1
JEE Main 2021 (Online) 27th July Morning Shift
Numerical
+4
-1
Let the domain of the function

$$f(x) = {\log _4}\left( {{{\log }_5}\left( {{{\log }_3}(18x - {x^2} - 77)} \right)} \right)$$ be (a, b). Then the value of the integral $$\int\limits_a^b {{{{{\sin }^3}x} \over {({{\sin }^3}x + {{\sin }^3}(a + b - x)}}} dx$$ is equal to _____________.
2
JEE Main 2021 (Online) 27th July Morning Shift
Numerical
+4
-1
Let $$F:[3,5] \to R$$ be a twice differentiable function on (3, 5) such that

$$F(x) = {e^{ - x}}\int\limits_3^x {(3{t^2} + 2t + 4F'(t))dt}$$. If $$F'(4) = {{\alpha {e^\beta } - 224} \over {{{({e^\beta } - 4)}^2}}}$$, then $$\alpha$$ + $$\beta$$ is equal to _______________.
3
JEE Main 2021 (Online) 18th March Evening Shift
Numerical
+4
-1
Let P(x) be a real polynomial of degree 3 which vanishes at x = $$-$$3. Let P(x) have local minima at x = 1, local maxima at x = $$-$$1 and $$\int\limits_{ - 1}^1 {P(x)dx}$$ = 18, then the sum of all the coefficients of the polynomial P(x) is equal to _________.
4
JEE Main 2021 (Online) 18th March Morning Shift
Numerical
+4
-1
Let f(x) and g(x) be two functions satisfying f(x2) + g(4 $$-$$ x) = 4x3 and g(4 $$-$$ x) + g(x) = 0, then the value of $$\int\limits_{ - 4}^4 {f{{(x)}^2}dx}$$ is