The number of isosceles (but not equilateral) triangles with integer sides and no side exceeding 10 is

Find the value of the parameter $m$ for which the sum of the squares of the roots of the equation $x^2+(m-2)x-(m+3)=0$ has the least value.

Prove that if $x>0,y>0$ and $x+y=1$, then $\left(1+\frac{1}{x}\right)\left(1+\frac{1}{y}\right)\ge 9$.

Given that the definite integral ${\int }_0^3\frac{dx}{\sqrt[3]{x-2}}$ has a finite value, find the value of the integral.

Evaluate the value of ${\int }_1^2\frac{x}{x^2+1}(2\log x+1)dx$.

Ram and Shyam stand in a queue at random along with 10 other persons. Find the probability that there are exactly three persons between them.

A fair coin is tossed once. If it turns up heads, a balanced die is rolled twice. Then the sum of the number of dots on the faces that show up is recorded. If the coin turns up tails, the same die is rolled once. In this case the number of dots on the face that shows up is recorded. What is the probability that the recorded number is larger than 5?

Consider a large village, where only two newspapers $P_1$ and $P_2$ are available to the families. It is known that the proportion of families

$$(i)\text{not taking }{P}_1\text{ is }0.48,$$

$$(ii)\text{not taking }{P}_2\text{ is }0.58,$$

$$(iii)\text{taking only }{P}_2\text{ is }\mathrm{0.30.}$$

The probability that a randomly chosen family from the village takes only $P_1$ is

Prove that the rectangle of greatest area inscribed in a circle of radius 1 is a square. What is the area of the square?