Question

A and B throw a pair of dice alternatively, till one of them gets a total of 10 and wins the game. Find their respective probabilities of winning, if A starts first.

Find the points on the curve x² + y² – 2x – 3 = 0 at which tangent is parallel to x-axis.

Find the particular solution of the differential equation $\frac{d y}{d x}\ \ +\ 2y\ \tan\ x\ =\ \sin\ x,$ given that y = 0 when $x={\frac{\pi}{3}}\,.$

Using properties of determinants, prove that : $\begin{vmatrix}(a+1)(a+2) & a+2 & 1 \\ (a+2)(a+3) & a+3 & 1 \\ (a+3)(a+4) & a+4 & 1\end{vmatrix}$ = -2

The total cost associated with provision of free mid-day meals to x students of a school in primary classes is given by C(x) = 0·005x³ – 0·02x² + 30x + 50. If the marginal cost is given by rate of change ${\frac{d c}{d x}}$ of total cost, write the marginal cost of food for 300 students.

Find the solution of the differential equation $(x d y-y d x)\,y\,\mathrm{sin}{\left({\frac{y}{x}}\right)}=(y d x+x d y)x\cos\biggl({\frac{y}{x}}\biggr).$

Evaluate : $\large\textstyle\int{\frac{3x+1}{(x^{3}-x^{2}-x+1)}}d x\,.$