JEE Main 2021 (Online) 1st September Evening Shift

Let the acute angle bisector of the two planes x \(-\) 2y \(-\) 2z + 1 = 0 and 2x \(-\) 3y \(-\) 6z + 1 = 0 be the plane P. Then which of the following points lies on P?

The distance of line \(3y - 2z - 1 = 0 = 3x - z + 4\) from the point (2, \(-\)1, 6) is :

The function \(f(x) = {x^3} - 6{x^2} + ax + b\) is such that \(f(2) = f(4) = 0\). Consider two statements :Statement 1 : there exists x1, x2 \(\in\)(2, 4), x1 < x2, such that f'(x1) = \(-\)1 and f'(x2) = 0.Statement 2 : there exists x3, x4 ...

The area, enclosed by the curves \(y = \sin x + \cos x\) and \(y = \left| {\cos x - \sin x} \right|\) and the lines \(x = 0,x = {\pi \over 2}\), is :

If the sum of the coefficients in the expansion of (x + y)n is 4096, then the greatest coefficient in the expansion is _____________.

If for the complex numbers z satisfying | z \(-\) 2 \(-\) 2i | \(\le\) 1, the maximum value of | 3iz + 6 | is attained at a + ib, then a + b is equal to ______________.

Let f : R \(\to\) R be a continuous function. Then \(\mathop {\lim }\limits_{x \to {\pi \over 4}} {{{\pi \over 4}\int\limits_2^{{{\sec }^2}x} {f(x)\,dx} } \over {{x^2} - {{{\pi ^2}} \over {16}}}}\) is equal to :

Let \({J_{n,m}} = \int\limits_0^{{1 \over 2}} {{{{x^n}} \over {{x^m} - 1}}dx}\), \(\forall\) n > m and n, m \(\in\) N. Consider a matrix \(A = {[{a_{ij}}]_{3 \times 3}}\) where $${a_{ij}} = \left\{ {\matrix{
{{j_{6 + i,3}} - {j_{i + 3,3...

The function f(x), that satisfies the condition \(f(x) = x + \int\limits_0^{\pi /2} {\sin x.\cos y\,f(y)\,dy}\), is :

If y = y(x) is the solution curve of the differential equation \({x^2}dy + \left( {y - {1 \over x}} \right)dx = 0\) ; x > 0 and y(1) = 1, then \(y\left( {{1 \over 2}} \right)\) is equal to :

Let \(\theta\) be the acute angle between the tangents to the ellipse \({{{x^2}} \over 9} + {{{y^2}} \over 1} = 1\) and the circle \({x^2} + {y^2} = 3\) at their point of intersection in the first quadrant. Then tan\(\theta\) is equal to :

The range of the function, $$f(x) = {\log _{\sqrt 5 }}\left( {3 + \cos \left( {{{3\pi } \over 4} + x} \right) + \cos \left( {{\pi \over 4} + x} \right) + \cos \left( {{\pi \over 4} - x} \right) - \cos \left( {{{3\pi } \over 4} - x} \right...

\({\cos ^{ - 1}}(\cos ( - 5)) + {\sin ^{ - 1}}(\sin (6)) - {\tan ^{ - 1}}(\tan (12))\) is equal to :(The inverse trigonometric functions take the principal values)

Let \(f(x) = {x^6} + 2{x^4} + {x^3} + 2x + 3\), x \(\in\) R. Then the natural number n for which \(\mathop {\lim }\limits_{x \to 1} {{{x^n}f(1) - f(x)} \over {x - 1}} = 44\) is __________.

Let [t] denote the greatest integer \(\le\) t. The number of points where the function \(f(x) = [x]\left| {{x^2} - 1} \right| + \sin \left( {{\pi \over {[x] + 3}}} \right) - [x + 1],x \in ( - 2,2)\) is not continuous is _____________.

Which of the following is equivalent to the Boolean expression p \(\wedge\) \(\sim\) q ?

Consider the system of linear equations\(-\)x + y + 2z = 03x \(-\) ay + 5z = 12x \(-\) 2y \(-\) az = 7Let S1 be the set of all a\(\in\)R for which the system is inconsistent and S2 be the set of all a\(\in\)R for which the system has infini...

Consider the parabola with vertex \(\left( {{1 \over 2},{3 \over 4}} \right)\) and the directrix \(y = {1 \over 2}\). Let P be the point where the parabola meets the line \(x = - {1 \over 2}\). If the normal to the parabola at P intersects...

Let P1, P2, ......, P15 be 15 points on a circle. The number of distinct triangles formed by points Pi, Pj, Pk such that i +j + k \(\ne\) 15, is :

All the arrangements, with or without meaning, of the word FARMER are written excluding any word that has two R appearing together. The arrangements are listed serially in the alphabetic order as in the English dictionary. Then the serial n...

Two squares are chosen at random on a chessboard (see figure). The probability that they have a side in common is :

Let X be a random variable with distribution.
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The numbers of pairs (a, b) of real numbers, such that whenever \(\alpha\) is a root of the equation x2 + ax + b = 0, \(\alpha\)2 \(-\) 2 is also a root of this equation, is :

Let f(x) be a polynomial of degree 3 such that \(f(k) = - {2 \over k}\) for k = 2, 3, 4, 5. Then the value of 52 \(-\) 10f(10) is equal to :

Let Sn = 1 . (n \(-\) 1) + 2 . (n \(-\) 2) + 3 . (n \(-\) 3) + ..... + (n \(-\) 1) . 1, n \(\ge\) 4.The sum \(\sum\limits_{n = 4}^\infty {\left( {{{2{S_n}} \over {n!}} - {1 \over {(n - 2)!}}} \right)}\) is equal to :

Let a1, a2, ..........., a21 be an AP such that \(\sum\limits_{n = 1}^{20} {{1 \over {{a_n}{a_{n + 1}}}} = {4 \over 9}}\). If the sum of this AP is 189, then a6a16 is equal to :

Let the points of intersections of the lines x \(-\) y + 1 = 0, x \(-\) 2y + 3 = 0 and 2x \(-\) 5y + 11 = 0 are the mid points of the sides of a triangle \(\Delta\)ABC. Then, the area of the \(\Delta\)ABC is _____________.

A man starts walking from the point P(\(-\)3, 4), touches the x-axis at R, and then turns to reach at the point Q(0, 2). The man is walking at a constant speed. If the man reaches the point Q in the minimum time, then $$50\left( {{{(PR)}^2}...

If n is the number of solutions of the equation \(2\cos x\left( {4\sin \left( {{\pi \over 4} + x} \right)\sin \left( {{\pi \over 4} - x} \right) - 1} \right) = 1,x \in [0,\pi ]\) and S is the sum of all these solutions, then the ordered p...

Let \(\overrightarrow a = 2\widehat i - \widehat j + 2\widehat k\) and \(\overrightarrow b = \widehat i + 2\widehat j - \widehat k\). Let a vector \(\overrightarrow v\) be in the plane containing \(\overrightarrow a\) and $$\overrightar...