https://www.kalvikadal.in https://material.kalvikadal.in
JESUS THE KING (JTK) MATHS TUITION JESUS THE KING (JTK) MATHS TUITION
STD – XII – MATHEMATICS
SLOW LEARNERS QUESTION BANK
CHAPTER – 1
APPLICATIONS OF MATRICES AND DETERMINANTS
2 and 3 Marks:
−1 2 2
1. If adj A = [ 1 1 2], find A−1.
2 2 1
0 −2 0
2. If adjA = [ 6 2 −6], find A−1.
−3 0 6
7 7 −7
3. Find a matrix A if adj(A) = [−1 11 7 ].
11 5 7
2 −4 2
4. If adjA = [−3 12 −7], find A.
−2 0 2
n
1 0 1
l.i
5. Find adj(adj(A)) if adj A = [ 0 2 0]. da
−1 0 1
6. If A is symmetric, prove that then adj A is also symmetric.
ka
7. If A is a non-singular matrix of odd order, prove that |adj A| is positive.
2 9
8. Verify the property (AT)-1 = (A-1)T with A = [ ].
vi
1 7
0 −3 −2 −3
al
9. Verify (AB)-1 = B-1A-1 with A = [ ], B = [ ]
1 4 0 −1
.k
3 2 −1 −3
10. If A = [ ] and B = [ ], verify that (AB)-1 = B-1A-1.
7 5 5 2
w
8 −4
11. If A = [ ], verify that A (adj A) = (adj A)A = |A|I2.
w
−5 3
cosθ −sinθ
w
12. Prove that [ ] is orthogonal.
sinθ cosθ
1 −2 −1 0
12. Find the rank of [ ] by minor method.
3 −6 −3 1
1 −2 3
13. Find the rank of [2 4 −6] by minor method.
5 1 −1
0 1 2 1
14. Find the rank of [0 2 4 3] by minor method.
8 1 0 2
1 1 1 3
15. Find the rank of [2 −1 3 4 ] by row reduction method.
5 −1 7 11
1 2 −1
16. Find the rank of [3 −1 2] by row reduction method.
1 −2 3
1 −1 1
M. IJUSPlease
VIJAY, M.Sc.,
send your B.Ed,
Materials 915 917
& Question Papers to (or) 67 66 - 9385336929.
Whatsapp 86 80 99 2001
, https://www.kalvikadal.in https://material.kalvikadal.in
JESUS THE KING (JTK) MATHS TUITION JESUS THE KING (JTK) MATHS TUITION
3 −8 5 2
17. Find the rank of [ 2 −5 1 4 ] by row reduction method.
−1 2 3 −2
2 −2 4 3
18. Find the rank of [−3 4 −2 −1] by row reduction method.
6 2 −1 7
19. Solve the following system of linear equations, using matrix inversion
method: 5x + 2y = 3, 3x + 2y = 5.
20. Solve the following system of linear equations, using matrix inversion
method: 2x – y = 8, 3x + 2y = –2.
21. Solve the following systems of linear equations by Cramer’s rule:
5x – 2y + 16 = 0, x + 3y – 7 = 0.
5Marks:
1. 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.
2. Find the value of k for which the equations kx – 2y + z = 1,
n
x – 2ky + z = –2, x – 2y + kz = 1 have (i) no solution (ii) unique solution
l.i
(iii) infinitely many solution
da
3. Investigate the values of 𝜆 and 𝜇 the system of linear equations
2x + 3y + 5z = 9, 7x + 3y – 5z = 8, 2x + 3y + 𝜆z = 𝜇 have
ka
(i) no solution (ii) a unique solution (iii) an infinite number of solutions.
4. Test the consistency of the following system of linear equations
vi
x – y + z = –9, 2x – y + z = 4, 3x – y + z = 6, 4x – y + 2z = 7.
al
5. Test the consistency of the following system of linear equations
.k
2x + 2y + z = 5, x – y + z = 1, 3x + y + 2z = 4.
w
6. Test the consistency of the following system of linear equations
2x – y + z = 2, 6x – 3y + 3z = 6, 4x – 2y + 2z = 4.
w
7. Determine the values of 𝜆 for which the following system of equations
w
x + y + 3z = 0, 4x + 3y + 𝜆z = 0, 2x + y + 2z = 0 has (i) a unique solution
(ii) a non-trivial solution.
8. Solve, by Cramer’s rule, the system of equations
x1 – x2 = 3, 2x1 + 3x2 + 4x3 = 17, x2 + 2x3 = 7.
9. Solve the following systems of linear equations by Cramer’s rule:
3x + 3y – z = 11, 2x – y + 2z = 9, 4x + 3y + 2z = 25.
10. Solve the following systems of linear equations by Cramer’s rule:
3 4 2 1 2 1 2 5 4
– – – 1 = 0, + + – 2 = 0, – – + 1 = 0.
𝑥 𝑦 𝑧 𝑥 𝑦 𝑧 𝑥 𝑦 𝑧
cos 𝛼 0 sin 𝛼
11. If F(𝛼) = [ 0 1 0 ], show that [𝐹(𝛼)]−1 = 𝐹(−𝛼).
− sin 𝛼 0 cos 𝛼
5 3
12. If A = [ ], show that A2 – 3A – 7I2 = O2. Hence find A-1.
−1 −2
13. If ax + bx + c is divided by x + 3, x – 5 and x – 1, the remainders are
2
21, 61 and 9 respectively. Find a, b and c. (Gaussian elimination method)
M. IJUSPlease
VIJAY, M.Sc.,
send your B.Ed,
Materials 915 917
& Question Papers to (or) 67 66 - 9385336929.
