Of all triangles with given perimeter, find the triangle with the maximum area. Justify your answer

A 40 feet high screen is put on a vertical wall 10 feet above your eye-level. How far should you stand to maximize the angle subtended by the screen (from top to bottom) at your eye?

Study the derivatives of the function
\(\mathbf{y=\sqrt[3]{x^3-4x}}\)
and sketch its graph on the real line.

Suppose P and Q are the centres of two disjoint circles \(\mathbf{C_1}\) and \(\mathbf{C_2}\) respectively, such that P lies outside \(\mathbf{C_2}\) and Q lies outside \(\mathbf{C_1}\). Two tangents are drawn from the point P to the circle \(\mathbf{C_2}\), which intersect the circle \(\mathbf{C_1}\) at point A and B. Similarly, two tangents are drawn from the point Q to the circle \(\mathbf{C_1}\), which intersect the circle \(\mathbf{C_2}\) at points M and N. Show that AB=MN

Suppose ABC is a triangle with inradius r. The incircle touches the sides BC, CA, and AB at D,E and F respectively. If BD=x, CE=y and AF=z, then show that \(\mathbf{r^2=\frac{xyz}{x+y+z}}\)

Consider the equation \(\mathbf{x^5+x=10}\). Show that
(a) the equation has only one real root;
(b) this root lies between 1 and 2;
(c) this root must be irrational.

In how many ways can you divide the set of eight numbers \(\mathbf{{2,3,\cdots,9}}\) into 4 pairs such that no pair of numbers has \(\mathbf{\text{gcd} }\) equal to 2?

Suppose S is the set of all positive integers. For \(\mathbf{a,b \in S}\), define
\(\mathbf{a * b=\frac{\text{lcm}[a,b]}{\text{gcd}(a,b)}}\)
For example 8*12=6.
Show that exactly two of the following three properties are satisfied:
(i) If \(\mathbf{a,b \in S}\), then \(\mathbf{a*b \in S}\).
(ii) \(\mathbf{(a*b)*c=a*(b*c)}\) for all \(\mathbf{a,b,c \in S}\).
(iii) There exists an element \(\mathbf{i \in S}\) such that \(\mathbf{a *i =a}\) for all \(\mathbf{a \in S}\)

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Two subsets A and B of the (x,y)-plane are said to be equivalent if there exists a function f: Ato B which is both one-to-one and onto.
(i) Show that any two line segments in the plane are equivalent.
(ii) Show that any two circles in the plane are equivalent.

Solution to 6: Let y be the given limit. Then e^y = (2nCn)^(1/2n). Now 2nCn = (2n)!/(n!)^2 . We divide both Nr and Dr by (2n)^2n. Hence e^y = [ {(2n)!/(2n)^2n}^(1/2n)]/[{n!/n^n}^2]^(1/2n)*2. Now limit n tends to infinity (n!/n^n)^(1/2n) is e^-1. Hence on simplification limit n tends to infinity e^y =2 and finally result is log 2. Do it yourself to understand the solution.

How to solve problem number 5 and 6?

Solution to 6: Let y be the given limit. Then e^y = (2nCn)^(1/2n). Now 2nCn = (2n)!/(n!)^2 . We divide both Nr and Dr by (2n)^2n. Hence e^y = [ {(2n)!/(2n)^2n}^(1/2n)]/[{n!/n^n}^2]^(1/2n)*2. Now limit n tends to infinity (n!/n^n)^(1/2n) is e^-1. Hence on simplification limit n tends to infinity e^y =2 and finally result is log 2. Do it yourself to understand the solution.