JEE Main 2019 (Online) 9th January Morning Slot

The plane through the intersection of the planes x + y + z = 1 and 2x + 3y – z + 4 = 0 and parallel to y-axis
also passes through the point :

The equation of the line passing through (–4, 3, 1), parallel
to the plane x + 2y – z – 5 = 0 and intersecting
the line \({{x + 1} \over { - 3}} = {{y - 3} \over 2} = {{z - 2} \over { - 1}}\) is :

The maximum volume (in cu.m) of the right circular cone having slant height 3 m is :

The area (in sq. units) bounded by the parabolae y = x2 – 1, the tangent at the point (2, 3) to it and the y-axis is :

If the fractional part of the number \(\left\{ {{{{2^{403}}} \over {15}}} \right\} is \, {k \over {15}}\), then k is equal to :

Three circles of radii a, b, c (a < b < c) touch each other externally. If they have x-axis as a common tangent, then :

Let \(\alpha\) and \(\beta\) be two roots of the equation x2 + 2x + 2 = 0 , then \(\alpha ^{15}\) + \(\beta ^{15}\) is equal to :

Let
A = \(\left\{ {\theta \in \left( { - {\pi \over 2},\pi } \right):{{3 + 2i\sin \theta } \over {1 - 2i\sin \theta }}is\,purely\,imaginary} \right\}\)
. Then the sum of the elements in A is :

The value of \(\int\limits_0^\pi {{{\left| {\cos x} \right|}^3}} \,dx\) is :

If y = y(x) is the solution of the differential equation,
x\(dy \over dx\) + 2y = x2, satisfying y(1) = 1, then y(\(1\over2\)) is equal
to :

For \(x \in R - \left\{ {0,1} \right\}\), Let f1(x) = \(1\over x\), f2 (x) = 1 – x and f3 (x) = \(1 \over {1 - x}\)
be three given
functions. If a function, J(x) satisfies
(f2 o J o f1) (x) = f3 (x) then J(x) is equal to :

Let \(0 < \theta < {\pi \over 2}\). If the eccentricity of the
hyperbola \({{{x^2}} \over {{{\cos }^2}\theta }} - {{{y^2}} \over {{{\sin }^2}\theta }}\) = 1 is greater
than 2, then the length of
its latus rectum lies in the interval :

For x2 \(\ne\) n\(\pi\) + 1, n \(\in\) N (the set of natural numbers), the integral \(\int {x\sqrt {{{2\sin ({x^2} - 1) - \sin 2({x^2} - 1)} \over {2\sin ({x^2} - 1) + \sin 2({x^2} - 1)}}} dx}\) is equal to :
(where c is a constant o...

If \({\cos ^{ - 1}}\left( {{2 \over {3x}}} \right) + {\cos ^{ - 1}}\left( {{3 \over {4x}}} \right) = {\pi \over 2}\) (x > \(3 \over 4\)), then x is equal to :

Let f : R \(\to\) R be a function defined as
\(f(x) = \left\{ {\matrix{ 5 & ; & {x \le 1} \cr {a + bx} & ; & {1 < x < 3} \cr {b + 5x} & ; & {3 \le x < 5} \cr {30} & ; & {x \ge 5} \cr } } \right.\)
Then, f is

\(\mathop {\lim }\limits_{y \to 0} {{\sqrt {1 + \sqrt {1 + {y^4}} } - \sqrt 2 } \over {{y^4}}}\)

If the Boolean expression
(p \(\oplus\) q) \(\wedge\) (~ p \(\odot\) q) is equivalent
to p \(\wedge\) q, where \(\oplus , \odot \in \left\{ { \wedge , \vee } \right\}\), then the
ordered pair \(\left( { \oplus , \odot } \right)\)...

If \(A = \left[ {\matrix{ {\cos \theta } & { - \sin \theta } \cr {\sin \theta } & {\cos \theta } \cr } } \right]\), then the matrix A–50 when \(\theta\) = \(\pi \over 12\), is equal to :

The system of linear equations
x + y + z = 2
2x + 3y + 2z = 5
2x + 3y + (a2 – 1) z = a + 1 then

Equation of a common tangent to the circle, x2 + y2 – 6x = 0 and the parabola, y2 = 4x is :

If \(\theta\) denotes the acute angle between the curves, y = 10 – x2 and y = 2 + x2 at a point of their intersection, the |tan \(\theta\)| is equal to :

Axis of a parabola lies along x-axis. If its vertex and focus are at distances 2 and 4 respectively from the
origin, on the positive x-axis then which of the following points does not lie on it?

Consider a class of 5 girls and 7 boys. The number of different teams consisting of 2 girls and 3 boys that can
be formed from this class, if there are two specific boys A and B, who refuse to be the members of the same
team, is :

Two cards are drawn successively with replacement from a well-shuffled deck of 52 cards. Let X denote the random variable of number of aces obtained in the two drawn cards. Then P(X = 1) + P (X = 2) equals :

Let \({a_1},{a_2},.......,{a_{30}}\) be an A.P.,
\(S = \sum\limits_{i = 1}^{30} {{a_i}}\) and \(T = \sum\limits_{i = 1}^{15} {{a_{\left( {2i - 1} \right)}}}\).
If \(a_5\) = 27 and S - 2T = 75, then \(a_{10}\) is equal to :

If a, b, c be three distinct real numbers in G.P. and a + b + c = xb , then x cannot be

5 students of a class have an average height 150 cm and variance 18 cm2. A new student, whose height is 156 cm, joined them. The variance (in cm2) of the height of these six students is :

Consider the set of all lines px + qy + r = 0 such that 3p + 2q + 4r = 0. Which one of the following statements
is true?

For any \(\theta \in \left( {{\pi \over 4},{\pi \over 2}} \right)\), the expression
\(3{(\cos \theta - \sin \theta )^4}\)\(+ 6{(\sin \theta + \cos \theta )^2} + 4{\sin ^6}\theta\)
equals :

Let \(\overrightarrow a\) = \(\widehat i - \widehat j\), \(\overrightarrow b\) = \(\widehat i + \widehat j + \widehat k\) and \(\overrightarrow c\)
be a vector such that \(\overrightarrow a\) × \(\overrightarrow c\) + $$\overrightarro...