JEE Main 2021 (Online) 25th July Evening Shift

If the lines \({{x - k} \over 1} = {{y - 2} \over 2} = {{z - 3} \over 3}\) and \({{x + 1} \over 3} = {{y + 2} \over 2} = {{z + 3} \over 1}\) are co-planar, then the value of k is _____________.

The sum of all those terms which are rational numbers in the expansion of (21/3 + 31/4)12 is :

If the greatest value of the term independent of 'x' in the expansion of \({\left( {x\sin \alpha + a{{\cos \alpha } \over x}} \right)^{10}}\) is \({{10!} \over {{{(5!)}^2}}}\), then the value of 'a' is equal to :

The lowest integer which is greater than \({\left( {1 + {1 \over {{{10}^{100}}}}} \right)^{{{10}^{100}}}}\) is ______________.

Let n\(\in\)N and [x] denote the greatest integer less than or equal to x. If the sum of (n + 1) terms \({}^n{C_0},3.{}^n{C_1},5.{}^n{C_2},7.{}^n{C_3},.....\) is equal to 2100 . 101, then \(2\left[ {{{n - 1} \over 2}} \right]\) is equal to ...

If the co-efficient of x7 and x8 in the expansion of \({\left( {2 + {x \over 3}} \right)^n}\) are equal, then the value of n is equal to _____________.

The equation of a circle is Re(z2) + 2(Im(z))2 + 2Re(z) = 0, where z = x + iy. A line which passes through the center of the given circle and the vertex of the parabola, x2 \(-\) 6x \(-\) y + 13 = 0, has y-intercept equal to ______________.

If \(f(x) = \left\{ {\matrix{ {\int\limits_0^x {\left( {5 + \left| {1 - t} \right|} \right)dt,} } & {x > 2} \cr {5x + 1,} & {x \le 2} \cr } } \right.\), then

The value of the integral \(\int\limits_{ - 1}^1 {\log \left( {x + \sqrt {{x^2} + 1} } \right)dx}\) is :

Let y = y(x) be the solution of the differential equation xdy = (y + x3 cosx)dx with y(\(\pi\)) = 0, then \(y\left( {{\pi \over 2}} \right)\) is equal to :

Let a curve y = f(x) pass through the point (2, (loge2)2) and have slope \({{2y} \over {x{{\log }_e}x}}\) for all positive real value of x. Then the value of f(e) is equal to ______________.

If a tangent to the ellipse x2 + 4y2 = 4 meets the tangents at the extremities of it major axis at B and C, then the circle with BC as diameter passes through the point :

Consider function f : A \(\to\) B and g : B \(\to\) C (A, B, C \(\subseteq\) R) such that (gof)\(-\)1 exists, then :

Consider the functionwhere P(x) is a polynomial such that P'' (x) is always a constant and P(3) = 9. If f(x) is continuous at x = 2, then P(5) is equal to _____________.

Consider the statement "The match will be played only if the weather is good and ground is not wet". Select the correct negation from the following :

The number of distinct real roots of \(\left| {\matrix{ {\sin x} & {\cos x} & {\cos x} \cr {\cos x} & {\sin x} & {\cos x} \cr {\cos x} & {\cos x} & {\sin x} \cr } } \right| = 0\) in the interval $$ - {\pi \over 4} \le x \l...

If \(P = \left[ {\matrix{ 1 & 0 \cr {{1 \over 2}} & 1 \cr } } \right]\), then P50 is :

If \({}^n{P_r} = {}^n{P_{r + 1}}\) and \({}^n{C_r} = {}^n{C_{r - 1}}\), then the value of r is equal to :

Let X be a random variable such that the probability function of a distribution is given by \(P(X = 0) = {1 \over 2},P(X = j) = {1 \over {{3^j}}}(j = 1,2,3,...,\infty )\). Then the mean of the distribution and P(X is positive and even) resp...

A fair coin is tossed n-times such that the probability of getting at least one head is at least 0.9. Then the minimum value of n is ______________.

If a rectangle is inscribed in an equilateral triangle of side length \(2\sqrt 2\) as shown in the figure, then the square of the largest area of such a rectangle is _____________.

If [x] be the greatest integer less than or equal to x, then \(\sum\limits_{n = 8}^{100} {\left[ {{{{{( - 1)}^n}n} \over 2}} \right]}\) is equal to :

The number of real solutions of the equation, x2 \(-\) |x| \(-\) 12 = 0 is :

If a + b + c = 1, ab + bc + ca = 2 and abc = 3, then the value of a4 + b4 + c4 is equal to ______________.

The first of the two samples in a group has 100 items with mean 15 and standard deviation 3. If the whole group has 250 items with mean 15.6 and standard deviation \(\sqrt {13.44}\), then the standard deviation of the second sample is :

Let the equation of the pair of lines, y = px and y = qx, can be written as (y \(-\) px) (y \(-\) qx) = 0. Then the equation of the pair of the angle bisectors of the lines x2 \(-\) 4xy \(-\) 5y2 = 0 is :

The value of \(\cot {\pi \over {24}}\) is :

Let a, b and c be distinct positive numbers. If the vectors \(a\widehat i + a\widehat j + c\widehat k,\widehat i+\widehat k\) and \(c\widehat i + c\widehat j + b\widehat k\) are co-planar, then c is equal to :

If \(\left| {\overrightarrow a } \right| = 2,\left| {\overrightarrow b } \right| = 5\) and \(\left| {\overrightarrow a \times \overrightarrow b } \right|\) = 8, then \(\left| {\overrightarrow a .\,\overrightarrow b } \right|\) is equal to ...

If \(\left( {\overrightarrow a + 3\overrightarrow b } \right)\) is perpendicular to \(\left( {7\overrightarrow a - 5\overrightarrow b } \right)\) and \(\left( {\overrightarrow a - 4\overrightarrow b } \right)\) is perpendicular to $$\lef...