1
JEE Advanced 2017 Paper 1 Offline
+3
-1
By appropriately matching the information given in the three columns of the following table.

Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.

Column - 1 Column - 2 Column - 3
(i) $${x^2} + {y^2} = a$$ $$my = {m^2}x + a$$ $$\left( {{a \over {{m^2}}},\,{{2a} \over m}} \right)$$
(ii) $${x^2}{a^2}{y^2} = {a^2}]$$ $$y = mx + a\sqrt {{m^2} + 1}$$ $$\left( {{{ - ma} \over {\sqrt {{m^2} + 1} }},\,{a \over {\sqrt {{m^2} + 1} }}} \right)$$
(iii) $${y^2} = 4ax$$ $$y = mx + \sqrt {{a^2}{m^2} - 1}$$ $$\left( {{{ - {a^2}m} \over {\sqrt {{a^2}{m^2} + 1} }},\,{1 \over {\sqrt {{a^2}{m^2} + 1} }}} \right)$$
(iv) $${x^2} - {a^2}{y^2} = {a^2}$$ $$y = mx + \sqrt {{a^2}{m^2} + 1}$$ $$\left( {{{ - {a^2}m} \over {\sqrt {{a^2}{m^2} - 1} }},\,{{ - 1} \over {\sqrt {{a^2}{m^2} - 1} }}} \right)$$
The tangent to a suitable conic (Column 1) at $$\left( {\sqrt 3 ,\,{1 \over 2}} \right)$$ is found to be $$\sqrt 3 x + 2y = 4$$, then which of the following options is the only CORRECT combination?
A
(IV) (iv) (S)
B
(II) (iv) (R)
C
(IV) (iii) (S)
D
(II) (ii) (R)
2
JEE Advanced 2017 Paper 1 Offline
+3
-1
By appropriately matching the information given in the three columns of the following table.

Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.

Column - 1 Column - 2 Column - 3
(i) $${x^2} + {y^2} = a$$ $$my = {m^2}x + a$$ $$\left( {{a \over {{m^2}}},\,{{2a} \over m}} \right)$$
(ii) $${x^2}{a^2}{y^2} = {a^2}]$$ $$y = mx + a\sqrt {{m^2} + 1}$$ $$\left( {{{ - ma} \over {\sqrt {{m^2} + 1} }},\,{a \over {\sqrt {{m^2} + 1} }}} \right)$$
(iii) $${y^2} = 4ax$$ $$y = mx + \sqrt {{a^2}{m^2} - 1}$$ $$\left( {{{ - {a^2}m} \over {\sqrt {{a^2}{m^2} + 1} }},\,{1 \over {\sqrt {{a^2}{m^2} + 1} }}} \right)$$
(iv) $${x^2} - {a^2}{y^2} = {a^2}$$ $$y = mx + \sqrt {{a^2}{m^2} + 1}$$ $$\left( {{{ - {a^2}m} \over {\sqrt {{a^2}{m^2} - 1} }},\,{{ - 1} \over {\sqrt {{a^2}{m^2} - 1} }}} \right)$$
If a tangent to a suitable conic (Column 1) is found to be y = x + 8 and its point of contact is (8, 16), then which of the following options is the only CORRECT combination?
A
(III) (i) (P)
B
(I) (ii) (Q)
C
(II) (iv) (R)
D
(III) (ii) (Q)
3
JEE Advanced 2017 Paper 1 Offline
+3
-1
By approximately matching the information given in the three columns of the following table.

Let f(x) = x + loge x $$-$$ x loge x, x$$\in$$(0, $$\infty$$)

Column 1 contains information about zeroes of f(x), f'(x) and f"(x).

Column 2 contains information about the limiting behaviour of f(x), f'(x) and f"(x) at infinity.

Column 3 contains information about increasing/decreasing nature of f(x) and f'(x).

Column - 1 Column - 2 Column - 3
(i) f(x) = 0 for some $$x \in (1,{e^2})$$ (i) $$\mathop {\lim }\limits_{x \to \infty } \,f(x) = 0$$ f is increasing in (0, 1)
(ii) f'(x) = 0 for some $$x \in (1,e)$$ $$\mathop {\lim }\limits_{x \to \infty } \,f(x) = - \infty$$ f is decreasing in (e, $${e^2}$$)
(iii) f'(x) = 0 for some $$x \in (0,1)$$ $$\mathop {\lim }\limits_{x \to \infty } \,f'(x) = - \infty$$ f' is increasing in (0, 1)
(iv) f'(x) = 0 for some $$x \in (1,e)$$ $$\mathop {\lim }\limits_{x \to \infty } \,f'(x) = 0$$ f' is decreasing in (e, $${e^2}$$)
Which of the following options is the only INCORRECT combination?
A
(I) (iii) (P)
B
(II) (iv) (Q)
C
(II) (ii) (P)
D
(III) (i) (R)
4
JEE Advanced 2017 Paper 1 Offline
+3
-1
By approximately matching the information given in the three columns of the following table.

Let f(x) = x + loge x $$-$$ x loge x, x$$\in$$(0, $$\infty$$)

Column 1 contains information about zeroes of f(x), f'(x) and f"(x).

Column 2 contains information about the limiting behaviour of f(x), f'(x) and f"(x) at infinity.

Column 3 contains information about increasing/decreasing nature of f(x) and f'(x).

Column - 1 Column - 2 Column - 3
(i) f(x) = 0 for some $$x \in (1,{e^2})$$ (i) $$\mathop {\lim }\limits_{x \to \infty } \,f(x) = 0$$ f is increasing in (0, 1)
(ii) f'(x) = 0 for some $$x \in (1,e)$$ $$\mathop {\lim }\limits_{x \to \infty } \,f(x) = - \infty$$ f is decreasing in (e, $${e^2}$$)
(iii) f'(x) = 0 for some $$x \in (0,1)$$ $$\mathop {\lim }\limits_{x \to \infty } \,f'(x) = - \infty$$ f' is increasing in (0, 1)
(iv) f'(x) = 0 for some $$x \in (1,e)$$ $$\mathop {\lim }\limits_{x \to \infty } \,f'(x) = 0$$ f' is decreasing in (e, $${e^2}$$)
Which of the following options is the only CORRECT combination?
A
(I) (ii) (R)
B
(III) (iv) (P)
C
(II) (iii) (S)
D
(IV) (i) (S)
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