JEE Main 2021 (Online) 27th August Evening Shift

The angle between the straight lines, whose direction cosines are given by the equations 2l + 2m \(-\) n = 0 and mn + nl + lm = 0, is :

The equation of the plane passing through the line of intersection of the planes \(\overrightarrow r .\left( {\widehat i + \widehat j + \widehat k} \right) = 1\) and $$\overrightarrow r .\left( {2\widehat i + 3\widehat j - \widehat k} \righ...

Let S be the mirror image of the point Q(1, 3, 4) with respect to the plane 2x \(-\) y + z + 3 = 0 and let R(3, 5, \(\gamma\)) be a point of this plane. Then the square of the length of the line segment SR is ___________.

A box open from top is made from a rectangular sheet of dimension a \(\times\) b by cutting squares each of side x from each of the four corners and folding up the flaps. If the volume of the box is maximum, then x is equal to :

The area of the region bounded by the parabola (y \(-\) 2)2 = (x \(-\) 1), the tangent to it at the point whose ordinate is 3 and the x-axis is :

3 \(\times\) 722 + 2 \(\times\) 1022 \(-\) 44 when divided by 18 leaves the remainder __________.

Let Z be the set of all integers,\(A = \{ (x,y) \in Z \times Z:{(x - 2)^2} + {y^2} \le 4\}\)\(B = \{ (x,y) \in Z \times Z:{x^2} + {y^2} \le 4\}\)\(C = \{ (x,y) \in Z \times Z:{(x - 2)^2} + {(y - 2)^2} \le 4\}\)If the total number of rela...

Two circles each of radius 5 units touch each other at the point (1, 2). If the equation of their common tangent is 4x + 3y = 10, and C1(\(\alpha\), \(\beta\)) and C2(\(\gamma\), \(\delta\)), C1 \(\ne\) C2 are their centres, then |($$\alpha...

Let z1 and z2 be two complex numbers such that \(\arg ({z_1} - {z_2}) = {\pi \over 4}\) and z1, z2 satisfy the equation | z \(-\) 3 | = Re(z). Then the imaginary part of z1 + z2 is equal to ___________.

The value of the integral \(\int\limits_0^1 {{{\sqrt x dx} \over {(1 + x)(1 + 3x)(3 + x)}}}\) is :

A differential equation representing the family of parabolas with axis parallel to y-axis and whose length of latus rectum is the distance of the point (2, \(-\)3) from the line 3x + 4y = 5, is given by :

If the solution curve of the differential equation (2x \(-\) 10y3)dy + ydx = 0, passes through the points (0, 1) and (2, \(\beta\)), then \(\beta\) is a root of the equation :

If \(y(x) = {\cot ^{ - 1}}\left( {{{\sqrt {1 + \sin x} + \sqrt {1 - \sin x} } \over {\sqrt {1 + \sin x} - \sqrt {1 - \sin x} }}} \right),x \in \left( {{\pi \over 2},\pi } \right)\), then \({{dy} \over {dx}}\) at \(x = {{5\pi } \over 6}\)...

Two poles, AB of length a metres and CD of length a + b (b \(\ne\) a) metres are erected at the same horizontal level with bases at B and D. If BD = x and tan\(\angle\)ACB = \({1 \over 2}\), then :

Let A (sec\(\theta\), 2tan\(\theta\)) and B (sec\(\phi\), 2tan\(\phi\)), where \(\theta\) + \(\phi\) = \(\pi\)/2, be two points on the hyperbola 2x2 \(-\) y2 = 2. If (\(\alpha\), \(\beta\)) is the point of the intersection of the normals to...

If \(\int {{{2{e^x} + 3{e^{ - x}}} \over {4{e^x} + 7{e^{ - x}}}}dx = {1 \over {14}}(ux + v{{\log }_e}(4{e^x} + 7{e^{ - x}})) + C}\), where C is a constant of integration, then u + v is equal to _____________.

Let M and m respectively be the maximum and minimum values of the function f(x) = tan\(-\)1 (sin x + cos x) in \(\left[ {0,{\pi \over 2}} \right]\), then the value of tan(M \(-\) m) is equal to :

If \(\mathop {\lim }\limits_{x \to \infty } \left( {\sqrt {{x^2} - x + 1} - ax} \right) = b\), then the ordered pair (a, b) is :

The Boolean expression (p \(\wedge\) q) \(\Rightarrow\) ((r \(\wedge\) q) \(\wedge\) p) is equivalent to :

Let \(A = \left( {\matrix{ {[x + 1]} & {[x + 2]} & {[x + 3]} \cr {[x]} & {[x + 3]} & {[x + 3]} \cr {[x]} & {[x + 2]} & {[x + 4]} \cr } } \right)\), where [t] denotes the greatest integer less than or equal to t. If det(A) =...

Let A(a, 0), B(b, 2b + 1) and C(0, b), b \(\ne\) 0, |b| \(\ne\) 1, be points such that the area of triangle ABC is 1 sq. unit, then the sum of all possible values of a is :

Let [\(\lambda\)] be the greatest integer less than or equal to \(\lambda\). The set of all values of \(\lambda\) for which the system of linear equations x + y + z = 4, 3x + 2y + 5z = 3, 9x + 4y + (28 + [\(\lambda\)])z = [\(\lambda\)] has ...

If two tangents drawn from a point P to the parabola y2 = 16(x \(-\) 3) are at right angles, then the locus of point P is :

Let S = {1, 2, 3, 4, 5, 6, 9}. Then the number of elements in the set T = {A \(\subseteq\) S : A \(\ne\) \(\phi\) and the sum of all the elements of A is not a multiple of 3} is _______________.

Each of the persons A and B independently tosses three fair coins. The probability that both of them get the same number of heads is :

The probability distribution of random variable X is given by :
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The set of all values of K > \(-\)1, for which the equation \({(3{x^2} + 4x + 3)^2} - (k + 1)(3{x^2} + 4x + 3)(3{x^2} + 4x + 2) + k{(3{x^2} + 4x + 2)^2} = 0\) has real roots, is :

If 0 < x < 1 and \(y = {1 \over 2}{x^2} + {2 \over 3}{x^3} + {3 \over 4}{x^4} + ....\), then the value of e1 + y at \(x = {1 \over 2}\) is :

An online exam is attempted by 50 candidates out of which 20 are boys. The average marks obtained by boys is 12 with a variance 2. The variance of marks obtained by 30 girls is also 2. The average marks of all 50 candidates is 15. If $$\mu$...

Let S be the sum of all solutions (in radians) of the equation \({\sin ^4}\theta + {\cos ^4}\theta - \sin \theta \cos \theta = 0\) in [0, 4\(\pi\)]. Then \({{8S} \over \pi }\) is equal to ____________.