JEE Main 2021 (Online) 18th March Evening Shift

Let P be a plane containing the line \({{x - 1} \over 3} = {{y + 6} \over 4} = {{z + 5} \over 2}\) and parallel to the line \({{x - 1} \over 4} = {{y - 2} \over { - 3}} = {{z + 5} \over 7}\). If the point (1, \(-\)1, \(\alpha\)) lies on the...

Let the mirror image of the point (1, 3, a) with respect to the plane \(\overrightarrow r .\left( {2\widehat i - \widehat j + \widehat k} \right) - b = 0\) be (\(-\)3, 5, 2). Then, the value of | a + b | is equal to ____________.

The area bounded by the curve 4y2 = x2(4 \(-\) x)(x \(-\) 2) is equal to :

The term independent of x in the expansion of \({\left[ {{{x + 1} \over {{x^{2/3}} - {x^{1/3}} + 1}} - {{x - 1} \over {x - {x^{1/2}}}}} \right]^{10}}\), x \(\ne\) 1, is equal to ____________.

Let \({}^n{C_r}\) denote the binomial coefficient of xr in the expansion of (1 + x)n.
If \(\sum\limits_{k = 0}^{10} {({2^2} + 3k)} {}^{10}{C_k} = \alpha {.3^{10}} + \beta {.2^{10}},\alpha ,\beta \in R\), then \(\alpha\) + \(\beta\) is eq...

Let S1 : x2 + y2 = 9 and S2 : (x \(-\) 2)2 + y2 = 1. Then the locus of center of a variable circle S which touches S1 internally and S2 externally always passes through the points :

Let a complex number be w = 1 \(-\) \({\sqrt 3 }\)i. Let another complex number z be such that |zw| = 1 and arg(z) \(-\) arg(w) = \({\pi \over 2}\). Then the area of the triangle with vertices origin, z and w is equal to :

Let P(x) be a real polynomial of degree 3 which vanishes at x = \(-\)3. Let P(x) have local minima at x = 1, local maxima at x = \(-\)1 and \(\int\limits_{ - 1}^1 {P(x)dx}\) = 18, then the sum of all the coefficients of the polynomial P(x)...

Let g(x) = \(\int_0^x {f(t)dt}\), where f is continuous function in [ 0, 3 ] such that \({1 \over 3}\) \(\le\) f(t) \(\le\) 1 for all t\(\in\) [0, 1] and 0 \(\le\) f(t) \(\le\) \({1 \over 2}\) for all t\(\in\) (1, 3]. The largest p...

Let y = y(x) be the solution of the differential equation xdy \(-\) ydx = \(\sqrt {({x^2} - {y^2})} dx\), x \(\ge\) 1, with y(1) = 0. If the area bounded by the line x = 1, x = e\(\pi\), y = 0 and y = y(x) is \(\alpha\)e2\(\pi\) + $$\beta...

Let y = y(x) be the solution of the differential equation \({{dy} \over {dx}} = (y + 1)\left( {(y + 1){e^{{x^2}/2}} - x} \right)\), 0 < x < 2.1, with y(2) = 0. Then the value of \({{dy} \over {dx}}\) at x = 1 is equal to :

Let a tangent be drawn to the ellipse \({{{x^2}} \over {27}} + {y^2} = 1\) at \((3\sqrt 3 \cos \theta ,\sin \theta )\) where \(0 \in \left( {0,{\pi \over 2}} \right)\). Then the value of \(\theta\) such that the sum of intercepts on axes m...

If f(x) and g(x) are two polynomials such that the polynomial P(x) = f(x3) + x g(x3) is divisible by x2 + x + 1, then P(1) is equal to ___________.

Let f : R \(-\) {3} \(\to\) R \(-\) {1} be defined by f(x) = \({{x - 2} \over {x - 3}}\).Let g : R \(\to\) R be given as g(x) = 2x \(-\) 3. Then, the sum of all the values of x for which f\(-\)1(x) + g\(-\)1(x) = \({{13} \over 2}\) is e...

