IIT-JEE 2006

Match the following:

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Let \({\overrightarrow A }\) be vector parallel to line of intersection of planes \({P_1}\) and \({P_2}.\) Planes \({P_1}\) is parallel to the vectors \(2\widehat j + 3\widehat k\) and \(4\widehat j - 3\widehat k\) and that \({P_2}\) is pa...

A plane passes through $(1,-2,1)$ and is perpendicular to two planes $2 x-2 y+z=0$ and $x-y+2 z=4$. The distance of the plane from the point $(1,2,2)$ is:

If \(f(x)\) is a twice differentiable function such that \(f(A)=0, f(B)=2, f(C)=-1, f(D)=2\), \(f(e)=0\), where \(a < b < c < d < e\), then the minimum number of zeroes of \(g(x)=\left(f'(x)\right)^{2}+f''(x) f(x)\) in the interval $$[a, e]...

A tangent drawn to the curve $y=f(x)$ at $\mathrm{P}(x, y)$ cuts the X -axis and Y -axis at A and B respectively such that $\mathrm{BP}: \mathrm{AP}=3: 1$, given that $f(1)=1$, then

$f(x)$ is cubic polynomial which has local maximum at $x=-1$. If $f(2)=18, f(1)=-1$ and $f(x)$ has local minima at $x=0$, then

$$ \begin{aligned} & f(x)=\left\{\begin{array}{cc} e^x, & 0 \leq x \leq 1 \\ 2-e^{x-1}, & 1 < x \leq 2 \\ x-e, & 2 < x \leq 3 \end{array} \quad\right. \text { and } \\ & g(x)=\int_0^x f(t) d t, x \in[1,3] \text { then } g(x) \text { has } \...

If $f(x)$ is a twice differentiable function such that $f(A)=0, f(B)=2, f(C)=-1, f(D)=2$, $f(e)=0$, where $a < b < c < d < e$, then the minimum number of zeroes of $g(x)=\left(f^{\prime}(x)\right)^2 +f^{\prime \prime}(x) f(x)$ in the interv...

\(\int_\limits{0}^{\pi / 2} \sin x d x\) is equal to:

If \(\lim_\limits{t \rightarrow a} \frac{\int_{a}^{t} f(x) d x-\frac{(t-a)}{2}\{f(t)+f(a)\}}{(t-a)^{3}}=0\) then the degree of polynomial function \(f(x)\) almost is:

\(f''(x) < 0 \forall x \in(a, b)\) 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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