IIT-JEE 2001

Let \(- 1 \le p \le 1\). Show that the equation \(4{x^3} - 3x - p = 0\)
has a unique root in the interval \(\left[ {1/2,\,1} \right]\) and identify it.

Let \(b \ne 0\) and for \(j=0, 1, 2, ..., n,\) let \({S_j}\) be the area of
the region bounded by the \(y\)-axis and the curve \(x{e^{ay}} = \sin\) by,
\({{jr} \over b} \le y \le {{\left( {j + 1} \right)\pi } \over b}.\) Show that $${S_...

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.

Let \(\,2{x^2}\, + \,{y^2} - \,3xy = 0\) be the equation of a pair of tangents drawn from the origin O to a circle of radius 3 with centre in the first quadrant. If A is one of the points of contact, find the length of OA.

A hemispherical tank of radius \(2\) metres is initially full of water and has an outlet of \(12\) cm2 cross-sectional area at the bottom. The outlet is opened at some instant. The flow through the outlet is according to the law $$v(t)=0.6$...

Let \(P\) be a point on the ellipse \({{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1,0 < b < a\). Let the line parallel to \(y\)-axis passing through \(P\) meet the circle \({x^2} + {y^2} = {a^2}\) at the point \(Q\) such that \(P\) ...

Evaluate \(\int {{{\sin }^{ - 1}}\left( {{{2x + 2} \over {\sqrt {4{x^2} + 8x + 13} }}} \right)} \,dx.\)

An urn contains \(m\) white and \(n\) black balls. A ball is drawn at random and is put back into the urn along with \(k\) additional balls of the same colour as that of the ball drawn. A ball is again drawn at random. What is the probabil...

An unbiased die, with faces numbered \(1,2,3,4,5,6,\) is thrown \(n\) times and the list of \(n\) numbers showing up is noted. What is the probability that, among the numbers \(1,2,3,4,5,6,\) only three numbers appear in this list?

If \(\Delta\) is the area of a triangle with side lengths \(a, b, c,\) then show that \(\Delta \le {1 \over 4}\sqrt {\left( {a + b + c} \right)abc}\). Also show that the equality occurs in the above inequality if and only if \(a=b=c\).

Let \(a,\,b,\,c\) be real numbers with \(a \ne 0\) and let \(\alpha ,\,\beta\) be the roots of the equation \(a{x^2} + bx + c = 0\). Express the roots of \({a^3}{x^2} + abcx + {c^3} = 0\) in terms of \(\alpha ,\,\beta \,\).

Let \({a_1}\), \({a_2}\),.....,\({a_n}\) be positive real numbers in geometric progression. For each n, let \({A_n}\), \({G_n}\), \({H_n}\) be respectively, the arithmetic mean , geometric mean, and harmonic mean of \({a_1}\),\({a_2}\)........

Let \(a, b, c\) be real numbers with \({a^2} + {b^2} + {c^2} = 1.\) Show that
the equation $$\left| {\matrix{
{ax - by - c} & {bx + ay} & {cx + a} \cr
{bx + ay} & { - ax + by - c} & {cy + b} \cr
{cx + a} & {cy + b} & { - ax -...

Find \(3-\)dimensional vectors \({\overrightarrow v _1},{\overrightarrow v _2},{\overrightarrow v _3}\) satisfying
$$\,{\overrightarrow v _1}.{\overrightarrow v _1} = 4,\,{\overrightarrow v _1}.{\overrightarrow v _2} = - 2,\,{\overrighta...

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.

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 \lef...