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1
JEE Main 2021 (Online) 18th March Evening Shift
Numerical
+4
-1
Let f : R $$\to$$ R satisfy the equation f(x + y) = f(x) . f(y) for all x, y $$\in$$R and f(x) $$\ne$$ 0 for any x$$\in$$R. If the function f is differentiable at x = 0 and f'(0) = 3, then

$$\mathop {\lim }\limits_{h \to 0} {1 \over h}(f(h) - 1)$$ is equal to ____________.
2
JEE Main 2021 (Online) 17th March Morning Shift
Numerical
+4
-1
If the function $$f(x) = {{\cos (\sin x) - \cos x} \over {{x^4}}}$$ is continuous at each point in its domain and $$f(0) = {1 \over k}$$, then k is ____________.
3
JEE Main 2021 (Online) 16th March Evening Shift
Numerical
+4
-1
Let f : R $$\to$$ R and g : R $$\to$$ R be defined as

$$f(x) = \left\{ {\matrix{ {x + a,} & {x < 0} \cr {|x - 1|,} & {x \ge 0} \cr } } \right.$$ and

$$g(x) = \left\{ {\matrix{ {x + 1,} & {x < 0} \cr {{{(x - 1)}^2} + b,} & {x \ge 0} \cr } } \right.$$,

where a, b are non-negative real numbers. If (gof) (x) is continuous for all x $$\in$$ R, then a + b is equal to ____________.
Let f : (0, 2) $$\to$$ R be defined as f(x) = log2$$\left( {1 + \tan \left( {{{\pi x} \over 4}} \right)} \right)$$. Then, $$\mathop {\lim }\limits_{n \to \infty } {2 \over n}\left( {f\left( {{1 \over n}} \right) + f\left( {{2 \over n}} \right) + ... + f(1)} \right)$$ is equal to ___________.