78. Determinants Problems Part 2

51. \(\begin{vmatrix}x & 2 & 3 \\ 4 & x & 1 \\ x & 2 & 5\end{vmatrix} = 0\)

52. \(\begin{vmatrix}x + a & b & c \\ a & x + b & c \\ a & b & x + c\end{vmatrix} = 0\)

53. \(\begin{vmatrix}3 + x & 5 & 2 \\ 1 & 7 + x & 6 \\ 2 & 5 & 3 + x\end{vmatrix} = 0\)

Show without expanding at any stage that:

54. \(\begin{vmatrix}a + b & b + c & c + a\\ b + c & c + a & a + b \\ c + a & a + b & b + c\end{vmatrix} = 2\begin{vmatrix}a & b & c \\ b & c & a \\ c & a & b\end{vmatrix}\)

55. \(\begin{vmatrix}b + c & c + a & a + b \\ q + r & r + p & p + q \\ y + z & z + x & x + y \end{vmatrix} = 2\begin{vmatrix}a & b & c \\ p & q & r \\ x & y & z\end{vmatrix}\)

56. \(\begin{vmatrix}1 & \cos\alpha - \sin\alpha & \cos\alpha + \sin\alpha \\ 1 & \cos\beta - \sin\beta & \cos\beta + \sin\beta \\ 1 & \cos\gamma - \sin\gamma & \cos\gamma + \sin\gamma \end{vmatrix} = 2\begin{vmatrix}1 & \cos\alpha & \cos\alpha \\ 1 & \cos\beta & \sin\beta \\ 1 & \cos\gamma & \sin\gamma \end{vmatrix}\)

57. \(\begin{vmatrix}(a - 1)^2 & a^2 + 1 & a \\ (b - 1)^2 & b^2 + 1 & b \\ (c - 1)^2 & c^2 + 1 & c\end{vmatrix} = 0\)

58. \(\begin{vmatrix}0 & c & b \\ -c & 0 & a \\ -b & -a & 0\end{vmatrix} = 0\)

59. \(\begin{vmatrix}1 & a & bc \\ 1 & b & ca \\ 1 & c & ab\end{vmatrix} = \begin{vmatrix}1 & a & a^2 \\ 1 & b & b^2 \\ 1 & c & c^2\end{vmatrix}\)

60. \(\begin{vmatrix}a & b & c \\ x & y & z \\ yz & zx & xy\end{vmatrix} = \begin{vmatrix}ax & by & ca \\ x^2 & y^2 & z^2 \\ 1 & 1 & 1\end{vmatrix}\)

61. \(\begin{vmatrix}a & b & c \\ x & y & z \\ p & q & r\end{vmatrix} = \begin{vmatrix}y & b & q \\ x & a & p \\ z & c & r\end{vmatrix} = \begin{vmatrix}x & y & z \\ p & q & r \\ a & b & c\end{vmatrix}\)

62. Find the value of the fiollowing determinant

    \(\begin{vmatrix}m! & (m + 1)! & (m + 2)! \\ (m + 1)! & (m + 2)! & (m + 3)! \\ (m + 2)! & (m + 3)! & (m + 4)!\end{vmatrix}\)

63. Solve the following system of equations by Cramer’s rule:

    \(x + y = 4; 2x - 3y = 9\)

64. Solve the following equations by Cramer’s rule:

    \(2x - y + 3z = 0; x + y + z = 6; x - y + z = 2\)

65. Determine whether the following system of equations has no solution, unique solutionor infinite number of solutions:

    \(2x + 3y = 6; 4x + 6y = 10\)

66. Show that following system of equations is consistent:

    \(x + y - z = 1; 2x + 3y + z = 4; 4x + 3y + z = 16\)

67. Determine whether the following system of equations has no solution, unique solutionor infinite number of solutions:

    \(x + y = 2; 2x + 2y = 4\)

68. Determine whether the following system of equations is consistent:

    \(2x + y = 13; 6x + 3y = 18; x - y = -3\)

