Bihar Board 12th Maths Objective Answers Chapter 4 Determinants

Bihar Board 12th Maths Objective Questions and Answers

Bihar Board 12th Maths Objective Answers Chapter 4 Determinants

Question 1.
Evaluate the determinant \(\Delta=\left|\begin{array}{ll}
\log _{3} 512 & \log _{4} 3 \\
\log _{3} 8 & \log _{4} 9
\end{array}\right|\)
(a) \(\frac { 15 }{ 2 }\)
(b) 12
(c) \(\frac { 14 }{ 3 }\)
(d) 6
Answer:
(a) \(\frac { 15 }{ 2 }\)

Direction (2 – 3): Evaluate the following determinants.

Question 2.
\(\left|\begin{array}{cc}
x & -7 \\
x & 5 x+1
\end{array}\right|\)
(a) 3x² + 4
(b) x(5x + 8)
(c) 3x + 4x²
(d) x(3x + 4)
Answer:
(b) x(5x + 8)

Question 3.
\(\left|\begin{array}{cc}
\cos 15^{\circ} & \sin 15^{\circ} \\
\sin 75^{\circ} & \cos 75^{\circ}
\end{array}\right|\)
(a) 0
(b) 5
(c) 3
(d) 7
Answer:
(a) 0

Question 4.

Answer:
(b) 1

Question 5.

Answer:
(c) -1

Question 6.
\(\left|\begin{array}{ccc}
2 x y & x^{2} & y^{2} \\
x^{2} & y^{2} & 2 x y \\
y^{2} & 2 x y & x^{2}
\end{array}\right|=\)
(a) (x³ + y³)²
(b) (x² + y²)³
(c) -(x² + y²)³
(d) -(x³ + y³)²
Answer:
(d) -(x³ + y³)²

Question 7.
The value of \(\left|\begin{array}{ccc}
\cos (\alpha+\beta) & -\sin (\alpha+\beta) & \cos 2 \beta \\
\sin \alpha & \cos \alpha & \sin \beta \\
-\cos \alpha & \sin \alpha & \cos \beta
\end{array}\right|\) is independent of
(a) α
(b) β
(c) α, β
(d) none of these
Answer:
(a) α

Question 8.
Let \(\Delta=\left|\begin{array}{ccc}
x & y & z \\
x^{2} & y^{2} & z^{2} \\
x^{3} & y^{3} & z^{3}
\end{array}\right|\), then the value of ∆ is
(a) (x – y) (y – z) (z – x)
(b) xyz
(c) (x² + y² + z²)²
(d) xyz (x – y) (y – z) (z – x)
Answer:
(d) xyz (x – y) (y – z) (z – x)

Question 9.
The value of the determinant \(\left|\begin{array}{ccc}
\alpha & \beta & \gamma \\
\alpha^{2} & \beta^{2} & \gamma^{2} \\
\beta+\gamma & \gamma+\alpha & \alpha+\beta
\end{array}\right|=\)
(a) (α + β)(β + γ)(γ + α)
(b) (α – β)(β – γ)(γ – α)(α + β + γ)
(c) (α + β + γ)² (α – β – γ)²
(d) αβγ (α + β + γ)
Answer:
(b) (α – β)(β – γ)(γ – α)(α + β + γ)

Question 10.
Using properties of determinants, \(\left|\begin{array}{ccc}
1 & a & a^{2}-b c \\
1 & b & b^{2}-c a \\
1 & c & c^{2}-a b
\end{array}\right|=\)
(a) 0
(b) 1
(c) 2
(d) 3
Answer:
(a) 0

Question 11.
If a, b, c are the roots of the equation x³ – 3x² + 3x + 7 = 0, then the value of \(\left|\begin{array}{ccc}
2 b c-a^{2} & c^{2} & b^{2} \\
c^{2} & 2 a c-b^{2} & a^{2} \\
b^{2} & a^{2} & 2 a b-c^{2}
\end{array}\right|\) is
(a) 9
(b) 27
(c) 81
(d) 0
Answer:
(d) 0

