For the equation

$${{{d^3}y} \over {d{x^3}}} + x{\left( {{{dy} \over {dx}}} \right)^{3/2}} + {x^2}y = 0$$

the correct description is

The matrix M is defined as

$$M = \left[ {\matrix{ 1 & 3 \cr 4 & 2 \cr } } \right]$$

and has eigenvalues 5 and $$-$$2. The matrix Q is formed as

Q = M³ $$-$$ 4M² $$-$$ 2M

Which of the following is/are the eigenvalue(s) of matrix Q?

Consider the following recursive iteration scheme for different values of variable P with the initial guess x₁ = 1:

$${x_{n + 1}} = {1 \over 2}\left( {{x_n} + {P \over {{x_n}}}} \right)$$, n = 1, 2, 3, 4, 5

For P = 2, x₅ is obtained to be 1.414, rounded-off to three decimal places. For P = 3, x₅ is obtained to be 1.732, rounded-off to three decimal places. If P = 10, the numerical value of x₅ is __________. (round off to three decimal places)

The Fourier cosine series of a function is given by :

$$f(x) = \sum\limits_{n = 0}^\infty {{f_n}\cos nx} $$

For f(x) = cos⁴x, the numerical value of (f₄ + f₅) is _________. (round off to three decimal places)