Prove that \(\begin{vmatrix} {1\over a}&bc&b+c\\{1\over b}&ca&c+a\\{1\over c}&ab&a+b \end{vmatrix}=0\)

Resolve into partial fractions for the following : \(\frac{1}{x^2-1}\)

Show that the equation 12x² - 10xy + 2y² + 14x - 5y + 2 = 0 represents a pair of straight lines and also find the separate equations of the straight lines.

Prove that:  \(\sin { \theta } \cos { \theta } \left\{ \sin { \left( \frac { \pi }{ 2 } -\theta \right) } \csc { \theta } +\cos { \left( \frac { \pi }{ 2 } -\theta \right) \sec { \theta } } \right\} =1\)

Examine the following functions for continuity at indicated points
\(f(x)=\left\{\begin{array}{cl} \frac{x^2-4}{x-2}, & \text { if } x \neq 2 \\ 0, & \text { if } x=2 \end{array} \right.\) at x = 2

The cost function of a firm is \(C={1\over3}x^3-3x^2+9x\). Find the level of output (x > 0) when average cost is minimum.

Let u = log\(\frac { { x }^{ 4 }+{ y }^{ 4 } }{ x+y } \). By using Euler’s theorem show that \(x.\frac { \partial u }{ \partial x } +y.\frac { \partial u }{ \partial y } =3\) .

A person buy 20 shares of par value of Rs. 10 of a company which pays 9% dividend at such a price that he gets 12% on his money. Find the market value of a share.

Compute Quartile deviation from the following data

CI	10-20	20-30	30-40	40-50	50-60	60-70	70-80
f	12	19	5	10	9	6	6

An examination of 11 applicants for a accountant post was taken by a finance company. The marks obtained by the applicants in the reasoning and aptitude tests are given below.

Applicant	A	B	C	D	E	F	G	H	I	J	K
Reasoning test	20	50	28	25	70	90	76	45	30	19	26
Aptitude test	30	60	50	40	85	90	56	82	42	31	49

Calculate Spearman’s rank correlation coefficient from the data given above.