Question

Prove the following : $\begin{vmatrix}a² & bc & ac+c² \\ a²+ab & b² & ac \\ ab & b²+bc & c²\end{vmatrix}$ = $4a^{2}b^{2}c^{2}.$

A speaks truth in 60% of the cases, while B in 90% of cases. In what percent of cases are they likely to contradict each other in stating the same fact ? In the cases of contradiction do you think, the statement of B will carry more weight as he speaks truth in more number of cases than A ?

Using differentials, find the approximate value of $(3.968)^{1/2}$

Form the differential equation of the family of circles in the second quadrant and touching the co-ordinate axes.

Find the general solution of the differential equation (1 + tan y)(dx – dy) + 2xdy = 0.

Using properties of determinants, prove the following : $\begin{vmatrix}a & a² & bc \\ b & b² & ca \\ c & c² & ab\end{vmatrix}$ = $(a-b)(b-c)(c-a)(b c+c a+a b).$

Using properties of determinants, prove that : $\begin{vmatrix}a+b+2c & a & b \\ c & b+c+2a & b \\ c & a & c+a+2b\end{vmatrix}$ = $2(a+b+c)^{3}.$