12th Maths Model Exam -1

Minar Hr. Sec. School - Madurai

12th Maths Model Exam -1-Aug 2081

Time : 2.30 Hours

Part - A

Multiple Choice Question - Answer 16 QuestionQuestion No 3 compualsary

If A = \(\left[ \begin{matrix} 3 & 5 \\ 1 & 2 \end{matrix} \right] \), B = adj A and C = 3A, then \(\frac { \left| adjB \right| }{ \left| C \right| } \) =

\(\frac { 1 }{ 3 } \)
\(\frac { 1 }{ 9 } \)
\(\frac { 1 }{ 4 } \)
1

If A\(\left[ \begin{matrix} 1 & -2 \\ 1 & 4 \end{matrix} \right] =\left[ \begin{matrix} 6 & 0 \\ 0 & 6 \end{matrix} \right] \), then A =

\(\left[ \begin{matrix} 1 & -2 \\ 1 & 4 \end{matrix} \right] \)
\(\left[ \begin{matrix} 1 & 2 \\ -1 & 4 \end{matrix} \right] \)
\(\left[ \begin{matrix} 4 & 2 \\ -1 & 1 \end{matrix} \right] \)
\(\left[ \begin{matrix} 4 & -1 \\ 2 & 1 \end{matrix} \right] \)

If |adj(adj A)| = |A|⁹, then the order of the square matrix A is

If A = \(\left[ \begin{matrix} 2 & 0 \\ 1 & 5 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} 1 & 4 \\ 2 & 0 \end{matrix} \right] \) then |adj (AB)| =

If A = \(\left[ \begin{matrix} 3 & 1 & -1 \\ 2 & -2 & 0 \\ 1 & 2 & -1 \end{matrix} \right] \) and A^-1 = \(\left[ \begin{matrix} { a }_{ 11 } & { a }_{ 12 } & { a }_{ 13 } \\ { a }_{ 21 } & { a }_{ 22 } & { a }_{ 23 } \\ { a }_{ 31 } & { a }_{ 32 } & { a }_{ 33 } \end{matrix} \right] \) then the value of a₂₃ is

0
-2
-3
-1

If (AB)^-1 = \(\left[ \begin{matrix} 12 & -17 \\ -19 & 27 \end{matrix} \right] \) and A^-1 = \(\left[ \begin{matrix} 1 & -1 \\ -2 & 3 \end{matrix} \right] \), then B^-1 =

\(\left[ \begin{matrix} 2 & -5 \\ -3 & 8 \end{matrix} \right] \)
\(\left[ \begin{matrix} 8 & 5 \\ 3 & 2 \end{matrix} \right] \)
\(\left[ \begin{matrix} 3 & 1 \\ 2 & 1 \end{matrix} \right] \)
\(\left[ \begin{matrix} 8 & -5 \\ -3 & 2 \end{matrix} \right] \)

If A is a non-singular matrix such that A^-1 = \(\left[ \begin{matrix} 5 & 3 \\ -2 & -1 \end{matrix} \right] \), then (A^T)⁻¹ =

\(\left[ \begin{matrix} -5 & 3 \\ 2 & 1 \end{matrix} \right] \)
\(\left[ \begin{matrix} 5 & 3 \\ -2 & -1 \end{matrix} \right] \)
\(\left[ \begin{matrix} -1 & -3 \\ 2 & 5 \end{matrix} \right] \)
\(\left[ \begin{matrix} 5 & -2 \\ 3 & -1 \end{matrix} \right] \)

If A = \(\left[ \begin{matrix} 1 & \tan { \frac { \theta }{ 2 } } \\ -\tan { \frac { \theta }{ 2 } } & 1 \end{matrix} \right] \) and AB = I , then B =

\(\left( \cos ^{ 2 }{ \frac { \theta }{ 2 } } \right) A\)
\(\left( \cos ^{ 2 }{ \frac { \theta }{ 2 } } \right) { A }^{ T }\)
\(\left( \cos ^{ 2 }{ \theta } \right) I\)
(Sin²\(\frac { \theta }{ 2 } \))A

