1
JEE Advanced 2020 Paper 1 Offline
Numerical
+4
-0
Let f : [0, 2] $$\to$$ R be the function defined by

$$f(x) = (3 - \sin (2\pi x))\sin \left( {\pi x - {\pi \over 4}} \right) - \sin \left( {3\pi x + {\pi \over 4}} \right)$$

If $$\alpha ,\,\beta \in [0,2]$$ are such that $$\{ x \in [0,2]:f(x) \ge 0\} = [\alpha ,\beta ]$$, then the value of $$\beta - \alpha$$ is ..........
2
JEE Advanced 2020 Paper 1 Offline
Numerical
+4
-0
For a polynomial g(x) with real coefficients, let mg denote the number of distinct real roots of g(x). Suppose S is the set of polynomials with real coefficients defined by

$$S = \{ {({x^2} - 1)^2}({a_0} + {a_1}x + {a_2}{x^2} + {a_3}{x^3}):{a_0},{a_1},{a_2},{a_3} \in R\}$$;

For a polynomial f, let f' and f'' denote its first and second order derivatives, respectively. Then the minimum possible value of (mf' + mf''), where f $$\in$$ S, is ..............
3
JEE Advanced 2018 Paper 2 Offline
Numerical
+3
-0
Let X be a set with exactly 5 elements and Y be a set with exactly 7 elements. If $$\alpha$$ is the number of one-one functions from X to Y and $$\beta$$ is the number of onto functions from Y to X, then the value of $${1 \over {5!}}(\beta - \alpha )$$ is ..................
4
IIT-JEE 2009 Paper 2 Offline
Numerical
+3
-1

If the function $$f(x) = {x^3} + {e^{x/2}}$$ and $$g(x) = {f^{ - 1}}(x)$$, then the value of $$g'(1)$$ is _________.