1
JEE Main 2022 (Online) 30th June Morning Shift
Numerical
+4
-1

Suppose $$\mathop {\lim }\limits_{x \to 0} {{F(x)} \over {{x^3}}}$$ exists and is equal to L, where

$$F(x) = \left| {\matrix{ {a + \sin {x \over 2}} & { - b\cos x} & 0 \cr { - b\cos x} & 0 & {a + \sin {x \over 2}} \cr 0 & {a + \sin {x \over 2}} & { - b\cos x} \cr } } \right|$$.

Then, $$-$$112 L is equal to ___________.

2
JEE Main 2022 (Online) 28th June Evening Shift
Numerical
+4
-1

If $$\mathop {\lim }\limits_{x \to 1} {{\sin (3{x^2} - 4x + 1) - {x^2} + 1} \over {2{x^3} - 7{x^2} + ax + b}} = - 2$$, then the value of (a $$-$$ b) is equal to ___________.

3
JEE Main 2022 (Online) 25th June Evening Shift
Numerical
+4
-1

Let $$f(x) = \left[ {2{x^2} + 1} \right]$$ and $$g(x) = \left\{ {\matrix{ {2x - 3,} & {x < 0} \cr {2x + 3,} & {x \ge 0} \cr } } \right.$$, where [t] is the greatest integer $$\le$$ t. Then, in the open interval ($$-$$1, 1), the number of points where fog is discontinuous is equal to ______________.

4
JEE Main 2021 (Online) 1st September Evening Shift
Numerical
+4
-1
Let $$f(x) = {x^6} + 2{x^4} + {x^3} + 2x + 3$$, x $$\in$$ R. Then the natural number n for which $$\mathop {\lim }\limits_{x \to 1} {{{x^n}f(1) - f(x)} \over {x - 1}} = 44$$ is __________.
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