1
JEE Advanced 2020 Paper 2 Offline
Numerical
+4
-0 Let the function $$f:(0,\pi ) \to R$$ be defined by $$f(\theta ) = {(\sin \theta + \cos \theta )^2} + {(\sin \theta - \cos \theta )^4}$$

Suppose the function f has a local minimum at $$\theta$$ precisely when $$\theta \in \{ {\lambda _1}\pi ,....,{\lambda _r}\pi \}$$, where $$0 < {\lambda _1} < ...{\lambda _r} < 1$$. Then the value of $${\lambda _1} + ... + {\lambda _r}$$ is .............
2
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 ..........
3
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 ..............
4
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 ..................
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