1
JEE Advanced 2021 Paper 2 Online
Numerical
+2
-0
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 all $$x \in \left[ {{\pi \over 8},{{3\pi } \over 8}} \right]$$. Define $${S_i} = \int\limits_{{\pi \over 8}}^{{{3\pi } \over 8}} {f(x).{g_i}(x)dx}$$, i = 1, 2

The value of $${{16{S_1}} \over \pi }$$ is _____________.
2
JEE Advanced 2021 Paper 2 Online
Numerical
+2
-0
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 all $$x \in \left[ {{\pi \over 8},{{3\pi } \over 8}} \right]$$. Define $${S_i} = \int\limits_{{\pi \over 8}}^{{{3\pi } \over 8}} {f(x).{g_i}(x)dx}$$, i = 1, 2

The value of $${{48{S_2}} \over {{\pi ^2}}}$$ is ___________.
3
JEE Advanced 2021 Paper 2 Online
+3
-1
Let $$M = \{ (x,y) \in R \times R:{x^2} + {y^2} \le {r^2}\}$$, where r > 0. Consider the geometric progression $${a_n} = {1 \over {{2^{n - 1}}}}$$, n = 1, 2, 3, ...... . Let S0 = 0 and for n $$\ge$$ 1, let Sn denote the sum of the first n terms of this progression. For n $$\ge$$ 1, let Cn denote the circle with center (Sn$$-$$1, 0) and radius an, and Dn denote the circle with center (Sn$$-$$1, Sn$$-$$1) and radius an.
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
A
k + 2l = 22
B
2k + l = 26
C
2k + 3l = 34
D
3k + 2l = 40
4
JEE Advanced 2021 Paper 2 Online
+3
-1
Let $$M = \{ (x,y) \in R \times R:{x^2} + {y^2} \le {r^2}\}$$, where r > 0. Consider the geometric progression $${a_n} = {1 \over {{2^{n - 1}}}}$$, n = 1, 2, 3, ...... . Let S0 = 0 and for n $$\ge$$ 1, let Sn denote the sum of the first n terms of this progression. For n $$\ge$$ 1, let Cn denote the circle with center (Sn$$-$$1, 0) and radius an, and Dn denote the circle with center (Sn$$-$$1, Sn$$-$$1) and radius an.
Consider M with $$r = {{({2^{199}} - 1)\sqrt 2 } \over {{2^{198}}}}$$. The number of all those circles Dn that are inside M is
A
198
B
199
C
200
D
201
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