IIT-JEE 1999

In a triangle \(PQR,\angle R = \pi /2\). If \(\,\,\tan \left( {P/2} \right)\) and \(\tan \left( {Q/2} \right)\) are the roots of the equation \(a{x^2} + bx + c = 0\left( {a \ne 0} \right)\) then.

For a positive integer \(\,n\), let
$${f_n}\left( \theta \right) = \left( {\tan {\theta \over 2}} \right)\,\left( {1 + \sec \theta } \right)\,\left( {1 + \sec 2\theta } \right)\,\left( {1 + \sec 4\theta } \right).....\left( {1 + \sec {...

Let \(u\) and \(v\) be units vectors. If \(w\) is a vector such that \(w + \left( {w \times u} \right) = v,\) then prove that \(\left| {\left( {u \times v} \right) \cdot w} \right| \le 1/2\) and that the equality holds if and only if \(u\) ...

Let \(a=2i+j+k, b=i+2j-k\) and a unit vector \(c\) be coplanar. If \(c\) is perpendicular to \(a,\) then \(c =\)

Let \(a=2i+j-2k\) and \(b=i+j.\) If \(c\) is a vector such that \(a.\) \(c = \left| c \right|,\left| {c - a} \right| = 2\sqrt 2\) and the angle between \(\left( {a \times b} \right)\) and \(c\) is \({30^ \circ },\) then $$\left| {\left( {...

Let \(a\) and \(b\) two non-collinear unit vectors. If \(u = a - \left( {a\,.\,b} \right)\,b\) and \(v = a \times b,\) then \(\left| v \right|\) is