class 12

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JEE Advanced

Which statements is/are correct

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Let RS be the diameter of the circle $x_{2}+y_{2}=1,$ where S is the point $(1,0)$ Let P be a variable apoint (other than $RandS$) on the circle and tangents to the circle at $SandP$ meet at the point Q.The normal to the circle at P intersects a line drawn through Q parallel to RS at point E. then the locus of E passes through the point(s)- (A) $(31 ,3 1 )$ (B) $(41 ,21 )$ (C) $(31 ,−3 1 )$ (D) $(41 ,−21 )$

The number of real solutions of the equation $sin_{−1}(i=1∑∞ x_{i+1}−xi=1∑∞ (2x )_{i})=2π −cos_{−1}(i=1∑∞ (−2x )_{i}−i=1∑∞ (−x)_{i})$lying in the interval $(−21 ,21 )$is ____. (Here, the inverse trigonometric function $=sin_{−1}x$and $cos_{−1}x$assume values in $[2π ,2π ]$and $[0,π]$, respectively.)

Let $n_{1},andn_{2}$, be the number of red and black balls, respectively, in box I. Let $n_{3}andn_{4}$,be the number one red and b of red and black balls, respectively, in box II. One of the two boxes, box I and box II, was selected at random and a ball was drawn randomly out of this box. The ball was found to be red. If the probablity that this red ball was drawn from box II is $31 $ then the correct option(s) with the possible values of $n_{1},n_{2},n_{3},andn_{4}$, is(are)

Let $a,b,andc$ be three non coplanar unit vectors such that the angle between every pair of them is $3π $. If $a×b+b×x=pa+qb+rc$ where p,q,r are scalars then the value of $q_{2}p_{2}+2q_{2}+r_{2} $ is

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) $(I)(i)(P)$

A curve passes through the point $(1,6π )$ . Let the slope of the curve at each point $(x,y)$ be $xy +sec(xy ),x>0.$ Then the equation of the curve is

Box 1 contains three cards bearing numbers 1, 2, 3; box 2 contains five cards bearing numbers 1, 2, 3,4, 5; and box 3 contains seven cards bearing numbers 1, 2, 3, 4, 5, 6, 7. A card is drawn from each of the boxes. Let $x_{i}$ be the number on the card drawn from the ith box, i = 1, 2, 3.The probability that $x_{1}+x_{2}+x_{3}$ is odd isThe probability that $x_{1},x_{2},x_{3}$ are in an aritmetic progression is

For a point $P$in the plane, let $d_{1}(P)andd_{2}(P)$be the distances of the point $P$from the lines $x−y=0andx+y=0$respectively. The area of the region $R$consisting of all points $P$lying in the first quadrant of the plane and satisfying $2≤d_{1}(P)+d_{2}(P)≤4,$is