If A = \(\left[ \begin{matrix} \cos { \theta } & \sin { \theta } \\ -\sin { \theta } & \cos { \theta } \end{matrix} \right] \) and A(adj A) = \(\left[ \begin{matrix} k & 0 \\ 0 & k \end{matrix} \right] \) then adj (AB) is

0
sin θ
cos θ
1

If A = \(\left[ \begin{matrix} 2 & 3 \\ 5 & -2 \end{matrix} \right] \) be such that λA⁻¹ =A, then λ is

If adj A = \(\left[ \begin{matrix} 2 & 3 \\ 4 & -1 \end{matrix} \right] \) and adj B = \(\left[ \begin{matrix} 1 & -2 \\ -3 & 1 \end{matrix} \right] \) then adj (AB) is

\(\left[ \begin{matrix} -7 & -1 \\ 7 & -9 \end{matrix} \right] \)
\(\left[ \begin{matrix} -6 & 5 \\ -2 & -10 \end{matrix} \right] \)
\(\left[ \begin{matrix} -7 & 7 \\ -1 & -9 \end{matrix} \right] \)
\(\left[ \begin{matrix} -6 & -2 \\ 5 & -10 \end{matrix} \right] \)

The rank of the matrix \(\left[ \begin{matrix} 1 \\ \begin{matrix} 2 \\ -1 \end{matrix} \end{matrix}\begin{matrix} 2 \\ \begin{matrix} 4 \\ -2 \end{matrix} \end{matrix}\begin{matrix} 3 \\ \begin{matrix} 6 \\ -3 \end{matrix} \end{matrix}\begin{matrix} 4 \\ \begin{matrix} 8 \\ -4 \end{matrix} \end{matrix} \right] \) is

If x^ay^b = e^m, x^cy^d = eⁿ, Δ₁ = \(\left| \begin{matrix} m & b \\ n & d \end{matrix} \right| \), Δ₂ = \(\left| \begin{matrix} a & m \\ c & n \end{matrix} \right| \), Δ₃ = \(\left| \begin{matrix} a & b \\ c & d \end{matrix} \right| \), then the values of x and y are respectively,

e^{(Δ₂/Δ₁)}, e^{(Δ₃/Δ₁)}
log (Δ₁/Δ₃), log (Δ₂/Δ₃)
log (Δ₂/Δ₁), log(Δ₃/Δ₁)
e^{(Δ₁/Δ₃)},e^{(Δ₂/Δ₃)}

Which of the following is/are correct?
(i) Adjoint of a symmetric matrix is also a symmetric matrix.
(ii) Adjoint of a diagonal matrix is also a diagonal matrix.
(iii) If A is a square matrix of order n and λ is a scalar, then adj(λA) = λⁿ adj(A).
(iv) A(adjA) = (adjA)A = |A|I

Only (i)
(ii) and (iii)
(iii) and (iv)
(i), (ii) and (iv)

If ρ(A) = ρ([A | B]), then the system AX = B of linear equations is

consistent and has a unique solution
consistent
consistent and has infinitely many solution
inconsistent

If P = \(\left[ \begin{matrix} 1 & x & 0 \\ 1 & 3 & 0 \\ 2 & 4 & -2 \end{matrix} \right] \) is the adjoint of 3 × 3 matrix A and |A| = 4, then x is

If A, B and C are invertible matrices of some order, then which one of the following is not true?

adj A = |A|A^-1
adj(AB) = (adj A)(adj B)
det A^-1 = (det A)^-1
(ABC)^-1 = C^-1B^-1A^-1

Part - B

Generic 2 - Answer 5 QuestionQuestion No 3 compualsary

Find the adjoint of the following:
\(\left[ \begin{matrix} -3 & 4 \\ 6 & 2 \end{matrix} \right] \)

Find the inverse (if it exists) of the following:
\(\left[ \begin{matrix} -2 & 4 \\ 1 & -3 \end{matrix} \right] \)

If F(α) = \(\left[ \begin{matrix} \cos { \alpha } & 0 & \sin { \alpha } \\ 0 & 1 & 0 \\ -\sin { \alpha } & 0 & \cos { \alpha } \end{matrix} \right] \), show that [F(α)]^-1 = F(-α).

