Vidyamanohar

Thursday, May 02, 2013

Puzzle 47

Semi circles are drawn on each side of a square. An elastic band is stretched tightly around the shape formed (see the figure). Find the length of the elastic band in terms of the length x, of the side of the square.

Thursday, April 25, 2013

Puzzle 46 - Squares

Square ABCD, of side length 26, is centered at O. With D as center, draw another square EFGH with side length 22, none of its sides parallel to those of ABCD, so that OE is kept perpendicular to DA. Find the area of the square ABCD that is not inside the square EFGH.

Tuesday, April 23, 2013

Puzzle 45 - Similar Triangles

Let A is the set of all triangles in a plane. Define a function $f:A \times A \to {R^ + }$ as $f\left( {{T_1},\,{T_2}} \right) = a$ implies that triangle ${T_2}$ is similar to ${T_1}$ with the dilation constant or scale factor ‘a’. Find all the possible triangles ${T_1}$ and ${T_2}$ such that

a. Lengths of the sides of ${T_1}$ and ${T_2}$ are integers
b. The perimeter and area of ${T_2}$ are same.

Monday, April 22, 2013

Puzzle 44

W(a, b) is the centre of the circle. TE is a tangent to the circle at E, which lies on the y-axis. The circle touches the x-axis at D. The equation of TE is $y = \frac{3}{4}x - 9$ . Calculate the values of a and b.

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 $T$ . Consider points $A$ on ${C_1}$ and $B$ on ${C_1}$ , such that $AB$ subtends right angle at $T$ . Find the locus of midpoints of all such segments $AB$ .

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 $A$ and $B$ 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 $[0,\,\pi ]$

Saturday, February 25, 2012

Puzzle 36 - Functional Equation

Find the function f(x) if f(x) is a non constant and satisfies the equation $f(x) = {x^2} - \int\limits_0^1 {{{(f(t) + x)}^2}} dt$

Puzzle 35 - Complex system of equations

Find the complex roots of the system of equations

$(x + y)({x^2} - {y^2}) = 1176$

$(x - y)({x^2} + {y^2}) = 696$

Thursday, February 23, 2012

Puzzle 34 - Analytical Geometry

Vertex A of the triangle ABC is (14, 8). If the orthocenter of the triangle is H(8, 1) and the midpoint of the side BC is (3, -2), find the coordinates of the points B and C.

Puzzle 33 - Number Grid

In the grid shown, a number is to be placed in each smaller square so that the product of all the three numbers in any row, column, or diagonal is the same positive number. Find the sum of x and y.

Sunday, February 19, 2012

Puzzle 32 - Trigonometry

If $\alpha + \beta + \gamma = \pi$ and $\prod {\tan \left( {\frac{{\alpha + \beta - \gamma }}{4}} \right)} = 1$ find the value of $\sum {\sin \alpha } + \prod {\sin \alpha }$

Saturday, February 18, 2012

Puzzle 31 - Limit

Evaluate $\mathop {\lim }\limits_{n \to \infty } \left( {\frac{1}{{\sqrt 2 }}\sqrt {\frac{1}{2} + \frac{1}{2}\frac{1}{{\sqrt 2 }}} \sqrt {\frac{1}{2} + \frac{1}{2}\sqrt {\frac{1}{2} + \frac{1}{2}\frac{1}{{\sqrt 2 }}} } ...n\,terms} \right)$

Friday, February 17, 2012

Puzzle 30 - Parabola and Circle

The tangent and normal at the point $P(at^2, 2at)$ to the curve $y^2 = 4ax$ meet the x-axis and y-axis in T and G respectively, then find the angle at which the tangent at P to the curve is inclined to the tangent at P to the circle through the points T, P, G.

Tuesday, February 14, 2012

Puzzle 29 - Integration

If $\int {\frac{{1 - 7{{\cos }^2}x}}{{{{\sin }^7}x{{\cos }^2}x}}dx} = \frac{{f(x)}}{{{{\sin }^7}x}} + C$ , then find the value of $f'(0) + f''(\frac{\pi }{4})$

Monday, February 13, 2012

Puzzle 28 - Integration

Let ${\rm{f}}\left( {\rm{n}} \right)$ denotes the coefficient of ${{\rm{x}}^{{\rm{2}}0{\rm{12}}}}$ in the expansion of ${\left( {{\rm{1}} + {\rm{x}}} \right)^{\rm{n}}}$ . Then evaluate $\int\limits_0^1 {f( - t - 1)\left[ {\frac{1}{{t + 1}} + \frac{1}{{t + 2}} + \frac{1}{{t + 3}}...\frac{1}{{t + 2012}}} \right]} dt$