69. Show that the system of following equations has non-trivial solution.

    \(x + y - 6z = 0; 3x -y - 2z = 0; x -y + 2z = 0\)

70. For what value of \(k\) the following system of equations possess non-trivial solution. Also find all all the solutions of the system for that value of \(k\)

    \(x + y - kz = 0; 3x - y - 2z = 0; x - y + 2z = 0\)

Solve the following equations by Cramer’s rule:

71. \(x - 2y = 0; 7x + 6y = 40\)

72. \(x + y + z = 9; 3x + 2y - 3z = 0, z - x = 2\)

73. \(x - y + z = 0; 2x + 3y - 5z = -1; 3x - 4y + 2z = -1\)

74. \(2x + 3y - 3z = 0; 5x - 2y + 2z = 19; x + 7y - 5z = 5\)

75. \(x + y + z = 1; ax + by + cz = k; a^2x + b^2y + c^2z = k^2\)

    where \(a\neq b\neq c\)

76. \(3x + 2y - 2z = 1; -x + y - 4z = 1; 2x - 3y + 4z = 8\)

Determine whether the following system of equation has no solution, unique solution or infinite number of solutions:

77. \(3x + 9y = 5; 9x + 27y = 10\)

78. \(5x 3y = 3; x + y = 7\)

79. \(x + 2y = 5; 3x + 6y = 15\)

80. \(2x + 3y + z = 5; 3x + y + 5z = 7; x + 4y - 2z = 3\)

81. \(x + y - z = -2; 6x + 4y + 6z = 26; 2z + 7y + 4z = 31\)

82. \(x + 4y = 9; 2x + 8y = 18; y - 2x = 0\)

83. Find the value of \(k\) such that following system ofequations possess a non-trivial solution over the set of rationals \(Q\). For that value of \(k\) find all the solutions of the systems.

    \(x + ky + 3z = 0; 3x + ky - 2z = 0; 2x + 3y - 4z =0\)

84. If \(a, b, c\) are different, show that the following systen of equations has non-trivial solution only when \(a + b + c = 0\)

    \(ax + by + cz = 0; bx + cy + az = 0; cx + ay + bz = 0\)

85. For what value of \(\lambda\) the following system of equations has non-trivial solution.

    \(3x - y + 4z =0; x + 2y - 3z = 0; 6x + 5y - \lambda z = 0\)

86. For a positive integer \(n\), if

    \(D = \begin{vmatrix}n! & (n + 1)! & (n + 2)! \\ (n + 1)! & (n + 2)! & (n + 3)! \\ (n + 2)! & (n + 3)! & (n + 4)!\end{vmatrix}\)

    then show that \(\frac{D}{(n!)^3} - 4\) is divisible by \(n\).

87. Let the three digit numbers \(A28, 3B9, 62C,\) where \(A, B, C\) are integers between \(0\) and \(9\), be divisible by a fixed integer \(k\), show that the determinant

    \(\begin{vmatrix}A & 2 & 6 \\ 8 & 9 & C \\ 2 & B & 2\end{vmatrix}\) is divisible by \(k\).

88. Evaluate \(\begin{vmatrix}{}^xC_1 & {}^xC_2 & {}^xC_3 \\ {}^yC_1 & {}^yC_2 & {}^yC_3 \\ {}^zC_1 & {}^zC_2 & {}^zC_3\end{vmatrix}\)

89. If \(a\neq p, b\neq q, c\neq r\) and \(\begin{vmatrix}p & b & c \\ a & q & c \\ a & b & r\end{vmatrix} = 0\), then find the value of \(\frac{p}{p - a} + \frac{q}{q - b} + \frac{r}{r - c}\)

90. Show that \(\begin{vmatrix}(x - a)^2 & b^2 & c^2 \\ a^2 & (x - b)^2 & c^2 \\ a^2 & b^2 & (x - c)^2\end{vmatrix} = x^2(x - 2a)(x - 2b)(x - 2c)\left(x + \frac{a^2}{x - 2a} + \frac{b^2}{x - 2b} + \frac{c^2}{x - 2c}\right)\)

