JEE Advanced 2013 Paper 1 Offline

Perpendiculars are drawn from points on the line $\frac{x+2}{2}=\frac{y+1}{-1}=\frac{z}{3}$ to the plane $x+y+$ $z=3$. The foot of perpendiculars lie on the line

A line \(l\) passing through the origin is perpendicular to the lines
\($\,{l_1}:\left( {3 + t} \right)\widehat i + \left( { - 1 + 2t} \right)\widehat j + \left( {4 + 2t} \right)\widehat k,\,\,\,\,\, - \infty < t < \infty\)$
$$${l_2}:\le...

A rectangular sheet of fixed perimeter with sides having their lengths in the ratio \(8:15\) is converted into an open rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed squar...

The area enclosed by the curves \(y = \sin x + {\mathop{\rm cosx}\nolimits}\) and \(y = \left| {\cos x - \sin x} \right|\) over the interval \(\left[ {0,{\pi \over 2}} \right]\) is

Let complex numbers \(\alpha \,and\,{1 \over {\overline \alpha }}\,\) lie on circles \({\left( {x - {x_0}} \right)^2} + \,\,{\left( {y - {y_0}} \right)^2} = {r^2}\) and $$\,{\left( {x - {x_0}} \right)^2} + \,\,{\left( {y - {y_0}} \right)^2...

Let \(f\) \(:\,\,\left[ {{1 \over 2},1} \right] \to R\) (the set of all real number) be a positive,
non-constant and differentiable function such that
\(f'\left( x \right) < 2f\left( x \right)\) and \(f\left( {{1 \over 2}} \right) = 1.\) ...

A curve passes through the point \(\left( {1,{\pi \over 6}} \right)\). Let the slope of
the curve at each point \((x,y)\) be \({y \over x} + \sec \left( {{y \over x}} \right),x > 0.\)
Then the equation of the curve is

A vertical line passing through the point \((h,0)\) intersects the ellipse \({{{x^2}} \over 4} + {{{y^2}} \over 3} = 1\) at the points \(P\) and \(Q\). Let the tangents to the ellipse at \(P\) and \(Q\) meet at the point \(R\). If $$\Delta...

The value of \(\cot \left( {\sum\limits_{n = 1}^{23} {{{\cot }^{ - 1}}} \left( {1 + \sum\limits_{k = 1}^n {2k} } \right)} \right)\) is

The coefficient of three consecutive terms of \({\left( {1 + x} \right)^{n + 5}}\) are in the ratio \(5:10:14.\) Then \(n\) =

For 3 × 3 matrices M and N, which of the following statement(s)
is(are) NOT correct?

Consider the set of eight vectors \(V = \left\{ {a\,\hat i + b\,\hat j + c\hat k:a,\,b,\,c\, \in \left\{ { - 1,\,1} \right\}} \right\}\). Three non-coplanar vectors can be chosen from v in \({2^p}\) ways. Then p is

Of the three independent events \({E_1},{E_2}\) and \({E_3},\) the probability that only \({E_1}\) occurs is \(\alpha ,\) only \({E_2}\) occurs is \(\beta\) and only \({E_3}\) occurs is \(\gamma .\) Let the probability \(p\) that none of e...

Four persons independently solve a certain problem correctly with probabilities \({1 \over 2},{3 \over 4},{1 \over 4},{1 \over 8}.\) Then the probability that the problem is solved correctly by at least one of them is

Let \({S_n} = {\sum\limits_{k = 1}^{4n} {\left( { - 1} \right)} ^{{{k\left( {k + 1} \right)} \over 2}}}{k^2}.\) Then \({S_n}\)can take value(s)

A pack contains \(n\) cards numbered from \(1\) to \(n.\) Two consecutive numbered cards are removed from the pack and the sum of the numbers on the remaining cards is \(1224.\) If the smaller of the numbers on the removed cards is \(k,\) t...

For \(a > b > c > 0,\) the distance between \((1, 1)\) and the point of intersection of the lines \(ax + by + c = 0\) and \(bx + ay + c = 0\) is less than \(\left( {2\sqrt 2 } \right)\). Then

Let \(f\left( x \right) = x\sin \,\pi x,\,x > 0.\) Then for all natural numbers \(n,\,f'\left( x \right)\) vanishes at

The number of points in \(\left( { - \infty \,\infty } \right),\) for which \({x^2} - x\sin x - \cos x = 0,\) is

Let $\overrightarrow{\mathrm{PR}}=3 \hat{i}+\hat{j}-2 \hat{k}$ and $ \overrightarrow{\mathrm{SQ}}=\hat{i}-3 \hat{j}-4 \hat{k}$ determine diagonals of a parallelogram $P Q R S$ and $\overrightarrow{\mathrm{PT}}=\hat{i}+2 \hat{j}+3 \hat{k}$ b...