JEE Advanced 2019 Paper 1 Offline

Three lines are given by \(r = \lambda \widehat i,\,\lambda \in R\), \(r = \mu (\widehat i + \widehat j),\,\mu \in R\) and \(r = v(\widehat i + \widehat j + \widehat k),\,v\, \in R\)Let the lines cut the plane x + y + z = 1 at the points ...

Let L1 and L2 denote the lines\(r = \widehat i + \lambda ( - \widehat i + 2\widehat j + 2\widehat k)\), \(\lambda\)\(\in\) R
and \(r = \mu (2\widehat i - \widehat j + 2\widehat k),\,\mu \in R\) respectively. If L3 is a line which is pe...

The area of the region{(x, y) : xy \(\le\) 8, 1 \(\le\) y \(\le\) x2} is

Let the point B be the reflection of the point A(2, 3) with respect to the line \(8x - 6y - 23 = 0\). Let \(\Gamma_{A}\) and \(\Gamma_{B}\) be circles of radii 2 and 1 with centres A and B respectively. Let T be a common tangent to the ci...

A line y = mx + 1 intersects the circle \({(x - 3)^2} + {(y + 2)^2}\) = 25 at the points P and Q. If the midpoint of the line segment PQ has x-coordinate \(- {3 \over 5}\), then which one of the following options is correct?

Let S be the set of all complex numbers z satisfying \(\left| {z - 2 + i} \right| \ge \sqrt 5\). If the complex number z0 is such that \({1 \over {\left| {{z_0} - 1} \right|}}\) is the maximum of the set $$\left\{ {{1 \over {\left| {{z_0} ...

Let \(\omega \ne 1\) be a cube root of unity. Then the minimum of the set \(\{ {\left| {a + b\omega + c{\omega ^2}} \right|^2}:a,b,c\) distinct non-zero integers} equals ..................

If \(I = {2 \over \pi }\int\limits_{ - \pi /4}^{\pi /4} {{{dx} \over {(1 + {e^{\sin x}})(2 - \cos 2x)}}}\), then 27I2 equals .................

Let \(\Gamma\) denote a curve y = y(x) which is in the first quadrant and let the point (1, 0) lie on it. Let the tangent to I` at a point P intersect the y-axis at YP. If PYP has length 1 for each point P on I`, then which of the followin...

Define the collections {E1, E2, E3, ...} of ellipses and {R1, R2, R3.....} of rectangles as follows :\({E_1}:{{{x^2}} \over 9} + {{{y^2}} \over 4} = 1\)R1 : rectangle of largest area, with sides parallel to the axes, inscribed in E1;En : el...

Let f : R \(\to\) R be given by$$f(x) = \left\{ {\matrix{
{{x^5} + 5{x^4} + 10{x^3} + 10{x^2} + 3x + 1,} & {x < 0;} \cr
{{x^2} - x + 1,} & {0 \le x < 1;} \cr
{{2 \over 3}{x^3} - 4{x^2} + 7x - {8 \over 3},} & {1 \le x < 3;} \...

Let \(M = \left[ {\matrix{ {{{\sin }^4}\theta } \cr {1 + {{\cos }^2}\theta } \cr } \matrix{ { - 1 - {{\sin }^2}\theta } \cr {{{\cos }^4}\theta } \cr } } \right] = \alpha I + \beta {M^{ - 1}}\),where \(\alpha\) = $$\...

Let \(M = \left[ {\matrix{ 0 & 1 & a \cr 1 & 2 & 3 \cr 3 & b & 1 \cr } } \right]\) andadj \(M = \left[ {\matrix{ { - 1} & 1 & { - 1} \cr 8 & { - 6} & 2 \cr { - 5} & 3 & { - 1} \cr } } \right]\)where a and b...

There are three bags B1, B2 and B3. The bag B1 contains 5 red and 5 green balls, B2 contains 3 red and 5 green balls, and B3 contains 5 red and 3 green balls. Bags B1, B2 and B3 have probabilities \({3 \over {10}}\), \({3 \over {10}}\) and ...

Let S be the sample space of all 3 \(\times\) 3 matrices with entries from the set {0, 1}. Let the events E1 and E2 be given byE1 = {A\(\in\)S : det A = 0} andE2 = {A\(\in\)S : sum of entries of A is 7}.If a matrix is chosen at random...

In a non-right-angled triangle \(\Delta\)PQR, let p, q, r denote the lengths of the sides opposite to the angles At P, Q, R respectively. The median from R meets the side PQ at S, the perpendicular from P meets the side QR at E, and RS and...

Let \(\alpha\) and \(\beta\) be the roots of\({x^2} - x - 1 = 0\), with \(\alpha\) > \(\beta\). For all positive integers n, define\({a_n} = {{{\alpha ^n} - {\beta ^n}} \over {\alpha - \beta }},\,n \ge 1\)$${b_1} = 1\,and\,{b_n} = {a_{...

Let AP(a; d) denote the set of all the terms of an infinite arithmetic progression with first term a and common difference d > 0. If \(AP(1;3) \cap AP(2;5) \cap AP(3;7)\) = AP(a ; d), then a + d equals ..............