IIT-JEE 2001 Screening

The triangle formed by the tangent to the curve \(f\left( x \right) = {x^2} + bx - b\) at the point \((1, 1)\) and the coordinate axex, lies in the first quadrant. If its area is \(2\), then the value of \(b\) is

Let \(f\left( x \right) = \left( {1 + {b^2}} \right){x^2} + 2bx + 1\) and let \(m(b)\) be the minimum value of \(f(x)\). As \(b\) varies, the range of \(m(b)\) is

If \(f\left( x \right) = x{e^{x\left( {1 - x} \right)}},\) then \(f(x)\) is

Let PQ and RS be tangents at the extremities of the diameter PR of a circle of radius r. If PS and RQ intersect at a point X on the circumference of the circle, then 2r equals

Let A B be a chord of the circle \({x^2} + {y^2} = {r^2}\) subtending a right angle at the centre. Then the locus of the centriod of the triangle PAB as P moves on the circle is

Let \({z_1}\) and \({z_2}\) be \({n^{th}}\) roots of unity which subtend a right angle at the origin. Then \(n\) must be of the form

The complex numbers \({z_1},\,{z_2}\) and \({z_3}\) satisfying \({{{z_1} - {z_3}} \over {{z_2} - {z_3}}} = {{1 - i\sqrt 3 } \over 2}\,\) are the vertices of a triangle which is

The value of \(\int\limits_{ - \pi }^\pi {{{{{\cos }^2}x} \over {1 + {a^x}}}dx,\,a > 0,}\) is

Let \(f:\left( {0,\infty } \right) \to R\) and \(F\left( x \right) = \int\limits_0^x {f\left( t \right)dt.}\) If \(F\left( {{x^2}} \right) = {x^2}\left( {1 + x} \right)\), then \(f(4)\) equals

If \({\sin ^{ - 1}}\left( {x - {{{x^2}} \over 2} + {{{x^3}} \over 4} - ....} \right)\)
\($ + {\cos ^{ - 1}}\left( {{x^2} - {{{x^4}} \over 2} + {{{x^6}} \over 4} - ....} \right) = {\pi \over 2}\)$
for \(0 < \left| x \right| < \sqrt 2 ,\) ...

In the binomial expansion of \({\left( {a - b} \right)^n},\,n \ge 5,\) the sum of the \({5^{th}}\) and \({6^{th}}\) terms is zero. Then \(a/b\) equals

The equation of the directrix of the parabola \({y^2} + 4y + 4x + 2 = 0\)

The equation of the common tangent touching the circle \({\left( {x - 3} \right)^2} + {y^2} = 9\) and the parabola \({y^2} = 4x\) above the \(x\)-axis is

Let \({T_n}\) denote the number of triangles which can be formed using the vertices of a regular polygon of n sides. If \({T_{n + 1}} - {T_n} = 21\), then n equals

A man from the top of a \(100\) metres high tower sees a car moving towards the tower at an angle of depression of \({30^ \circ }\). After some time,the angle of depression becomes \({60^ \circ }\). The distance (in metres) travelled by the...

Let \(\alpha\), \(\beta\) be the roots of \({x^2} - x + p = 0\) and \(\gamma ,\delta\) be the roots of \({x^2} - 4x + q = 0.\) If \(\alpha ,\beta ,\gamma ,\delta\) are in G.P., then the integral values of \(p\) and \(q\) respectively, a...

If the sum of the first \(2n\) terms of the A.P.\(2,5,8,......,\) is equal to the sum of the first \(n\) terms of the A.P.\(57,59,61,.....,\) then \(n\) equals

Let the positive numbers \(a,b,c,d\) be in A.P. Then \(abc,\) \(abd,\) \(acd,\) \(bcd,\) are

Area of the parallelogram formed by the lines \(y = mx\), \(y = mx + 1\), \(y = nx\) and \(y = nx + 1\) equals

The number of integer values of \(m\), for which the \(x\)-coordinate of the point of intersection of the lines \(3x + 4y = 9\) and \(y = mx + 1\) is also an integer, is

The number of distinct real roots of \(\left| {\matrix{ {\sin x} & {\cos x} & {\cos x} \cr {\cos x} & {\sin x} & {\cos x} \cr {\cos x} & {\cos x} & {\sin x} \cr } } \right|\,\)
\(\, = 0\) in the interval $$ - {\pi \over ...

If \(\alpha + \beta = \pi /2\) and \(\beta + \gamma = \alpha ,\) then \(\tan \,\alpha \,\) equals

The maximum value of \(\left( {\cos {\alpha _1}} \right).\left( {\cos {\alpha _2}} \right).....\left( {\cos {\alpha _n}} \right),\) under the restrictions \(0 \le {\alpha _1},{\alpha _2},....,{\alpha _n} \le {\pi \over 2}\) vand $$\left( ...

Let \(\overrightarrow a = \overrightarrow i - \overrightarrow k ,\overrightarrow b = x\overrightarrow i + \overrightarrow j + \left( {1 - x} \right)\overrightarrow k\) and
$$\overrightarrow c = y\overrightarrow i - x\overrightarrow...

If \(\overrightarrow a \,,\,\overrightarrow b\) and \(\overrightarrow c\) are unit vectors, then $${\left| {\overrightarrow a - \overrightarrow b } \right|^2} + {\left| {\overrightarrow b - \overrightarrow c } \right|^2} + {\left| {\ove...