Question 12.
If \(\left|\begin{array}{ccc}
1+a^{2} x & \left(1+b^{2}\right) x & \left(1+c^{2}\right) x \\
\left(1+a^{2}\right) x & 1+b^{2} x & \left(1+c^{2}\right) x \\
\left(1+a^{2}\right) x & \left(1+B^{2}\right) x & 1+c^{2} x
\end{array}\right|\), then f(x) is apolynomial of degree
(a) 2
(b) 3
(c) 0
(d) 1
Answer:
(a) 2

Question 13.
\(\left|\begin{array}{lll}
a^{2} & 2 a b & b^{2} \\
b^{2} & a^{2} & 2 a b \\
2 a b & b^{2} & a^{2}
\end{array}\right|\) is equal to
(a) a³ – b³
(b) a³ + b³
(c) (a³ – b³)²
(d) (a³ + b³)²
Answer:
(d) (a³ + b³)²

Question 14.
If α, β, γ are in A.P., then \(\left|\begin{array}{ccc}
x-3 & x-4 & x-\alpha \\
x-2 & x-3 & x-\beta \\
x-1 & x-2 & x-\gamma
\end{array}\right|=\)
(a) 0
(b) (x – 2)(x – 3)(x – 4)
(c) (x – α)(x – β)(x – γ)
(d) αβγ (α – β)(β – γ)²
Answer:
(a) 0

Direction (15 – 19): Find the value of the following determinants.

Question 15.
\(\left|\begin{array}{ccc}
1 & a^{2}+b c & a^{3} \\
1 & b^{2}+c a & b^{3} \\
1 & c^{2}+a b & c^{3}
\end{array}\right|\)
(a) -(a – b)(b – c)(c – a)(a² + b² + c²)
(b) (a – b)(b – c)(c – a)
(c) (a² + b² + c²)
(d) (a – b)(b – c)(c – a)(a² + b² + c²)
Answer:
(a) -(a – b)(b – c)(c – a)(a² + b² + c²)

Question 16.
\(\left|\begin{array}{ccc}
(b+c)^{2} & a^{2} & b c \\
(c+a)^{2} & b^{2} & c a \\
(a+b)^{2} & c^{2} & a b
\end{array}\right|=\)
(a) (a – b)(b – c)(c – a)(a² + b² + c²)
(b) -(a – b)(b – c)(c – a)
(c) (a – b)(b – c)(c – a)(a + b + c)(a² + b² + c²)
(d) 0
Answer:
(c) (a – b)(b – c)(c – a)(a + b + c)(a² + b² + c²)

Question 17.
Find the area of the triangle with vertices P(4, 5), Q(4, -2) and R(-6, 2).
(a) 21 sq. units
(b) 35 sq. units
(c) 30 sq. units
(d) 40 sq. units
Answer:
(b) 35 sq. units

Question 18.
If the points (a₁, b₁), (a₂, b₂) and(a₁ + a₂, b₁ + b₂) are collinear, then
(a) a₁b₂ = a₂b₁
(b) a₁ + a₂ = b₁ + b₂
(c) a₂b₂ = a₁b₁
(d) a₁ + b₁ = a₂ + b₂
Answer:
(a) a₁b₂ = a₂b₁

Question 19.
If the points (2, -3), (k, -1) and (0, 4) are collinear, then find the value of 4k.
(a) 4
(b) 7/140
(c) 47
(d) 40/7
Answer:
(d) 40/7

Question 20.
Find the area of the triangle whose vertices are (-2, 6), (3, -6) and (1, 5).
(a) 30 sq. units
(b) 35 sq. units
(c) 40 sq. units
(d) 15.5 sq. units
Answer:
(d) 15.5 sq. units

Question 21.
If the points (3, -2), (x, 2), (8, 8) are collinear, then find the value of x.
(a) 2
(b) 3
(c) 4
(d) 5
Answer:
(d) 5

Question 22.
Using determinants, find the equation of the line joining the points (1, 2) and (3, 6).
(a) y = 2x
(b) x = 3y
(c) y = x
(d) 4x – y = 5
Answer:
(a) y = 2x

Question 23.
Find the minor of the element of second row and third column in the following determinant \(\left[\begin{array}{ccc}
2 & -3 & 5 \\
6 & 0 & 4 \\
1 & 5 & -7
\end{array}\right]\)
(a) 13
(b) 4
(c) 5
(d) 0
Answer:
(a) 13

Question 24.
If \(\Delta=\left|\begin{array}{lll}
5 & 3 & 8 \\
2 & 0 & 1 \\
1 & 2 & 3
\end{array}\right|\), then write the minor of the element a₂₃.
(a) 7
(b) -7
(c) 4
(d) 8
Answer:
(a) 7

Direction (25 – 27): Write the cofactors of each element of the first column of the following matrices.

