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12th Maths Monthly Test -1 (Applications Of Matrices And Determinants)

St. Britto Hr. Sec. School - Madurai

12th Maths Monthly Test -1 (Applications Of Matrices And Determinants)-Aug 2020

Time : 2.30 Hours

Part - A

Multiple Choice Question - Answer All Question

If A is a matrix of order m x n, then \(\rho\)(A) is

m
n
≤ min (m,n)
≥ min (m,n)

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 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)}

If A^TA⁻¹ is symmetric, then A² =

A^-1
(A^T)²
A^T
(A^-1)²

Test question 2 By bala

Option A
Option B Option D
Option C
Option D

In a homogeneous system if \(\rho\) (A) =\(\rho\)([A|0]) < the number of unknouns then the system has ________

trivial solution
only non - trivial solution
no solution
trivial solution and infinitely many non - trivial solutions

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 A = \(\left[ \begin{matrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{matrix} \right] \), then adj(adj A) is

\(\left[ \begin{matrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{matrix} \right] \)
\(\left[ \begin{matrix} 6 & -6 & 8 \\ 4 & -6 & 8 \\ 0 & -2 & 2 \end{matrix} \right] \)
\(\left[ \begin{matrix} -3 & 3 & -4 \\ -2 & 3 & -4 \\ 0 & 1 & -1 \end{matrix} \right] \)
\(\left[ \begin{matrix} 3 & -3 & 4 \\ 0 & -1 & 1 \\ 2 & -3 & 4 \end{matrix} \right] \)

If A = [2 0 1] then the rank of AA^T is ______

1
2
3
0

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

17
14
19
21

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, 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

If A is a 3 × 3 non-singular matrix such that AA^T = A^TA and B = A^-1A^T, then BB^T =

A
B
I
B^T

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

In the non - homogeneous system of equations with 3 unknowns if \(\rho\)(A) = \(\rho\)([AIB]) = 2, then the system has _______

unique solution
one parameter family of solution
two parameter family of solutions
in consistent

In the system of equations with 3 unknowns, if Δ = 0, and one of Δ_x, Δ_y of Δ_z is non zero then the system is ______

Consistent
inconsistent
consistent with one parameter family of solutions
consistent with two parameter family of solutions

Part - B

Generic 2 - Answer All Question

A family of 3 people went out for dinner in a restaurant. The cost of two dosai, three idlies and two vadais is Rs.150. The cost of the two dosai, two idlies and four vadais is Rs.200. The cost of five dosai, four idlies and two vadais is Rs.250. The family has Rs.350 in hand and they ate 3 dosai and six idlies and six vadais. Will they be able to manage to pay the bill within the amount they had ?

If A = \(\left[ \begin{matrix} a & b \\ c & d \end{matrix} \right] \) is non-singular, find A⁻¹.

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).

Find the rank of the following matrices by minor method:
\(\left[ \begin{matrix} 1 \\ 3 \end{matrix}\begin{matrix} -2 \\ -6 \end{matrix}\begin{matrix} -1 \\ -3 \end{matrix}\begin{matrix} 0 \\ 1 \end{matrix} \right] \)

If A is symmetric, prove that then adj Ais also symmetric.

Solve the following systems of linear equations by Cramer’s rule:
\(\frac { 3 }{ x } \) + 2y = 12, \(\frac { 2 }{ x } \) + 3y = 13

Test for consistency and if possible, solve the following systems of equations by rank method.
2x + 2y + z = 5, x - y + z = 1, 3x + y + 2z = 4

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

Find the matrix A for which A\(\left[ \begin{matrix} 5 & 3 \\ -1 & -2 \end{matrix} \right] =\left[ \begin{matrix} 14 & 7 \\ 7 & 7 \end{matrix} \right] \).

Find the inverse (if it exists) of the following:
\(\left[ \begin{matrix} 2 & 3 & 1 \\ 3 & 4 & 1 \\ 3 & 7 & 2 \end{matrix} \right] \)

For any 2 x 2 matrix, if A (adj A) =\(\left[ \begin{matrix} 10 & 0 \\ 0 & 10 \end{matrix} \right] \) then find |A|.

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

Find the rank of the matrix A =\(\left[ \begin{matrix} 4 \\ 7 \end{matrix}\begin{matrix} 5 \\ -3 \end{matrix}\begin{matrix} -6 \\ 0 \end{matrix}\begin{matrix} 1 \\ 8 \end{matrix} \right] \).
A boy is walking along the path y = ax² + bx + c through the points (−6, 8),(−2, −12) , and (3, 8) . He wants to meet his friend at P(7,60) . Will he meet his friend? (Use Gaussian elimination method.)

Find the rank of the following matrices by minor method:
\(\left[ \begin{matrix} 1 & -2 & 3 \\ 2 & 4 & -6 \\ 5 & 1 & -1 \end{matrix} \right] \)
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.

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.

Part - C

Generic 3 - Answer All Question

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

Solve: 2x + 3y = 10, x + 6y = 4 using Cramer's rule.

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.

If A = \(\left[ \begin{matrix} 3 & 2 \\ 7 & 5 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} -1 & -3 \\ 5 & 2 \end{matrix} \right] \), verify that (AB)^-1 = B^-1A^-1

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.

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

Verify (AB)^-1 =B^-1 A^-1 for A=\(\left[ \begin{matrix} 2 & 1 \\ 5 & 3 \end{matrix} \right] \) and B=\(\left[ \begin{matrix} 4 & 5 \\ 3 & 4 \end{matrix} \right] \).

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.
Solve 2x - 3y = 7, 4x - 6y = 14 by Gaussian Jordan method.

Solve the following system:
x + 2y + 3z = 0, 3x + 4y + 4z = 0, 7x + 10y + 12z = 0.
Solve: x + y + 3z = 4, 2x + 2y + 6z = 7, 2x + y + z = 10.

Solve the following system of linear equations, using matrix inversion method:
5x + 2y = 3, 3x + 2y = 5.

Part - D

Generic 5 - Answer All Question

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

Find the condition on a, b and c so that the following system of linear equations has one parameter family of solutions: x + y + z = a, x + 2y + 3z = b, 3x + 5y + 7z = c.

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
Test for consistency of the following system of linear equations and if possible solve:
x + 2y - z = 3, 3x - y + 2z = 1, x - 2y + 3z = 3, x - y + z + 1 = 0

If A = \(\left[ \begin{matrix} 8 & -6 & 2 \\ -6 & 7 & 4 \\ 2 & -4 & 3 \end{matrix} \right] \), verify thatA(adj A)=(adj A)A = |A| I₃.

By using Gaussian elimination method, balance the chemical reaction equation: C₅H₈ + O₂ u27f6 CO₂ + H₂O. (The above is the reaction that is taking place in the burning of organic compound called isoprene.)

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