12th Maths Monthly Test - 2 (Applications Of Matrices And Determinants)

St. Britto Hr. Sec. School - Madurai

12th Maths Monthly Test - 2 (Applications Of Matrices And Determinants)-Aug 2020

Time : 1.30 Hours

Part - A

Generic 2 - Answer All Question

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

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

Find adj(adj A) if adj A = \(\left[ \begin{matrix} 1 & 0 & 1 \\ 0 & 2 & 0 \\ -1 & 0 & 1 \end{matrix} \right] \).

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

If A = \(\left[ \begin{matrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{matrix} \right] \), show that A^-1 = \(\frac {1}{2}\) (A² - 3I).

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

Find the inverse of each of the following by Gauss-Jordan method:
\(\left[ \begin{matrix} 2 & -1 \\ 5 & -2 \end{matrix} \right] \)

If A = \(\left[ \begin{matrix} -5 & 1 & 3 \\ 7 & 1 & -5 \\ 1 & -1 & 1 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} 1 & 1 & 2 \\ 3 & 2 & 1 \\ 2 & 1 & 3 \end{matrix} \right] \), find the products AB and BA and hence solve the system of equations x + y + 2z = 1, 3x + 2y + z = 7, 2x + y + 3z = 2.

4 men and 4 women can finish a piece of work jointly in 3 days while 2 men and 5 women can finish the same work jointly in 4 days. Find the time taken by one man alone and that of one woman alone to finish the same work by using matrix inversion method.

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

A chemist has one solution which is 50% acid and another solution which is 25% acid. How much each should be mixed to make 10 litres of a 40% acid solution ? (Use Cramer’s rule to solve the problem).

Solve the following systems of linear equations by Gaussian elimination method:
2x − 2y + 3z = 2, x + 2y − z = 3,3x − y + 2z = 1

An amount of Rs.65,000 is invested in three bonds at the rates of 6%,8% and 10% per annum respectively. The total annual income is Rs.4,800. The income from the third bond is Rs.600 more than that from the second bond. Determine the price of each bond. (Use Gaussian elimination method.)

Test for consistency and if possible, solve the following systems of equations by rank method.
x - y + 2z = 2, 2x + y + 4z = 7, 4x - y + z = 4

Investigate the values of λ and m the system of linear equations 2x + 3y + 5z = 9, 7x + 3y - 5z = 8, 2x + 3y + λz = μ, have
(i) no solution
(ii) a unique solution
(iii) an infinite number of solutions.

Determine the values of λ for which the following system of equations x + y + 3z = 0, 4x + 3y + λz = 0, 2x + y + 2z = 0 has
(i) a unique solution
(ii) a non-trivial solution

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

Find the inverse (if it exists) of the following:
\(\left[ \begin{matrix} 5 & 1 & 1 \\ 1 & 5 & 1 \\ 1 & 1 & 5 \end{matrix} \right] \)

Find the rank of the following matrices by minor method:
\(\left[ \begin{matrix} -1 & 3 \\ 4 & -7 \\ 3 & -4 \end{matrix} \right] \)

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

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

Find the inverse of each of the following by Gauss-Jordan method:
\(\left[ \begin{matrix} 1 & -1 & 0 \\ 1 & 0 & -1 \\ 6 & -2 & -3 \end{matrix} \right] \)

Solve the following system of linear equations by matrix inversion method:
2x − y = 8, 3x + 2y = −2

Solve the following system of linear equations by matrix inversion method:
x + y + z − 2 = 0, 6x − 4y + 5z − 31 = 0, 5x + 2y + 2z = 13.

Solve the following systems of linear equations by Cramer’s rule:
3x + 3y − z = 11, 2x − y + 2z = 9, 4x + 3y + 2z = 25.

Part - B

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.

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.

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

If A = \(\frac { 1 }{ 7 } \left[ \begin{matrix} 6 & -3 & a \\ b & -2 & 6 \\ 2 & c & 3 \end{matrix} \right] \) is orthogonal, find a, b and c , and hence A⁻¹.