If adj(A) = \(\left[ \begin{matrix} 2 & -4 & 2 \\ -3 & 12 & -7 \\ -2 & 0 & 2 \end{matrix} \right] \), find A.

If adj(A) = \(\left[ \begin{matrix} 0 & -2 & 0 \\ 6 & 2 & -6 \\ -3 & 0 & 6 \end{matrix} \right] \), find A⁻¹.

Given A = \(\left[ \begin{matrix} 1 & -1 \\ 2 & 0 \end{matrix} \right] \), B = \(\left[ \begin{matrix} 3 & -2 \\ 1 & 1 \end{matrix} \right] \) and C = \(\left[ \begin{matrix} 1 & 1 \\ 2 & 2 \end{matrix} \right] \), find a matrix X such that AXB = C.
A = \(\left[ \begin{matrix} 1 & \tan { x } \\ -\tan { x } & 1 \end{matrix} \right] \), show that A^TA^-1 = \(\left[ \begin{matrix} \cos { 2x } & -\sin { 2x } \\ \sin { 2x } & \cos { 2x } \end{matrix} \right] \)

Part - C

Generic 3 - Answer All QuestionsQuestion No 3 compualsary

Under what co.nditions will the rank of the matrix \(\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & h-2 & 2 \\ \begin{matrix} 0 \\ 0 \end{matrix} & \begin{matrix} 0 \\ 0 \end{matrix} & \begin{matrix} h+2 \\ 3 \end{matrix} \end{matrix} \right] \) be less than 3?

Solve: x + y + 3z = 4, 2x + 2y + 6z = 7, 2x + y + z = 10.

If the rank of the matrix \(\left[ \begin{matrix} \lambda & -1 & 0 \\ 0 & \lambda & -1 \\ -1 & 0 & \lambda \end{matrix} \right] \) is 2, then find ⋋.

Verify the property (A^T)^-1 = (A^-1) with A = \(\left[ \begin{matrix} 2 & 9 \\ 1 & 7 \end{matrix} \right] \).

Reduce the matrix \(\left[ \begin{matrix} 0 \\ -1 \\ 4 \end{matrix}\begin{matrix} 3 \\ 0 \\ 2 \end{matrix}\begin{matrix} 1 \\ 2 \\ 0 \end{matrix}\begin{matrix} 6 \\ 5 \\ 0 \end{matrix} \right] \) to row-echelon form.

Find the rank of the matrix \(\left[ \begin{matrix} 2 \\ \begin{matrix} -3 \\ 6 \end{matrix} \end{matrix}\begin{matrix} -2 \\ \begin{matrix} 4 \\ 2 \end{matrix} \end{matrix}\begin{matrix} 4 \\ \begin{matrix} -2 \\ -1 \end{matrix} \end{matrix}\begin{matrix} 3 \\ \begin{matrix} -1 \\ 7 \end{matrix} \end{matrix} \right] \) by reducing it to an echelon form.

Find the inverse of the non-singular matrix A = \(\left[ \begin{matrix} 0 & 5 \\ -1 & 6 \end{matrix} \right] \), by Gauss-Jordan method.

Solve the system of linear equations, by Gaussian elimination method 4x + 3y + 6z = 25, x + 5y + 7z = 13, 2x + 9y + z = 1.

Test for consistency of the following system of linear equations and if possible solve:
x - y + z = -9, 2x - 2y + 2z = -18, 3x - 3y + 3z + 27 = 0.

Solve the following system:
x + 2y + 3z = 0, 3x + 4y + 4z = 0, 7x + 10y + 12z = 0.

