JEE Advanced 2019 Paper 2 Offline

Three lines \({L_1}:r = \lambda \widehat i\), \(\lambda\) \(\in\) R,\({L_2}:r = \widehat k + \mu \widehat j\), \(\mu\) \(\in\) R and \({L_3}:r = \widehat i + \widehat j + v\widehat k\), v \(\in\) R are given.For which point(s) Q on ...

Let f : R \(\to\) R be given by\(f(x) = (x - 1)(x - 2)(x - 5)\). Define\(F(x) = \int\limits_0^x {f(t)dt}\), x > 0Then which of the following options is/are correct?

Let, \(f(x) = {{\sin \pi x} \over {{x^2}}}\), x > 0Let x1 < x2 < x3 < ... < xn < ... be all the points of local maximum of f and y1 < y2 < y3 < ... < yn < ... be all the points of local minimum of f.Then which of the following options is/ar...

The value of the integral \(\int\limits_0^{\pi /2} {{{3\sqrt {\cos \theta } } \over {{{(\sqrt {\cos \theta } + \sqrt {\sin \theta } )}^5}}}} d\theta\) equals ..............

The value of $${\sec ^{ - 1}}\left( \matrix{
{1 \over 4}\sum\limits_{k = 0}^{10} {\sec \left( {{{7\pi } \over {12}} + {{k\pi } \over 2}} \right)}
\sec \left( {{{7\pi } \over {12}} + {{(k + 1)\pi } \over 2}} \right) \hfill \cr} \right)...

Let f : R be a function. We say that f has PROPERTY 1 if \(\mathop {\lim }\limits_{h \to 0} {{f(h) - f(0)} \over {\sqrt {|h|} }}\) exists and is finite, and PROPERTY 2 if \(\mathop {\lim }\limits_{h \to 0} {{f(h) - f(0)} \over {{h^2}}}\) ex...

For \(a \in R,\,|a|\, > 1\), let $$\mathop {\lim }\limits_{n \to \infty } \left( {{{1 + \root 3 \of 2 + ...\root 3 \of n } \over {{n^{7/3}}\left( {{1 \over {{{(an + 1)}^2}}} + {1 \over {{{(an + 2)}^2}}} + ... + {1 \over {{{(an + n)}^2}}}} ...

Let x \(\in\) R and let \(P = \left[ {\matrix{ 1 & 1 & 1 \cr 0 & 2 & 2 \cr 0 & 0 & 3 \cr } } \right]\), \(Q = \left[ {\matrix{ 2 & x & x \cr 0 & 4 & 0 \cr x & x & 6 \cr } } \right]\) and R = PQP\(-\)1, w...

Suppose
det\(\left| {\matrix{ {\sum\limits_{k = 0}^n k } & {\sum\limits_{k = 0}^n {{}^n{C_k}{k^2}} } \cr {\sum\limits_{k = 0}^n {{}^n{C_k}.k} } & {\sum\limits_{k = 0}^n {{}^n{C_k}{3^k}} } \cr } } \right| = 0\)
holds for some p...

$${P_1} = I = \left[ {\matrix{
1 & 0 & 0 \cr
0 & 1 & 0 \cr
0 & 0 & 1 \cr

} } \right],\,{P_2} = \left[ {\matrix{
1 & 0 & 0 \cr
0 & 0 & 1 \cr
0 & 1 & 0 \cr

} } \right],\,{P_3} = \left[ {\matrix{
0 & 1 & 0 ...

Let the circle C1 : x2 + y2 = 9 and C2 : (x \(-\) 3)2 + (y \(-\) 4)2 = 16, intersect at the points X and Y. Suppose that another circle C3 : (x \(-\) h)2 + (y \(-\) k)2 = r2 satisfies the following conditions :(i) centre of C3 is collinear ...

Let the circles C1 : x2 + y2 = 9 and C2 : (x \(-\) 3)2 + (y \(-\) 4)2 = 16, intersect at the points X and Y. Suppose that another circle C3 : (x \(-\) h)2 + (y \(-\) k)2 = r2 satisfies the following conditions :(i) Centre of C3 is collinear...

Five persons A, B, C, D and E are seated in a circular arrangement. If each of them is given a hat of one of the three colours red, blue and green, then the number of ways of distributing the hats such that the persons seated in adjacent se...

Let |X| denote the number of elements in a set X. Let S = {1, 2, 3, 4, 5, 6} be a sample space, where each element is equally likely to occur. If A and B are independent events associated with S, then the number of ordered pairs (A, B) such...

For non-negative integers n, let$$f(n) = {{\sum\limits_{k = 0}^n {\sin \left( {{{k + 1} \over {n + 2}}\pi } \right)} \sin \left( {{{k + 2} \over {n + 2}}\pi } \right)} \over {\sum\limits_{k = 0}^n {{{\sin }^2}\left( {{{k + 1} \over {n + 2}}...

Let f(x) = sin(\(\pi\) cos x) and g(x) = cos(2\(\pi\) sin x) be two functions defined for x > 0. Define the following sets whose elements are written in the increasing order:X = {x : f(x) = 0}, Y = {x : f'(x) = 0}Z = {x : g(x) = 0}, W = {...

Let f(x) = sin(\(\pi\) cos x) and g(x) = cos(2\(\pi\) sin x) be two functions defined for x > 0. Define the following sets whose elements are written in the increasing order :X = {x : f(x) = 0}, Y = {x : f'(x) = 0}Z = {x : g(x) = 0}, W = ...

Let \(\overrightarrow a = 2\widehat i + \widehat j - \widehat k\) and \(\overrightarrow b = \widehat i + 2\widehat j + \widehat k\) be two vectors. Consider a vector c = \(\alpha\)\(\overrightarrow a\) + \(\beta\)\(\overrightarrow b\), $$...