class 12

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JEE Advanced

STATEMENT - 1 A cloth covers a table. Some dishes are kept on it. The cloth can be pulled out without dislodging the dishes from the table. because STATEMENT - 2 For every action there is an equal and opposite reaction.

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Let $f:(0,∞)R$ be given by $f(x)=∫_{x1}te_{−(t+t1)}dt ,$ then (a)$f(x)$ is monotonically increasing on $[1,∞)$(b)$f(x)$ is monotonically decreasing on $(0,1)$(c)$f(2_{x})$ is an odd function of $x$ on $R$

Â·If the normals of the parabola $y_{2}=4x$ drawn at the end points of its latus rectum are tangents to the circle $(x−3)_{2}(y+2)_{2}=r_{2}$ , then the value of $r_{2}$ is

For $a>b>c>0$, if the distance between $(1,1)$ and the point of intersection of the line $ax+by−c=0$ is less than $22 $ then,

Column 1,2 and 3 contains conics, equations of tangents to the conics and points of contact, respectively.Column I, Column 2, Column 3I, $x_{2}+y_{2}=a$, (i), $my=m_{2}x+a$, (P), $(m_{2}a ,m2a )$II, $x_{2}+a_{2}y_{2}=a$, (ii), $y=mx+am_{2}+1 $, (Q), $(m_{2}+1 −ma ,m_{2}+1 a )$III, $y_{2}=4ax$, (iii), $y=mx+a_{2}m_{2}−1 $, (R), $(a_{2}m_{2}+1 −a_{2}m ,a_{2}m_{2}+1 1 )$IV, $x_{2}−a_{2}y_{2}=a_{2}$, (iv), $y=mx+a_{2}m_{2}+1 $, (S), $(a_{2}m_{2}+1 −a_{2}m ,a_{2}m_{2}+1 −1 )$For $a=2 ,if$a tangent is drawn to a suitable conic (Column 1) at the point of contact $(−1,1),$then which of the following options is the only CORRECT combination for obtaining its equation?(I) (ii) (Q) (b) (III) (i) (P)(II) (ii) (Q) (d) $(I)(i)(P)$

Column 1,2 and 3 contains conics, equations of tangents to the conics and points of contact, respectively.Column I, Column 2, Column 3I, $x_{2}+y_{2}=a$, (i), $my=m_{2}x+a$, (P), $(m_{2}a ,m2a )$II, $x_{2}+a_{2}y_{2}=a$, (ii), $y=mx+am_{2}+1 $, (Q), $(m_{2}+1 −ma ,m_{2}+1 a )$III, $y_{2}=4ax$, (iii), $y=mx+a_{2}m_{2}−1 $, (R), $(a_{2}m_{2}+1 −a_{2}m ,a_{2}m_{2}+1 1 )$IV, $x_{2}−a_{2}y_{2}=a_{2}$, (iv), $y=mx+a_{2}m_{2}+1 $, (S), $(a_{2}m_{2}+1 −a_{2}m ,a_{2}m_{2}+1 −1 )$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 followingoptions is the only CORRECT combination?(III) (ii) (Q) (b) (II) (iv) (R)(I) (ii) (Q) (d) (III) (i) (P)

If $f(x)∣cos(2x)cos(2x)sin(2x)−cosxcosx−sinxsinxsinxcosx∣,then:$$f_{prime}(x)=0$at exactly three point in $(−π,π)$$f_{prime}(x)=0$at more than three point in $(−π,π)$$f(x)$attains its maximum at $x=0$$f(x)$attains its minimum at $x=0$

Let $f:(0,π)→R$be a twice differentiable function such that $(lim)_{t→x}t−xf(x)sint−f(x)sinx =sin_{2}x$for all $x∈(0,π)$. If $f(6π )=−12π $, then which of the following statement(s) is (are) TRUE?$f(4π )=42 π $(b) $f(x)<6x_{4} −x_{2}$for all $x∈(0,π)$(c) There exists $α∈(0,π)$such that $f_{prime}(α)=0$(d) $f(2π )+f(2π )=0$

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 $ΔOPQ$ is $32$ , then which of the following is (are) the coordinates of $P?$