Part - D

Generic 5 - Answer All QuestionQuestion No 3 compualsary

If A = \(\left[ \begin{matrix} 5 & 3 \\ -1 & -2 \end{matrix} \right] \), show that A² - 3A - 7I₂ = O₂. Hence find A⁻¹.
If A = \(\frac { 1 }{ 9 } \left[ \begin{matrix} -8 & 1 & 4 \\ 4 & 4 & 7 \\ 1 & -8 & 4 \end{matrix} \right] \), prove that A⁻¹ = A^T.
If A = \(\left[ \begin{matrix} 8 & -4 \\ -5 & 3 \end{matrix} \right] \), verify that A(adj A) = |A|I₂.

Using determinants; find the quadratic defined by f(x) =ax² + bx + c, if f(1) =0, f(2) =-2 and f(3) = -6.

Solve: \(\frac { 2 }{ x } +\frac { 3 }{ y } +\frac { 10 }{ z } =4,\frac { 4 }{ x } -\frac { 6 }{ y } +\frac { 5 }{ z } =1,\frac { 6 }{ x } +\frac { 9 }{ y } -\frac { 20 }{ z } \)=2

The sum of three numbers is 20. If we multiply the third number by 2 and add the first number to the result we get 23. By adding second and third numbers to 3 times the first number we get 46. Find the numbers using Cramer's rule.

For what value of λ, the system of equations x+y+z=1, x+2y+4z=λ, x+4y+10z=λ² is consistent.

Show that the equations -2x + y + z = a, x - 2y + z = b, x + y -2z = c are consistent only if a + b + c =0.

Investigate for what values of λ and μ the system of linear equations x + 2y + z = 7, x + y + λz = μ, x + 3y − 5z = 5 has
(i) no solution
(ii) a unique solution
(iii) an infinite number of solutions

InstaQB

Institutions

12th Standard English Medium

12th Standard Tamil Medium

11th Standard English Medium

11th Standard Tamil Medium

10th Standard English Medium

9th Standard English Medium

8th Standard English Medium

7th Standard English Medium

Class XII

Class XI

12Th Standard

11Th Standard

10Th Standard

9Th Standard

12th Maths Model Exam -1

Minar Hr. Sec. School - Madurai

12th Maths Model Exam -1-Aug 2081

Part - A

Multiple Choice Question - Answer 16 QuestionQuestion No 3 compualsary

If A = \(\left[ \begin{matrix} 3 & 5 \\ 1 & 2 \end{matrix} \right] \), B = adj A and C = 3A, then \(\frac { \left| adjB \right| }{ \left| C \right| } \) =

If A\(\left[ \begin{matrix} 1 & -2 \\ 1 & 4 \end{matrix} \right] =\left[ \begin{matrix} 6 & 0 \\ 0 & 6 \end{matrix} \right] \), then A =

If |adj(adj A)| = |A|9, then the order of the square matrix A is

If A = \(\left[ \begin{matrix} 2 & 0 \\ 1 & 5 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} 1 & 4 \\ 2 & 0 \end{matrix} \right] \) then |adj (AB)| =

If (AB)-1 = \(\left[ \begin{matrix} 12 & -17 \\ -19 & 27 \end{matrix} \right] \) and A-1 = \(\left[ \begin{matrix} 1 & -1 \\ -2 & 3 \end{matrix} \right] \), then B-1 =

If A is a non-singular matrix such that A-1 = \(\left[ \begin{matrix} 5 & 3 \\ -2 & -1 \end{matrix} \right] \), then (AT)−1 =

If A = \(\left[ \begin{matrix} 1 & \tan { \frac { \theta }{ 2 } } \\ -\tan { \frac { \theta }{ 2 } } & 1 \end{matrix} \right] \) and AB = I , then B =

If A = \(\left[ \begin{matrix} \cos { \theta } & \sin { \theta } \\ -\sin { \theta } & \cos { \theta } \end{matrix} \right] \) and A(adj A) = \(\left[ \begin{matrix} k & 0 \\ 0 & k \end{matrix} \right] \) then adj (AB) is

