JEE Advanced 2013 Paper 2 Offline

Two lines \({L_1}:x = 5,{y \over {3 - \alpha }} = {z \over { - 2}}\) and \({L_2}:x = \alpha ,{y \over { - 1}} = {z \over {2 - \alpha }}\) are coplanar. Then \(\alpha\) can take value(s)

Consider the lines \({L_1}:{{x - 1} \over 2} = {y \over { - 1}} = {{z + 3} \over 1},{L_2} : {{x - 4} \over 1} = {{y + 3} \over 1} = {{z + 3} \over 2}\) and the planes \({P_1}:7x + y + 2z = 3,{P_2} = 3x + 5y - 6z = 4.\) Let \(ax+by+cz=d\) be...

The function \(f(x) = 2\left| x \right| + \left| {x + 2} \right| - \left| {\left| {x + 2} \right| - 2\left| x \right|} \right|\) has a local minimum or a local maximum at x =

Let \(f:\left[ {0,1} \right] \to R\) (the set of all real numbers) be a function. Suppose the function \(f\) is twice differentiable, \(f(0) = f(1)=0\) and satisfies $$f''\left( x \right) - 2f'\left( x \right) + f\left( x \right) \ge .{e^x}...

Let \(f:\left[ {0,1} \right] \to R\) (the set of all real numbers) be a function. Suppose the function \(f\) is twice differentiable, \(f(0) = f(1)=0\) and satisfies $$f''\left( x \right) - 2f'\left( x \right) + f\left( x \right) \ge .{e^x}...

Circle (s) touching x-axis at a distance 3 from the origin and having an intercept of length \(2\sqrt 7\) on y-axis is (are)

Let $\omega=\frac{\sqrt{3}+i}{2}$ and $P=\left\{\omega^n: n=1,2,3, \ldots\right\}$. Further
$\mathrm{H}_1=\left\{z \in \mathrm{C}: \operatorname{Re} z<\frac{1}{2}\right\}$ and
$\mathrm{H}_2=\left\{z \in \mathrm{C}: \operatorname{Re} z<\fr...

Let \(S = {S_1} \cap {S_2} \cap {S_3}\), where \({S_1} = \left\{ {z \in C:\left| z \right| < 4} \right\},{S_2} = \left\{ {z \in C:{\mathop{\rm Im}\nolimits} \left[ {{{z - 1 + \sqrt 3 i} \over {1 - \sqrt 3 i}}} \right] > 0} \right\}\) and $$...

Let \(S = {S_1} \cap {S_2} \cap {S_3}\), where \({S_1} = \left\{ {z \in C:\left| z \right| < 4} \right\},{S_2} = \left\{ {z \in C:{\mathop{\rm Im}\nolimits} \left[ {{{z - 1 + \sqrt 3 i} \over {1 - \sqrt 3 i}}} \right] > 0} \right\}\) and $$...

Match List \(I\) with List \(II\) and select the correct answer using the code given below the lists:
List \(I\)
\(P.\)\(\,\,\,\,\,\) $${\left( {{1 \over {{y^2}}}{{\left( {{{\cos \left( {{{\tan }^{ - 1}}y} \right) + y\sin \left( {{{\tan }...

\(a \in R\) (the set of all real numbers), a \(\ne\) \(-\)1, \(\mathop {\lim }\limits_{n \to \infty } {{({1^a} + {2^a} + ... + {n^a})} \over {{{(n + 1)}^{a - 1}}[(na + 1) + (na + 2) + ... + (na + n)]}} = {1 \over {60}}\), Then a = ?

Let \(\omega\) be a complex cube root of unity with \(\omega\) \(\ne\) 1 and P = [pij] be a n \(\times\) n matrix with pij = \(\omega\)i + j. Then P2 \(\ne\) 0, when n = ?

Let \(PQ\) be a focal chord of the parabola \({y^2} = 4ax\). The tangents to the parabola at \(P\) and \(Q\) meet at a point lying on the line \(y=2x+a\), \(a>0\).
Length of chord \(PQ\) is

A line \(L:y=mx+3\) meets \(y\)-axis at R\((0, 3)\) and the arc of the parabola \({y^2} = 16x,\) \(0 \le y \le 6\) at the point \(F\left( {{x_0},{y_0}} \right)\). The tangent to the parabola at \(F\left( {{x_0},{y_0}} \right)\) intersects t...

Let \(PQ\) be a focal chord of the parabola \({y^2} = 4ax\). The tangents to the parabola at \(P\) and \(Q\) meet at a point lying on the line \(y=2x+a\), \(a>0\).
If chord \(PQ\) subtends an angle \(\theta\) at the vertex of $${y^2} = 4a...

A box \({B_1}\) contains \(1\) white ball, \(3\) red balls and \(2\) black balls. Another box \({B_2}\) contains \(2\) white balls, \(3\) red balls and \(4\) black balls. A third box \({B_3}\) contains \(3\) white balls, \(4\) red balls and...

A box \({B_1}\) contains \(1\) white ball, \(3\) red balls and \(2\) black balls. Another box \({B_2}\) contains \(2\) white balls, \(3\) red balls and \(4\) black balls. A third box \({B_3}\) contains \(3\) white balls, \(4\) red balls and...

In a triangle \(PQR\), \(P\) is the largest angle and \(\cos P = {1 \over 3}\). Further the incircle of the triangle touches the sides \(PQ\), \(QR\) and \(RP\) at \(N,L\) and \(M\) respectively, such that the lengths of \(PN, QL\) and $$RM...

If \({3^x}\, = \,{4^{x - 1}},\) then \(x\, =\)

match List \(I\) with List \(II\) and select the correct answer using the code given below the lists:
\(\,\,\,\,\) \(\,\,\,\,\) \(\,\,\,\,\) List \(I\)
(P.)\(\,\,\,\,\) Volume of parallelopiped determined by vectors $$\overrightarrow a ,\...