JEE Main 2020 (Online) 9th January Evening Slot

If the distance between the plane,
23x – 10y – 2z + 48 = 0 and the plane
containing the lines
\({{x + 1} \over 2} = {{y - 3} \over 4} = {{z + 1} \over 3}\) and
$${{x + 3} \over 2} = {{y + 2} \over 6} = {{z - 1} \over \lambda }\left( {\lambd...

Given : \(f(x) = \left\{ {\matrix{ {x\,\,\,\,\,,} & {0 \le x < {1 \over 2}} \cr {{1 \over 2}\,\,\,\,,} & {x = {1 \over 2}} \cr {1 - x\,\,\,,} & {{1 \over 2} < x \le 1} \cr } } \right.\)
and $$g(x) = \left( {x - {1 \over 2}}...

In the expansion of \({\left( {{x \over {\cos \theta }} + {1 \over {x\sin \theta }}} \right)^{16}}\), if \({\ell _1}\) is
the least value of the term independent of x
when \({\pi \over 8} \le \theta \le {\pi \over 4}\) and \({\ell _2}\) ...

If Cr \(\equiv\) 25Cr and
C0 + 5.C1 + 9.C2 + .... + (101).C25 = 225.k, then k is equal to _____.

If the curves, x2 – 6x + y2 + 8 = 0 and
x2 – 8y + y2 + 16 – k = 0, (k > 0) touch each other
at a point, then the largest value of k is ______.

If z be a complex number satisfying
|Re(z)| + |Im(z)| = 4, then |z| cannot be :

Let a function ƒ : [0, 5] \(\to\) R be continuous,
ƒ(1) = 3 and F be defined as :
\(F(x) = \int\limits_1^x {{t^2}g(t)dt}\) , where \(g(t) = \int\limits_1^t {f(u)du}\) Then for the function F, the point x = 1 is :

If \({{dy} \over {dx}} = {{xy} \over {{x^2} + {y^2}}}\); y(1) = 1; then a value of x
satisfying y(x) = e is :

Let ƒ and g be differentiable functions on R
such that fog is the identity function. If for some
a, b \(\in\) R, g'(a) = 5 and g(a) = b, then ƒ'(b) is
equal to :

If \(x = 2\sin \theta - \sin 2\theta\) and \(y = 2\cos \theta - \cos 2\theta\),
\(\theta \in \left[ {0,2\pi } \right]\), then \({{{d^2}y} \over {d{x^2}}}\) at \(\theta\) = \(\pi\) is :

The length of the minor axis (along y-axis) of
an ellipse in the standard form is \({4 \over {\sqrt 3 }}\). If this
ellipse touches the line, x + 6y = 8; then its
eccentricity is :

Let a – 2b + c = 1.
If \(f(x)=\left| {\matrix{ {x + a} & {x + 2} & {x + 1} \cr {x + b} & {x + 3} & {x + 2} \cr {x + c} & {x + 4} & {x + 3} \cr } } \right|\), then:

If \(\int {{{d\theta } \over {{{\cos }^2}\theta \left( {\tan 2\theta + \sec 2\theta } \right)}}} = \lambda \tan \theta + 2{\log _e}\left| {f\left( \theta \right)} \right| + C\)
where C is a constant of integration, then the
ordered pair...

Let [t] denote the greatest integer \(\le\) t
and \(\mathop {\lim }\limits_{x \to 0} x\left[ {{4 \over x}} \right] = A\). Then the function,
f(x) = [x2]sin(\(\pi\)x) is discontinuous, when x is
equal to :

If p \(\to\) (p \(\wedge\) ~q) is false, then the truth values
of p and q are respectively :

The following system of linear equations
7x + 6y – 2z = 0
3x + 4y + 2z = 0
x – 2y – 6z = 0, has

If one end of a focal chord AB of the parabola
y2 = 8x is at \(A\left( {{1 \over 2}, - 2} \right)\), then the equation of
the tangent to it at B is :

A random variable X has the following
probability distribution :

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If 10 different balls are to be placed in 4 distinct
boxes at random, then the probability that two
of these boxes contain exactly 2 and 3 balls is :

Let a, b \(\in\) R, a \(\ne\) 0 be such that the equation,
ax2 – 2bx + 5 = 0 has a repeated root \(\alpha\), which
is also a root of the equation, x2 – 2bx – 10 = 0.
If \(\beta\) is the other root of this equation, then
\(\alpha\)2 +...

Let an be the nth term of a G.P. of positive terms.
\(\sum\limits_{n = 1}^{100} {{a_{2n + 1}} = 200}\) and \(\sum\limits_{n = 1}^{100} {{a_{2n}} = 100}\),

then \(\sum\limits_{n = 1}^{200} {{a_n}}\) is equal to :

The number of terms common to the two A.P.'s
3, 7, 11, ....., 407 and 2, 9, 16, ....., 709 is ______.

If A = {x \(\in\) R : |x| < 2} and B = {x \(\in\) R : |x – 2| \(\ge\) 3};
then :

If \(x = \sum\limits_{n = 0}^\infty {{{\left( { - 1} \right)}^n}{{\tan }^{2n}}\theta }\) and \(y = \sum\limits_{n = 0}^\infty {{{\cos }^{2n}}\theta }\) for
0 < \(\theta\) < \({\pi \over 4}\), then :

Let \(\overrightarrow a\), \(\overrightarrow b\) and \(\overrightarrow c\) be three vectors such that \(\left| {\overrightarrow a } \right| = \sqrt 3\),
$$\left| {\overrightarrow b } \right| = 5,\overrightarrow b .\overrightarrow c = 1...