12th Maths Monthly Test - 3 (Applications Of Matrices And Determinants)

St. Britto Hr. Sec. School - Madurai

12th Maths Monthly Test - 3 (Applications Of Matrices And Determinants)-Aug 2020

Time : 1.30 Hours

Part - A

Generic 2 - Answer All Question

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

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

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 = \(\left[ \begin{matrix} \cos { 2x } & -\sin { 2x } \\ \sin { 2x } & \cos { 2x } \end{matrix} \right] \)

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.

Decrypt the received encoded message \(\left[ \begin{matrix} 2 & -3 \end{matrix} \right] \left[ \begin{matrix} 20 & 4 \end{matrix} \right] \) with the encryption matrix \(\left[ \begin{matrix} -1 & -1 \\ 2 & 1 \end{matrix} \right] \)
and the decryption matrix as its inverse, where the system of codes are described by the numbers 1 - 26 to the letters A - Z respectively, and the number 0 to a blank space.

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

A man is appointed in a job with a monthly salary of certain amount and a fixed amount of annual increment. If his salary was Rs.19,800 per month at the end of the first month after 3 years of service and Rs.23,400 per month at the end of the first month after 9 years of service, find his starting salary and his annual increment. (Use matrix inversion method to solve the problem.)
If A = \(\left[ \begin{matrix} 5 & 3 \\ -1 & -2 \end{matrix} \right] \), show that A² - 3A - 7I₂ = O₂. Hence find A⁻¹.

The prices of three commodities A, B and C are Rs.x, y and z per units respectively. A person P purchases 4 units of B and sells two units of A and 5 units of C . Person Q purchases 2 units of C and sells 3 units of A and one unit of B . Person R purchases one unit of A and sells 3 unit of B and one unit of C . In the process, P, Q and R earn Rs.15,000, Rs.1,000 and Rs.4,000 respectively. Find the prices per unit of A, B and C . (Use matrix inversion method to solve the problem.)

In a competitive examination, one mark is awarded for every correct answer while \(\frac { 1 }{ 4 }\) mark is deducted for every wrong answer. A student answered 100 questions and got 80 marks. How many questions did he answer correctly ? (Use Cramer’s rule to solve the problem).

A fish tank can be filled in 10 minutes using both pumps A and B simultaneously. However, pump B can pump water in or out at the same rate. If pump B is inadvertently run in reverse, then the tank will be filled in 30 minutes. How long would it take each pump to fill the tank by itself ? (Use Cramer’s rule to solve the problem).
Find the rank of the following matrices by row reduction method:
\(\left[ \begin{matrix} 1 \\ \begin{matrix} 2 \\ 5 \end{matrix} \end{matrix}\begin{matrix} 1 \\ \begin{matrix} -1 \\ -1 \end{matrix} \end{matrix}\begin{matrix} 1 \\ \begin{matrix} 3 \\ 7 \end{matrix} \end{matrix}\begin{matrix} 3 \\ \begin{matrix} 4 \\ 11 \end{matrix} \end{matrix} \right] \)

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.

Part - B

Generic 3 - Answer All Question

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

Solve: 3x+ay =4, 2x+ay=2, a≠0 by Cramer's rule.

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.

Find the inverse of the matrix \(\left[ \begin{matrix} 2 & -1 & 3 \\ -5 & 3 & 1 \\ -3 & 2 & 3 \end{matrix} \right] \).

Verify (AB)^-1 = B^-1A^-1 with A = \(\left[ \begin{matrix} 0 & -3 \\ 1 & 4 \end{matrix} \right] \), B = \(\left[ \begin{matrix} -2 & -3 \\ 0 & -1 \end{matrix} \right] \).

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.
Verify that (A^-1)^T = (A^T)^-1 for A=\(\left[ \begin{matrix} -2 & -3 \\ 5 & -6 \end{matrix} \right] \).

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.

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

Part - C

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 = \(\frac { 1 }{ 7 } \left[ \begin{matrix} 6 & -3 & a \\ b & -2 & 6 \\ 2 & c & 3 \end{matrix} \right] \) 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.)

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12th Maths Monthly Test - 3 (Applications Of Matrices And Determinants)

St. Britto Hr. Sec. School - Madurai

12th Maths Monthly Test - 3 (Applications Of Matrices And Determinants)-Aug 2020

Part - A

Generic 2 - Answer All Question

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

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

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

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.

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

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

In a competitive examination, one mark is awarded for every correct answer while \(\frac { 1 }{ 4 }\) mark is deducted for every wrong answer. A student answered 100 questions and got 80 marks. How many questions did he answer correctly ? (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.

Part - B

Generic 3 - Answer All Question

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

Solve: 3x+ay =4, 2x+ay=2, a≠0 by Cramer's rule.

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.

Find the inverse of the matrix \(\left[ \begin{matrix} 2 & -1 & 3 \\ -5 & 3 & 1 \\ -3 & 2 & 3 \end{matrix} \right] \).

Verify (AB)-1 = B-1A-1 with A = \(\left[ \begin{matrix} 0 & -3 \\ 1 & 4 \end{matrix} \right] \), B = \(\left[ \begin{matrix} -2 & -3 \\ 0 & -1 \end{matrix} \right] \).

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.

Verify that (A-1)T = (AT)-1 for A=\(\left[ \begin{matrix} -2 & -3 \\ 5 & -6 \end{matrix} \right] \).

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.

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

Part - C

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 = \(\frac { 1 }{ 7 } \left[ \begin{matrix} 6 & -3 & a \\ b & -2 & 6 \\ 2 & c & 3 \end{matrix} \right] \) 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.

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

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

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 = \(\left[ \begin{matrix} \cos { 2x } & -\sin { 2x } \\ \sin { 2x } & \cos { 2x } \end{matrix} \right] \)

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

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

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.

Verify (AB)^-1 = B^-1A^-1 with A = \(\left[ \begin{matrix} 0 & -3 \\ 1 & 4 \end{matrix} \right] \), B = \(\left[ \begin{matrix} -2 & -3 \\ 0 & -1 \end{matrix} \right] \).

Verify that (A^-1)^T = (A^T)^-1 for A=\(\left[ \begin{matrix} -2 & -3 \\ 5 & -6 \end{matrix} \right] \).

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

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