JEE Main 2019 (Online) 10th April Morning Slot

If the length of the perpendicular from the point (\(\beta\), 0, \(\beta\)) (\(\beta\) \(\ne\) 0) to the line,
\({x \over 1} = {{y - 1} \over 0} = {{z + 1} \over { - 1}}\) is \(\sqrt {{3 \over 2}}\), then
\(\beta\) is equal to :

If Q(0, –1, –3) is the image of the point P in the plane 3x – y + 4z = 2 and R is the point (3, –1, –2), then the
area (in sq. units) of \(\Delta\)PQR is :

If the coefficients of x2
and x3
are both zero, in the expansion of the expression (1 + ax + bx2
) (1 – 3x)15 in
powers of x, then the ordered pair (a,b) is equal to :

If the circles x2
+ y2
+ 5Kx + 2y + K = 0 and 2(x2
+ y2) + 2Kx + 3y –1 = 0, (K\(\in\)R), intersect at the points
P and Q, then the line 4x + 5y – K = 0 passes through P and Q, for :

The line x = y touches a circle at the point (1,1). If the circle also passes through the point (1, – 3), then its
radius is :

If a > 0 and z = \({{{{\left( {1 + i} \right)}^2}} \over {a - i}}\), has magnitude \(\sqrt {{2 \over 5}}\), then \(\overline z\) is equal to :

The value of \(\int\limits_0^{2\pi } {\left[ {\sin 2x\left( {1 + \cos 3x} \right)} \right]} dx\),
where [t] denotes the greatest integer function is :

\(\mathop {\lim }\limits_{n \to \infty } \left( {{{{{(n + 1)}^{1/3}}} \over {{n^{4/3}}}} + {{{{(n + 2)}^{1/3}}} \over {{n^{4/3}}}} + ....... + {{{{(2n)}^{1/3}}} \over {{n^{4/3}}}}} \right)\)
is equal to :

If y = y(x) is the solution of the differential equation
\({{dy} \over {dx}} = \left( {\tan x - y} \right){\sec ^2}x\), \(x \in \left( { - {\pi \over 2},{\pi \over 2}} \right)\),
such that y (0) = 0, then $$y\left( { - {\pi \over 4}} \r...

If the line x – 2y = 12 is tangent to the ellipse \({{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1\) at the point \(\left( {3, - {9 \over 2}} \right)\) , then the length of the latus
rectum of the ellipse is :

Let f(x) = x2
, x \(\in\) R. For any A \(\subseteq\) R, define g (A) = { x \(\in\) R : f(x) \(\in\) A}. If S = [0,4], then which one of the
following statements is not true ?

Let f(x) = ex – x and g(x) = x2 – x, \(\forall\) x \(\in\) R. Then the set of all x \(\in\) R, where the function h(x) = (fog) (x) is increasing, is :

ABC is a triangular park with AB = AC = 100 metres. A vertical tower is situated at the mid-point of BC. If
the angles of elevation of the top of the tower at A and B are cot–1
(3\(\sqrt 2\) ) and cosec–1
(2\(\sqrt 2\) ) respectively,
th...

If a directrix of a hyperbola centred at the origin and passing through the point (4, –2\(\sqrt 3\) ) is 5x = 4\(\sqrt 5\) and
its eccentricity is e, then :

If \(\int {{{dx} \over {{{\left( {{x^2} - 2x + 10} \right)}^2}}}} = A\left( {{{\tan }^{ - 1}}\left( {{{x - 1} \over 3}} \right) + {{f\left( x \right)} \over {{x^2} - 2x + 10}}} \right) + C\)
where C is a constant of integration then :

If \(\mathop {\lim }\limits_{x \to 1} {{{x^4} - 1} \over {x - 1}} = \mathop {\lim }\limits_{x \to k} {{{x^3} - {k^3}} \over {{x^2} - {k^2}}}\), then k is :

Let f : R \(\to\) R be differentiable at c \(\in\) R and f(c) = 0. If g(x) = |f(x)| , then at x = c, g is :

If$$f(x) = \left\{ {\matrix{
{{{\sin (p + 1)x + \sin x} \over x}} & {,x < 0} \cr
q & {,x = 0} \cr
{{{\sqrt {x + {x^2}} - \sqrt x } \over {{x^{{\raise0.5ex\hbox{$\scriptstyle 3$}
\kern-0.1em/\kern-0.15em
\lower0.25ex\hbox{$\scr...

Which one of the following Boolean expressions is a tautology?

If \({\Delta _1} = \left| {\matrix{ x & {\sin \theta } & {\cos \theta } \cr { - \sin \theta } & { - x} & 1 \cr {\cos \theta } & 1 & x \cr } } \right|\) and
$${\Delta _2} = \left| {\matrix{
x & {\sin 2\theta } & {\cos 2\...

If the system of linear equations
x + y + z = 5
x + 2y + 2z = 6
x + 3y + \(\lambda\)z = \(\mu\), (\(\lambda\), \(\mu\) \(\in\) R), has infinitely many solutions, then the value of \(\lambda\) + \(\mu\) is :

The number of 6 digit numbers that can be formed using the digits 0, 1, 2, 5, 7 and 9 which are divisible by
11 and no digit is repeated is :

Assume that each born child is equally likely to be a boy or a girl. If two families have two children each,
then the conditional probability that all children are girls given that at least two are girls is :

All the pairs (x, y) that satisfy the inequality
\({2^{\sqrt {{{\sin }^2}x - 2\sin x + 5} }}.{1 \over {{4^{{{\sin }^2}y}}}} \le 1\)
also satisfy the equation

If \(\alpha\) and \(\beta\) are the roots of the quadratic equation,
x2 + x sin \(\theta\) - 2 sin \(\theta\) = 0, \(\theta \in \left( {0,{\pi \over 2}} \right)\), then
$${{{\alpha ^{12}} + {\beta ^{12}}} \over {\left( {{\alpha ^{ - ...

The sum
\({{3 \times {1^3}} \over {{1^3}}} + {{5 \times ({1^3} + {2^3})} \over {{1^2} + {2^2}}} + {{7 \times \left( {{1^3} + {2^3} + {3^3}} \right)} \over {{1^2} + {2^2} + {3^2}}} + .....\) upto 10 terms is:

If a1, a2, a3, ............... an are in A.P. and a1 + a4 + a7 + ........... + a16 = 114, then a1 + a6 + a11 + a16 is equal to :

If for some x \(\in\) R, the frequency distribution of the marks obtained by 20 students in a test is :

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.tg td{font-family:Arial, sans-serif;font-size:14px;padding:10px 5px;border-style:s...

The region represented by| x – y | \(\le\) 2 and | x + y| \(\le\) 2 is bounded by a :

Let A (3, 0, –1), B(2, 10, 6) and C(1, 2, 1) be the vertices of a triangle and M be the midpoint of AC. If G
divides BM in the ratio, 2 : 1, then cos (\(\angle\)GOA) (O being the origin) is equal to :