1
JEE Main 2022 (Online) 25th July Morning Shift
+4
-1

Let the locus of the centre $$(\alpha, \beta), \beta>0$$, of the circle which touches the circle $$x^{2}+(y-1)^{2}=1$$ externally and also touches the $$x$$-axis be $$\mathrm{L}$$. Then the area bounded by $$\mathrm{L}$$ and the line $$y=4$$ is:

A
$$\frac{32 \sqrt{2}}{3}$$
B
$$\frac{40 \sqrt{2}}{3}$$
C
$$\frac{64}{3}$$
D
$$\frac{32}{3}$$
2
JEE Main 2022 (Online) 30th June Morning Shift
+4
-1

$$\mathop {\lim }\limits_{n \to \infty } \sum\limits_{r = 1}^n {{r \over {2{r^2} - 7rn + 6{n^2}}}}$$ is equal to :

A
$${\log _e}\left( {{{\sqrt 3 } \over 2}} \right)$$
B
$${\log _e}\left( {{{3\sqrt 3 } \over 4}} \right)$$
C
$${\log _e}\left( {{{27} \over 4}} \right)$$
D
$${\log _e}\left( {{4 \over 3}} \right)$$
3
JEE Main 2022 (Online) 30th June Morning Shift
+4
-1

Let $${{dy} \over {dx}} = {{ax - by + a} \over {bx + cy + a}},\,a,b,c \in R$$, represents a circle with center ($$\alpha$$, $$\beta$$). Then, $$\alpha$$ + 2$$\beta$$ is equal to :

A
$$-$$1
B
0
C
1
D
2
4
JEE Main 2022 (Online) 29th June Evening Shift
+4
-1

Let f be a real valued continuous function on [0, 1] and $$f(x) = x + \int\limits_0^1 {(x - t)f(t)dt}$$.

Then, which of the following points (x, y) lies on the curve y = f(x) ?

A
(2, 4)
B
(1, 2)
C
(4, 17)
D
(6, 8)
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