1
JEE Main 2021 (Online) 27th July Evening Shift
+4
-1
Let f : R $$\to$$ R be defined as $$f(x + y) + f(x - y) = 2f(x)f(y),f\left( {{1 \over 2}} \right) = - 1$$. Then, the value of $$\sum\limits_{k = 1}^{20} {{1 \over {\sin (k)\sin (k + f(k))}}}$$ is equal to :
A
cosec2(21) cos(20) cos(2)
B
sec2(1) sec(21) cos(20)
C
cosec2(1) cosec(21) sin(20)
D
sec2(21) sin(20) sin(2)
2
JEE Main 2021 (Online) 25th July Evening Shift
+4
-1
Consider function f : A $$\to$$ B and g : B $$\to$$ C (A, B, C $$\subseteq$$ R) such that (gof)$$-$$1 exists, then :
A
f and g both are one-one
B
f and g both are onto
C
f is one-one and g is onto
D
f is onto and g is one-one
3
JEE Main 2021 (Online) 25th July Morning Shift
+4
-1
Let g : N $$\to$$ N be defined as

g(3n + 1) = 3n + 2,

g(3n + 2) = 3n + 3,

g(3n + 3) = 3n + 1, for all n $$\ge$$ 0.

Then which of the following statements is true?
A
There exists an onto function f : N $$\to$$ N such that fog = f
B
There exists a one-one function f : N $$\to$$ N such that fog = f
C
gogog = g
D
There exists a function : f : N $$\to$$ N such that gof = f
4
JEE Main 2021 (Online) 20th July Evening Shift
+4
-1
Let $$f:R - \left\{ {{\alpha \over 6}} \right\} \to R$$ be defined by $$f(x) = {{5x + 3} \over {6x - \alpha }}$$. Then the value of $$\alpha$$ for which (fof)(x) = x, for all $$x \in R - \left\{ {{\alpha \over 6}} \right\}$$, is :
A
No such $$\alpha$$ exists
B
5
C
8
D
6
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