IIT-JEE 2006

\(f''(x) < 0 \forall x \in(a, b)\) and \(c\) is a point such that \(a < c < b\), and \((c, f(C))\) is the point lying on the curve for which \(\mathrm{F}(C)\) is maximum, then \(f'(C)\) is equal to:

\(\text { Match the following : }\)


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A line $M$ through $A$ is drawn parallel to $B D$. Point $S$ moves such that its distances from
the line BD and the vertex A are equal. If locus of S cuts M at $\mathrm{T}_2$ and $\mathrm{T}_3$ and AC at $\mathrm{T}_1$, then area of $\Delta...

A circle touches the line $L$ and the circle $C_1$ externally such that both the circles are on the same side of the line, then the locus of center of the circle is:

If \(w=\alpha+\mathrm{i} \beta\), where \(\beta \neq 0\) and \(z \neq 1\), satisfies the condition that \(\left(\frac{w-\bar{w} z}{1-z}\right)\) is purely real, then the set of values of \(z\) is:

If $P$ is a point on $C_1$ and $Q$ in another point on $\mathrm{C}_2$, then $\frac{\mathrm{PA}^2+\mathrm{PB}^2+\mathrm{PC}^2+\mathrm{PD}^2}{\mathrm{QA}^2+\mathrm{QB}^2+\mathrm{QC}^2+\mathrm{QD}^2}$ is equal to :

\(\text { The value of } 5050 \frac{\int_0^1\left(1-x^{50}\right)^{100} d x}{\int_0^{\frac{1}{1}}\left(1-x^{50}\right)^{101} d x} \text { is : }\)

If $a_n=\frac{3}{4}-\left(\frac{3}{4}\right)^2+\left(\frac{3}{4}\right)^3+\cdots \cdots(-1)^{n-1}\left(\frac{3}{4}\right)^n$ and $b_n=1-a_n$, then find the minimum natural number $n_0$ such that $b_n>a_n \forall n>n_0$

If \(f''(x)=-f(x)\) and \(g(x)=f'(x)\) and \(\mathrm{F}(x)=\left(f\left(\frac{x}{2}\right)\right)^{2}+\left(g\left(\frac{x}{2}\right)\right)^{2}\) and given that \(\mathrm{F}(5)=5\), then \(\mathrm{F}(10)\) is equal to :

If a hyperbola passes through the focus of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ and its transverse and conjugate axes coincide with the major and minor axes of the ellipse, and the product of eccentricities is 1 , then

$\int \frac{x^2-1}{x^3 \sqrt{2 x^4-2 x^2+1}} d x$ is equal to

If $f(x)=\min \left\{1, x^2, x^3\right\}$, then

For $x>0, \mathop {\lim }\limits_{x \to 0}\left((\sin x)^{1 / x}+(1 / x)^{\sin x}\right)$ is :

The value of \(|U|\) is :

The value of $\left[\begin{array}{lll}3 & 2 & 0\end{array}\right] U\left[\begin{array}{l}3 \\ 2 \\ 0\end{array}\right]$ is :

The sum of the elements of $\mathrm{U}^{-1}$ is:

\(\text { Normals are drawn at points } \mathrm{P}, \mathrm{Q} \text { and } \mathrm{R} \text { lying on the parabola } y^2=4 x \text { which intersect at }(3,0) \text {. Then }\)
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The axis of a parabola is along the line \(y = x\) and the distances of its vertex and focus from origin are \(\sqrt 2\) and \(2\sqrt 2\) respectively. If vertex and focus both lie in the first quadrant, then the equation of the parabola ...

The equations of the common tangents to the parabola \(y = {x^2}\) and \(y = - {\left( {x - 2} \right)^2}\) is/are

If \(\mathrm{P}\left(u_{i}\right) \propto i\), where \(i=1,2,3, \ldots n\), then \(\lim_\limits{n \rightarrow \infty} \mathrm{P}(w)\) is equal to:

If \(\mathrm{P}\left(u_{i}\right)=c\), where \(c\) is a constant then \(\mathrm{P}\left(u_{n} / w\right)\) is equal to:

If \(n\) is even and E denotes the event of choosing even numbered urn \(\left(\mathrm{P}\left(u_{i}\right)=\frac{1}{n}\right)\),
then the value of \(\mathrm{P}(w / \mathrm{E})\) is :

Given an isosceles triangle, whose one angle is $120^{\circ}$ and radius of its incircle $=\sqrt{3}$. Then the area of the triangle in sq. units is

