Find the values of x if \(\begin{vmatrix} 2 & 4 \\5 & 1 \end{vmatrix}=\begin{vmatrix} 2x & 4\\6 & x \end{vmatrix}.\)

Show that 10P₃ = 9 P₃ + 3. 9P₂

Find the number of diagonals that can be drawn by joining the angular points of octagon ?

Find the angle between the pair of lines represented by the equation 3x²+10xy+8y²+14x+22y+15=0.

Find the value of \(\cos\left(\frac{5\pi}{12}\right)\)

Prove that \(sin^2\left(\frac{\pi}{8}+\frac x2\right)-sin^2\left(\frac{\pi}{8}-\frac x2\right)=\frac{1}{\sqrt2}\sin x.\)

Show that the function f(x) = 5x -3 is continous at x = +3

Prove that the function given by f(x) = \(\left| x-1 \right| \), x \(\in\) R is not differentiable at x =1

If f(x,y) = 3x² + 4y³ + 6xy - x²y³ + 6. Find f_yy(1,1)

A factory owner purchases two types of machines A and B for his factory. The requirements and the limitations for the machines are as follows:

Machine	Area occupied	Labour Force	Daily Output (in units)
A	1000 m2	12 men	60
B	1200 m2	8 men	40

He has maximum area of 9000 m² available, and 72 skilled labourers who can operate both the machines. How many machines of each type should he buy to maximize the daily output. Formulate the above as LPP.