JEE Main 2021 (Online) 27th August Morning Shift

The distance of the point (1, \(-\)2, 3) from the plane x \(-\) y + z = 5 measured parallel to a line, whose direction ratios are 2, 3, \(-\)6 is :

Equation of a plane at a distance \(\sqrt {{2 \over {21}}}\) from the origin, which contains the line of intersection of the planes x \(-\) y \(-\) z \(-\) 1 = 0 and 2x + y \(-\) 3z + 4 = 0, is :

A wire of length 20 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a regular hexagon. Then the length of the side (in meters) of the hexagon, so that the combined area of the square and the ...

The number of distinct real roots of the equation 3x4 + 4x3 \(-\) 12x2 + 4 = 0 is _____________.

\(\sum\limits_{k = 0}^{20} {{{\left( {{}^{20}{C_k}} \right)}^2}}\) is equal to :

Let the equation x2 + y2 + px + (1 \(-\) p)y + 5 = 0 represent circles of varying radius r \(\in\) (0, 5]. Then the number of elements in the set S = {q : q = p2 and q is an integer} is __________.

If \(S = \left\{ {z \in C:{{z - i} \over {z + 2i}} \in R} \right\}\), then :

If \({U_n} = \left( {1 + {1 \over {{n^2}}}} \right)\left( {1 + {{{2^2}} \over {{n^2}}}} \right)^2.....\left( {1 + {{{n^2}} \over {{n^2}}}} \right)^n\), then \(\mathop {\lim }\limits_{n \to \infty } {({U_n})^{{{ - 4} \over {{n^2}}}}}\) is eq...

\(\int\limits_6^{16} {{{{{\log }_e}{x^2}} \over {{{\log }_e}{x^2} + {{\log }_e}({x^2} - 44x + 484)}}dx}\) is equal to :

Let y = y(x) be the solution of the differential equation \({{dy} \over {dx}} = 2(y + 2\sin x - 5)x - 2\cos x\) such that y(0) = 7. Then y(\(\pi\)) is equal to :

Let us consider a curve, y = f(x) passing through the point (\(-\)2, 2) and the slope of the tangent to the curve at any point (x, f(x)) is given by f(x) + xf'(x) = x2. Then :

If \({y^{1/4}} + {y^{ - 1/4}} = 2x\), and \(({x^2} - 1){{{d^2}y} \over {d{x^2}}} + \alpha x{{dy} \over {dx}} + \beta y = 0\), then | \(\alpha\) \(-\) \(\beta\) | is equal to __________.

If x2 + 9y2 \(-\) 4x + 3 = 0, x, y \(\in\) R, then x and y respectively lie in the intervals :

If the minimum area of the triangle formed by a tangent to the ellipse \({{{x^2}} \over {{b^2}}} + {{{y^2}} \over {4{a^2}}} = 1\) and the co-ordinate axis is kab, then k is equal to _______________.

If \(\int {{{dx} \over {{{({x^2} + x + 1)}^2}}} = a{{\tan }^{ - 1}}\left( {{{2x + 1} \over {\sqrt 3 }}} \right) + b\left( {{{2x + 1} \over {{x^2} + x + 1}}} \right) + C}\), x > 0 where C is the constant of integration, then the value of $$...

If \({({\sin ^{ - 1}}x)^2} - {({\cos ^{ - 1}}x)^2} = a\); 0 < x < 1, a \(\ne\) 0, then the value of 2x2 \(-\) 1 is :

If \(\alpha\), \(\beta\) are the distinct roots of x2 + bx + c = 0, then \(\mathop {\lim }\limits_{x \to \beta } {{{e^{2({x^2} + bx + c)}} - 1 - 2({x^2} + bx + c)} \over {{{(x - \beta )}^2}}}\) is equal to :

The statement (p \(\wedge\) (p \(\to\) q) \(\wedge\) (q \(\to\) r)) \(\to\) r is :

If the matrix \(A = \left( {\matrix{ 0 & 2 \cr K & { - 1} \cr } } \right)\) satisfies \(A({A^3} + 3I) = 2I\), then the value of K is :

If the system of linear equations2x + y \(-\) z = 3x \(-\) y \(-\) z = \(\alpha\)3x + 3y + \(\beta\)z = 3has infinitely many solution, then \(\alpha\) + \(\beta\) \(-\) \(\alpha\)\(\beta\) is equal to _____________.

A tangent and a normal are drawn at the point P(2, \(-\)4) on the parabola y2 = 8x, which meet the directrix of the parabola at the points A and B respectively. If Q(a, b) is a point such that AQBP is a square, then 2a + b is equal to :

A number is called a palindrome if it reads the same backward as well as forward. For example 285582 is a six digit palindrome. The number of six digit palindromes, which are divisible by 55, is ____________.

When a certain biased die is rolled, a particular face occurs with probability \({1 \over 6} - x\) and its opposite face occurs with probability \({1 \over 6} + x\). All other faces occur with probability \({1 \over 6}\). Note that opposite...

Let \({{\sin A} \over {\sin B}} = {{\sin (A - C)} \over {\sin (C - B)}}\), where A, B, C are angles of triangle ABC. If the lengths of the sides opposite these angles are a, b, c respectively, then :

If 0 < x < 1, then \({3 \over 2}{x^2} + {5 \over 3}{x^3} + {7 \over 4}{x^4} + .....\), is equal to :

If for x, y \(\in\) R, x > 0, y = log10x + log10x1/3 + log10x1/9 + ...... upto \(\infty\) terms and \({{2 + 4 + 6 + .... + 2y} \over {3 + 6 + 9 + ..... + 3y}} = {4 \over {{{\log }_{10}}x}}\), then the ordered pair (x, y) is equal to :

If A = {x \(\in\) R : |x \(-\) 2| > 1}, B = {x \(\in\) R : \(\sqrt {{x^2} - 3}\) > 1}, C = {x \(\in\) R : |x \(-\) 4| \(\ge\) 2} and Z is the set of all integers, then the number of subsets of the set (A \(\cap\) B \(\cap\) C)c \(\cap\) Z ...

Let n be an odd natural number such that the variance of 1, 2, 3, 4, ......, n is 14. Then n is equal to _____________.

Let A be a fixed point (0, 6) and B be a moving point (2t, 0). Let M be the mid-point of AB and the perpendicular bisector of AB meets the y-axis at C. The locus of the mid-point P of MC is :

Let \(\overrightarrow a = \widehat i + 5\widehat j + \alpha \widehat k\), \(\overrightarrow b = \widehat i + 3\widehat j + \beta \widehat k\) and \(\overrightarrow c = - \widehat i + 2\widehat j - 3\widehat k\) be three vectors such tha...