JEE Main 2019 (Online) 12th January Morning Slot

A tetrahedron has vertices P(1, 2, 1), Q(2, 1, 3), R(–1, 1, 2) and O(0, 0, 0). The angle between the faces OPQ and PQR is :

The perpendicular distance from the origin to the plane containing the two lines, \({{x + 2} \over 3} = {{y - 2} \over 5} = {{z + 5} \over 7}\) and \({{x - 1} \over 1} = {{y - 4} \over 4} = {{z + 4} \over 7},\) is :

The area (in sq. units) of the region bounded by the parabola, y = x2 + 2 and the lines, y = x + 1, x = 0 and x = 3, is

A ratio of the 5th term from the beginning to the 5th term from the end in the binomial expansion of \({\left( {{2^{1/3}} + {1 \over {2{{\left( 3 \right)}^{1/3}}}}} \right)^{10}}\) is :

Let C1 and C2 be the centres of the circles x2 + y2 – 2x – 2y – 2 = 0 and x2 + y2 – 6x – 6y + 14 = 0 respectively. If P and Q are the points of intersection of these circles, then the area (in sq. units) of the quadrilateral PC1QC2 is :

If a variable line, 3x + 4y – \(\lambda\) = 0 is such that the two circles x2 + y2 – 2x – 2y + 1 = 0 and x2 + y2 – 18x – 2y + 78 = 0 are on its opposite sides, then the set of all values of \(\lambda\) is the interval :

If \({{z - \alpha } \over {z + \alpha }}\left( {\alpha \in R} \right)\) is a purely imaginary number and | z | = 2, then a value of \(\alpha\) is :

Let f and g be continuous functions on [0, a] such that f(x) = f(a – x) and g(x) + g(a – x) = 4, then \(\int\limits_0^a \,\)f(x) g(x) dx is equal to :

Let y = y(x) be the solution of the differential equation, x\({{dy} \over {dx}}\) + y = x loge x, (x > 1). If 2y(2) = loge 4 \(-\) 1, then y(e) is equal to :

For x > 1, if (2x)2y = 4e2x\(-\)2y,
then (1 + loge 2x)2 \({{dy} \over {dx}}\) is equal to :

If the vertices of a hyperbola be at (–2, 0) and (2, 0) and one of its foci be at (–3, 0), then which one of the following points does not lie on this hyperbola?

The integral \(\int \,\)cos(loge x) dx is equal to : (where C is a constant of integration)

Considering only the principal values of inverse functions, the set
A = { x \(\ge\) 0: tan\(-\)1(2x) + tan\(-\)1(3x) = \({\pi \over 4}\)}

Let S be the set of all points in (–\(\pi\), \(\pi\)) at which the function, f(x) = min{sin x, cos x} is not differentiable. Then S is a subset of which of the following ?

\(\mathop {\lim }\limits_{x \to \pi /4} {{{{\cot }^3}x - \tan x} \over {\cos \left( {x + {\pi \over 4}} \right)}}\) is :

The Boolean expression ((p \(\wedge\) q) \(\vee\) (p \(\vee\) \(\sim\) q)) \(\wedge\) (\(\sim\) p \(\wedge\) \(\sim\) q) is equivalent to :

An ordered pair (\(\alpha\), \(\beta\)) for which the system of linear equations
(1 + \(\alpha\)) x + \(\beta\)y + z = 2
\(\alpha\)x + (1 + \(\beta\))y + z = 3
\(\alpha\)x + \(\beta\)y + 2z = 2
has a unique solution, is :

Let P = \(\left[ {\matrix{ 1 & 0 & 0 \cr 3 & 1 & 0 \cr 9 & 3 & 1 \cr } } \right]\) and Q = [qij] be two 3 \(\times\) 3 matrices such that Q – P5 = I3.
Then \({{{q_{21}} + {q_{31}}} \over {{q_{32}}}}\) is equal to :

The maximum area (in sq. units) of a rectangle having its base on the x-axis and its other two vertices on the parabola, y = 12 – x2 such that the rectangle lies inside the parabola, is :

Let P(4, –4) and Q(9, 6) be two points on the parabola, y2 = 4x and let x be any point on the arc POQ of this parabola, where O is the vertex of this parabola, such that the area of \(\Delta\)PXQ is maximum. Then this maximum area (in sq. ...

Consider three boxes, each containing, 10 balls labelled 1, 2, … , 10. Suppose one ball is randomly drawn from each of the boxes. Denote by ni, the label of the ball drawn from the ith box, (i = 1, 2, 3). Then, the number of ways in which t...

In a random experiment, a fair die is rolled until two fours are obtained in succession. The probability that the experiment will end in the fifth throw of the die is equal to :

If \(\lambda\) be the ratio of the roots of the quadratic equation in x, 3m2x2 + m(m – 4)x + 2 = 0, then the least value of m for which \(\lambda + {1 \over \lambda } = 1,\) is

The product of three consecutive terms of a G.P. is 512. If 4 is added to each of the first and the second of these terms, the three terms now form an A.P. Then the sum of the original three terms of the given G.P. is :

Let  Sk = \({{1 + 2 + 3 + .... + k} \over k}.\) If   \(S_1^2 + S_2^2 + .....\, + S_{10}^2 = {5 \over {12}}\)A,  then A is equal to :

Let S = {1, 2, 3, … , 100}. The number of non-empty subsets A of S such that the product of elements in A is even is :

If the sum of the deviations of 50 observations from 30 is 50, then the mean of these observations is :

If the straight line, 2x – 3y + 17 = 0 is perpendicular to the line passing through the points (7, 17) and (15, \(\beta\)), then \(\beta\) equals :

The maximum value of 3cos\(\theta\) + 5sin \(\left( {\theta - {\pi \over 6}} \right)\) for any real value of \(\theta\) is :

The sum of the distinct real values of \(\mu\), for which the vectors, \(\mu \widehat i + \widehat j + \widehat k,\)   \(\widehat i + \mu \widehat j + \widehat k,\)   \(\widehat i + \widehat j + \mu \widehat k\)  are co-planar, is :