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**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 m ^{3}) when the depth of water at the centre of the tank is 6 m? [2019 : 2 Marks, Set-ll]**

(b) 396Ï€

(c) 468Ï€

(d) 78Ï€

Number of non zero rows = 2

rank of A = 2

âˆ´** **Complex Eigen values and complex Eigen vectors.**Question 5: Which one of the following matrices is singular? [2018 : 1 Marks, Set-I]****Answer: **(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 6:**** For the given orthogonal matrix Q, [2018 : 1 Marks, Set-I]The inverse is**

Or âˆµ Q is orthogonal

âˆ´ Q

|A - Î»I| = 0

(3 - Î») (1 - Î») - 8 = 0

3 - 4Î» + Î»

Î»

Characteristics equation is |A â€” Î»I| = 0

5 - 5Î», - Î», + Î»^{2} + 4 = 0

Î»2 - 6Î» + 9 = 0

Î» = 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 â‰ I ****(b) QP = I but PQ â‰ I ****(c) PQ = I and QP = I ****(d) PQ = QP = I****Answer: (c)****Solution: **Given that P is inverse of Q.

P = Q^{-1}

PQ = Q^{-1}Q, QP = QQ^{-1}

PQ = I , QP = I

âˆ´ 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****Answer: **(b)**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]****(a) 4 ****(b) 3 ****(c) 2 ****(d) 1****Answer:** (d)**Solution: Consider the â€˜2 x 2â€™ square matrix **

â‡’ Î»^{2}- (a + d)Î» + (ad - be) = 0 .......(i)

Putting Î» = 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.

âˆ´ Î» = 1 satisfied the eq. (i) but Î» = 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]****(a) -2 ****(b) 1 ****(c) 7/3 ****(d) 14/3****Answer:** (d)**Solution: **

Let Î»_{1} and Î»_{2} be the eigen value of matrix A

âˆ´

Sum of eigen value

= Î»_{1} + Î»_{2} = 2 + p .......(i)

Product of eigen value

Î»_{1}Î»_{2} = 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 a_{ij} = i.j. The rank of A is [2015 : 1 Mark, Set-II]**

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.

Only 1 and 2 satisfy this equation.

Î» = 1, 1,2

Hence,Smallest eigen value = 1 and

Largest eigen value = 2

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