91. If \(a>0, d>0,\) find the value of the determinant

    \(\begin{vmatrix}\frac{1}{a} & \frac{1}{a(a + d)} & \frac{1}{(a + d)(a + 2d)} \\ \frac{1}{a + )} & \frac{1}{(a + d)(a + 2d)} & \frac{1}{(a + 2d)(a + 3d)} \\ \frac{1}{a + 2d} & \frac{1}{(a + 2d)(a + 3d)} & \frac{1}{(a + 3d)(a + 4d)}\end{vmatrix}\)

92. Show that \(\begin{vmatrix}\frac{1}{a + x} & \frac{1}{a + y} & \frac{1}{a + z} \\ \frac{1}{b + x} & \frac{1}{b + y} & \frac{1}{b + z} \\ \frac{1}{c + x} & \frac{1}{c + y} & \frac{1}{c + z}\end{vmatrix} = \frac{(a - b)(b - c)(c - a)(x - )(y - z)(z - x)}{(a + x)(b + x)(c + x)(b + x)(b + y)(b + z)(c + x)(c + y)(c + z)}\)

93. If \(2s = a + b + c,\) show that

    \(\begin{vmatrix}a^2 & (s - a)^2 & (s - a)^2 \\ (s - b)^2 & s^2 & (s - b)^2 \\ (s - c)^2 & (s - c)^2 & c^2\end{vmatrix} = 2s^3(s - a)(s - b)(s - c)\)

94. Show that \(\begin{vmatrix}ax - by - cz & ay + bx & cx + az \\ ay + bx & by -cz - ax & bz + cy \\ cx + az & bz + cy & cz - ax - by\end{vmatrix} = (x^2 + y^2 + z^2)(a^2 + b^2 + c^2)(ax + by + cz)\)

95. Find the value of \(\theta\) lying between \(0\) and \(\pi/2\) and staisfying the equation:

    \(\begin{vmatrix}1 + \cos^2\theta & \sin^2\theta & 4\sin\theta \\ \cos^2\theta & 1 + \sin^2\theta & 4\sin\theta \\ \cos^2\theta & \sin^2\theta & 1 + 4\sin\theta\end{vmatrix} = 0\)

96. If \(a^2 + b^2 + c^2 = 1.\) then prove that

    \(\begin{vmatrix}a^2 + (b^2 + c^2)\cos\phi & ab(1 - \cos\phi) & ac(1 - \cos\phi) \\ ab(1 - \cos\phi) & b^2 + (c^2 + a^2)\cos\phi & bc(1 - \cos\phi) \\ ca(1 - \cos\phi) & bc(1 - \cos\phi) & c^2 + (a^2 + b^2)\cos\phi\end{vmatrix} = \cos^2\phi\)

97. If non of \(a, b, c\) is zero, show that

    \(\begin{vmatrix}-bc & b^2 + bc & c^2 + bc \\ a^2 + ac & -ac & c^2 + ac \\ a^2 + ab & b^2 + ab & -ab\end{vmatrix} = (ab + bc + ca)^3\)

98. If \(u, v\) are functions of \(x\) and \(y = \frac{u}{v},\) show that

    \(v^2\frac{d^2x}{dy^2} = \begin{vmatrix}u & v & 0 \\ u^{\prime} & v^{\prime} & v \\ u'' & v'' & 2v'\end{vmatrix}\) where primes denotes derivatives.

99. If \(a\neq 0\) and \(a\neq 1\) show that

    \(\begin{vmatrix}x + 1 & x & x \\ x & x + a & x \\ x & x & x + a^2\end{vmatrix} = a^3\left[1 + \frac{x(a^3 - 1)}{a^2(a - 1)}\right]\)

100. If \(p + q + r = 0,\) prove that

     \(\begin{vmatrix}pq & qb & rc \\ qc & ra & pb \\ rb & pc & qa\end{vmatrix} = pqr\begin{vmatrix}a & b & c \\ c & a & b \\ b & c & a\end{vmatrix}\)