Question

Find the particular solution of differential equation : ${\frac{d y}{d x}}=-{\frac{x+y\cos x}{1+\sin x}}$ given that y = 1 when x = 0.

Find the position vector of a point R, which divides the line joining two points P and Q whose position vectors are $2\stackrel{\rightarrow}{a}+\stackrel{\rightarrow}{b}$ and $\stackrel{\rightarrow}{a}-3\stackrel{\rightarrow}{b}$ respectively, externally in the ratio 1 : 2 Also, show that P is the mid-point, of line segment RQ.

Show that the function f in A = $R-\left\{{\frac{2}{3}}\right\}$ defined as f(x) =$\frac{4x+3}{6x-4}$ is one-one and onto. Hence find $f^{-1}.$

Find the equation of the tangent to the curve y = x⁴ – 6x³ + 13x² – 10x + 5 at point x = 1, y = 0.

Using properties of determinants, prove that following : $\begin{vmatrix}a+x & y & z \\ x & a+y & z \\ x & y & a+z\end{vmatrix}$ = $a^{2}(a+x+y+z).$

Let A = R – {2}, B = R – {1}. If f : A → B is a function defined by $f(x)\,=\,\left({\frac{x-1}{x-2}}\right)\!,$ show that f is one-one and onto. Hence find $f^{-1}.$

A and B throw a pair of dice alternately. A wins the game if he gets a total of 7 and B wins the game if he gets a total of 10. If A starts the game, then find the probability that B wins.