JEE Main 2019 (Online) 8th April Morning Slot

The magnitude of the projection of the vector
\(\mathop {2i}\limits^ \wedge + \mathop {3j}\limits^ \wedge + \mathop k\limits^ \wedge\) on the vector perpendicular to the plane
containing the vectors $$\mathop {i}\limits^ \wedge + \m...

The length of the perpendicular from the point
(2, –1, 4) on the straight line,
\({{x + 3} \over {10}}\)= \({{y - 2} \over {-7}}\) = \({{z} \over {1}}\)
is :

The equation of a plane containing the line of
intersection of the planes 2x – y – 4 = 0 and
y + 2z – 4 = 0 and passing through the point
(1, 1, 0) is :

If S1 and S2 are respectively the sets of local
minimum and local maximum points of the function,
ƒ(x) = 9x4 + 12x3 – 36x2 + 25, x \(\in\) R,
then :

Let ƒ : [0, 2] \(\to\) R be a twice differentiable
function such that ƒ''(x) > 0, for all x \(\in\) (0, 2).
If \(\phi\)(x) = ƒ(x) + ƒ(2 – x), then \(\phi\) is :

The area (in sq. units) of the region
A = { (x, y) \(\in\) R × R|  0 \(\le\) x \(\le\) 3, 0 \(\le\) y \(\le\) 4,
y \(\le\) x2 + 3x} is :

The sum of the co-efficients of all even
degree terms in x in the expansion of
\({\left( {x + \sqrt {{x^3} - 1} } \right)^6}\) + \({\left( {x - \sqrt {{x^3} - 1} } \right)^6}\), (x > 1) is equal to:

The sum of the series
2.20C0
+ 5.20C1 + 8.20C2 + 11.20C3 + ... +62.20C20 is equal to :

The sum of the squares of the lengths of the chords
intercepted on the circle, x2 + y2 = 16, by the lines,
x + y = n, n \(\in\) N, where N is the set of all natural
numbers, is :

If \(\alpha\) and \(\beta\) be the roots of the equation
x2 – 2x + 2 = 0, then the least value of n for which \({\left( {{\alpha \over \beta }} \right)^n} = 1\) is :

If \(f(x) = {{2 - x\cos x} \over {2 + x\cos x}}\) and g(x) = logex, (x > 0) then
the value of integral \(\int\limits_{ - {\pi \over 4}}^{{\pi \over 4}} {g\left( {f\left( x \right)} \right)dx{\rm{ }}}\) is

Let y = y(x) be the solution of the differential equation, \({({x^2} + 1)^2}{{dy} \over {dx}} + 2x({x^2} + 1)y = 1\) such
that y(0) = 0. If \(\sqrt ay(1)\) = \(\pi \over 32\) , then the value of
'a' is :

If \(2y = {\left( {{{\cot }^{ - 1}}\left( {{{\sqrt 3 \cos x + \sin x} \over {\cos x - \sqrt 3 \sin x}}} \right)} \right)^2}\),
x \(\in\) \(\left( {0,{\pi \over 2}} \right)\) then \(dy \over dx\) is equal to:

If the tangents on the ellipse 4x2 + y2 = 8 at the
points (1, 2) and (a, b) are perpendicular to each
other, then a2 is equal to :

If \(f(x) = {\log _e}\left( {{{1 - x} \over {1 + x}}} \right)\), \(\left| x \right| < 1\) then \(f\left( {{{2x} \over {1 + {x^2}}}} \right)\) is equal to

\(\int {{{\sin {{5x} \over 2}} \over {\sin {x \over 2}}}dx}\) is equal to
(where c is a constant of integration)

If \(\alpha = {\cos ^{ - 1}}\left( {{3 \over 5}} \right)\), \(\beta = {\tan ^{ - 1}}\left( {{1 \over 3}} \right)\) where \(0 < \alpha ,\beta < {\pi \over 2}\) , then \(\alpha\) - \(\beta\) is equal to :

\(\mathop {\lim }\limits_{x \to 0} {{{{\sin }^2}x} \over {\sqrt 2 - \sqrt {1 + \cos x} }}\) equals:

The contrapositive of the statement "If you are
born in India, then you are a citizen of India", is :

Let \(A = \left( {\matrix{ {\cos \alpha } & { - \sin \alpha } \cr {\sin \alpha } & {\cos \alpha } \cr } } \right)\), (\(\alpha\) \(\in\) R) such that $${A^{32}} = \left( {\matrix{
0 & { - 1} \cr
1 & 0 \cr

} } \rig...

The greatest value of c \(\in\) R for which the system
of linear equations
x – cy – cz = 0
cx – y + cz = 0
cx + cy – z = 0
has a non-trivial solution, is :

The shortest distance between the line y = x and
the curve y2 = x – 2 is :

All possible numbers are formed using the digits
1, 1, 2, 2, 2, 2, 3, 4, 4 taken all at a time. The number
of such numbers in which the odd digits occupy
even places is :

Let A and B be two non-null events such that
A \(\subset\) B . Then, which of the following statements
is always correct?

The sum of the solutions of the equation
\(\left| {\sqrt x - 2} \right| + \sqrt x \left( {\sqrt x - 4} \right) + 2 = 0\)
(x > 0) is equal to:

The sum of all natural numbers 'n' such that
100 < n < 200 and H.C.F. (91, n) > 1 is :

The mean and variance of seven observations are
8 and 16, respectively. If 5 of the observations are
2, 4, 10, 12, 14, then the product of the remaining
two observations is :

Let O(0, 0) and A(0, 1) be two fixed points. Then
the locus of a point P such that the perimeter of
\(\Delta\)AOP is 4, is :

A point on the straight line, 3x + 5y = 15 which is equidistant from the coordinate axes will lie only
in :

If cos(\(\alpha\) + \(\beta\)) = 3/5 ,sin ( \(\alpha\) - \(\beta\)) = 5/13 and
0 < \(\alpha , \beta\) < \(\pi \over 4\), then tan(2\(\alpha\)) is equal to :