If A is a non-singular matrix of odd order, prove that |adj A| is positive

Find the rank of the matrix \(\left[ \begin{matrix} 1 & 2 & 3 \\ 2 & 1 & 4 \\ 3 & 0 & 5 \end{matrix} \right] \) by reducing it to a row-echelon form.

Solve the following system of homogenous equations.
2x + 3y − z = 0, x − y − 2z = 0, 3x + y + 3z = 0

Show that the system of equations is inconsistent. 2x + 5y= 7, 6x + 15y = 13.

Find z⁻¹, if z = (2 + 3i) (1− i).

Simplify the following
\({ i }^{ 59 }+\frac { 1 }{ { i }^{ 59 } } \)

If z =\(\left( \frac { \sqrt { 3 } }{ 2 } +\frac { i }{ 2 } \right) ^{ 107 }+\left( \frac { \sqrt { 3 } }{ 2 } -\frac { i }{ 2 } \right) ^{ 107 }\), then show that Im (z) = 0

Determine the number of positive and negative roots of the equation x⁹- 5x⁸-14x⁷= 0.

If sin ∝, cos ∝ are the roots of the equation ax² + bx + c-0 (c ≠ 0), then prove that (n + c)² - b² + c²

Find the value of sec⁻¹\(\left( -\frac { 2\sqrt { 3 } }{ 3 } \right) \)

Find the principal value of \({sin }^{ -1 }\left( sin\left( \frac { 5\pi }{ 6 } \right) \right) \)

Evaluate \(sin\left( { cos }^{ -1 }\left( \frac { 3 }{ 5 } \right) \right) \)

Find centre and radius of the following circles.
x²+y²−x+2y−3 = 0

For the ellipse x² + 3y² = a², find the length of major and minor axis.

Find the volume of the parallelepiped whose coterminous edges are represented by the vectors \(-6\hat { i } +14\hat { j } +10\hat { k } ,14\hat { i } -10\hat { j } -6\hat { k } \) and \(2\hat { i } +4\hat { j } -2\hat { k } \)

Find the parametric form of vector equation of a line passing through a point (2, -1, 3) and parallel to line \({ \overset { \rightarrow }{ r } }=\left( \overset { \wedge }{ i } +\overset { \wedge }{ j } \right) +t\left( 2\overset { \wedge }{ i } +\overset { \wedge }{ j } -2\overset { \wedge }{ k } \right) \)

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p=-1

A manufacturer wants to design an open box having a square base and a surface area of 108 sq. cm. Determine the dimensions of the box for the maximum volume.

Find the maximum and minimum values of f(x) = |x+3| ∀ \(x\in R\).

Evaluate: \(\int ^{log 2}_{-log 2} e ^{-|x|}\) dx.

Evaluate \(\int _{ 1 }^{ 2 }{ \frac { 3x }{ { 9x }^{ 2 }-1 } dx } \)

Show that y = a cos bx is a solution of the differential equation \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +{ b }^{ 2 }y=0\).

Suppose X is the number of tails occurred when three fair coins are tossed once simultaneously. Find the values of the random variable X and number of points in its reverse images.

Examine the binary operation (closure property) of the following operations on the respective sets (if it is not, make it binary)
\(a*b=\left( \frac { a-1 }{ b-1 } \right) ,\forall a,b\in Q\)

Let \(A=\left( \begin{matrix} 1 & 0 \\ 0 & 1 \\ 1 & 0 \end{matrix}\begin{matrix} 1 & 0 \\ 0 & 1 \\ 0 & 1 \end{matrix} \right) ,B=\left( \begin{matrix} 0 & 1 \\ 1 & 0 \\ 1 & 0 \end{matrix}\begin{matrix} 0 & 1 \\ 1 & 0 \\ 0 & 1 \end{matrix} \right) ,C=\left( \begin{matrix} 1 & 1 \\ 0 & 1 \\ 1 & 1 \end{matrix}\begin{matrix} 0 & 1 \\ 1 & 0 \\ 1 & 1 \end{matrix} \right) \)be any three boolean matrices of the same type.
Find (A∨B)∧C 

In the set of integers under the operation * defined by a * b = a + b - 1. Find the identity element.