Question

Find : $\large\textstyle\int$$\large\frac{x^{2}\,d x}{x^{4}+x^{2}-12}\;.$

A bag contains (2n + 1) coins. It is known that (n – 1) of these coins have a head on both sides, whereas the rest of the coins are fair. A coin is picked up at random from the bag and is tossed. If the probability that the toss results in a head is 31/42, determine the value of n.

Solve the following differential equation : ${\frac{d y}{d x}}\ +\,\sec x{y}=\tan x,\left(0\leq x<{\frac{\pi}{2}}\right)\!.$

Show that the relation R on the set Z of all integers defined by (x, y) ∈ $R\Leftrightarrow(x-y)$ is divisible by 3 is an equivalence relation.

Evaluate : $\large\textstyle\int(x-3){\sqrt{x^{2}+3x-18}}\,d x\,.$

Find the angle of intersection of the curves x² + y² = 4 and (x – 2)² + y²= 4, at the point in the first quadrant.

Three schools A, B and C organised a fete (mela) collecting funds for flood victims in which they sold hand-helds fans, mats and toys made from recycled material, the sale price of each being ₹ 25, ₹ 100 and ₹ 50 respectively. The following table shows the number of articles of each type sold :