Question 25.
\(A=\left[\begin{array}{lll}
1 & b c & a \\
1 & c a & b \\
1 & a b & c
\end{array}\right]\)
(a) a(c² – b²), b(a² – c²), c(b² – a²)
(b) a(c² – b²), b(c² – a²), c(b² – a²)
(c) bc, ab, 2b
(d) None of these
Answer:
(a) a(c² – b²), b(a² – c²), c(b² – a²)

Question 26.
\(A=\left[\begin{array}{lll}
0 & 2 & 6 \\
1 & 5 & 0 \\
7 & 3 & 1
\end{array}\right]\)
(a) 5, 16, 30
(b) 5, -16, -30
(c) 5, 16, -30
(d) -5, -16, -30
Answer:
(c) 5, 16, -30

Question 27.
\(A=\left[\begin{array}{ccc}
2 & 5 & -1 \\
-3 & 0 & 1 \\
1 & 1 & -1
\end{array}\right]\)
(a) -1, 4, 5
(b) -4, 5, -1
(c) 4, 5, 1
(d) -4, -5, 1
Answer:
(a) -1, 4, 5

Question 28.
Find cofactors of a₂₁ and a₃₁ of the matrix A = [a_ij] = \(\left[\begin{array}{ccc}
1 & 3 & -2 \\
4 & -5 & 6 \\
3 & 5 & 2
\end{array}\right]\)
(a) -16, 8
(b) -16, -8
(c) 16, 8
(d) 16, -8
Answer:
(a) -16, 8

Question 29.
Find the minor of 6 and cofactor of 4 respectively in the determinant \(\Delta=\left|\begin{array}{lll}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{array}\right|\)
(a) 6, 6
(b) 6, -6
(c) -6, -6
(d) -6, 6
Answer:
(d) -6, 6

Question 30.
Find the cofactors of the element of third row and second column of the following determinant \(\left|\begin{array}{ccc}
1 & x & y+z \\
1 & y & z+x \\
1 & z & x+y
\end{array}\right|\)
(a) x – y
(b) y – x
(c) x – z
(d) z – x
Answer:
(b) y – x

Question 31.

Answer:
(b) \(\left[\begin{array}{cc}
4 & -2 \\
-3 & 1
\end{array}\right]\)

Question 32.

Answer:
(b) \(\left[\begin{array}{ccc}
15 & 6 & -15 \\
0 & -3 & 0 \\
-10 & 0 & 5
\end{array}\right]\)

Question 33.
Find x, if \(\left[\begin{array}{ccc}
1 & 2 & x \\
1 & 1 & 1 \\
2 & 1 & -1
\end{array}\right]\) is singular
(a) 1
(b) 2
(c) 3
(d) 4
Answer:
(d) 4

Question 34.
Find the value of x for which the matrix \(A=\left[\begin{array}{ccc}
3-x & 2 & 2 \\
2 & 4-x & 1 \\
-2 & -4 & -1-x
\end{array}\right]\) is singular.
(a) 0, 1
(b) 1, 3
(c) 0, 3
(d) 3, 2
Answer:
(c) 0, 3

Question 35.

Answer:
(b) \(-\frac{25}{13}\)

Question 36.
For what value of x, matrix \(\left[\begin{array}{ll}
6-x & 4 \\
3-x & 1
\end{array}\right]\) is a singularmatrix?
(a) 1
(b) 2
(c) -1
(d) -2
Answer:
(b) 2

Question 37.
Compute (AB)^-1, If

Answer:
(a) \(\frac{1}{19}\left[\begin{array}{ccc}
16 & 12 & 1 \\
21 & 11 & -7 \\
10 & -2 & 3
\end{array}\right]\)

Question 38.

