Question

The position vectors of points A, B and C are $λ\stackrel{\wedge}{i}+3\stackrel{\wedge}{j}$,$12\stackrel{\wedge}{i}+μ\stackrel{\wedge}{j}$ and $11\stackrel{\wedge}{i}-3\stackrel{\wedge}{j}$ respectively. If C divides the line segment joining A and B in the ratio 3 : 1, find the values of λ and μ.

Find the general solution of the differential equation ${\frac{d y}{d x}}+{\frac{2}{x}}y=x$

Find the particular solution of the differential equation ${\frac{d y}{d x}}={\frac{1+y^{2}}{1+x^{2}}}$ given that y(0) = ${\sqrt{3}}.$

Find : $\textstyle\int$$\left({\frac{1-x}{1+x^{2}}}\right)^{2}e^{x}d x~.$

From the differential equation of equation y = a cos 2x + b sin 2x, where a and b are constant.

If E and F are two events such that P(E) = 1/4, P(F) = 1/2 and P(E ∩ F) = 1/8, find (i) P(E or F) (ii) P(not E and not F).

A box contains 50 bolts and 50 nuts. Half of the bolts and nuts are rusted. If two items are drawn with replacement, what is the probability that either both are rusted or both are bolts.