JEE Advanced 2021 Paper 2 Online

For any real numbers \(\alpha\) and \(\beta\), let \({y_{\alpha ,\beta }}(x)\), x\(\in\)R, be the solution of the differential equation \({{dy} \over {dx}} + \alpha y = x{e^{\beta x}},y(1) = 1\). Let $$S = \{ {y_{\alpha ,\beta }}(x):\alpha ...

Let f1 : (0, \(\infty\)) \(\to\) R and f2 : (0, \(\infty\)) \(\to\) R be defined by \({f_1}(x) = \int\limits_0^x {\prod\limits_{j = 1}^{21} {{{(t - j)}^j}dt} }\), x > 0 and \({f_2}(x) = 98{(x - 1)^{50}} - 600{(x - 1)^{49}} + 2450,x > 0\), ...

Let f1 : (0, \(\infty\)) \(\to\) R and f2 : (0, \(\infty\)) \(\to\) R be defined by \({f_1}(x) = \int\limits_0^x {\prod\limits_{j = 1}^{21} {{{(t - j)}^j}dt} }\), x > 0 and \({f_2}(x) = 98{(x - 1)^{50}} - 600{(x - 1)^{49}} + 2450,x > 0\), ...

Consider the region R = {(x, y) \(\in\) R \(\times\) R : x \(\ge\) 0 and y2 \(\le\) 4 \(-\) x}. Let F be the family of all circles that are contained in R and have centers on the x-axis. Let C be the circle that has largest radius among the...

Consider the region R = {(x, y) \(\in\) R \(\times\) R : x \(\ge\) 0 and y2 \(\le\) 4 \(-\) x}. Let F be the family of all circles that are contained in R and have centers on the x-axis. Let C be the circle that has largest radius among the...

Consider M with \(r = {{1025} \over {513}}\). Let k be the number of all those circles Cn that are inside M. Let l be the maximum possible number of circles among these k circles such that no two circles intersect. Then

Consider M with \(r = {{({2^{199}} - 1)\sqrt 2 } \over {{2^{198}}}}\). The number of all those circles Dn that are inside M is

Let \(f:\left[ { - {\pi \over 2},{\pi \over 2}} \right] \to R\) be a continuous function such that \(f(0) = 1\) and \(\int_0^{{\pi \over 3}} {f(t)dt = 0}\). Then which of the following statements is(are) TRUE?

Let \({g_i}:\left[ {{\pi \over 8},{{3\pi } \over 8}} \right] \to R,i = 1,2\), and \(f:\left[ {{\pi \over 8},{{3\pi } \over 8}} \right] \to R\) be functions such that \({g_1}(x) = 1,{g_2}(x) = |4x - \pi |\) and \(f(x) = {\sin ^2}x\), for a...

Let \({g_i}:\left[ {{\pi \over 8},{{3\pi } \over 8}} \right] \to R,i = 1,2\), and \(f:\left[ {{\pi \over 8},{{3\pi } \over 8}} \right] \to R\) be functions such that \({g_1}(x) = 1,{g_2}(x) = |4x - \pi |\) and \(f(x) = {\sin ^2}x\), for a...

Which of the following statements is TRUE?

Which of the following statements is TRUE?

For any real number x, let [ x ] denote the largest integer less than or equal to x. If \(I = \int\limits_0^{10} {\left[ {\sqrt {{{10x} \over {x + 1}}} } \right]dx}\), then the value of 9I is __________.

Let E be the ellipse \({{{x^2}} \over {16}} + {{{y^2}} \over 9} = 1\). For any three distinct points P, Q and Q' on E, let M(P, Q) be the mid-point of the line segment joining P and Q, and M(P, Q') be the mid-point of the line segment joini...

Let E denote the parabola y2 = 8x. Let P = (\(-\)2, 4), and let Q and Q' be two distinct points on E such that the lines PQ and PQ' are tangents to E. Let F be the focus of E. Then which of the following statements is(are) TRUE?

Let \({S_1} = \left\{ {(i,j,k):i,j,k \in \{ 1,2,....,10\} } \right\}\),\({S_2} = \left\{ {(i,j):1 \le i < j + 2 \le 10,i,j \in \{ 1,2,...,10\} } \right\}\),$${S_3} = \left\{ {(i,j,k,l):1 \le i < j < k < l,i,j,k,l \in \{ 1,2,...,10\} } \righ...

A number of chosen at random from the set {1, 2, 3, ....., 2000}. Let p be the probability that the chosen number is a multiple of 3 or a multiple of 7. Then the value of 500p is __________.

Consider a triangle PQR having sides of lengths p, q and r opposite to the angles P, Q and R, respectively. Then which of the following statements is (are) TRUE?

Let O be the origin and \(\overrightarrow {OA} = 2\widehat i + 2\widehat j + \widehat k\) and \(\overrightarrow {OB} = \widehat i - 2\widehat j + 2\widehat k\) and $$\overrightarrow {OC} = {1 \over 2}\left( {\overrightarrow {OB} - \lamb...