If A = \(\left[ \begin{matrix} 2 & 3 \\ 5 & -2 \end{matrix} \right] \) be such that λA−1 =A, then λ is

If adj A = \(\left[ \begin{matrix} 2 & 3 \\ 4 & -1 \end{matrix} \right] \) and adj B = \(\left[ \begin{matrix} 1 & -2 \\ -3 & 1 \end{matrix} \right] \) then adj (AB) is

If xayb = em, xcyd = en, Δ1 = \(\left| \begin{matrix} m & b \\ n & d \end{matrix} \right| \), Δ2 = \(\left| \begin{matrix} a & m \\ c & n \end{matrix} \right| \), Δ3 = \(\left| \begin{matrix} a & b \\ c & d \end{matrix} \right| \), then the values of x and y are respectively,

Which of the following is/are correct? (i) Adjoint of a symmetric matrix is also a symmetric matrix. (ii) Adjoint of a diagonal matrix is also a diagonal matrix. (iii) If A is a square matrix of order n and λ is a scalar, then adj(λA) = λn adj(A). (iv) A(adjA) = (adjA)A = |A|I

If ρ(A) = ρ([A | B]), then the system AX = B of linear equations is

If P = \(\left[ \begin{matrix} 1 & x & 0 \\ 1 & 3 & 0 \\ 2 & 4 & -2 \end{matrix} \right] \) is the adjoint of 3 × 3 matrix A and |A| = 4, then x is

If A, B and C are invertible matrices of some order, then which one of the following is not true?

Part - B

Generic 2 - Answer 5 QuestionQuestion No 3 compualsary

Find the adjoint of the following: \(\left[ \begin{matrix} -3 & 4 \\ 6 & 2 \end{matrix} \right] \)

Find the inverse (if it exists) of the following: \(\left[ \begin{matrix} -2 & 4 \\ 1 & -3 \end{matrix} \right] \)

If F(α) = \(\left[ \begin{matrix} \cos { \alpha } & 0 & \sin { \alpha } \\ 0 & 1 & 0 \\ -\sin { \alpha } & 0 & \cos { \alpha } \end{matrix} \right] \), show that [F(α)]-1 = F(-α).

If adj(A) = \(\left[ \begin{matrix} 2 & -4 & 2 \\ -3 & 12 & -7 \\ -2 & 0 & 2 \end{matrix} \right] \), find A.

If adj(A) = \(\left[ \begin{matrix} 0 & -2 & 0 \\ 6 & 2 & -6 \\ -3 & 0 & 6 \end{matrix} \right] \), find A−1.

Given A = \(\left[ \begin{matrix} 1 & -1 \\ 2 & 0 \end{matrix} \right] \), B = \(\left[ \begin{matrix} 3 & -2 \\ 1 & 1 \end{matrix} \right] \) and C = \(\left[ \begin{matrix} 1 & 1 \\ 2 & 2 \end{matrix} \right] \), find a matrix X such that AXB = C.

A = \(\left[ \begin{matrix} 1 & \tan { x } \\ -\tan { x } & 1 \end{matrix} \right] \), show that ATA-1 = \(\left[ \begin{matrix} \cos { 2x } & -\sin { 2x } \\ \sin { 2x } & \cos { 2x } \end{matrix} \right] \)

Part - C

Generic 3 - Answer All QuestionsQuestion No 3 compualsary

Under what co.nditions will the rank of the matrix \(\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & h-2 & 2 \\ \begin{matrix} 0 \\ 0 \end{matrix} & \begin{matrix} 0 \\ 0 \end{matrix} & \begin{matrix} h+2 \\ 3 \end{matrix} \end{matrix} \right] \) be less than 3?

Solve: x + y + 3z = 4, 2x + 2y + 6z = 7, 2x + y + z = 10.

