Vidyamanohar: March 2012

Wednesday, March 28, 2012

Puzzle 43 - Locus of midpoints

Two circles ${C_1},\,{C_2}$ with different radii are given in the plane touching each other externally at

. Consider points

on ${C_1}$ and

on ${C_1}$ , such that

subtends right angle at

. Find the locus of midpoints of all such segments

Tuesday, March 27, 2012

Puzzle 42

Let $\inline f:R \to R$ is a function such that $\inline f(f(x)) = {x^2} - x + 1$ . Find the value of f(0).

Thursday, March 15, 2012

Puzzlee 41 - Polynomials + Trigonometry

If $\alpha < \beta < \gamma$ are three real roots of the equation ${x^3} - 3x - 1 = 0$ , show that ${\gamma ^2} - \gamma = {\beta ^2} - \alpha$ .

Wednesday, March 14, 2012

Puzzle 40 - Matrices

Let

and

are two different $3 \times 3$ matrices such that ${A^3} = {B^3}$ and ${A^2}B = {B^2}A$ . Does the matrix ${A^2} + {B^2}$ have inverse?

Tuesday, March 13, 2012

Puzzle 39 - 2012 Integration

Find the value of $\frac{{6539}}{{101}}\int\limits_{ - 1}^1 {\frac{{{t^{12}} + 31}}{{1 + {{2011}^t}}}dt}$

Thursday, March 08, 2012

Puzzle 38 - Point inside the Circle

Find the values of $\alpha$ for which the point $(\alpha - 1,\;\alpha + 1)$ lies in the larger segment of the circle ${x^2} + {y^2} - x - y - 6 = 0$ made by the chord whose equation is x + y – 2 = 0

Tuesday, March 06, 2012

Puzzle 37 - Extreme Values of Integral

Define the function $f(x) = \int\limits_0^{\frac{\pi }{2}} {\frac{{\cos |t - x|}}{{1 + \sin |t - x|}}} dt;\,0 \le x \le \pi$ , find the maximum and minimum values of f(x) in the interval $[0,\,\pi ]$

Vidyamanohar