IIT-JEE 1995

Let \((h, k)\) be a fixed point, where \(h > 0,k > 0.\). A straight line passing through this point cuts the possitive direction of the coordinate axes at the points \(P\) and \(Q\). Find the minimum area of the triangle \(OPQ\), \(O\) bein...

Consider a square with vertices at \((1,1), (-1,1), (-1,-1)\) and \((1, -1)\). Let \(S\) be the region consisting of all points inside the square which are nearer to the origin than to any edge. Sketch the region \(S\) and find its area.

If \(i{z^3} + {z^2} - z + i = 0\) , then show that \(\left| z \right| = 1\).

If \(\left| {Z - W} \right| \le 1,\left| W \right| \le 1\), show that \({\left| {Z - W} \right|^2} \le {(\left| Z \right| - \left| W \right|)^2} + {(ArgZ - Arg\,W)^2}\)

Let \({I_m} = \int\limits_0^\pi {{{1 - \cos mx} \over {1 - \cos x}}} dx.\) Use mathematical induction to prove that \({I_m} = m\,\pi ,m = 0,1,2,........\)

Evaluate the definite integral :
\($\int\limits_{ - 1/\sqrt 3 }^{1/\sqrt 3 } {\left( {{{{x^4}} \over {1 - {x^4}}}} \right){{\cos }^{ - 1}}\left( {{{2x} \over {1 + {x^2}}}} \right)} dx\)$

Let \(y=f(x)\) be a curve passing through \((1,1)\) such that the triangle formed by the coordinate axes and the tangent at any point of the curve lies in the first quadrant and has area \(2.\) From the differential equation and determine a...

Let '\(d\)' be the perpendicular distance from the centre of the ellipse \({{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1\) to the tangent drawn at a point \(P\) on the ellipse. If \({F_1}\) and \({F_2}\) are the two foci of the elli...

Show that the locus of a point that divides a chord of slope \(2\) of the parabola \({y^2} = 4x\) internally in the ratio \(1:2\) is a parabola. Find the vertex of this parabola.

Let \(a,\,b,\,c\) be real. If \(a{x^2} + bx + c = 0\) has two real roots \(\alpha\) and \(\beta ,\) where \(\alpha < - 1\) and \(\beta > 1,\) then show that \(1 + {c \over a} + \left| {{b \over a}} \right| < 0.\)

The orthocentre of the triangle formed by the lines \(xy=0\) and \(x+y=1\) is

Find the smallest positive number \(p\) for which the equation \(\cos \left( {p\,\sin x} \right) = \sin \left( {p\cos x} \right)\) has a solution \(x\, \in \,\left[ {0,2\pi } \right]\).

The minimum value of the expression \(\sin \,\alpha + \sin \,\beta \, + \sin \,\gamma ,\,\) where \(\alpha ,\,\beta ,\,\gamma\) are real numbers satisfying \(\alpha + \beta + \gamma = \pi\) is