Answer:
(a) A^-1

Question 39.

Answer:
(a) \(\frac{1}{11}\left[\begin{array}{cc}
14 & 5 \\
5 & 1
\end{array}\right]\)

Question 40.

Answer:
(b) \(\frac{1}{17}\left[\begin{array}{cc}
4 & 3 \\
-3 & 2
\end{array}\right]\)

Question 41.

Answer:
(a) \(\left[\begin{array}{cc}
4 & 2 \\
-1 & 1
\end{array}\right]\)

Question 42.
If for the non-singular matrix A, A² = I, then find A^-1.
(a) A
(b) I
(c) O
(d) None of these
Answer:
(a) A

Question 43.
If the equation a(y + z) = x, b(z + x) = y, c(x + y) = z have non-trivial solutions then the value of \(\frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c}\) is
(a) 1
(b) 2
(c) -1
(d) -2
Answer:
(b) 2

Question 44.
A non-trivial solution of the system of equations x + λy + 2z = 0, 2x + λz = 0, 2λx – 2y + 3z = 0 is given by x : y : z =
(a) 1 : 2 : -2
(b) 1: -2 : 2
(c) 2 : 1 : 2
(d) 2 : 1 : -2
Answer:
(d) 2 : 1 : -2

Question 45.
If 4x + 3y + 6z = 25, x + 5y + 7z = 13, 2x + 9y + z = 1, then z = ________
(a) 1
(b) 3
(c) -2
(d) 2
Answer:
(d) 2

Question 46.
If the equations 2x + 3y + z = 0, 3x + y – 2z = 0 and ax + 2y – bz = 0 has non-trivial solution, then
(a) a – b = 2
(b) a + b + 1 = 0
(c) a + b = 3
(d) a – b – 8 = 0
Answer:
(a) a – b = 2

Question 47.
Solve the following system of equations x – y + z = 4, x – 2y + 2z = 9 and 2x + y + 3z = 1.
(a) x = -4, y = -3, z = 2
(b) x = -1, y = -3, z = 2
(c) x = 2, y = 4, z = 6
(d) x = 3, y = 6, z = 9
Answer:
(b) x = -1, y = -3, z = 2

Question 48.
If the system of equations x + ky – z = 0, 3x – ky – z = 0 & x – 3y + z = 0 has non-zero solution, then k is equal to
(a) -1
(b) 0
(c) 1
(d) 2
Answer:
(c) 1

Question 49.
If the system of equations 2x + 3y + 5 = 0, x + ky + 5 = 0, kx – 12y – 14 = 0 has non-trivial solution, then the value of k is
(a) -2, \(\frac{12}{5}\)
(b) -1, \(\frac{1}{5}\)
(c) -6, \(\frac{17}{5}\)
(d) 6, \(-\frac{12}{5}\)
Answer:
(c) -6, \(\frac{17}{5}\)

Question 50.
If \(\left|\begin{array}{cc}
2 x & 5 \\
8 & x
\end{array}\right|=\left|\begin{array}{cc}
6 & -2 \\
7 & 3
\end{array}\right|\), then the value of x is
(a) 3
(b) ±3
(c) ±6
(d) 6
Answer:
(c) ±6

Question 51.
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:
(b) 3

Question 52.
The number of distinct real roots of \(\left|\begin{array}{ccc}
\sin x & \cos x & \cos x \\
\cos x & \sin x & \cos x \\
\cos x & \cos x & \sin x
\end{array}\right|=0\) in the interval \(-\frac{\pi}{4} \leq x \leq \frac{\pi}{4}\) is
(a) 0
(b) 2
(c) 1
(d) 3
Answer:
(c) 1

Question 53.

(a) 0
(b) -1
(c) 2
(d) 3
Answer:
(a) 0

Question 54.

Answer:
(a) \(\frac{1}{2}\)

Question 55.
The value of the determinant \(\left|\begin{array}{ccc}
x & x+y & x+2 y \\
x+2 y & x & x+y \\
x+y & x+2 y & x
\end{array}\right|\) is
(a) 9x² (x + y)
(b) 9y² (x + y)
(c) 3y² (x + y)
(d) 7x² (x + y)
Answer:
(b) 9y² (x + y)