Internal bisector of $\angle A$ of triangle $A B C$ meets side BC at D . A line drawn through D perpendicular to AD intersects the side AC at E and the side AB at F . If $a, b, c$ represent sides of $\triangle \mathrm{ABC}$ then

Let \(a, b, c\) be the sides of a triangle. No two of them are equal and \(\lambda \in R\). If the roots of the equation \(x^{2}+2(a+b+c) x+3 \lambda(a b+b c+c a)=0\) are real, then,

If roots of the equation $x^2-10 c x-11 d=0$ are $a, b$ and those of $x^2-10 a x-11 b=0$ are $c, d$, then the value of $a+b+c+d$ is $(a, b, c$ and $d$ are distinct numbers)

Let \(\theta \in\left(0, \frac{\pi}{4}\right)\) and \(t_{1}=(\tan \theta)^{\tan \theta}, t_{2}=(\tan \theta)^{\cot \theta}, t_{3}=(\cot \theta)^{\tan \theta}\) and \(t_{4}=(\cot \theta)^{\cot \theta}\), then

If $0<\theta<2 \pi$, then the intervals of values of $\theta$ for which $2 \sin ^2 \theta-5 \sin \theta+2>0$, is

Let \(\overrightarrow a = \widehat i + 2\widehat j + \widehat k,\,\overrightarrow b = \widehat i - \widehat j + \widehat k\) and \(\overrightarrow c = \widehat i + \widehat j - \widehat k.\) A vector in the plane of $$\overrightarrow a ...

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Match the following columns.

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Initially, the capacitor was uncharged. Now, switch $S_1$ is closed and $S_2$ is kept open. If time constant of this circuit is $\tau$, then

If the total charge stored in the LC circuit.is $\mathrm{Q}_0$, then for $t \geq 0$,

After the capacitor gets fully charged, $\mathrm{S}_1$ is opened and $S_2$ is closed so that the inductor is connected in series with the capacitor. Then,

In hydrogen-like atom $(z=11)$, $n$th line of Lyman series has wavelength A equal to the de Broglie's wavelength of electron in the level from which it originated. What is the value of $n$ ?

\(\text { Match the following Columns. }\)


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In the following circuits, it is given that $\mathrm{R}_1=1 \Omega, \mathrm{R}_2=2 \Omega, \mathrm{C}_1=2 \mu \mathrm{~F}$ and $\mathrm{C}_2=4 \mu \mathrm{~F}$.The time constants (in $\mu \mathrm{s}$ ) for the circuits I, II, III are, rcspe...

In the given diagram, a line of force of a particular force field is shown. Out of the following options, it can never represent

Consider a cylindrical element as shown in the figure. The current that flows through the element is I and resistivity of material of the cylinder is $\rho$. Choose the correct option out the following

The graph between $\frac{1}{\lambda}$ and stopping potential (V) of three metals having work functions $\phi_1, \phi_2$, and $\phi_3$ in an experiment of photoelectric effect is plotted as shown in the figure. Which of the following stateme...

What is the advantage of this system?

What is the disadvantage of this system?

Which force causes the train to elevate upwards

The electrostatic potential $\left(\phi_r\right)$ of a spherical symmetric system, kept at origin, is shown in the adjacent figure, and given as
$$ \begin{array}{ll} \phi_r=\frac{q}{4 \pi \epsilon_0 r} & \left(r \geq \mathrm{R}_0\right) \\ ...

A simple telescope used to view distant objects has eyepiece and objective lens of focal lengths $f_e$ and $f_0$, respectively. Match Column I with Column II:
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Graph of position of image versus position of point object from a convex lens is shown. Then, the focal length of the lens is

A biconvex lens of focal length $f$ forms a circular image of sun of radius $r$ in focal plane. Then

A point object is placed at a distance of 20 cm from a thin plano-convex lens of focal length 15 cm , if the plane surface is silvered. The image will form at

A system of binary stars of masses $m_{\mathrm{A}}$ and $m_{\mathrm{B}}$ are moving in circular orbits of radii $r_{\mathrm{A}}$ and $r_R$, respectively. If $\mathrm{T}_A$ and $\mathrm{T}_B$ are the time periods of masses $m_A$ and $m_B$ re...

In a dark room with ambient temperature $\mathrm{T}_0$, a black body is kept at a temperature T . Keeping the temperature of the black body constant (at T), sunrays are allowed to fall on the black body through a hole in the roof of the dar...