Vidyamanohar: February 2012

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

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

to the curve

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 $\int\limits_0^1 {f( - t - 1)\left[ {\frac{1}{{t + 1}} + \frac{1}{{t + 2}} + \frac{1}{{t + 3}}...\frac{1}{{t + 2012}}} \right]} dt$

Puzzle 27 - Determinant

Define the function $f(x) = ({p_1} - x)({p_2} - x)({p_3} - x)...({p_n} - x);$ ${p_i} \in R$ . Let a and b are two unequal real numbers. Let M is an n×n matrix with all a’s above the diagonal and b’s below the diagonal and ${p_1},\,\,{p_2},\,\,{p_3},\,...{p_n}$ along the diagonal.
Show that the determinant of M is given by $\frac{{bf(a) - af(b)}}{{b - a}}$

Vidyamanohar