Question 1: Consider the hemi-spherical tank of radius 13 m as shown in the figure (not drawn to scale). What is the volume of water (in m3) when the depth of water at the centre of the tank is 6 m? [2019 : 2 Marks, Set-ll]
Solution: Volume of water
Question 2: The inverse of the matrix [2019 : 2 Marks, Set-ll]
Question 3: The rank of the following matrix is [2018 : 2 Marks, Set-II]
Number of non zero rows = 2
rank of A = 2
Question 4: The matrix has [2018 : 2 Marks, Set-II]
(a) real eigenvalues and eigenvectors
(b) real eigenvalues but complex eigenvectors
(c) complex eigenvalues but real eigenvectors
(d) complex eigenvalues and eigenvectors
∴ Complex Eigen values and complex Eigen vectors.
Question 5: Which one of the following matrices is singular? [2018 : 1 Marks, Set-I]
Solution: Option (a): |A| = 6 - 5 = 1
Option (b): |A| = 9 - 4 = 5
Option (c): |A| = 12-12 = 0
Option (d): |A| = 8 - 18 = -10
Hence matrix (c) is singular.
Question 6: For the given orthogonal matrix Q, [2018 : 1 Marks, Set-I]
The inverse is
Or ∵ Q is orthogonal
∴ Q-1 = QT
Question 7: is equal to [2017 : 2 Marks, Set-II]
Question 8: Consider the following simultaneous equations (with c1 and c2 being constants): [2017 : 1 Mark, Set-II]
3x1 + 2x2 = c1
4x1 + x2 = c2
The characteristics equation for these simultaneous equations is
(a) λ2 - 4λ - 5 = 0
(b) λ2 - 4λ + 5 = 0
(c) λ2 + 4λ - 5 = 0
(d) λ2 + 4λ + 5 = 0
|A - λI| = 0
(3 - λ) (1 - λ) - 8 = 0
3 - 4λ + λ2 - 8 = 0
λ2 - 4λ - 5 = 0
Question 9: Consider the matrix Which one of the following statements is TRUE for the eigenvalues and eigen vectors of this matrix? [2017 : 2 Marks, Set-I]
(a) Eigenvalue 3 has a multiplicity of 2, and only one independent eigenvector exists.
(b) Eigenvalue 3 has a multiplicity of 2, and two independent eigenvector exists.
(c) Eigenvalue 3 has a multiplicity of 2, and no independent eigenvector exists.
(d) Eigenvalue are 3 and -3, and two indepedent eigenvectors exist.
Characteristics equation is |A — λI| = 0
5 - 5λ, - λ, + λ2 + 4 = 0
λ2 - 6λ + 9 = 0
λ = 3, 3
Algebraic multiplicity of eigen value 3 is 2. It has only one independent eigen vector exists.
Question 10: The matrix P is the inverse of a matrix Q. If I denotes the identity matrix, which one of the following options is correct? [2017 : 1 Mark, Set-I]
(a) PQ = I but QP ≠ I
(b) QP = I but PQ ≠ I
(c) PQ = I and QP = I
(d) PQ = QP = I
Solution: Given that P is inverse of Q.
P = Q-1
PQ = Q-1Q, QP = QQ-1
PQ = I , QP = I
∴ PQ = QP = I
Question 11: Consider the following linear system. [2016 : 2 Marks, Set-Il]
x + 2y - 3z = a
2x + 3y + 3x = b
5x + 9y - 6z = c
This system is consistent if a, b and c satisfy the equation
(a) 7 a - b - c = 0
(b) 3a + b - c = 0
(c) 3 a - b + c = 0
(d) 7 a - b + c = 0
Question 12: If the entries in each column of a square matrix M add up to 1, then an eigen value of M is [2016 : 1 Mark, Set-I]
Solution: Consider the ‘2 x 2’ square matrix
⇒ λ2- (a + d)λ + (ad - be) = 0 .......(i)
Putting λ = 1, we get
1 - (a + d) + ad - bc = 0
1 - a - d + ad - (1 - d)(1 - a) = 0
1 - a - d + ad 1 + a+ d - ad = 0
0 = 0 which is true.
∴ λ = 1 satisfied the eq. (i) but λ = 2, 3, 4 does not satisfy the eq. (i). For all possible values of a, d.
Question 13: The two Eigen values of the matrix have a ratio of 3 : 1 for p = 2. What is another value of p for which the Eigen values have the same ratio of 3 : 1? [2015 : 2 Marks, Set-II]
Let λ1 and λ2 be the eigen value of matrix A
Sum of eigen value
= λ1 + λ2 = 2 + p .......(i)
Product of eigen value
λ1λ2 = 2p - 1 ........(ii)
By putting values of p from options.
By putting option (d) 14/3 in above equations gives value
Hence ratio of two eigen values
Question 14: with n > 3 and aij = i.j. The rank of A is [2015 : 1 Mark, Set-II]
(c) n - 1
Solution: Rank of A = 1
Because each row will be scalar multiple of first row. So we will get only one non-zero row in row Echeleaon form of A
Rank of A = 1
Because all the minors of order greater than 1 will be zero.
Question 15: The smallest and largest Eigen values of the following matrix are [2015 : 2 Marks, Set-I]
(a) 1.5 and 2.5
(b) 0.5 and 2.5
(c) 1.0 and 3.0
(d) 1.0 and 2.0
Solution: For eigen values |A - λI| = 0
Only 1 and 2 satisfy this equation.
λ = 1, 1,2
Hence,Smallest eigen value = 1 and
Largest eigen value = 2