class 12

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JEE Advanced

STATEMENT - 1 If there is no external torque on a body about its centre of mass, then the velocity of the center of mass remains constant. Because STATEMENT - 2 The linear momentum of an isolated system remains constant.

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Let $S={xϵ(−π,π):x=0,+2π }$The sum of all distinct solutions of the equation $3 secx+cosecx+2(tanx−cotx)=0$ in the set S is equal to

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 $C_{2}andC_{3}atR_{2}andR_{3},$ respectively. Suppose $C_{2}andC_{3}$ have equal radii $23 $ and centres at $Q_{2}$ and $Q_{3}$ respectively. If $Q_{2}$ and $Q_{3}$ lie on the y-axis, then (a)$Q2Q3=12$(b)$R2R3=46 $(c)area of triangle $OR2R3$ is $62 $(d)area of triangle $PQ2Q3is=42 $

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

Let $P=⎣⎡ 1416 014 001 ⎦⎤ $and $I$ be the identity matrix of order $3$. If $Q=[q_{()}ij]$ is a matrix, such that $P_{50}−Q=I$, then $q_{21}q_{31}+q_{32} $ equals

Three boys and two girls stand in a queue. The probability, that the number of boys ahead is at least one more than the number of girls ahead of her, is (A) $21 $ (B) $31 $ (C) $32 $ (D) $43 $

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), $(a_{2}m_{2}+1 −a_{2}m ,a_{2}m_{2}+1 −1 )$The tangent to a suitable conic (Column 1) at $(3 ,21 )$is found to be $3 x+2y=4,$then which of the following options is the only CORRECT combination?(IV) (iii) (S) (b) (II) (iii) (R)(II) (iv) (R) (d) (IV) (iv) (S)

Four person independently solve a certain problem correctly with probabilities $21 ,43 ,41 ,81 ˙$Then the probability that he problem is solve correctly by at least one of them is$256235 $b. $25621 $c. $2563 $d. $256253 $

Let $f:R→R$be a differentiable function with $f(0)=0$. If $y=f(x)$satisfies the differential equation $dxdy =(2+5x)(5x−2)1 $, then the value of $(lim)_{x→∞}f(x)$is ______