1
JEE Advanced 2024 Paper 1 Online
+3
-1

Let the straight line $y=2 x$ touch a circle with center $(0, \alpha), \alpha>0$, and radius $r$ at a point $A_1$. Let $B_1$ be the point on the circle such that the line segment $A_1 B_1$ is a diameter of the circle. Let $\alpha+r=5+\sqrt{5}$.

Match each entry in List-I to the correct entry in List-II.

List-I List-II
(P) $\alpha$ equals (1) $(-2, 4)$
(Q) $r$ equals (2) $\sqrt{5}$
(R) $A_1$ equals (3) $(-2, 6)$
(S) $B_1$ equals (4) $5$
(5) $(2, 4)$

The correct option is
A
$(\mathrm{P}) \rightarrow(4) \quad(\mathrm{Q}) \rightarrow(2) \quad(\mathrm{R}) \rightarrow(1) \quad(\mathrm{S}) \rightarrow(3)$
B
$(\mathrm{P}) \rightarrow(2) \quad(\mathrm{Q}) \rightarrow(4) \quad(\mathrm{R}) \rightarrow(1) \quad(\mathrm{S}) \rightarrow(3)$
C
$(\mathrm{P}) \rightarrow(4) \quad(\mathrm{Q}) \rightarrow(2) \quad(\mathrm{R}) \rightarrow(5) \quad(\mathrm{S}) \rightarrow(3)$
D
$(\mathrm{P}) \rightarrow(2) \quad(\mathrm{Q}) \rightarrow(4) \quad(\mathrm{R}) \rightarrow(3) \quad(\mathrm{S}) \rightarrow(5)$
2
JEE Advanced 2021 Paper 2 Online
+3
-1
Let $$M = \{ (x,y) \in R \times R:{x^2} + {y^2} \le {r^2}\}$$, where r > 0. Consider the geometric progression $${a_n} = {1 \over {{2^{n - 1}}}}$$, n = 1, 2, 3, ...... . Let S0 = 0 and for n $$\ge$$ 1, let Sn denote the sum of the first n terms of this progression. For n $$\ge$$ 1, let Cn denote the circle with center (Sn$$-$$1, 0) and radius an, and Dn denote the circle with center (Sn$$-$$1, Sn$$-$$1) and radius an.
Consider M with $$r = {{1025} \over {513}}$$. Let k be the number of all those circles Cn that are inside M. Let l be the maximum possible number of circles among these k circles such that no two circles intersect. Then
A
k + 2l = 22
B
2k + l = 26
C
2k + 3l = 34
D
3k + 2l = 40
3
JEE Advanced 2021 Paper 2 Online
+3
-1
Let $$M = \{ (x,y) \in R \times R:{x^2} + {y^2} \le {r^2}\}$$, where r > 0. Consider the geometric progression $${a_n} = {1 \over {{2^{n - 1}}}}$$, n = 1, 2, 3, ...... . Let S0 = 0 and for n $$\ge$$ 1, let Sn denote the sum of the first n terms of this progression. For n $$\ge$$ 1, let Cn denote the circle with center (Sn$$-$$1, 0) and radius an, and Dn denote the circle with center (Sn$$-$$1, Sn$$-$$1) and radius an.
Consider M with $$r = {{({2^{199}} - 1)\sqrt 2 } \over {{2^{198}}}}$$. The number of all those circles Dn that are inside M is
A
198
B
199
C
200
D
201
4
JEE Advanced 2021 Paper 1 Online
+3
-1
Consider a triangle $$\Delta$$ whose two sides lie on the x-axis and the line x + y + 1 = 0. If the orthocenter of $$\Delta$$ is (1, 1), then the equation of the circle passing through the vertices of the triangle $$\Delta$$ is
A
x2 + y2 $$-$$ 3x + y = 0
B
x2 + y2 + x + 3y = 0
C
x2 + y2 + 2y $$-$$ 1 = 0
D
x2 + y2 + x + y = 0
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