1
JEE Advanced 2016 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-2
The circle $${C_1}:{x^2} + {y^2} = 3,$$ with centre at $$O$$, intersects the parabola $${x^2} = 2y$$ at the point $$P$$ in the first quadrant, Let the tangent to the circle $${C_1}$$, at $$P$$ touches other two circles $${C_2}$$ and $${C_3}$$ at $${R_2}$$ and $${R_3}$$, respectively. Suppose $${C_2}$$ and $${C_3}$$ have equal radil $${2\sqrt 3 }$$ and centres $${Q_2}$$ and $${Q_3}$$, respectively. If $${Q_2}$$ and $${Q_3}$$ lie on the $$y$$-axis, then
A
$${Q_2}{Q_3} = 12$$
B
$${R_2}{R_3} = 4\sqrt 6$$
C
area of the triangle $$O{R_2}{R_3}$$ is $$6\sqrt 2$$
D
area of the triangle $$P{Q_2}{Q_3}$$ is $$4\sqrt 2$$
2
JEE Advanced 2015 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-1
Let $${E_1}$$ and $${E_2}$$ be two ellipses whose centres are at the origin. The major axes of $${E_1}$$ and $${E_2}$$ lie along the $$x$$-axis and the $$y$$-axis, respectively. Let $$S$$ be the circle $${x^2} + {\left( {y - 1} \right)^2} = 2$$. The straight line $$x+y=3$$ touches the curves $$S$$, $${E_1}$$ and $${E_2}$$ at $$P, Q$$ and $$R$$ respectively. Suppose that $$PQ = PR = {{2\sqrt 2 } \over 3}$$. If $${e_1}$$ and $${e_2}$$ are the eccentricities of $${E_1}$$ and $${E_2}$$, respectively, then the correct expression(s) is (are)
A
$$\mathop e\nolimits_1^2 + \mathop e\nolimits_2^2 = {{43} \over {40}}$$
B
$${e_1}{e_2} = {{\sqrt 7 } \over {2\sqrt {10} }}$$
C
$$\left| {\mathop e\nolimits_1^2 + \mathop e\nolimits_2^2 } \right| = {5 \over 8}$$
D
$${e_1}{e_2} = {{\sqrt 3 } \over 4}$$
3
JEE Advanced 2015 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-1
Consider the hyperbola $$H:{x^2} - {y^2} = 1$$ and a circle $$S$$ with center $$N\left( {{x_2},0} \right)$$. Suppose that $$H$$ and $$S$$ touch each other at a point $$P\left( {{x_1},{y_1}} \right)$$ with $${{x_1} > 1}$$ and $${{y_1} > 0}$$. The common tangent to $$H$$ and $$S$$ at $$P$$ intersects the $$x$$-axis at point $$M$$. If $$(l, m)$$ is the centroid of the triangle $$PMN$$, then the correct expressions(s) is(are)
A
$${{dl} \over {d{x_1}}} = 1 - {1 \over {3x_1^2}}$$ for $${x_1} > 1$$
B
$${{dm} \over {d{x_1}}} = {{{x_1}} \over {3\left( {\sqrt {x_1^2 - 1} } \right)}}$$ for $${x_1} > 1$$
C
$${{dl} \over {d{x_1}}} = 1 + {1 \over {3x_1^2}}$$ for $${x_1} > 1$$
D
$${{dm} \over {d{y_1}}} = {1 \over 3}$$ for $${y_1} > 0$$
4
JEE Advanced 2015 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-1
Let $$P$$ and $$Q$$ be distinct points on the parabola $${y^2} = 2x$$ such that a circle with $$PQ$$ as diameter passes through the vertex $$O$$ of the parabola. If $$P$$ lies in the first quadrant and the area of the triangle $$\Delta OPQ$$ is $${3\sqrt 2 ,}$$ then which of the following is (are) the coordinates of $$P$$?
A
$$\left( {4,2\sqrt 2 } \right)$$
B
$$\left( {9,3\sqrt 2 } \right)$$
C
$$\left( {{1 \over 4},{1 \over {\sqrt 2 }}} \right)$$
D
$$\left( {1,\sqrt 2 } \right)$$
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