JEE Main 2021 (Online) 25th February Morning Shift

The equation of the line through the point (0, 1, 2) and perpendicular to the line \({{x - 1} \over 2} = {{y + 1} \over 3} = {{z - 1} \over { - 2}}\) is :

Let \(\alpha\) be the angle between the lines whose direction cosines satisfy the equations l + m \(-\) n = 0 and l2 + m2 \(-\) n2 = 0. Then the value of sin4\(\alpha\) + cos4\(\alpha\) is :

Let f(x) be a polynomial of degree 6 in x, in which the coefficient of x6 is unity and it has extrema at x = \(-\)1 and x = 1. If \(\mathop {\lim }\limits_{x \to 0} {{f(x)} \over {{x^3}}} = 1\), then \(5.f(2)\) is equal to _________.

If Rolle's theorem holds for the function \(f(x) = {x^3} - a{x^2} + bx - 4\), \(x \in [1,2]\) with \(f'\left( {{4 \over 3}} \right) = 0\), then ordered pair (a, b) is equal to :

If the curves, \({{{x^2}} \over a} + {{{y^2}} \over b} = 1\) and \({{{x^2}} \over c} + {{{y^2}} \over d} = 1\) intersect each other at an angle of 90\(^\circ\), then which of the following relations is TRUE?

The graphs of sine and cosine functions, intersect each other at a number of points and between two consecutive points of intersection, the two graphs enclose the same area A. Then A4 is equal to __________.

Let the lines (2 \(-\) i)z = (2 + i)\(\overline z\) and (2 \(+\) i)z + (i \(-\) 2)\(\overline z\) \(-\) 4i = 0, (here i2 = \(-\)1) be normal to a circle C. If the line iz + \(\overline z\) + 1 + i = 0 is tangent to this circle C, then it...

The value of \(\int\limits_{ - 1}^1 {{x^2}{e^{[{x^3}]}}} dx\), where [ t ] denotes the greatest integer \(\le\) t, is :

If a curve passes through the origin and the slope of the tangent to it at any point (x, y) is \({{{x^2} - 4x + y + 8} \over {x - 2}}\), then this curve also passes through the point :

Let f, g : N \(\to\) N such that f(n + 1) = f(n) + f(1) \(\forall\) n\(\in\)N and g be any arbitrary function. Which of the following statements is NOT true?

A man is observing, from the top of a tower, a boat speeding towards the lower from a certain point A, with uniform speed. At that point, angle of depression of the boat with the man's eye is 30\(^\circ\) (Ignore man's height). After sailin...

The locus of the point of intersection of the lines \(\left( {\sqrt 3 } \right)kx + ky - 4\sqrt 3 = 0\) and \(\sqrt 3 x - y - 4\left( {\sqrt 3 } \right)k = 0\) is a conic, whose eccentricity is _________.

The value of the integral \(\int {{{\sin \theta .\sin 2\theta ({{\sin }^6}\theta + {{\sin }^4}\theta + {{\sin }^2}\theta )\sqrt {2{{\sin }^4}\theta + 3{{\sin }^2}\theta + 6} } \over {1 - \cos 2\theta }}} \,d\theta\) is :

The number of points, at which the function f(x) = | 2x + 1 | \(-\) 3| x + 2 | + | x2 + x \(-\) 2 |, x\(\in\)R is not differentiable, is __________.

\(\mathop {\lim }\limits_{n \to \infty } {\left( {1 + {{1 + {1 \over 2} + ........ + {1 \over n}} \over {{n^2}}}} \right)^n}\) is equal to :

The statement A \(\to\) (B \(\to\) A) is equivalent to :

Let \(A = \left[ {\matrix{ x & y & z \cr y & z & x \cr z & x & y \cr } } \right]\), where x, y and z are real numbers such that x + y + z > 0 and xyz = 2. If \({A^2} = {I_3}\), then the value of \({x^3} + {y^3} + {z^3}\) is...

If the system of equationskx + y + 2z = 13x \(-\) y \(-\) 2z = 2\(-\)2x \(-\)2y \(-\)4z = 3has infinitely many solutions, then k is equal to __________.

If \(A = \left[ {\matrix{ 0 & { - \tan \left( {{\theta \over 2}} \right)} \cr {\tan \left( {{\theta \over 2}} \right)} & 0 \cr } } \right]\) and $$({I_2} + A){({I_2} - A)^{ - 1}} = \left[ {\matrix{
a & { - b} \cr
b & a...

A tangent is drawn to the parabola y2 = 6x which is perpendicular to the line 2x + y = 1. Which of the following points does NOT lie on it?

The total number of numbers, lying between 100 and 1000 that can be formed with the digits 1, 2, 3, 4, 5, if the repetition of digits is not allowed and numbers are divisible by either 3 or 5, is _____________.

The total number of positive integral solutions (x, y, z) such that xyz = 24 is :

When a missile is fired from a ship, the probability that it is intercepted is \({1 \over 3}\) and the probability that the missile hits the target, given that it is not intercepted, is \({3 \over 4}\). If three missiles are fired independe...

The coefficients a, b and c of the quadratic equation, ax2 + bx + c = 0 are obtained by throwing a dice three times. The probability that this equation has equal roots is :

The integer 'k', for which the inequality x2 \(-\) 2(3k \(-\) 1)x + 8k2 \(-\) 7 > 0 is valid for every x in R, is :

Let A1, A2, A3, ....... be squares such that for each n \(\ge\) 1, the length of the side of An equals the length of diagonal of An+1. If the length of A1 is 12 cm, then the smallest value of n for which area of An is less than one, is __...

If \(0 < \theta ,\phi < {\pi \over 2},x = \sum\limits_{n = 0}^\infty {{{\cos }^{2n}}\theta } ,y = \sum\limits_{n = 0}^\infty {{{\sin }^{2n}}\phi }\) and \(z = \sum\limits_{n = 0}^\infty {{{\cos }^{2n}}\theta .{{\sin }^{2n}}\phi }\) t...

The image of the point (3, 5) in the line x \(-\) y + 1 = 0, lies on :

All possible values of \(\theta\) \(\in\) [0, 2\(\pi\)] for which sin 2\(\theta\) + tan 2\(\theta\) > 0 lie in :

Let \(\overrightarrow a = \widehat i + 2\widehat j - \widehat k\), \(\overrightarrow b = \widehat i - \widehat j\) and \(\overrightarrow c = \widehat i - \widehat j - \widehat k\) be three given vectors. If \(\overrightarrow r\) is a ve...