IIT JEE 2001

Let \(C_1\) and \(C_2\) be two circles with \(C_2\) lying inside \(C_1\). A circle C lying inside \(C_1\) touches \(C_1\) internally and \(C_2\) externally. Identify the locus of the centre of C.

The rate of a first order reaction is 0.04 mol litre^-1 s^-1 at 10 minutes and 0.03 mol litre^-1 s^-1 at 20 minutes after initiation. Find the half-life of the reaction.

One quarter section is cut from a uniform circular disc of radius $R$. This section has a mass $M$. It is made to rotate about a line perpendicular to its plane and passing through the centre of the original disc. Its moment of inertia about the axis of rotation is

Show, by vector methods, that the angular bisectors of a triangle are concurrent and find an expression for the position vector of the point of concurrency in terms of the position vectors of the vertices.

Hydrogen peroxide solution (20 ml) reacts quantitatively with a solution of KMnO₄ solution is just decolourised by 10 ml of MnSO₄ in neutral medium simultaneously forming a dark brown precipitate of hydrated MnO₂. The brown precipitated is dissolved in 10 ml of 0.2 M sodium oxalate under boiling condition in the presence of dilute H₂SO₄. Write the balanced equations involved in the reactions and calculate the molarity of H₂O₂.

Let \(\overrightarrow A \left( t \right) = {f_1}\left( t \right)\widehat i + {f_2}\left( t \right)\widehat j\) and \(\overrightarrow B \left( t \right) = {g_1}\left( t \right)\overrightarrow i + {g_2}\left( t \right)\widehat j,t \in \left[ {0,1} \right],\)
where \({f_1},{f_2},{g_1},{g_2}\) are continuous functions. If \(\overrightarrow A \left( t \right)\) and \(\overrightarrow B \left( t \right)\) are nonzero vectors for all \(t\) and \(\overrightarrow A \left( 0 \right) = 2\widehat i + 3\widehat j,\) \(\,\overrightarrow A \left( 1 \right) = 6\widehat i + 2\widehat j,\) \(\,\overrightarrow B \left( 0 \right) = 3\widehat i + 2\widehat j\) and \(\,\overrightarrow B \left( 1 \right) = 2\widehat i + 6\widehat j.\) Then show that \(\,\overrightarrow A \left( t \right)\) and \(\,\overrightarrow B \left( t \right)\) are parallel for some \(t.\)