1
JEE Advanced 2013 Paper 2 Offline
+4
-1
Let $$PQ$$ be a focal chord of the parabola $${y^2} = 4ax$$. The tangents to the parabola at $$P$$ and $$Q$$ meet at a point lying on the line $$y=2x+a$$, $$a>0$$.

If chord $$PQ$$ subtends an angle $$\theta$$ at the vertex of $${y^2} = 4ax$$, then tan $$\theta =$$

A
$${2 \over 3}\sqrt 7$$
B
$${-2 \over 3}\sqrt 7$$
C
$${2 \over 3}\sqrt 5$$
D
$${-2 \over 3}\sqrt 5$$
2
JEE Advanced 2013 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-1
In a triangle $$PQR$$, $$P$$ is the largest angle and $$\cos P = {1 \over 3}$$. Further the incircle of the triangle touches the sides $$PQ$$, $$QR$$ and $$RP$$ at $$N,L$$ and $$M$$ respectively, such that the lengths of $$PN, QL$$ and $$RM$$ are consecutive even integers. Then possible length(s) of the side(s) of the triangle is (are)
A
$$16$$
B
$$18$$
C
$$24$$
D
$$22$$
3
JEE Advanced 2013 Paper 2 Offline
+4
-1
Match List $$I$$ with List $$II$$ and select the correct answer using the code given below the lists:

List $$I$$
$$P.$$$$\,\,\,\,\,$$ $${\left( {{1 \over {{y^2}}}{{\left( {{{\cos \left( {{{\tan }^{ - 1}}y} \right) + y\sin \left( {{{\tan }^{ - 1}}y} \right)} \over {\cot \left( {{{\sin }^{ - 1}}y} \right) + \tan \left( {{{\sin }^{ - 1}}y} \right)}}} \right)}^2} + {y^4}} \right)^{1/2}}$$ takes value

$$Q.$$ $$\,\,\,\,$$ If $$\cos x + \cos y + \cos z = 0 = \sin x + \sin y + \sin z$$ then
possible value of $$\cos {{x - y} \over 2}$$ is

$$R.$$ $$\,\,\,\,\,$$ If $$\cos \left( {{\pi \over 4} - x} \right)\cos 2x + \sin x\sin 2\sec x = \cos x\sin 2x\sec x +$$
$$\cos \left( {{\pi \over 4} + x} \right)\cos 2x$$ then possible value of $$\sec x$$ is

$$S.$$ $$\,\,\,\,\,$$ If $$\cot \left( {{{\sin }^{ - 1}}\sqrt {1 - {x^2}} } \right) = \sin \left( {{{\tan }^{ - 1}}\left( {x\sqrt 6 } \right)} \right),\,\,x \ne 0,$$
Then possible value of $$x$$ is

List $$II$$
$$1.$$ $$\,\,\,\,\,$$ $${1 \over 2}\sqrt {{5 \over 3}}$$

$$2.$$ $$\,\,\,\,\,$$ $$\sqrt 2$$

$$3.$$ $$\,\,\,\,\,$$ $${1 \over 2}$$

$$1.$$ $$\,\,\,\,$$ $$1$$

A
$$P = 4,Q = 3,R = 1,S = 2$$
B
$$P = 4,Q = 3,R = 2,S = 1$$
C
$$P = 3,Q = 4,R = 2,S = 1$$
D
$$P = 3,Q = 4,R = 1,S = 2$$
4
JEE Advanced 2013 Paper 2 Offline
+4
-1
Let $$f:\left[ {0,1} \right] \to R$$ (the set of all real numbers) be a function. Suppose the function $$f$$ is twice differentiable,
$$f(0) = f(1)=0$$ and satisfies $$f''\left( x \right) - 2f'\left( x \right) + f\left( x \right) \ge .{e^x},x \in \left[ {0,1} \right]$$.

Which of the following is true for $$0 < x < 1?$$

A
$$0 < f\left( x \right) < \infty$$
B
$$- {1 \over 2} < f\left( x \right) < {1 \over 2}$$
C
$$- {1 \over 4} < f\left( x \right) < 1$$
D
$$- \infty < f\left( x \right) < 0$$
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