IIT-JEE 2004 Screening

If the lines \({{x - 1} \over 2} = {{y + 1} \over 3} = {{z - 1} \over 4}\) and \(\,{{x - 3} \over 1} = {{y - k} \over 2} = {z \over 1}\) intersect, then the value of \(k\) is

If \(f\left( x \right) = {x^a}\log x\) and \(f\left( 0 \right) = 0,\) then the value of \(\alpha\) for which Rolle's theorem can be applied in \(\left[ {0,1} \right]\) is

If \(f\left( x \right) = {x^3} + b{x^2} + cx + d\) and \(0 < {b^2} < c,\) then in \(\left( { - \infty ,\infty } \right)\)

The area enclosed between the curves \(y = a{x^2}\) and
\(x = a{y^2}\left( {a > 0} \right)\) is \(1\) sq. unit, then the value of \(a\) is

If one of the diameters of the circle \({x^2} + {y^2} - 2x - 6y + 6 = 0\) is a chord to the circle with centre (2, 1), then the radius of the circle is

If \(\omega\) \(\left( { \ne 1} \right)\) be a cube root of unity and \({\left( {1 + {\omega ^2}} \right)^n} = {\left( {1 + {\omega ^4}} \right)^n},\) then the least positive value of n is

The value of the integral \(\int\limits_0^1 {\sqrt {{{1 - x} \over {1 + x}}} dx}\) is

If \(f(x)\) is differentiable and \(\int\limits_0^{{t^2}} {xf\left( x \right)dx = {2 \over 5}{t^5},}\) then \(f\left( {{4 \over {25}}} \right)\) equals

If \(y=y(x)\) and \({{2 + \sin x} \over {y + 1}}\left( {{{dy} \over {dx}}} \right) = - \cos x,y\left( 0 \right) = 1,\)
then \(y\left( {{\pi \over 2}} \right)\) equals

If \(y\) is a function of \(x\) and log \((x+y)-2xy=0\), then the value of \(y'(0)\) is equal to

If tangents are drawn to the ellipse \({x^2} + 2{y^2} = 2,\) then the locus of the mid-point of the intercept made by the tangents between the coordinate axes is

If the line \(62x + \sqrt 6 y = 2\) touches the hyperbola \({x^2} - 2{y^2} = 4\), then the point of contact is

The value of \(x\) for which \(sin\left( {{{\cot }^{ - 1}}\left( {1 + x} \right)} \right) = \cos \left( {{{\tan }^{ - 1}}\,x} \right)\) is

If \({}^{n - 1}{C_r} = \left( {{k^2} - 3} \right)\,{}^n{C_{r + 1,}}\) then \(k \in\)

The angle between the tangents drawn from the point \((1, 4)\) to the parabola \({y^2} = 4x\) is

If three distinct numbers are chosen randomly from the first \(100\) natural numbers, then the probability that all three of them are divisible by both \(2\) and \(3\) is

The sides of a triangle are in the ratio \(1:\sqrt 3 :2\), then the angles of the triangle are in the ratio

If one root is square of the other root of the equation \({x^2} + px + q = 0\), then the realation between \(p\) and \(q\) is

For all \('x',{x^2} + 2ax + 10 - 3a > 0,\) then the interval in which '\(a\)' lies is

An infinite G.P. has first term '\(x\)' and sum '\(5\)', then \(x\) belongs to

Area of the triangle formed by the line \(x + y = 3\) and angle bisectors of the pair of straight line \({x^2} - {y^2} + 2y = 1\) is

Given both \(\theta\) and \(\phi\) are acute angles and \(\sin \,\theta = {1 \over 2},\,\) \(\cos \,\phi = {1 \over 3},\) then the value of \(\theta + \phi\) belongs to

The unit vector which is orthogonal to the vector \(3\overrightarrow i + 2\overrightarrow j + 6\overrightarrow k\) and is coplanar with the vectors \(\,2\widehat i + \widehat j + \widehat k\) and $$\,\widehat i - \widehat j + \widehat k$...

If \(\overrightarrow a = \left( {\widehat i + \widehat j + \widehat k} \right),\overrightarrow a .\overrightarrow b = 1\) and \(\overrightarrow a \times \overrightarrow b = \widehat j - \widehat k,\) then \(\overrightarrow b\) is