JEE Main 2021 (Online) 31st August Morning Shift

Let the equation of the plane, that passes through the point (1, 4, \(-\)3) and contains the line of intersection of the planes 3x \(-\) 2y + 4z \(-\) 7 = 0 and x + 5y \(-\) 2z + 9 = 0, be \(\alpha\)x + \(\beta\)y + \(\gamma\)z + 3 = 0, the...

The square of the distance of the point of intersection of the line \({{x - 1} \over 2} = {{y - 2} \over 3} = {{z + 1} \over 6}\) and the plane \(2x - y + z = 6\) from the point (\(-\)1, \(-\)1, 2) is __________.

The number of real roots of the equation \({e^{4x}} + 2{e^{3x}} - {e^x} - 6 = 0\) is :

If 'R' is the least value of 'a' such that the function f(x) = x2 + ax + 1 is increasing on [1, 2] and 'S' is the greatest value of 'a' such that the function f(x) = x2 + ax + 1 is decreasing on [1, 2], then the value of |R \(-\) S| is ____...

If \(\left( {{{{3^6}} \over {{4^4}}}} \right)k\) is the term, independent of x, in the binomial expansion of \({\left( {{x \over 4} - {{12} \over {{x^2}}}} \right)^{12}}\), then k is equal to ___________.

If the variable line 3x + 4y = \(\alpha\) lies between the two circles (x \(-\) 1)2 + (y \(-\) 1)2 = 1 and (x \(-\) 9)2 + (y \(-\) 1)2 = 4, without intercepting a chord on either circle, then the sum of all the integral values of \(\alpha\)...

A point z moves in the complex plane such that \(\arg \left( {{{z - 2} \over {z + 2}}} \right) = {\pi \over 4}\), then the minimum value of \({\left| {z - 9\sqrt 2 - 2i} \right|^2}\) is equal to _______________.

Let f be a non-negative function in [0, 1] and twice differentiable in (0, 1). If \(\int_0^x {\sqrt {1 - {{(f'(t))}^2}} dt = \int_0^x {f(t)dt} }\), \(0 \le x \le 1\) and f(0) = 0, then $$\mathop {\lim }\limits_{x \to 0} {1 \over {{x^2}}}\i...

Let [t] denote the greatest integer \(\le\) t. Then the value of \(8.\int\limits_{ - {1 \over 2}}^1 {([2x] + |x|)dx}\) is ___________.

If \(x\phi (x) = \int\limits_5^x {(3{t^2} - 2\phi '(t))dt}\), x > \(-\)2, and \(\phi\)(0) = 4, then \(\phi\)(2) is __________.

If \({{dy} \over {dx}} = {{{2^{x + y}} - {2^x}} \over {{2^y}}}\), y(0) = 1, then y(1) is equal to :

The line \(12x\cos \theta + 5y\sin \theta = 60\) is tangent to which of the following curves?

A vertical pole fixed to the horizontal ground is divided in the ratio 3 : 7 by a mark on it with lower part shorter than the upper part. If the two parts subtend equal angles at a point on the ground 18 m away from the base of the pole, th...

The integral \(\int {{1 \over {\root 4 \of {{{(x - 1)}^3}{{(x + 2)}^5}} }}} \,dx\) is equal to : (where C is a constant of integration)

The function \(f(x) = \left| {{x^2} - 2x - 3} \right|\,.\,{e^{\left| {9{x^2} - 12x + 4} \right|}}\) is not differentiable at exactly :

If the function $$f(x) = \left\{ {\matrix{
{{1 \over x}{{\log }_e}\left( {{{1 + {x \over a}} \over {1 - {x \over b}}}} \right)} & , & {x < 0} \cr
k & , & {x = 0} \cr
{{{{{\cos }^2}x - {{\sin }^2}x - 1} \over {\sqrt {{x^2} + 1} ...

\(\mathop {\lim }\limits_{x \to 0} {{{{\sin }^2}\left( {\pi {{\cos }^4}x} \right)} \over {{x^4}}}\) is equal to :

Let *, ▢ \(\in\){\(\wedge\), \(\vee\)} be such that the Boolean expression (p * \(\sim\) q) \(\Rightarrow\) (p ▢ q) is a tautology. Then :

If the following system of linear equations2x + y + z = 5x \(-\) y + z = 3x + y + az = bhas no solution, then :

If \({a_r} = \cos {{2r\pi } \over 9} + i\sin {{2r\pi } \over 9}\), r = 1, 2, 3, ....., i = \(\sqrt { - 1}\), then the determinant $$\left| {\matrix{
{{a_1}} & {{a_2}} & {{a_3}} \cr
{{a_4}} & {{a_5}} & {{a_6}} \cr
{{a_7}} & {{a...

The length of the latus rectum of a parabola, whose vertex and focus are on the positive x-axis at a distance R and S (> R) respectively from the origin, is :

The number of six letter words (with or without meaning), formed using all the letters of the word 'VOWELS', so that all the consonants never come together, is ___________.

An electric instrument consists of two units. Each unit must function independently for the instrument to operate. The probability that the first unit functions is 0.9 and that of the second unit is 0.8. The instrument is switched on and it...

cosec18\(^\circ\) is a root of the equation :

The sum of 10 terms of the series
\({3 \over {{1^2} \times {2^2}}} + {5 \over {{2^2} \times {3^2}}} + {7 \over {{3^2} \times {4^2}}} + ....\) is :

Three numbers are in an increasing geometric progression with common ratio r. If the middle number is doubled, then the new numbers are in an arithmetic progression with common difference d. If the fourth term of GP is 3 r2, then r2 \(-\) d...

Which of the following is not correct for relation R on the set of real numbers ?

The mean of 10 numbers 7 \(\times\) 8, 10 \(\times\) 10, 13 \(\times\) 12, 16 \(\times\) 14, ....... is ____________.

If p and q are the lengths of the perpendiculars from the origin on the lines,x cosec \(\alpha\) \(-\) y sec \(\alpha\) = k cot 2\(\alpha\) andx sin\(\alpha\) + y cos\(\alpha\) = k sin2\(\alpha\)respectively, then k2 is equal to :

Let \(\overrightarrow a\) and \(\overrightarrow b\) be two vectors such that \(\left| {2\overrightarrow a + 3\overrightarrow b } \right| = \left| {3\overrightarrow a + \overrightarrow b } \right|\) and the angle between $$\overrightarro...