Whatsapp 86 80 99 2001
JESUS THE KING (JTK) MATHS TUITION JESUS THE KING (JTK) MATHS TUITION
STD – XII – MATHEMATICS
SLOW LEARNERS QUESTION BANK
CHAPTER – 1
APPLICATIONS OF MATRICES AND DETERMINANTS
2 and 3 Marks:
−1 2 2
1. If adj A = [ 1 1 2], find A−1.
2 2 1
0 −2 0
2. If adjA = [ 6 2 −6], find A−1.
−3 0 6
7 7 −7
3. Find a matrix A if adj(A) = [−1 11 7 ].
11 5 7
2 −4 2
4. If adjA = [−3 12 −7], find A.
−2 0 2
n
1 0 1
l.i
5. Find adj(adj(A)) if adj A = [ 0 2 0]. da
−1 0 1
6. If A is symmetric, prove that then adj A is also symmetric.
ka
7. If A is a non-singular matrix of odd order, prove that |adj A| is positive.
2 9
8. Verify the property (AT)-1 = (A-1)T with A = [ ].
vi
1 7
0 −3 −2 −3
al
9. Verify (AB)-1 = B-1A-1 with A = [ ], B = [ ]
1 4 0 −1
.k
3 2 −1 −3
10. If A = [ ] and B = [ ], verify that (AB)-1 = B-1A-1.
7 5 5 2
w
8 −4
11. If A = [ ], verify that A (adj A) = (adj A)A = |A|I2.
w
−5 3
cosθ −sinθ
w
12. Prove that [ ] is orthogonal.
sinθ cosθ
1 −2 −1 0
12. Find the rank of [ ] by minor method.
3 −6 −3 1
1 −2 3
13. Find the rank of [2 4 −6] by minor method.
5 1 −1
0 1 2 1
14. Find the rank of [0 2 4 3] by minor method.
8 1 0 2
1 1 1 3
15. Find the rank of [2 −1 3 4 ] by row reduction method.
5 −1 7 11
1 2 −1
16. Find the rank of [3 −1 2] by row reduction method.
1 −2 3
1 −1 1
M. IJUSPlease
VIJAY, M.Sc.,
send your B.Ed,
Materials 915 917
& Question Papers to (or) 67 66 - 9385336929.
Whatsapp 86 80 99 2001
, https://www.kalvikadal.in https://material.kalvikadal.in
JESUS THE KING (JTK) MATHS TUITION JESUS THE KING (JTK) MATHS TUITION
3 −8 5 2
17. Find the rank of [ 2 −5 1 4 ] by row reduction method.
−1 2 3 −2
2 −2 4 3
18. Find the rank of [−3 4 −2 −1] by row reduction method.
6 2 −1 7
19. Solve the following system of linear equations, using matrix inversion
method: 5x + 2y = 3, 3x + 2y = 5.
20. Solve the following system of linear equations, using matrix inversion
method: 2x – y = 8, 3x + 2y = –2.
21. Solve the following systems of linear equations by Cramer’s rule:
5x – 2y + 16 = 0, x + 3y – 7 = 0.
5Marks:
1. 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.
2. Find the value of k for which the equations kx – 2y + z = 1,
n
x – 2ky + z = –2, x – 2y + kz = 1 have (i) no solution (ii) unique solution
l.i
(iii) infinitely many solution
da
3. Investigate the values of 𝜆 and 𝜇 the system of linear equations
2x + 3y + 5z = 9, 7x + 3y – 5z = 8, 2x + 3y + 𝜆z = 𝜇 have
ka
(i) no solution (ii) a unique solution (iii) an infinite number of solutions.
4. Test the consistency of the following system of linear equations
vi
x – y + z = –9, 2x – y + z = 4, 3x – y + z = 6, 4x – y + 2z = 7.
al
5. Test the consistency of the following system of linear equations
.k
2x + 2y + z = 5, x – y + z = 1, 3x + y + 2z = 4.
w
6. Test the consistency of the following system of linear equations
2x – y + z = 2, 6x – 3y + 3z = 6, 4x – 2y + 2z = 4.
w
7. Determine the values of 𝜆 for which the following system of equations
w
x + y + 3z = 0, 4x + 3y + 𝜆z = 0, 2x + y + 2z = 0 has (i) a unique solution
(ii) a non-trivial solution.
8. Solve, by Cramer’s rule, the system of equations
x1 – x2 = 3, 2x1 + 3x2 + 4x3 = 17, x2 + 2x3 = 7.
9. Solve the following systems of linear equations by Cramer’s rule:
3x + 3y – z = 11, 2x – y + 2z = 9, 4x + 3y + 2z = 25.
10. Solve the following systems of linear equations by Cramer’s rule:
3 4 2 1 2 1 2 5 4
– – – 1 = 0, + + – 2 = 0, – – + 1 = 0.
𝑥 𝑦 𝑧 𝑥 𝑦 𝑧 𝑥 𝑦 𝑧
cos 𝛼 0 sin 𝛼
11. If F(𝛼) = [ 0 1 0 ], show that [𝐹(𝛼)]−1 = 𝐹(−𝛼).
− sin 𝛼 0 cos 𝛼
5 3
12. If A = [ ], show that A2 – 3A – 7I2 = O2. Hence find A-1.
−1 −2
13. If ax + bx + c is divided by x + 3, x – 5 and x – 1, the remainders are
2
21, 61 and 9 respectively. Find a, b and c. (Gaussian elimination method)
M. IJUSPlease
VIJAY, M.Sc.,
send your B.Ed,
Materials 915 917
& Question Papers to (or) 67 66 - 9385336929.
Whatsapp 86 80 99 2001
