-trivial solution for
(a) Exactly one value of λ
(b) Exactly two values of λ
(c) Exactly three values of λ
(d) Infinitely many values of λ.
Answer
Answer: (c) Exactly three values of λ
Hint:
The system AX = O has non-trivial solution if det A = 0
i.,e if \(\left|\begin{array}{rrr}
1 & \lambda & -1 \\
\lambda & -1 & -1 \\
1 & 1 & -\lambda
\end{array}\right|\) = 0
⇒ (1)(λ + 1) -λ(-λ² + 1) + (-1)(λ + 1) = 0
⇒ λ + 1 + λ³ – λ – λ – 1 = 0
⇒ λ³ – λ = 0
⇒ λ(λ² – 1) = 0
⇒ λ = 0, 1, -1.
Hence, λ = -1, 0, 1.
Question 22.
Let on be a complex number such that 2ω + 1 = z, where z = √-3
If \(\left|\begin{array}{ccc}
1 & 1 & 1 \\
1 & -\omega^{2}-1 & \omega^{2} \\
1 & \omega^{2} & \omega
\end{array}\right|\) = 3k the k is equal to
(a) -1
(b) 1
(c) z
(d) -z
Answer
Answer: (c) z
Hint:
[Operating R1 → R1 + R2 + R3]
= 3[- ω (ω² + 1) – ω4]
= 3[- ω³ – ω – ω] = 3[- 1 – 2ω]
= – 3(1 + 2ω) = – 3z.
Thus 3k = – 3z.
Hence, k = -z.
Question 23.
If Sis the set of distinct values of ‘h’ for which the following system of linear equations:
x + y + z = 1,
x + ay + z = 1,
ax + by + z = 1
has no solution, then S is
(a) a finite set containing two or more elements
(b) a singleton
(c) an empty set
(d) an infinite set.
Answer
Answer: (b) a singleton
Hint:
D = \(\left|\begin{array}{lll}
1 & 1 & 1 \\
1 & a & 1 \\
a & b & 1
\end{array}\right|\) = 0
⇒ a – 1
⇒ x + y + z = 1
and x + by + z = 0.
The planes are parallel
⇒ b = 1.
Hence, S is a singleton.
Question 24.
If A = \(\left[\begin{array}{rr}
2 & -3 \\
-4 & 1
\end{array}\right]\), then adj. (3A² + 12A) is equal to
Answer
Answer:
Hint:
Question 25.
If the system of linear equations:
x + ky + 3z = 0
3x + ky – 2z = 0
2x + 4y – 3z = 0
has a non zero solution (x, y, z), then \(\frac {xz}{y^2}\) is equal to:
(a) -10
(b) 10
(c) -30
(d) 30
Answer
Answer: (b) 10
Hint:
The given system has non-zero solution
⇒ \(\left|\begin{array}{ccc}
1 & k & 3 \\
3 & k & -2 \\
2 & 4 & -3
\end{array}\right|\)
⇒ 1(-3k + 8)-k (-9 + 4) + 3 (12 – 2k) = 0
⇒ 44 – 4k = 0
⇒ k = 11
Let z = λ
Thus x + 11 y = -3λ
and 3x + 11 y = 2λ
Fill in the blanks
Question 1.
If \(\left|\begin{array}{ll}
x & 2 \\
8 & x
\end{array}\right|\) = \(\left|\begin{array}{ll}
3 & 2 \\
9 & 6
\end{array}\right|\), then the value of x is ……………..
Answer
Answer: ±4
Hint:
\(\left|\begin{array}{ll}
x & 2 \\
8 & x
\end{array}\right|\) = \(\left|\begin{array}{ll}
3 & 2 \\
9 & 6
\end{array}\right|\)
⇒ x² – 16 = 18 – 18
⇒ x² = 16
⇒ x = ±4.
Question 2.
Let A be a 3 x 3 determinant and |A| = 7. Then the value of |2A| is ……………….
Answer
Answer: 56
Hint:
|2A| = 2³ |A| = 8 x 7 = 56.
Question 3.
If A = \(\left[\begin{array}{ll}
1 & 2 \\
4 & 2
\end{array}\right]\) Then the value of k = ……………
if |2A| = k|A|
Answer
Answer: 4
Hint:
if |2A| = 2²|A| = 4|A|
k = 4
Question 4.
If A is a skew-symmetric matrix of order 3, then det A = ……………..
Answer
Answer: 0.
Question 5.
The value of \(\left[\begin{array}{ccc}
102 & 18 & 36 \\
1 & 3 & 4 \\
17 & 3 & 6
\end{array}\right]\) is ……………..
Answer
Answer: 0
Hint:
Δ = 6\(\left|\begin{array}{ccc}
17 & 3 & 6 \\
1 & 3 & 4 \\
17 & 3 & 6
\end{array}\right|\) = 6(0) = 0
Question 6.
If Δ = \(\left|\begin{array}{ll}
1 & a \\
1 & b
\end{array}\right|\), then minor of ‘b’ is ………………
Answer
Answer: 1
Question 7.
Minor of ‘d’ is = \(\left|\begin{array}{ll}
a & c \\
b & d
\end{array}\right|\), is ………………
Answer
Answer: a
Question 8.
A square matrix A has inverse if and only if A is ………………
Answer
Answer: Invertible.
Question 9.
Co-factor of ‘b’ in \(\left|\begin{array}{ll}
a & c \\
b & d
\end{array}\right|\) is ……………
Answer
Answer: -c
Question 10.
If Δ = \(\left|\begin{array}{lll}
1 & 2 & 3 \\
2 & 0 & 1 \\
5 & 3 & 8
\end{array}\right|\) then minor of a22 is …………….
Answer
Answer: -7
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