IIT-JEE 2007 Paper 2 Offline

Consider the planes \(3 x-6 y-2 z=15\) and \(2 x+y-2 z=5\).
STATEMENT - 1 : The parametric equations of the line of intersection of the given planes are \(x=3+14 t, y=1+2 t, z=15 t\)
STATEMENT - 2 : The vectors $$14 \hat{i}+2 \hat{j}+15 \ha...

Let \(\mathrm{ABCD}\) be a quadrilateral with area 18 , with side \(\mathrm{A B}\) parallel to the side \(\mathrm{C D}\) and \(\mathrm{A B}=2 \mathrm{CD}\). Let \(\mathrm{AD}\) be perpendicular to \(\mathrm{AB}\) and \(\mathrm{CD}\). If a c...

Match the statements in Column I with the properties Column II.

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If \(|z|=1\) and \(z \neq \pm 1\), then all the values of \(\frac{z}{1-z^{2}}\) lie on

The differential equation \(\frac{d y}{d x}=\frac{\sqrt{1-y^{2}}}{y}\) determines a family of circles with :

\(\frac{d^{2} x}{d y^{2}}\) equals :

Let \(f(x)=\frac{x}{\left(1+x^{n}\right)^{1 / n}}\) for \(n \geq 2\) and \(g(x)=\underbrace{(f o f o \ldots . o f)}_{f \text { occurs } n \text { times }}(x)\). Then \(\int x^{n-2} g(x) d x\) equals :

Let \((x,y)\) be such that \({\sin ^{ - 1}}(ax) + {\cos ^{ - 1}}(y) + {\cos ^{ - 1}}(bxy) = {\pi \over 2}\).
Match the statements in Column I with the statements in Column II.

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Let \(f(x)=2+\cos x\) for all real \(x\).
STATEMENT - 1 : For each real \(t\), there exists a point \(c\) in \([t, t+\pi]\) such that \(f^{\prime}(C)=0\).
STATEMENT - 2 : \(f(t)=f(t+2 \pi)\) for each real \(t\).

The line \(y=x\) meets \(y=k e^{\mathrm{x}}\) for \(k \leq 0\) at

The positive value of \(k\) for which \(k e^{x}-x=0\) has only one root is

For \(k > 0\), the set of all values of \(k\) for which \(k e^{x}-x=0\) has two distinct roots is

Let \(f(x) = {{{x^2} - 6x + 5} \over {{x^2} - 5x + 6}}\).
Match the conditions/expressions in Column I with statements in Column II.

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STATEMENT - 1 : The curve \(y=\frac{-x^{2}}{2}+x+1\) is symmetric with respect to the line \(x=1\).
STATEMENT - 2 : A parabola is symmetric about its axis.

The letters of the word COCHIN are permuted and all the permutations are arranged in an alphabetical order as in an English dictionary. The number of words that appear before the word COCHIN is :

Let \({E^c}\) denote the complement of an event \(E.\) Let \(E, F, G\) be pairwise independent events with \(P\left( G \right) > 0\) and \(P\left( {E \cap F \cap G} \right) = 0.\) Then \(P\left( {{E^c} \cap {F^c}|G} \right)\) equals

Which one of the following statements is correct?

Which one of the following statements is correct?

Which one of the following statements is correct?

Let \(\mathrm{O(0,0), P(3,4), Q(6,0)}\) be the vertices of the triangle OPQ. The point R inside the triangle OPQ is such that the triangles OPR, PQR, OQR are of equal area. The coordinates of R are

Lines \(\mathrm{L}_{1}: y-x=0\) and \(\mathrm{L}_{2}: 2 x+y=0\) intersect the line \(\mathrm{L}_{3}: y+2=0\) at \(\mathrm{P}\) and \(\mathrm{Q}\), respectively. The bisector of the acute angle between \(L_{1}\) and \(L_{2}\) intersects $$L_...

Let \(\vec{a}, \vec{b}, \vec{c}\) be unit vectors such that \(\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}\). Which one of the following is correct?