If the rank of the matrix \(\left[ \begin{matrix} \lambda & -1 & 0 \\ 0 & \lambda & -1 \\ -1 & 0 & \lambda \end{matrix} \right] \) is 2, then find ⋋.

Verify the property (AT)-1 = (A-1) with A = \(\left[ \begin{matrix} 2 & 9 \\ 1 & 7 \end{matrix} \right] \).

Reduce the matrix \(\left[ \begin{matrix} 0 \\ -1 \\ 4 \end{matrix}\begin{matrix} 3 \\ 0 \\ 2 \end{matrix}\begin{matrix} 1 \\ 2 \\ 0 \end{matrix}\begin{matrix} 6 \\ 5 \\ 0 \end{matrix} \right] \) to row-echelon form.

Find the inverse of the non-singular matrix A = \(\left[ \begin{matrix} 0 & 5 \\ -1 & 6 \end{matrix} \right] \), by Gauss-Jordan method.

Solve the system of linear equations, by Gaussian elimination method 4x + 3y + 6z = 25, x + 5y + 7z = 13, 2x + 9y + z = 1.

Test for consistency of the following system of linear equations and if possible solve: x - y + z = -9, 2x - 2y + 2z = -18, 3x - 3y + 3z + 27 = 0.

Solve the following system: x + 2y + 3z = 0, 3x + 4y + 4z = 0, 7x + 10y + 12z = 0.

Part - D

Generic 5 - Answer All QuestionQuestion No 3 compualsary

If A = \(\left[ \begin{matrix} 5 & 3 \\ -1 & -2 \end{matrix} \right] \), show that A2 - 3A - 7I2 = O2. Hence find A−1.

If A = \(\frac { 1 }{ 9 } \left[ \begin{matrix} -8 & 1 & 4 \\ 4 & 4 & 7 \\ 1 & -8 & 4 \end{matrix} \right] \), prove that A−1 = AT.

If A = \(\left[ \begin{matrix} 8 & -4 \\ -5 & 3 \end{matrix} \right] \), verify that A(adj A) = |A|I2.

Using determinants; find the quadratic defined by f(x) =ax2 + bx + c, if f(1) =0, f(2) =-2 and f(3) = -6.

Solve: \(\frac { 2 }{ x } +\frac { 3 }{ y } +\frac { 10 }{ z } =4,\frac { 4 }{ x } -\frac { 6 }{ y } +\frac { 5 }{ z } =1,\frac { 6 }{ x } +\frac { 9 }{ y } -\frac { 20 }{ z } \)=2

The sum of three numbers is 20. If we multiply the third number by 2 and add the first number to the result we get 23. By adding second and third numbers to 3 times the first number we get 46. Find the numbers using Cramer's rule.

For what value of λ, the system of equations x+y+z=1, x+2y+4z=λ, x+4y+10z=λ2 is consistent.

Show that the equations -2x + y + z = a, x - 2y + z = b, x + y -2z = c are consistent only if a + b + c =0.

Investigate for what values of λ and μ the system of linear equations x + 2y + z = 7, x + y + λz = μ, x + 3y − 5z = 5 has (i) no solution (ii) a unique solution (iii) an infinite number of solutions

Part - E

Match the following - Answer 5 QuestionsQuestion No 3 compualsary

Trivial solution of AX=0

Non - Trivial solution of AX=0

\(\rho\)(A) = \(\rho\)[(A/0)] < n

\(\rho\)(A) = \(\rho\)[(A/0)] = n

\(\rho\)(A) = \(\rho\)[(A|B]) =3 = number of unknowns

Comment

Other Study material

12th Maths Monthly Test - 1 ( Ordinary Differentia

12th Maths Monthly Test - 2 ( Ordinary Differentia

12th Maths Monthly Test - 3 (Ordinary Differential

12th Maths Monthly Test - 1 (Probability Distribut

If |adj(adj A)| = |A|⁹, then the order of the square matrix A is

If (AB)^-1 = \(\left[ \begin{matrix} 12 & -17 \\ -19 & 27 \end{matrix} \right] \) and A^-1 = \(\left[ \begin{matrix} 1 & -1 \\ -2 & 3 \end{matrix} \right] \), then B^-1 =