Find the inverse of A = \(\left[ \begin{matrix} 2 & 1 & 1 \\ 3 & 2 & 1 \\ 2 & 1 & 2 \end{matrix} \right] \) by Gauss-Jordan method.

If A = \(\left[ \begin{matrix} -4 & 4 & 4 \\ -7 & 1 & 3 \\ 5 & -3 & -1 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} 1 & -1 & 1 \\ 1 & -2 & -2 \\ 2 & 1 & 3 \end{matrix} \right] \), find the productsAB and BAand hence solve the system of equations x - y + z = 4, x - 2y - 2z = 9, 2x + y + 3z = 1.

The upward speed v(t)of a rocket at time t is approximated by v(t) = at² + bt + c ≤ t ≤ 100 where a, b and c are constants. It has been found that the speed at times t = 3, t = 6, and t = 9 seconds are respectively, 64, 133, and 208 miles per second respectively. Find the speed at time t = 15 seconds. (Use Gaussian elimination method.)

Test for consistency of the following system of linear equations and if possible solve:
4x − 2y + 6z = 8, x + y − 3z = −1, 15x − 3y + 9z = 21.

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.

Solve the system: x + 3y - 2z = 0, 2x - y + 4z = 0, x - 11y + 14z = 0

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12th Maths Monthly Test - 2 (Applications Of Matrices And Determinants)

St. Britto Hr. Sec. School - Madurai

12th Maths Monthly Test - 2 (Applications Of Matrices And Determinants)-Aug 2020

Part - A

Generic 2 - Answer All Question

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

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

Find adj(adj A) if adj A = \(\left[ \begin{matrix} 1 & 0 & 1 \\ 0 & 2 & 0 \\ -1 & 0 & 1 \end{matrix} \right] \).

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

If A = \(\left[ \begin{matrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{matrix} \right] \), show that A-1 = \(\frac {1}{2}\) (A2 - 3I).

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

Find the inverse of each of the following by Gauss-Jordan method: \(\left[ \begin{matrix} 2 & -1 \\ 5 & -2 \end{matrix} \right] \)

4 men and 4 women can finish a piece of work jointly in 3 days while 2 men and 5 women can finish the same work jointly in 4 days. Find the time taken by one man alone and that of one woman alone to finish the same work by using matrix inversion method.

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

A chemist has one solution which is 50% acid and another solution which is 25% acid. How much each should be mixed to make 10 litres of a 40% acid solution ? (Use Cramer’s rule to solve the problem).

Solve the following systems of linear equations by Gaussian elimination method: 2x − 2y + 3z = 2, x + 2y − z = 3,3x − y + 2z = 1

An amount of Rs.65,000 is invested in three bonds at the rates of 6%,8% and 10% per annum respectively. The total annual income is Rs.4,800. The income from the third bond is Rs.600 more than that from the second bond. Determine the price of each bond. (Use Gaussian elimination method.)

Test for consistency and if possible, solve the following systems of equations by rank method. x - y + 2z = 2, 2x + y + 4z = 7, 4x - y + z = 4

Investigate the values of λ and m the system of linear equations 2x + 3y + 5z = 9, 7x + 3y - 5z = 8, 2x + 3y + λz = μ, have (i) no solution (ii) a unique solution (iii) an infinite number of solutions.

Determine the values of λ for which the following system of equations x + y + 3z = 0, 4x + 3y + λz = 0, 2x + y + 2z = 0 has (i) a unique solution (ii) a non-trivial solution

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

Find the inverse (if it exists) of the following: \(\left[ \begin{matrix} 5 & 1 & 1 \\ 1 & 5 & 1 \\ 1 & 1 & 5 \end{matrix} \right] \)

Find the rank of the following matrices by minor method: \(\left[ \begin{matrix} -1 & 3 \\ 4 & -7 \\ 3 & -4 \end{matrix} \right] \)

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

Find the inverse of each of the following by Gauss-Jordan method: \(\left[ \begin{matrix} 1 & -1 & 0 \\ 1 & 0 & -1 \\ 6 & -2 & -3 \end{matrix} \right] \)

Solve the following system of linear equations by matrix inversion method: 2x − y = 8, 3x + 2y = −2

Solve the following system of linear equations by matrix inversion method: x + y + z − 2 = 0, 6x − 4y + 5z − 31 = 0, 5x + 2y + 2z = 13.

