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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?
[2019: 2 Marks, Setll]
(a) 156π
(b) 396π
(c) 468π
(d) 78π
Ans. (b)
Solution. Volume of water
Q.2. The inverse of the matrix
[2019: 2 Marks, Setll]
Ans. (d)
Q.3. The rank of the following matrix is
[2018 : 2 Marks, SetII]
(a) 1
(b) 2
(c) 3
(d) 4
Ans. (b)
Solution.
► Number of non zero rows = 2
► Rank of A = 2
Q.4. The matrix has
[2018 : 2 Marks, SetII]
(a) real eigenvalues and eigenvectors
(b) real eigenvalues but complex eigenvectors
(c) complex eigenvalues but real eigenvectors
(d) complex eigenvalues and eigenvectors
Ans. (d)
Solution.
∴ Complex Eigenvalues and complex Eigen vectors.
Q.5. Which one of the following matrices is singular?
[2018: 1 Marks, SetI]
Ans. (c)
Solution. Option (a): A = 6  5 = 1
Option (b): A = 9  4 = 5
Option (c): A = 1212 = 0
Option (d): A = 8  18 = 10
Hence matrix (c) is singular.
Q.7. If A = and B = AB^{T} is equal to
[2017 : 2 Marks, SetII]
Ans. (a)
Solution.
Q.8. Consider the following simultaneous equations (with c_{1} and c_{2} being constants):
[2017 : 1 Mark, SetII]
3x_{1} + 2x_{2} = c_{1}
4x_{1} + x_{2} = c_{2}
The characteristics equation for these simultaneous equations is
(a) λ^{2}  4λ  5 = 0
(b) λ^{2}  4λ + 5 = 0
(c) λ^{2} + 4λ  5 = 0
(d) λ^{2} + 4λ + 5 = 0
Ans. (a)
Solution.
A  λI = 0
(3  λ) (1  λ)  8 = 0
3  4λ + λ^{2}  8 = 0
λ^{2}  4λ  5 = 0
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, SetI]
(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 16: For what value of p the following set of equations will have no solution? [2015 : 1 Mark, SetI]
2x + 3y = 5
3x + py = 10
Solution: Given system of equations has no solution if the lines are parallel i.e., their slopes are equal
2/3 = 3/p
⇒ p = 4.5
Question 17: The rank of the matrix is _____. [2014 : 2 Marks, SetII]
Solution:
Determinant of matrix is not zero.
∴ Rank is 2
Question 18: The determinant of matrix [2014 : 1 Mark, SetII]
Solution:
Interchanging column 1 and column 2 and taking transpose,
Question 19: With reference to the conventional Cartesian (x, y) coordinate system, the vertices of a triangle have the following coordinates; (x_{1}, y_{1}) = (1, 0); (x_{2}, y_{2}) = (2, 2); (x_{3}, y_{3}) = (4, 3). The area of the triangle is equal to [2014 : 1 Mark, SetI]
(a) 3/2
(b) 3/4
(c) 4/5
(d) 5/2
Answer: (a)
Solution:
Area of triangle is
Question 20: The sum of Eigen values of matrix, [M] is where [2014 : 1 Mark, SetI]
(a) 915
(b) 1355
(c) 1640
(d) 2180
Answer: (a)
Solution: Sum of eigen values = trace of matrix
= 215 + 150 + 550 = 915
Question 21: Given the matrices the product K^{T} JK is ____. [2014 : 1 Mark, SetI]
Solution:
Question 22: There are three matrixes P(4 x 2), Q(2 x 4) and R(4 x 1). The minimum of multiplication required to compute the matrix PQR is [2013 : 1 Mark]
Solution: If we multiply QR first then,
Q_{2x4} x R_{(4x1)} having multiplication number 8.
There fore P_{(4 x 2)} QR_{(2 x 1)} will have minimum number of multiplication = (8 + 8) = 16.
Question 23: The eigen values of matrix [2011 : 2 Marks]
(a) 2.42 and 6.86
(b) 3.48 and 13.53
(c) 4.70 and 6.86
(d) 6.86 and 9.50
Answer: (b)
Solution: We need eigen values of
The characteristic equation is,
So eigen values are,
λ = 3.48, 13.53
Question 24: [A] is square matix which is neither symmetric nor skewsymmetric and [A]^{T} is its transpose. The sum and difference of these matrices are defined as [S] = [A] + [A]^{T} and [D] = [A]  [A]^{T}, respectively. Which of the following statements is TRUE? [2011 : 1 Mark]
(a) Both [S] and [D] are symmetric
(b) Both [S] and [D] are skewsymmetric
(c) [S] is skewsymmetric and [D] is symmetric
(d) [S] is symmetric and [D] is skewsymmetric
Answer: (d)
Solution: Since (A + A^{t}) = A^{t} + (A^{t})^{t}
= A^{t} + A
i.e. S^{t} = S
∴ S is symmetric
Since (A  A^{t})^{t} = A^{t}  (A^{t})^{t}
= A^{t}  A = (A  A^{t})
i.e. D^{t} =  D
So D is SkewSymmetric.
Question 25: The inverse of the matrix [2010 : 2 Marks]
Answer: (b)
Solution:
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