If A is a non-singular matrix such that A^-1 = \(\left[ \begin{matrix} 5 & 3 \\ -2 & -1 \end{matrix} \right] \), then (A^T)⁻¹ =

If A = \(\left[ \begin{matrix} 2 & 3 \\ 5 & -2 \end{matrix} \right] \) be such that λA⁻¹ =A, then λ is

If x^ay^b = e^m, x^cy^d = eⁿ, Δ₁ = \(\left| \begin{matrix} m & b \\ n & d \end{matrix} \right| \), Δ₂ = \(\left| \begin{matrix} a & m \\ c & n \end{matrix} \right| \), Δ₃ = \(\left| \begin{matrix} a & b \\ c & d \end{matrix} \right| \), then the values of x and y are respectively,

Which of the following is/are correct?
(i) Adjoint of a symmetric matrix is also a symmetric matrix.
(ii) Adjoint of a diagonal matrix is also a diagonal matrix.
(iii) If A is a square matrix of order n and λ is a scalar, then adj(λA) = λⁿ adj(A).
(iv) A(adjA) = (adjA)A = |A|I

Find the adjoint of the following:
\(\left[ \begin{matrix} -3 & 4 \\ 6 & 2 \end{matrix} \right] \)

Find the inverse (if it exists) of the following:
\(\left[ \begin{matrix} -2 & 4 \\ 1 & -3 \end{matrix} \right] \)

If F(α) = \(\left[ \begin{matrix} \cos { \alpha } & 0 & \sin { \alpha } \\ 0 & 1 & 0 \\ -\sin { \alpha } & 0 & \cos { \alpha } \end{matrix} \right] \), show that [F(α)]^-1 = F(-α).

If adj(A) = \(\left[ \begin{matrix} 0 & -2 & 0 \\ 6 & 2 & -6 \\ -3 & 0 & 6 \end{matrix} \right] \), find A⁻¹.

A = \(\left[ \begin{matrix} 1 & \tan { x } \\ -\tan { x } & 1 \end{matrix} \right] \), show that A^TA^-1 = \(\left[ \begin{matrix} \cos { 2x } & -\sin { 2x } \\ \sin { 2x } & \cos { 2x } \end{matrix} \right] \)

Verify the property (A^T)^-1 = (A^-1) with A = \(\left[ \begin{matrix} 2 & 9 \\ 1 & 7 \end{matrix} \right] \).

Test for consistency of the following system of linear equations and if possible solve:
x - y + z = -9, 2x - 2y + 2z = -18, 3x - 3y + 3z + 27 = 0.

Solve the following system:
x + 2y + 3z = 0, 3x + 4y + 4z = 0, 7x + 10y + 12z = 0.

If A = \(\left[ \begin{matrix} 5 & 3 \\ -1 & -2 \end{matrix} \right] \), show that A² - 3A - 7I₂ = O₂. Hence find A⁻¹.

If A = \(\frac { 1 }{ 9 } \left[ \begin{matrix} -8 & 1 & 4 \\ 4 & 4 & 7 \\ 1 & -8 & 4 \end{matrix} \right] \), prove that A⁻¹ = A^T.

If A = \(\left[ \begin{matrix} 8 & -4 \\ -5 & 3 \end{matrix} \right] \), verify that A(adj A) = |A|I₂.

Using determinants; find the quadratic defined by f(x) =ax² + bx + c, if f(1) =0, f(2) =-2 and f(3) = -6.

For what value of λ, the system of equations x+y+z=1, x+2y+4z=λ, x+4y+10z=λ² is consistent.

Investigate for what values of λ and μ the system of linear equations x + 2y + z = 7, x + y + λz = μ, x + 3y − 5z = 5 has
(i) no solution
(ii) a unique solution
(iii) an infinite number of solutions