JEE Main 2021 (Online) 18th March Morning Shift

The equation of the planes parallel to the plane x \(-\) 2y + 2z \(-\) 3 = 0 which are at unit distance from the point (1, 2, 3) is ax + by + cz + d = 0. If (b \(-\) d) = k(c \(-\) a), then the positive value of k is :

Let the plane ax + by + cz + d = 0 bisect the line joining the points (4, \(-\)3, 1) and (2, 3, \(-\)5) at the right angles. If a, b, c, d are integers, then the minimum value of (a2 + b2 + c2 + d2) is _________.

Let (1 + x + 2x2)20 = a0 + a1x + a2x2 + .... + a40x40. Then a1 + a3 + a5 + ..... + a37 is equal to

Choose the correct statement about two circles whose equations are given below :x2 + y2 \(-\) 10x \(-\) 10y + 41 = 0x2 + y2 \(-\) 22x \(-\) 10y + 137 = 0

For the four circles M, N, O and P, following four equations are given :Circle M : x2 + y2 = 1Circle N : x2 + y2 \(-\) 2x = 0Circle O : x2 + y2 \(-\) 2x \(-\) 2y + 1 = 0Circle P : x2 + y2 \(-\) 2y = 0If the centre of circle M is joined with...

Let z1, z2 be the roots of the equation z2 + az + 12 = 0 and z1, z2 form an equilateral triangle with origin. Then, the value of |a| is :

If the equation \(a|z{|^2} + \overline {\overline \alpha z + \alpha \overline z } + d = 0\) represents a circle where a, d are real constants then which of the following condition is correct?

Let f(x) and g(x) be two functions satisfying f(x2) + g(4 \(-\) x) = 4x3 and g(4 \(-\) x) + g(x) = 0, then the value of \(\int\limits_{ - 4}^4 {f{{(x)}^2}dx}\) is

The differential equation satisfied by the system of parabolas y2 = 4a(x + a) is :

The real valued function \(f(x) = {{\cos e{c^{ - 1}}x} \over {\sqrt {x - [x]} }}\), where [x] denotes the greatest integer less than or equal to x, is defined for all x belonging to :

If the functions are defined as \(f(x) = \sqrt x\) and \(g(x) = \sqrt {1 - x}\), then what is the common domain of the following functions :f + g, f \(-\) g, f/g, g/f, g \(-\) f where $$(f \pm g)(x) = f(x) \pm g(x),(f/g)x = {{f(x)} \over ...

The integral \(\int {{{(2x - 1)\cos \sqrt {{{(2x - 1)}^2} + 5} } \over {\sqrt {4{x^2} - 4x + 6} }}} dx\) is equal to (where c is a constant of integration)

If \(f(x) = \int {{{5{x^8} + 7{x^6}} \over {{{({x^2} + 1 + 2{x^7})}^2}}}dx,(x \ge 0),f(0) = 0}\) and \(f(1) = {1 \over K}\), then the value of K is

If \(\mathop {\lim }\limits_{x \to 0} {{{{\sin }^{ - 1}}x - {{\tan }^{ - 1}}x} \over {3{x^3}}}\) is equal to L, then the value of (6L + 1) is

If \(f(x) = \left\{ {\matrix{ {{1 \over {|x|}}} & {;\,|x|\, \ge 1} \cr {a{x^2} + b} & {;\,|x|\, < 1} \cr } } \right.\) is differentiable at every point of the domain, then the values of a and b are respectively :

The solutions of the equation $$\left| {\matrix{
{1 + {{\sin }^2}x} & {{{\sin }^2}x} & {{{\sin }^2}x} \cr
{{{\cos }^2}x} & {1 + {{\cos }^2}x} & {{{\cos }^2}x} \cr
{4\sin 2x} & {4\sin 2x} & {1 + 4\sin 2x} \cr

} } \right| = 0...

Let \(\alpha\), \(\beta\), \(\gamma\) be the real roots of the equation, x3 + ax2 + bx + c = 0, (a, b, c \(\in\) R and a, b \(\ne\) 0). If the system of equations (in u, v, w) given by \(\alpha\)u + \(\beta\)v + \(\gamma\)w = 0, \(\beta\)u ...

Let \(A + 2B = \left[ {\matrix{ 1 & 2 & 0 \cr 6 & { - 3} & 3 \cr { - 5} & 3 & 1 \cr } } \right]\) and \(2A - B = \left[ {\matrix{ 2 & { - 1} & 5 \cr 2 & { - 1} & 6 \cr 0 & 1 & 2 \cr } } \right]\). If Tr(A) ...

The missing value in the following figure is

The sum of all the 4-digit distinct numbers that can be formed with the digits 1, 2, 2 and 3 is :

The number of times the digit 3 will be written when listing the integers from 1 to 1000 is :

The value of \(3 + {1 \over {4 + {1 \over {3 + {1 \over {4 + {1 \over {3 + ....\infty }}}}}}}}\) is equal to

\({1 \over {{3^2} - 1}} + {1 \over {{5^2} - 1}} + {1 \over {{7^2} - 1}} + .... + {1 \over {{{(201)}^2} - 1}}\) is equal to

If \(\alpha\), \(\beta\) are natural numbers such that 100\(\alpha\) \(-\) 199\(\beta\) = (100)(100) + (99)(101) + (98)(102) + ...... + (1)(199), then the slope of the line passing through (\(\alpha\), \(\beta\)) and origin is :

The mean age of 25 teachers in a school is 40 years. A teacher retires at the age of 60 years and a new teacher is appointed in his place. If the mean age of the teachers in this school now is 39 years, then the age (in years) of the newly ...

The equation of one of the straight lines which passes through the point (1, 3) and makes an angles \({\tan ^{ - 1}}\left( {\sqrt 2 } \right)\) with the straight line, y + 1 = 3\({\sqrt 2 }\) x is :

The number of integral values of m so that the abscissa of point of intersection of lines 3x + 4y = 9 and y = mx + 1 is also an integer, is :

A square ABCD has all its vertices on the curve x2y2 = 1. The midpoints of its sides also lie on the same curve. Then, the square of area of ABCD is _________.

The number of solutions of the equation \(|\cot x| = \cot x + {1 \over {\sin x}}\) in the interval [ 0, 2\(\pi\) ] is

A vector \(\overrightarrow a\) has components 3p and 1 with respect to a rectangular cartesian system. This system is rotated through a certain angle about the origin in the counter clockwise sense. If, with respect to new system, $$\overr...