JEE Advanced 2021 Paper 1 Online

The area of the region \(\left\{ {\matrix{ {(x,y):0 \le x \le {9 \over 4},} & {0 \le y \le 1,} & {x \ge 3y,} & {x + y \ge 2} \cr } } \right\}\) is

Consider a triangle \(\Delta\) whose two sides lie on the x-axis and the line x + y + 1 = 0. If the orthocenter of \(\Delta\) is (1, 1), then the equation of the circle passing through the vertices of the triangle \(\Delta\) is

Let $\theta_1, \theta_2, \ldots, \theta_{10}$ be positive valued angles (in radian) such that $\theta_1+\theta_2+\cdots+\theta_{10}=2 \pi$. Define the complex numbers $z_1=e^{i \theta_1}, z_k=z_{k-1} e^{i \theta_k}$ for $k=2,3, \ldots, 10$,...

For any complex number w = c + id, let \(\arg (w) \in ( - \pi ,\pi ]\), where \(i = \sqrt { - 1}\). Let \(\alpha\) and \(\beta\) be real numbers such that for all complex numbers z = x + iy satisfying $$\arg \left( {{{z + \alpha } \over {z...

Let f : R \(\to\) R be defined by \(f(x) = {{{x^2} - 3x - 6} \over {{x^2} + 2x + 4}}\)Then which of the following statements is (are) TRUE?

Let \(\alpha\), \(\beta\) and \(\gamma\) be real numbers such that the system of linear equationsx + 2y + 3z = \(\alpha\)4x + 5y + 6z = \(\beta\)7x + 8y + 9z = \(\gamma\) \(-\) 1is consistent. Let | M | represent the determinant of the mat...

Let \(\alpha\), \(\beta\) and \(\gamma\) be real numbers such that the system of linear equationsx + 2y + 3z = \(\alpha\)4x + 5y + 6z = \(\beta\)7x + 8y + 9z = \(\gamma\) \(-\) 1is consistent. Let | M | represent the determinant of the mat...

For any 3 \(\times\) 3 matrix M, let | M | denote the determinant of M. Let\(E = \left[ {\matrix{ 1 & 2 & 3 \cr 2 & 3 & 4 \cr 8 & {13} & {18} \cr } } \right]\), $$P = \left[ {\matrix{
1 & 0 & 0 \cr
0 & 0 & 1 \cr
...

For any 3 \(\times\) 3 matrix M, let |M| denote the determinant of M. Let I be the 3 \(\times\) 3 identity matrix. Let E and F be two 3 \(\times\) 3 matrices such that (I \(-\) EF) is invertible. If G = (I \(-\) EF)\(-\)1, then which of the...

Consider three sets E1 = {1, 2, 3}, F1 = {1, 3, 4} and G1 = {2, 3, 4, 5}. Two elements are chosen at random, without replacement, from the set E1, and let S1 denote the set of these chosen elements. Let E2 = E1 \(-\) S1 and F2 = F1 \(\cup\)...

Three numbers are chosen at random, one after another with replacement, from the set S = {1, 2, 3, ......, 100}. Let p1 be the probability that the maximum of chosen numbers is at least 81 and p2 be the probability that the minimum of chose...

Three numbers are chosen at random, one after another with replacement, from the set S = {1, 2, 3, ......, 100}. Let p1 be the probability that the maximum of chosen numbers is at least 81 and p2 be the probability that the minimum of chose...

Let E, F and G be three events having probabilities \(P(E) = {1 \over 8}\), \(P(F) = {1 \over 6}\) and \(P(G) = {1 \over 4}\), and let P (E \(\cap\) F \(\cap\) G) = \({1 \over {10}}\). For any event H, if Hc denotes the complement, then whi...

In a triangle ABC, let AB = \(\sqrt {23}\), BC = 3 and CA = 4. Then the value of \({{\cot A + \cot C} \over {\cot B}}\) is _________.

For x \(\in\) R, the number of real roots of the equation \(3{x^2} - 4\left| {{x^2} - 1} \right| + x - 1 = 0\) is ________.

For any positive integer n, let Sn : (0, \(\infty\)) \(\to\) R be defined by \({S_n}(x) = \sum\nolimits_{k = 1}^n {{{\cot }^{ - 1}}\left( {{{1 + k(k + 1){x^2}} \over x}} \right)}\), where for any x \(\in\) R, $${\cot ^{ - 1}}(x) \in (0,\pi...

Consider the lines L1 and L2 defined by \({L_1}:x\sqrt 2 + y - 1 = 0\) and \({L_2}:x\sqrt 2 - y + 1 = 0\)For a fixed constant \(\lambda\), let C be the locus of a point P such that the product of the distance of P from L1 and the distance...

Consider the lines L1 and L2 defined by \({L_1}:x\sqrt 2 + y - 1 = 0\) and \({L_2}:x\sqrt 2 - y + 1 = 0\)For a fixed constant \(\lambda\), let C be the locus of a point P such that the product of the distance of P from L1 and the distance...

Let \(\overrightarrow u\), \(\overrightarrow v\) and \(\overrightarrow w\) be vectors in three-dimensional space, where \(\overrightarrow u\) and \(\overrightarrow v\) are unit vectors which are not perpendicular to each other and $$\o...