1
JEE Main 2022 (Online) 30th June Morning Shift
MCQ (Single Correct Answer)
+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
$${\log _e}\left( {{4 \over 3}} \right)$$
2
JEE Main 2022 (Online) 30th June Morning Shift
MCQ (Single Correct Answer)
+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
3
JEE Main 2022 (Online) 29th June Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

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)
4
JEE Main 2022 (Online) 29th June Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

If $$\int\limits_0^2 {\left( {\sqrt {2x} - \sqrt {2x - {x^2}} } \right)dx = \int\limits_0^1 {\left( {1 - \sqrt {1 - {y^2}} - {{{y^2}} \over 2}} \right)dy + \int\limits_1^2 {\left( {2 - {{{y^2}} \over 2}} \right)dy + I} } } $$, then I equals

A
$$\int\limits_0^1 {\left( {1 + \sqrt {1 - {y^2}} } \right)dy} $$
B
$$\int\limits_0^1 {\left( {{{{y^2}} \over 2} - \sqrt {1 - {y^2}} + 1} \right)dy} $$
C
$$\int\limits_0^1 {\left( {1 - \sqrt {1 - {y^2}} } \right)dy} $$
D
$$\int\limits_0^1 {\left( {{{{y^2}} \over 2} + \sqrt {1 - {y^2}} + 1} \right)dy} $$
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