IIT-JEE 2010 Paper 1 Offline

If the distance between the plane \(Ax-2y+z=d\) and the plane containing the lines \({{x - 1} \over 2} = {{y - 2} \over 3} = {{z - 3} \over 4}\) and \({{x - 2} \over 3} = {{y - 3} \over 4} = {{z - 4} \over 5}\,\) is \(\sqrt 6 \,\,,\) then...

Equation of the plane containing the straight line \({x \over 2} = {y \over 3} = {z \over 4}\) and perpendicular to the plane containing the straight lines \({x \over 3} = {y \over 4} = {z \over 2}\) and $${x \over 4} = {y \over 2} = {z \ov...

Let \(f\) be a real-valued differentiable function on \(R\) (the set of all real numbers) such that \(f(1)=1\). If the \(y\)-intercept of the tangent at any point \(P(x,y)\) on the curve \(y=f(x)\) is equal to the cube of the abscissa of $$...

Let \(f\) be a real-valued function defined on the interval \(\left( {0,\infty } \right)\)
by \(\,f\left( x \right) = \ln x + \int\limits_0^x {\sqrt {1 + \sin t\,} dt.}\) then which of the following
statement(s) is (are) true?

Let \({{z_1}}\) and \({{z_2}}\) be two distinct complex number and let z =( 1 - t)\({{z_1}}\) + t\({{z_2}}\) for some real number t with 0 < t < 1. IfArg (w) denote the principal argument of a non-zero complex number w, then

Let $z_1$ and $z_2$ be two distinct complex numbers let $z=(1-t) z_1+t z_2$ for some real number t with $0 < t < 1$.
If $\operatorname{Arg}(w)$ denotes the principal argument of a nonzero complex number $w$, then :

For any real number \(x,\) let \(\left[ x \right]\) denote the largest integer less than or equal to \(x.\) Let \(f\) be a real valued function defined on the interval \(\left[ { - 10,10} \right]\) by
$$$f\left( x \right) = \left\{ {\matrix...

The value of \(\int\limits_0^1 {{{{x^4}{{\left( {1 - x} \right)}^4}} \over {1 + {x^2}}}dx}\) is (are)

The value of \(\mathop {\lim }\limits_{x \to 0} {1 \over {{x^3}}}\int\limits_0^x {{{t\ln \left( {1 + t} \right)} \over {{t^4} + 4}}} dt\) is

Let $f, g$ and $h$ be real valued functions defined on the interval $[0,1]$ by
$f(x)=e^{x^2}+e^{-x^2}$,
$g(x)=x e^{x^2}+e^{-x^2}$
and $h(x)=x^2 e^{x^2}+e^{-x^2}$.
If $a, b$ and $c$ denote, respectively, the absolute maximum of $f, g$ an...

The circle \({x^2} + {y^2} - 8x = 0\) and hyperbola \({{{x^2}} \over 9} - {{{y^2}} \over 4} = 1\) intersect at the points \(A\) and \(B\).
Equation of a common tangent with positive slope to the circle as well as to the hyperbola is

The line \(2x + y = 1\) is tangent to the hyperbola \({{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1\). If this line passes through the point of intersection of the nearest directrix and the \(x\)-axis, then the eccentricity of the h...

The circle \({x^2} + {y^2} - 8x = 0\) and hyperbola \({{{x^2}} \over 9} - {{{y^2}} \over 4} = 1\) intersect at the points \(A\) and \(B\).
Equation of the circle with \(AB\) as its diameter is

The number of $3 \times 3$ matrices A whose entries are either 0 or 1 and for which the system $\mathrm{A}\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]$ has exactly two distinct solu...

The number of $A$ in $T_p$ such that $A$ is either symmetric or skew-symmetric or both, and $\operatorname{det}(\mathrm{A}) \operatorname{divisible}$ by $p$ is :

The number of A in $\mathrm{T}_p$ such that the trace of A is not divisible by $p$ but $\operatorname{det}(\mathrm{A})$ is divisible by $p$ is
[Note : The trace of a matrix is the sum of its diagonal entries.]

The number of A in $\mathrm{T}_p$ such that $\operatorname{det}(\mathrm{A})$ is not divisible by $p$ is :

Let \(A\) and \(B\) be two distinct points on the parabola \({y^2} = 4x\). If the axis of the parabola touches a circle of radius \(r\) having \(AB\) as its diameter, then the slope of the line joining \(A\) and \(B\) can be

Let \(\omega\) be a complex cube root of unity with \(\omega \ne 1.\) A fair die is thrown three times. If \({r_1},\) \({r_2}\) and \({r_3}\) are the numbers obtained on the die, then the probability that $${\omega ^{{r_1}}} + {\omega ^{{...

If the angles \(A, B\) and \(C\) of a triangle are in an arithmetic progression and if \(a, b\) and \(c\) denote the lengths of the sides opposite to \(A, B\) and \(C\) respectively, then the value of the expression $${a \over c}\sin 2C + ...

Let \(ABC\) be a triangle such that \(\angle ACB = {\pi \over 6}\) and let \(a, b\) and \(c\) denote the lengths of the sides opposite to \(A\), \(B\) and \(C\) respectively. The value(s) of \(x\) for which $$a = {x^2} + x + 1,\,\,\,b = {x...

Let \(p\) and \(q\) be real numbers such that \(p \ne 0,\,{p^3} \ne q\) and \({p^3} \ne - q.\) If \({p^3} \ne - q.\) and \(\,\beta\) are nonzero complex numbers satisfying \(\alpha \, + \beta = - p\,\) and $${\alpha ^3} + {\beta ^3} =...

Let \({S_k}\)= 1, 2,....., 100, denote the sum of the infinite geometric series whose first term is \(\,{{k - 1} \over {k\,!}}\) and the common ratio is \({1 \over k}\). Then the value of $${{{{100}^2}} \over {100!}}\,\, + \,\,\sum\limits_{...

The maximum value of the expression \({1 \over {{{\sin }^2}\theta + 3\sin \theta \cos \theta + 5{{\cos }^2}\theta }}\) is

The number of values of \(\theta\) in the interval, \(\left( { - {\pi \over 2},\,{\pi \over 2}} \right)\) such
that\(\,\theta \ne {{n\pi } \over 5}\) for \(n = 0,\, \pm 1,\, \pm 2\) and \(\tan \,\theta = \cot \,5\theta \,\) as well a...

The number of all possible values of \(\theta\) where \(0 < \theta < \pi ,\) for which the system of equations
\($\left( {y + z} \right)\cos {\mkern 1mu} 3\theta = \left( {xyz} \right){\mkern 1mu} \sin 3\theta\)$
$$$x\sin 3\theta = {{2...

Let \(P,Q,R\) and \(S\) be the points on the plane with position vectors \({ - 2\widehat i - \widehat j,4\widehat i,3\widehat i + 3\widehat j}\) and \({ - 3\widehat i + 2\widehat j}\) respectively. The quadrilateral \(PQRS\) must be a

If \(\overrightarrow a\) and \(\overrightarrow b\) are vectors in space given by \(\overrightarrow a = {{\widehat i - 2\widehat j} \over {\sqrt 5 }}\) and $$\overrightarrow b = {{2\widehat i + \widehat j + 3\widehat k} \over {\sqrt {14}...