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1
JEE Main 2022 (Online) 29th June Morning Shift
Numerical
+4
-1
English
Hindi

Let c, k $$\in$$ R. If $$f(x) = (c + 1){x^2} + (1 - {c^2})x + 2k$$ and $$f(x + y) = f(x) + f(y) - xy$$, for all x, y $$\in$$ R, then the value of $$|2(f(1) + f(2) + f(3) + \,\,......\,\, + \,\,f(20))|$$ is equal to ____________.

2
JEE Main 2022 (Online) 27th June Evening Shift
Numerical
+4
-1
English
Hindi

Let S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Define f : S $$\to$$ S as

$$f(n) = \left\{ {\matrix{ {2n} & , & {if\,n = 1,2,3,4,5} \cr {2n - 11} & , & {if\,n = 6,7,8,9,10} \cr } } \right.$$.

Let g : S $$\to$$ S be a function such that $$fog(n) = \left\{ {\matrix{ {n + 1} & , & {if\,n\,\,is\,odd} \cr {n - 1} & , & {if\,n\,\,is\,even} \cr } } \right.$$.

Then $$g(10)g(1) + g(2) + g(3) + g(4) + g(5))$$ is equal to _____________.

3
JEE Main 2022 (Online) 27th June Evening Shift
Numerical
+4
-1
English
Hindi

Let [t] denote the greatest integer $$\le$$ t and {t} denote the fractional part of t. The integral value of $$\alpha$$ for which the left hand limit of the function

$$f(x) = [1 + x] + {{{\alpha ^{2[x] + {\{x\}}}} + [x] - 1} \over {2[x] + \{ x\} }}$$ at x = 0 is equal to $$\alpha - {4 \over 3}$$, is _____________.

Let f : R $$\to$$ R be a function defined by $$f(x) = {{2{e^{2x}}} \over {{e^{2x}} + e}}$$. Then $$f\left( {{1 \over {100}}} \right) + f\left( {{2 \over {100}}} \right) + f\left( {{3 \over {100}}} \right) + \,\,\,.....\,\,\, + \,\,\,f\left( {{{99} \over {100}}} \right)$$ is equal to ______________.