The value of x if \(\begin{vmatrix} 0 & 1 & 0 \\ x & 2 & x \\ 1 & 3 & x \end{vmatrix}=0\) is_________.

If A is square matrix of order 3, then |kA| is________.

If \(\triangle=\begin{vmatrix} {a}_{11} & {a}_{12} & {a}_{13} \\ {a}_{21} & {a}_{22} & {a}_{23} \\ {a}_{31} & {a}_{32} & {a}_{33} \end{vmatrix}\) and A_ij is cofactor of a_ij, then value of \(\triangle\) is given by ________.

The number of permutation of n different things taken r at a time, when the repetition is allowed is ________.

The distance between directrix and focus of a parabola y² = 4ax is _______.

The technology matrix of an economic system of two industries is\(\begin{bmatrix} 0.50 & 0.30 \\ 0.41 & 0.33 \end{bmatrix}\). Test whether the system is viable as per Hawkins Simon conditions.

How many permutations can be made out of the letters of the word "TRIANGLE" beginning with T?

In how many ways 10 identical keys can be arranged in a ring?

Convert the parabola y²=4x+4y into standard form.

The supply of a commodity is related to the price by the relation x = \(\sqrt{5p-15}\) . Show that the supply curve is a parabola.

Find the term independent of x in the expansion of \({ \left( x-\frac { 2 }{ { x }^{ 2 } } \right) }^{ 15 }\)

If tan \(\alpha={{1}\over{7}},\sin\beta{{1}\over{\sqrt{10}}},\) Prove that \(\alpha+2\beta{{\pi}\over4{}}\) where \(0<\alpha<{{\pi}\over{2}}\) and \(0<\beta<{{\pi}\over{}2}.\)

Find the locus of a point which moves in such a way that the square of its distance from the point (3, -2) is numerically equal to its distance from the line 5x - 12y = 13

Find the equation of the circle which touches the line x = 0, y = 0 and x = a.

For what values of a and b does the equation (a - 2)x² + by² + (b - 2)xy + 4x + 4y - 1 = 0 represents a circle? Write down the resulting equation of the circle.

Evaluate:\(\begin{vmatrix} 1&a&a^2-bc\\1&b&b^2-ca\\1&c&c^2-ab \end{vmatrix}\)

Show that the matrices A =\(\left[ \begin{matrix} 2 & 2 & 1 \\ 1 & 3 & 1 \\ 1 & 2 & 2 \end{matrix} \right] \)and B =\(\left[ \begin{matrix} \frac { 4 }{ 5 } & -\frac { 2 }{ 5 } & -\frac { 1 }{ 5 } \\ -\frac { 1 }{ 5 } & \frac { 3 }{ 5 } & -\frac { 1 }{ 5 } \\ -\frac { 1 }{ 5 } & -\frac { 2 }{ 5 } & \frac { 4 }{ 5 } \end{matrix} \right] \) are inverses of each other.

The average variable cost of a monthly output of x tonnes of a firm producing a valuable metal is  Rs. \(\frac { 1 }{ 5 } { x }^{ 2 }-6x+100\). Show that the average variable cost curve is a parabola. Also find the output and the  average cost at the vertex of the parabola.

If a parabolic reflector is 20 cm in diameter and 5 cm deep, find the focus.