IIT-JEE 2007 Paper 1 Offline

Let \(f(x)\) be differentiable on the interval (0, \(\infty\)) such that \(f(1)=1\), and \(\mathop {\lim }\limits_{t \to x} {{{t^2}f(x) - {x^2}f(t)} \over {t - x}} = 1\) for each \(x > 0\). Then \(f(x)\) is

Tangents are drawn from the point (17, 7) to the circle \(x^2+y^2=169\).
Statement 1 : The tangents are mutually perpendicular.
Statement 2 : The locus of the points from which mutually perpendicular tangents can be drawn to the given circl...

A man walks a distance of 3 units from the origin towards the north-east (N 45\(^\circ\)E) direction. From there, he walks a distance of 4 units towards the north-west (N 45\(^\circ\)W) direction to reach a point P. Then the position of P i...

\(\mathop {\lim }\limits_{x \to {\pi \over 4}} {{\int\limits_2^{{{\sec }^2}x} {f(t)\,dt} } \over {{x^2} - {{{\pi ^2}} \over {16}}}}\) equal

Match the integrals in Column I with the values in Column II.

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A hyperbola, having the transverse axis of the length \(2\sin \theta\), is confocal with the ellipse \(3{x^2} + 4{y^2} = 12\). Then its equation is

Let F(x) be an indefinite integral of \(\sin^2x\).
Statement 1 : The function F(x) satisfies F(\(x+\pi\)) = F(\(x\)) for all real x.
Statement 2 : \({\sin ^2}(x + \pi ) = {\sin ^2}x\) for all real x.

In the following [x] denotes the greatest integer less than or equal to x.
Match the functions in Column I with the properties Column II.

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The tangent to the curve \(y=e^x\) drawn at the point (\(c,e^c\)) intersects the line joining the points (\(c-1,e^{c-1}\)) and (\(c+1,e^{c+1}\))

The ratio of the areas of the triangles PQS and PQR is

The radius of the circumcircle of the triangle PRS is

The radius of the incircle of the triangle PQR is

One Indian and four American men and their wives are to be seated randomly around a circular table. Then the conditional probability that the Indian man is seated adjacent to his wife given that each American man is seated adjacent to his w...

Let H\(_1\), H\(_2\), ..., H\(_n\) be mutually exclusive and exhaustive events with P(H\(_i\)) > 0, i = 1, 2, ..., n. Let E be any other event with 0 < P(E) < 1.
Statement 1 : P(H\(_i\) | E) > P(E | H\(_i\)). P(H\(_i\)) for \(i=1,2,...,n\)....

Let \(\alpha,\beta\) be the roots of the equation \(x^2-px+r=0\) and \(\frac{\alpha}{2},2\beta\) be the roots of the equation \(x^2-qx+r=0\). Then the value of r is

The sum V\(_1\) + V\(_2\) + ... + V\(_n\) is

T\(_r\) is always

Which one of the following is a correct statement?

Consider the following linear equations
\(ax + by + cz = 0\)
\(bx + cy + az = 0\)
\(cx + ay + bz = 0\)
Match the conditions/expressions in Column I with statements in Column II.

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The number of solutions of the pair of equations
\(2{\sin ^2}\theta - \cos 2\theta = 0\)
\(2{\cos ^2}\theta - 3\sin \theta = 0\)
in the interval \([0,2\pi]\) is

The number of distinct real values of \(\lambda\), for which the vectors \(- {\lambda ^2}\widehat i + \widehat j + \widehat k,\widehat i - {\lambda ^2}\widehat j + \widehat k\) and \(\widehat i + \widehat j - {\lambda ^2}\widehat k\) are c...

Let the vector \(\overrightarrow {PQ} ,\overrightarrow {QR} ,\overrightarrow {RS} ,\overrightarrow {ST} ,\overrightarrow {TU}\) and \(\overrightarrow {UP}\), represent the sides of a regular hexagon.
Statement 1 : $$\overrightarrow {PQ} ...