IIT-JEE 2000

Suppose \(p\left( x \right) = {a_0} + {a_1}x + {a_2}{x^2} + .......... + {a_n}{x^n}.\) If
\(\left| {p\left( x \right)} \right| \le \left| {{e^{x - 1}} - 1} \right|\) for all \(x \ge 0\), prove that
$$\left| {{a_1} + 2{a_2} + ........ + n...

For \(x>0,\) let \(f\left( x \right) = \int\limits_e^x {{{\ln t} \over {1 + t}}dt.}\) Find the function
\(f\left( x \right) + f\left( {{1 \over x}} \right)\) and show that $$f\left( e \right) + f\left( {{1 \over e}} \right) = {1 \over 2}....

If \({x^2} + {y^2} = 1\) then

Let \(ABC\) be an equilateral triangle inscribed in the circle \({x^2} + {y^2} = {a^2}\). Suppose perpendiculars from \(A, B, C\) to the major axis of the ellipse \(x.{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1\), \((a>b)\) meets ...

A coin probability \(p\) of showing head when tossed. It is tossed \(n\) times. Let \({p_n}\) denote the probability that no two (or more) consecutive heads occur. Prove that \({p_1} = 1,\,\,{p_2} = 1 - {p^2}\) and $${p_n} = \left( {1 - p} ...

For any positive integer \(m\), \(n\) (with \(n \ge m\)), let \(\left( {\matrix{ n \cr m \cr } } \right) = {}^n{C_m}\)
Prove that $$\left( {\matrix{
n \cr
m \cr

} } \right) + \left( {\matrix{
{n - 1} \cr
m...

Let \(a,\,b,\,c\) be possitive real numbers such that \({b^2} - 4ac > 0\) and let \({\alpha _1} = c.\) Prove by induction that $${\alpha _{n + 1}} = {{a\alpha _n^2} \over {\left( {{b^2} - 2a\left( {{\alpha _1} + {\alpha _2} + ... + {\alpha ...

For every possitive integer \(n\), prove that
\(\sqrt {\left( {4n + 1} \right)} < \sqrt n + \sqrt {n + 1} < \sqrt {4n + 2}.\)
Hence or otherwise, prove that $$\left[ {\sqrt n + \sqrt {\left( {n + 1} \right)} } \right] = \left[ {\sqrt ...

Let \({C_1}\) and \({C_2}\) be respectively, the parabolas \({x^2} = y - 1\) and \({y^2} = x - 1\). Let \(P\) be any point on \({C_1}\) and \(Q\) be any point on \({C_2}\). Let \({P_1}\) and \({Q_1}\) be the reflections of \(P\) and \(Q\), ...

A coin has probability \(p\) of showing head when tossed. It is tossed \(n\) times. Let \({p_n}\) denote the probability that no two (or more) consecutive heads occur. Prove that \({p_1} = 1,{p_2} = 1 - {p^2}\) and $${p_n} = \left( {1 - p} ...

Let \(ABC\) be a triangle with incentre \(I\) and inradius \(r\). Let \(D,E,F\) be the feet of the perpendiculars from \(I\) to the sides \(BC\), \(CA\) and \(AB\) respectively. If \({r_1},{r_2}\) and \({r_3}\) are the radii of circles insc...

If \(\alpha ,\,\beta\) are the roots of \(a{x^2} + bx + c = 0\), \(\,\left( {a \ne 0} \right)\) and \(\alpha + \delta ,\,\,\beta + \delta\) are the roots of \(A{x^2} + Bx + c = 0,\) \(\left( {A \ne 0\,} \right)\,\) for some contant $...

The fourth power of the common difference of an arithmatic progression with integer entries is added to the product of any four consecutive terms of it. Prove that the resulting sum is the square of an integer.

For points \(P\,\,\, = \left( {{x_1},\,{y_1}} \right)\) and \(Q\,\,\, = \left( {{x_2},\,{y_2}} \right)\) of the co-ordinate plane, a new distance \(d\left( {P,\,Q} \right)\) is defined by \(d\left( {P,\,Q} \right)\)$$ = \left( {{x_2},\,{y_2...

Let \(ABC\) and \(PQR\) be any two triangles in the same plane. Assume that the prependiculars from the points \(A, B, C\) to the sides \(QR, RP, PQ\) respectively are concurrent. Using vector methods or otherwise, prove that the prependicu...

In any triangle \(ABC,\) prove that
\($\cot {A \over 2} + \cot {B \over 2} + \cot {C \over 2} = \cot {A \over 2}\cot {B \over 2}\cot {C \over 2}.\)$