JEE Main 2019 (Online) 8th April Evening Slot

If a point R(4, y, z) lies on the line segment joining
the points P(2, –3, 4) and Q(8, 0, 10), then the
distance of R from the origin is :

The vector equation of the plane through the line
of intersection of the planes x + y + z = 1 and 2x
+ 3y+ 4z = 5 which is perpendicular to the plane
x – y + z = 0 is :

The height of a right circular cylinder of maximum
volume inscribed in a sphere of radius 3 is

Given that the slope of the tangent to a curve y
= y(x) at any point (x, y) is
\(2y \over x^2\). If the curve passes through the centre of the circle x2 + y2 – 2x – 2y = 0, then its equation is :

Let S(\(\alpha\)) = {(x, y) : y2
\(\le\) x, 0 \(\le\) x \(\le\) \(\alpha\)} and A(\(\alpha\))
is area of the region S(\(\alpha\)). If for a \(\lambda\), 0 < \(\lambda\) < 4,
A(\(\lambda\)) : A(4) = 2 : 5, then \(\lambda\) e...

If the fourth term in the binomial expansion of
\({\left( {\sqrt {{x^{\left( {{1 \over {1 + {{\log }_{10}}x}}} \right)}}} + {x^{{1 \over {12}}}}} \right)^6}\) is equal to 200, and x > 1,
then the value of x is :

The tangent and the normal lines at the point
( \(\sqrt 3\), 1) to the circle x2
+ y2 = 4 and the x-axis form a triangle. The area of this triangle (in
square units) is :

If \(z = {{\sqrt 3 } \over 2} + {i \over 2}\left( {i = \sqrt { - 1} } \right)\),
then (1 + iz + z5 + iz8)9 is equal to :

Let \(f(x) = \int\limits_0^x {g(t)dt}\) where g is a non-zero even
function. If ƒ(x + 5) = g(x), then \(\int\limits_0^x {f(t)dt}\) equals-

If ƒ(1) = 1, ƒ'(1) = 3, then the derivative of
ƒ(ƒ(ƒ(x))) + (ƒ(x))2
at x = 1 is :

In an ellipse, with centre at the origin, if the
difference of the lengths of major axis and minor
axis is 10 and one of the foci is at (0,5\(\sqrt 3\)), then
the length of its latus rectum is :

Let ƒ(x) = ax
(a > 0) be written as
ƒ(x) = ƒ1
(x) + ƒ2
(x), where ƒ1
(x) is an even
function of ƒ2
(x) is an odd function. Then
ƒ1
(x + y) + ƒ1
(x – y) equals

Two vertical poles of heights, 20 m and 80 m stand
a part on a horizontal plane. The height (in meters)
of the point of intersection of the lines joining the
top of each pole to the foot of the other, from this
horizontal plane is :

If the eccentricity of the standard hyperbola
passing through the point (4,6) is 2, then the
equation of the tangent to the hyperbola at (4,6)
is :

If \(\int {{{dx} \over {{x^3}{{(1 + {x^6})}^{2/3}}}} = xf(x){{(1 + {x^6})}^{{1 \over 3}}} + C}\)
where C is a constant of integration, then the
function ƒ(x) is equal to

Let ƒ : R \(\to\) R be a differentiable function
satisfying ƒ'(3) + ƒ'(2) = 0.
Then \(\mathop {\lim }\limits_{x \to 0} {\left( {{{1 + f(3 + x) - f(3)} \over {1 + f(2 - x) - f(2)}}} \right)^{{1 \over x}}}\) is equal to

Let ƒ : [–1,3] \(\to\) R be defined as
$$f(x) = \left\{ {\matrix{
{\left| x \right| + \left[ x \right]} & , & { - 1 \le x < 1} \cr
{x + \left| x \right|} & , & {1 \le x < 2} \cr
{x + \left[ x \right]} & , & {2 \le x \le 3} \...

Which one of the following statements is not a
tautology?

Let the number 2,b,c be in an A.P. and
A = \(\left[ {\matrix{ 1 & 1 & 1 \cr 2 & b & c \cr 4 & {{b^2}} & {{c^2}} \cr } } \right]\). If det(A) \(\in\) [2, 16], then c
lies in the interval :

The tangent to the parabola y2
= 4x at the point
where it intersects the circle x2
+ y2
= 5 in the
first quadrant, passes through the point :

The number of four-digit numbers strictly greater
than 4321 that can be formed using the digits
0,1,2,3,4,5 (repetition of digits is allowed) is :

The minimum number of times one has to toss a
fair coin so that the probability of observing at least
one head is at least 90% is :

If the lengths of the sides of a triangle are in A.P.
and the greatest angle is double the smallest, then
a ratio of lengths of the sides of this triangle is :

The number of integral values of m for which the
equation
(1 + m2
)x2
– 2(1 + 3m)x + (1 + 8m) = 0
has no real root is :

The sum
\(\sum\limits_{k = 1}^{20} {k{1 \over {{2^k}}}}\) is equal to

If three distinct numbers a, b, c are in G.P. and the
equations ax2
+ 2bx + c = 0 and
dx2
+ 2ex + ƒ = 0 have a common root, then
which one of the following statements is
correct?

A student scores the following marks in five tests
:
45, 54, 41, 57, 43.
His score is not known for the
sixth test. If the mean score is 48 in the six tests,
then the standard deviation of the marks in six tests
is

If the system of linear equations
x – 2y + kz = 1
2x + y + z = 2
3x – y – kz = 3
has a solution (x,y,z), z \(\ne\) 0, then (x,y) lies on
the straight line whose equation is :

Suppose that the points (h,k), (1,2) and (–3,4) lie
on the line L1
. If a line L2
passing through the points
(h,k) and (4,3) is perpendicular to L1
, then
\(k \over h\)
equals :

Let \(\mathop a\limits^ \to = 3\mathop i\limits^ \wedge + 2\mathop j\limits^ \wedge + x\mathop k\limits^ \wedge\) and \(\mathop b\limits^ \to = \mathop i\limits^ \wedge - \mathop j\limits^ \wedge + \mathop k\limits^ \wedge\)...