IIT-JEE 2005 Screening

A variable plane at a distance of the one unit from the origin cuts the coordinates axes at \(A,\) \(B\) and \(C.\) If the centroid \(D\) \((x, y, z)\) of triangle \(ABC\) satisfies the relation $${1 \over {{x^2}}} + {1 \over {{y^2}}} + {1 ...

If \(P(x)\) is a polynomial of degree less than or equal to \(2\) and \(S\) is the set of all such polynomials so that \(P(0)=0\), \(P(1)=1\) and \(P'\left( x \right) > 0\,\,\forall x \in \left[ {0,1} \right],\) then

The area bounded by the parabola \(y = {\left( {x + 1} \right)^2}\) and
\(y = {\left( {x - 1} \right)^2}\) and the line \(y=1/4\) is

A circle is given by \({x^2}\, + \,{(y\, - \,1\,)^2}\, = \,1\), another circle C touches it externally and also the x-axis, then thelocus of its centre is

\(a,\,b,\,c\) are integers, not all simultaneously equal and \(\omega\) is cube root of unity \(\left( {\omega \ne 1} \right),\) then minimum value of \(\left| {a + b\omega + c{\omega ^2}} \right|\) is

\(\int\limits_{ - 2}^0 {\left\{ {{x^3} + 3{x^2} + 3x + 3 + \left( {x + 1} \right)\cos \left( {x + 1} \right)} \right\}\,\,dx}\) is equal to

If \(y=y(x)\) and it follows the relation \(x\cos \,y + y\,cos\,x = \pi\) then \(y''(0)=\)

For the primitive integral equation \(ydx + {y^2}dy = x\,dy;\)
\(x \in R,\,\,y > 0,y = y\left( x \right),\,y\left( 1 \right) = 1,\) then \(y(-3)\) is

The differential equation \({{dy} \over {dx}} = {{\sqrt {1 - {y^2}} } \over y}\) determines a family of circles with

The solution of primitive integral equation \(\left( {{x^2} + {y^2}} \right)dy = xy\)
\(dx\) is \(y=y(x),\) If \(y(1)=1\) and \(\left( {{x_0}} \right) = e\), then \({{x_0}}\) is equal to

If \(f(x)\) is a twice differentiable function and given that \(f\left( 1 \right) = 1;f\left( 2 \right) = 4,f\left( 3 \right) = 9\), then

The minimum area of triangle formed by the tangent to the \({{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1\) and coordinate axes is

If \(\int\limits_{\sin x}^1 {{t^2}f\left( t \right)dt = 1 - \sin x,}\) then f\(\left( {{1 \over {\sqrt 3 }}} \right)\) is

The value of $$$\left( {\matrix{
{30} \cr
0 \cr

} } \right)\left( {\matrix{
{30} \cr
{10} \cr

} } \right) - \left( {\matrix{
{30} \cr
1 \cr

} } \right)\left( {\matrix{
{30} \cr
{11} \cr

} } \r...

Tangent to the curve \(y = {x^2} + 6\) at a point \((1, 7)\) touches the circle \({x^2} + {y^2} + 16x + 12y + c = 0\) at a point \(Q\). Then the coordinates of \(Q\) are

A rectangle with sides of lenght (2m - 1) and (2n - 1) units is divided into squares of unit lenght by drawing parallel lines as shown in the diagram, then the number of rectangles possible with odd side lengths is

If the LCM of p, q is \({r^2}\,{r^4}\,{s^2}\), where r, s, t are prime numbers and p, q are the positive integers then number of ordered pair (p, q) is

A six faced fair dice is thrown until \(1\) comes, then the probability that \(1\) comes in even no. of trials is

In a triangle \(ABC\), \(a,b,c\) are the lengths of its sides and \(A,B,C\) are the angles of triangle \(ABC\). The correct relation is given by

In the quadratic equation \(\,\,a{x^2} + bx + c = 0,\) \(\Delta\) \(= {b^2} - 4ac\) and \(\alpha + \beta ,\,{\alpha ^2} + {\beta ^2},\,{\alpha ^3} + {\beta ^3},\) are in G.P. where \(\alpha ,\beta\) are the root of $$\,\,a{x^2} + bx + c...

\(\cos \left( {\alpha - \beta } \right) = 1\) and \(\,\cos \left( {\alpha + \beta } \right) = 1/e\) where \(\alpha ,\,\beta \in \left[ { - \pi ,\pi } \right].\)
Paris of \(\alpha ,\,\beta\) which satisfy both the equations is/are

If \(\overrightarrow a \,,\,\overrightarrow b ,\overrightarrow c\) are three non-zero, non-coplanar vectors and
$$\overrightarrow {{b_1}} = \overrightarrow b - {{\overrightarrow b .\,\overrightarrow a } \over {{{\left| {\overrightarrow...