Solve the following systems of linear equations by Cramer’s rule: 3x + 3y − z = 11, 2x − y + 2z = 9, 4x + 3y + 2z = 25.

Part - B

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.

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.

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

If A = \(\frac { 1 }{ 7 } \left[ \begin{matrix} 6 & -3 & a \\ b & -2 & 6 \\ 2 & c & 3 \end{matrix} \right] \) is orthogonal, find a, b and c , and hence A−1.

Find the inverse of A = \(\left[ \begin{matrix} 2 & 1 & 1 \\ 3 & 2 & 1 \\ 2 & 1 & 2 \end{matrix} \right] \) by Gauss-Jordan method.

Test for consistency of the following system of linear equations and if possible solve: 4x − 2y + 6z = 8, x + y − 3z = −1, 15x − 3y + 9z = 21.

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.

Solve the system: x + 3y - 2z = 0, 2x - y + 4z = 0, x - 11y + 14z = 0

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

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 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 = \(\left[ \begin{matrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{matrix} \right] \), show that A^-1 = \(\frac {1}{2}\) (A² - 3I).

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

Find the inverse of each of the following by Gauss-Jordan method:
\(\left[ \begin{matrix} 2 & -1 \\ 5 & -2 \end{matrix} \right] \)

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

Solve the following systems of linear equations by Gaussian elimination method:
2x − 2y + 3z = 2, x + 2y − z = 3,3x − y + 2z = 1

Test for consistency and if possible, solve the following systems of equations by rank method.
x - y + 2z = 2, 2x + y + 4z = 7, 4x - y + z = 4

Investigate the values of λ and m the system of linear equations 2x + 3y + 5z = 9, 7x + 3y - 5z = 8, 2x + 3y + λz = μ, have
(i) no solution
(ii) a unique solution
(iii) an infinite number of solutions.

Determine the values of λ for which the following system of equations x + y + 3z = 0, 4x + 3y + λz = 0, 2x + y + 2z = 0 has
(i) a unique solution
(ii) a non-trivial solution

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

Find the inverse (if it exists) of the following:
\(\left[ \begin{matrix} 5 & 1 & 1 \\ 1 & 5 & 1 \\ 1 & 1 & 5 \end{matrix} \right] \)

Find the rank of the following matrices by minor method:
\(\left[ \begin{matrix} -1 & 3 \\ 4 & -7 \\ 3 & -4 \end{matrix} \right] \)

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

Find the inverse of each of the following by Gauss-Jordan method:
\(\left[ \begin{matrix} 1 & -1 & 0 \\ 1 & 0 & -1 \\ 6 & -2 & -3 \end{matrix} \right] \)

Solve the following system of linear equations by matrix inversion method:
2x − y = 8, 3x + 2y = −2

Solve the following system of linear equations by matrix inversion method:
x + y + z − 2 = 0, 6x − 4y + 5z − 31 = 0, 5x + 2y + 2z = 13.

Solve the following systems of linear equations by Cramer’s rule:
3x + 3y − z = 11, 2x − y + 2z = 9, 4x + 3y + 2z = 25.

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

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.

If A = \(\frac { 1 }{ 7 } \left[ \begin{matrix} 6 & -3 & a \\ b & -2 & 6 \\ 2 & c & 3 \end{matrix} \right] \) is orthogonal, find a, b and c , and hence A⁻¹.

Test for consistency of the following system of linear equations and if possible solve:
4x − 2y + 6z = 8, x + y − 3z = −1, 15x − 3y + 9z = 21.