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**Q.1. Consider the hemispherical tank of radius 13 m as shown in the figure (not drawn to scale). What is the volume of water (in m ^{3}) when the depth of water at the centre of the tank is 6 m? **

(b) 396π

(c) 468π

(d) 78π

► Number of non zero rows = 2

► Rank of A = 2

∴** **Complex Eigenvalues and complex Eigen vectors.**Q.5. Which one of the following matrices is singular? ****[2018: 1 Marks, Set-I]****Ans. **(c)**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 1:For the given orthogonal matrix Q, [2018: 1 Marks, Set-I]

The inverse is

The inverse is

**Q.7. If A = and B = AB ^{T} is equal to **

|A - λI| = 0

(3 - λ) (1 - λ) - 8 = 0

3 - 4λ + λ

λ

Question 2:Consider the matrix Which one of the following statements is TRUE for the eigenvalues and eigenvectors of this matrix? [2017: 2 Marks, Set-I]

**Q.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****Ans. **(c)**Solution. **Given that P is inverse of Q.

► P = Q^{-1}

► PQ = Q^{-1}Q, QP = QQ^{-1}

► PQ = I , QP = I

∴ PQ = QP = I

Question 3: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** **

x + 2y - 3z = a

2x + 3y + 3x = b

5x + 9y - 6z = c

This system is consistent if a, b and c satisfy the equation

Question 4: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]

Question 5:The two Eigenvalues of the matrix have a ratio of 3 : 1 for p = 2. What is another value of p for which the Eigenvalues have the same ratio of 3 : 1?

[2015: 2 Marks, Set-II]

[2015: 2 Marks, Set-II]

Question 6:Let A = with n __>__ 3 and a_{ij} = i.j. The rank of A is [2015 : 1 Mark, Set-II]

Question 7:The smallest and largest Eigen values of the following matrix are [2015 : 2 Marks, Set-I]

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