A pole stands vertically inside a triangular park ABC. Let the angle of elevation of the top of the pole from each corner of the park be \({\pi \over 3}\). If the radius of the circumcircle of \(\Delta\)ABC is 2, then the height of the pol...

Consider a hyperbola H : x2 \(-\) 2y2 = 4. Let the tangent at a point P(4, \({\sqrt 6 }\)) meet the x-axis at Q and latus rectum at R(x1, y1), x1 > 0. If F is a focus of H which is nearer to the point P, then the area of \(\Delta\)QFR is eq...

Let f : R \(\to\) R satisfy the equation f(x + y) = f(x) . f(y) for all x, y \(\in\)R and f(x) \(\ne\) 0 for any x\(\in\)R. If the function f is differentiable at x = 0 and f'(0) = 3, then $$\mathop {\lim }\limits_{h \to 0} {1 \over h}(f(...

Let f : R \(\to\) R be a function defined as$$f(x) = \left\{ \matrix{
{{\sin (a + 1)x + \sin 2x} \over {2x}},if\,x < 0 \hfill \cr
b,\,if\,x\, = 0 \hfill \cr
{{\sqrt {x + b{x^3}} - \sqrt x } \over {b{x^{5/2}}}},\,if\,x > 0 \hfill ...

If P and Q are two statements, then which of the following compound statement is a tautology?

Let I be an identity matrix of order 2 \(\times\) 2 and P = \(\left[ {\matrix{ 2 & { - 1} \cr 5 & { - 3} \cr } } \right]\). Then the value of n\(\in\)N for which Pn = 5I \(-\) 8P is equal to ____________.

Let the system of linear equations 4x + \(\lambda\)y + 2z = 02x \(-\) y + z = 0\(\mu\)x + 2y + 3z = 0, \(\lambda\), \(\mu\)\(\in\)R.has a non-trivial solution. Then which of the following is true?

If \(\sum\limits_{r = 1}^{10} {r!({r^3} + 6{r^2} + 2r + 5) = \alpha (11!)}\), then the value of \(\alpha\) is equal to ___________.

Let in a Binomial distribution, consisting of 5 independent trials, probabilities of exactly 1 and 2 successes be 0.4096 and 0.2048 respectively. Then the probability of getting exactly 3 successes is equal to :

Let S1 be the sum of first 2n terms of an arithmetic progression. Let S2 be the sum of first 4n terms of the same arithmetic progression. If (S2 \(-\) S1) is 1000, then the sum of the first 6n terms of the arithmetic progression is equal to...

Define a relation R over a class of n \(\times\) n real matrices A and B as "ARB iff there exists a non-singular matrix P such that PAP\(-\)1 = B". Then which of the following is true?

Let in a series of 2n observations, half of them are equal to a and remaining half are equal to \(-\)a. Also by adding a constant b in each of these observations, the mean and standard deviation of new set become 5 and 20, respectively. The...

Let the centroid of an equilateral triangle ABC be at the origin. Let one of the sides of the equilateral triangle be along the straight line x + y = 3. If R and r be the radius of circumcircle and incircle respectively of \(\Delta\)ABC, th...

If 15sin4\(\alpha\) + 10cos4\(\alpha\) = 6, for some \(\alpha\)\(\in\)R, then the value of 27sec6\(\alpha\) + 8cosec6\(\alpha\) is equal to :

Let \(\overrightarrow a\) and \(\overrightarrow b\) be two non-zero vectors perpendicular to each other and \(|\overrightarrow a | = |\overrightarrow b |\). If \(|\overrightarrow a \times \overrightarrow b | = |\overrightarrow a |\), the...

In a triangle ABC, if \(|\overrightarrow {BC} | = 8,|\overrightarrow {CA} | = 7,|\overrightarrow {AB} | = 10\), then the projection of the vector \(\overrightarrow {AB}\) on \(\overrightarrow {AC}\) is equal to :