JEE Main 2020 (Online) 4th September Morning Slot

If the equation of a plane P, passing through the intersection of the planes, x + 4y - z + 7 = 0
and 3x + y + 5z = 8 is ax + by + 6z = 15 for some a, b \(\in\) R, then the distance of the point
(3, 2, -1) from the plane P is...........

Let f be a twice differentiable function on (1, 6). If f(2) = 8, f’(2) = 5, f’(x) \(\ge\) 1 and f''(x) \(\ge\) 4, for all x \(\in\) (1, 6), then :

Let \({\left( {2{x^2} + 3x + 4} \right)^{10}} = \sum\limits_{r = 0}^{20} {{a_r}{x^r}}\)
Then \({{{a_7}} \over {{a_{13}}}}\) is equal to ______.

The value of \(\sum\limits_{r = 0}^{20} {{}^{50 - r}{C_6}}\) is equal to:

Let \(u = {{2z + i} \over {z - ki}}\), z = x + iy and k > 0. If the curve represented by Re(u) + Im(u) = 1 intersects the y-axis at the points P and Q where PQ = 5, then the value of k is :

Let \(f(x) = \left| {x - 2} \right|\) and g(x) = f(f(x)), \(x \in \left[ {0,4} \right]\). Then \(\int\limits_0^3 {\left( {g(x) - f(x)} \right)} dx\) is equal to:

Let y = y(x) be the solution of the differential equation, xy'- y = x2(xcosx + sinx), x > 0. if y (\(\pi\)) = \(\pi\) then \(y''\left( {{\pi \over 2}} \right) + y\left( {{\pi \over 2}} \right)\) is equal to :

If \(\left( {a + \sqrt 2 b\cos x} \right)\left( {a - \sqrt 2 b\cos y} \right) = {a^2} - {b^2}\)
where a > b > 0, then \({{dx} \over {dy}}\,\,at\left( {{\pi \over 4},{\pi \over 4}} \right)\) is :

Let \({{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1\) (a > b) be a given ellipse, length of whose latus rectum is 10. If its eccentricity is the maximum value of the function, \(\phi \left( t \right) = {5 \over {12}} + t - {t^2}\), ...

Let P(3, 3) be a point on the hyperbola, \({{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1\). If the normal to it at P intersects the x-axis
at (9, 0) and e is its eccentricity, then the ordered pair (a2, e2) is equal to :

Let \(f\left( x \right) = \int {{{\sqrt x } \over {{{\left( {1 + x} \right)}^2}}}dx\left( {x \ge 0} \right)}\). Then f(3) – f(1) is eqaul to :

The integral \(\int {{{\left( {{x \over {x\sin x + \cos x}}} \right)}^2}dx}\) is equal to
(where C is a constant of integration):

Suppose a differentiable function f(x) satisfies the identity f(x+y) = f(x) + f(y) + xy2 + x2y, for all real x and y.
\(\mathop {\lim }\limits_{x \to 0} {{f\left( x \right)} \over x} = 1\), then f'(3) is equal to ______.

Given the following two statements:
\(\left( {{S_1}} \right):\left( {q \vee p} \right) \to \left( {p \leftrightarrow \sim q} \right)\) is a tautology
\(\left( {{S_2}} \right): \,\,\sim q \wedge \left( { \sim p \leftrightarrow q} \right)\)...

If \(A = \left[ {\matrix{ {\cos \theta } & {i\sin \theta } \cr {i\sin \theta } & {\cos \theta } \cr } } \right]\), \(\left( {\theta = {\pi \over {24}}} \right)\)
and $${A^5} = \left[ {\matrix{
a & b \cr
c & d \cr

}...

If the system of equations
x - 2y + 3z = 9
2x + y + z = b
x - 7y + az = 24, has infinitely many solutions, then a - b is equal to.........

The probability of a man hitting a target is \({1 \over {10}}\). The least number of shots required, so that the
probability of his hitting the target at least once is greater than \({1 \over {4}}\), is ____________.

A triangle ABC lying in the first quadrant has two vertices as A(1, 2) and B(3, 1). If \(\angle BAC = {90^o}\) and area\(\left( {\Delta ABC} \right) = 5\sqrt 5\) s units, then the abscissa of the vertex C is :

Let \(\alpha\) and \(\beta\) be the roots of x2 - 3x + p=0 and \(\gamma\) and \(\delta\) be the roots of x2 - 6x + q = 0. If \(\alpha, \beta, \gamma, \delta\)
form a geometric progression.Then ratio (2q + p) : (2q - p) is:

Let [t] denote the greatest integer \(\le\) t. Then the equation in x,
[x]2 + 2[x+2] - 7 = 0 has :

If 1+(1–22.1)+(1–42.3)+(1-62.5)+......+(1-202.19)= \(\alpha\) - 220\(\beta\), then an ordered pair \(\left( {\alpha ,\beta } \right)\) is equal to:

A survey shows that 63% of the people in a city read newspaper A whereas 76% read
newspaper B. If x% of the people read both the newspapers, then a possible value of x can be:

The mean and variance of 8 observations are 10 and 13.5, respectively. If 6 of these observations
are 5, 7, 10, 12, 14, 15, then the absolute difference of the remaining two observations is :

Two vertical poles AB = 15 m and CD = 10 m are standing apart on a horizontal ground with points A
and C on the ground. If P is the point of intersection of BC and AD, then the height of P (in m)
above the line AC is :

Let x0 be the point of Local maxima of \(f(x) = \overrightarrow a .\left( {\overrightarrow b \times \overrightarrow c } \right)\), where \(\overrightarrow a = x\widehat i - 2\widehat j + 3\widehat k\), $$\overrightarrow b = - 2\widehat ...