JEE Main 2021 (Online) 16th March Morning Shift

Let P be a plane lx + my + nz = 0 containing the line, \({{1 - x} \over 1} = {{y + 4} \over 2} = {{z + 2} \over 3}\). If plane P divides the line segment AB joining points A(\(-\)3, \(-\)6, 1) and B(2, 4, \(-\)3) in ratio k : 1 then the val...

If for a > 0, the feet of perpendiculars from the points A(a, \(-\)2a, 3) and B(0, 4, 5) on the plane lx + my + nz = 0 are points C(0, \(-\)a, \(-\)1) and D respectively, then the length of line segment CD is equal to :

Let the position vectors of two points P and Q be 3\(\widehat i\) \(-\) \(\widehat j\) + 2\(\widehat k\) and \(\widehat i\) + 2\(\widehat j\) \(-\) 4\(\widehat k\), respectively. Let R and S be two points such that the direction ratios of l...

If n is the number of irrational terms in the expansion of \({\left( {{3^{1/4}} + {5^{1/8}}} \right)^{60}}\), then (n \(-\) 1) is divisible by :

Let [ x ] denote greatest integer less than or equal to x. If for n\(\in\)N, \({(1 - x + {x^3})^n} = \sum\limits_{j = 0}^{3n} {{a_j}{x^j}}\), then $$\sum\limits_{j = 0}^{\left[ {{{3n} \over 2}} \right]} {{a_{2j}} + 4} \sum\limits_{j = 0}^{...

Let a complex number z, |z| \(\ne\) 1, satisfy \({\log _{{1 \over {\sqrt 2 }}}}\left( {{{|z| + 11} \over {{{(|z| - 1)}^2}}}} \right) \le 2\). Then, the largest value of |z| is equal to ____________.

Let z and \(\omega\) be two complex numbers such that \(\omega = z\overline z - 2z + 2,\left| {{{z + i} \over {z - 3i}}} \right| = 1\) and Re(\(\omega\)) has minimum value. Then, the minimum value of n \(\in\) N for which \(\omega\)n is re...

Let f : R \(\to\) R be a continuous function such that f(x) + f(x + 1) = 2, for all x\(\in\)R. If \({I_1} = \int\limits_0^8 {f(x)dx}\) and \({I_2} = \int\limits_{ - 1}^3 {f(x)dx}\), then the value of I1 + 2I2 is equal to ____________.

Let f : (0, 2) \(\to\) R be defined as f(x) = log2\(\left( {1 + \tan \left( {{{\pi x} \over 4}} \right)} \right)\). Then, $$\mathop {\lim }\limits_{n \to \infty } {2 \over n}\left( {f\left( {{1 \over n}} \right) + f\left( {{2 \over n}} \r...

If the normal to the curve y(x) = \(\int\limits_0^x {(2{t^2} - 15t + 10)dt}\) at a point (a, b) is parallel to the line x + 3y = \(-\)5, a > 1, then the value of | a + 6b | is equal to ___________.

If y = y(x) is the solution of the differential equation, \({{dy} \over {dx}} + 2y\tan x = \sin x,y\left( {{\pi \over 3}} \right) = 0\), then the maximum value of the function y(x) over R is equal to:

Let the curve y = y(x) be the solution of the differential equation, \({{dy} \over {dx}}\) = 2(x + 1). If the numerical value of area bounded by the curve y = y(x) and x-axis is \({{4\sqrt 8 } \over 3}\), then the value of y(1) is equal to ...

The range of a\(\in\)R for which the function f(x) = (4a \(-\) 3)(x + loge 5) + 2(a \(-\) 7) cot\(\left( {{x \over 2}} \right)\) sin2\(\left( {{x \over 2}} \right)\), x \(\ne\) 2n\(\pi\), n\(\in\)N has critical points, is :

The locus of the midpoints of the chord of the circle, x2 + y2 = 25 which is tangent to the hyperbola, \({{{x^2}} \over 9} - {{{y^2}} \over {16}} = 1\) is :

Let \({S_k} = \sum\limits_{r = 1}^k {{{\tan }^{ - 1}}\left( {{{{6^r}} \over {{2^{2r + 1}} + {3^{2r + 1}}}}} \right)}\). Then \(\mathop {\lim }\limits_{k \to \infty } {S_k}\) is equal to :

Let the functions f : R \(\to\) R and g : R \(\to\) R be defined as :\(f(x) = \left\{ {\matrix{ {x + 2,} & {x < 0} \cr {{x^2},} & {x \ge 0} \cr } } \right.\) and $$g(x) = \left\{ {\matrix{
{{x^3},} & {x < 1} \cr
{3x ...

If \(\mathop {\lim }\limits_{x \to 0} {{a{e^x} - b\cos x + c{e^{ - x}}} \over {x\sin x}} = 2\), then a + b + c is equal to ____________.

Which of the following Boolean expression is a tautology?

Let \(A = \left[ {\matrix{ i & { - i} \cr { - i} & i \cr } } \right],i = \sqrt { - 1}\). Then, the system of linear equations $${A^8}\left[ {\matrix{
x \cr
y \cr

} } \right] = \left[ {\matrix{
8 \cr
{64} \c...

The total number of 3 \(\times\) 3 matrices A having entries from the set {0, 1, 2, 3} such that the sum of all the diagonal entries of AAT is 9, is equal to _____________.

Let \(P = \left[ {\matrix{ { - 30} & {20} & {56} \cr {90} & {140} & {112} \cr {120} & {60} & {14} \cr } } \right]\) and $$A = \left[ {\matrix{
2 & 7 & {{\omega ^2}} \cr
{ - 1} & { - \omega } & 1 \cr
0 & { - \om...

If the three normals drawn to the parabola, y2 = 2x pass through the point (a, 0) a \(\ne\) 0, then 'a' must be greater than :

A pack of cards has one card missing. Two cards are drawn randomly and are found to be spades. The probability that the missing card is not a spade, is :

Let ABCD be a square of side of unit length. Let a circle C1 centered at A with unit radius is drawn. Another circle C2 which touches C1 and the lines AD and AB are tangent to it, is also drawn. Let a tangent line from the point C to the ci...

Consider an arithmetic series and a geometric series having four initial terms from the set {11, 8, 21, 16, 26, 32, 4}. If the last terms of these series are the maximum possible four digit numbers, then the number of common terms in these ...

The number of elements in the set {x \(\in\) R : (|x| \(-\) 3) |x + 4| = 6} is equal to :

Consider three observations a, b, and c such that b = a + c. If the standard deviation of a + 2, b + 2, c + 2 is d, then which of the following is true?

The number of roots of the equation, (81)sin2x + (81)cos2x = 30 in the interval [ 0, \(\pi\) ] is equal to :

If for x \(\in\) \(\left( {0,{\pi \over 2}} \right)\), log10sinx + log10cosx = \(-\)1 and log10(sinx + cosx) = \({1 \over 2}\)(log10 n \(-\) 1), n > 0, then the value of n is equal to :

Let a vector \(\alpha \widehat i + \beta \widehat j\) be obtained by rotating the vector \(\sqrt 3 \widehat i + \widehat j\) by an angle 45\(^\circ\) about the origin in counterclockwise direction in the first quadrant. Then the area of tri...