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JEE Advanced

A Uniformly charged thin spherical shell of radius R carries uniform surface charge density of $σ$ per unit area. It is made of two hemispherical shells, held together by pressing them with force F. F is proportional to

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Let $a,b,xandy$ be real numbers such that $a−b=1andy=0.$ If the complex number $z=x+iy$ satisfies $Im(z+1az+b )=y$ , then which of the following is (are) possible value9s) of x? (a)$−1−1−y_{2} $ (b) $1+1+y_{2} $(c)$−1+1−y_{2} $ (d) $−1−1+y_{2} $

A pack contains $n$cards numbered from 1 to $n$. Two consecutive numbered cards are removed from the pack and the sum of the numbers on the remaining cards is 1224. If the smaller of het numbers on the removed cards is $k,$then $k−20=$____________.

Suppose that $p ,q andr$ are three non-coplanar vectors in $R_{3}$. Let the components of a vector $s$ along $p ,q andr$ be 4, 3 and 5, respectively. If the components of this vector $s$ along $(−p +q +r),(p −q +r)and(−p −q +r)$ are x, y and z, respectively, then the value of $2x+y +z$ is

Let $f(x)=∣1−x∣1−x(1+∣1−x∣) cos(1−x1 )$ for $x=1.$ Then: (A)$(lim)_{n→1_{−}}f(x)$ does not exist (B)$(lim)_{n→1_{+}}f(x)$ does not exist (C)$(lim)_{n→1_{−}}f(x)=0$ (D)$(lim)_{n→1_{+}}f(x)=0$

Consider the cube in the first octant with sides OP,OQ and OR of length 1, along the x-axis, y-axis and z-axis, respectively, where $O(0,0,0)$ is the origin. Let $S(21 ,21 ,21 )$ be the centre of the cube and T be the vertex of the cube opposite to the origin O such that S lies on the diagonal OT. If $p =SP,q =SQ ,r=SR$ and $t=ST$ then the value of $∣(p ×q )×(r×(t)∣is$

Let $MandN$ be two $3×3$ matrices such that $MN=NM˙$ Further, if $M=N_{2}andM_{2}=N_{4},$ then Determinant of $(M_{2}+MN_{2})$ is 0 There is a $3×3$ non-zeero matrix $U$ such tht $(M_{2}+MN_{2})U$ is the zero matrix Determinant of $(M_{2}+MN_{2})≥1$For a $3×3$ matrix $U,if(M_{2}+MN_{2})U$ equal the zero mattix then $U$ is the zero matrix

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,

Let $S$be the circle in the $xy$-plane defined by the equation $x_{2}+y_{2}=4.$(For Ques. No 15 and 16)Let $P$be a point on the circle $S$with both coordinates being positive. Let the tangent to $S$at $P$intersect the coordinate axes at the points $M$and $N$. Then, the mid-point of the line segment $MN$must lie on the curve$(x+y)_{2}=3xy$(b) $x_{2/3}+y_{2/3}=2_{4/3}$(c) $x_{2}+y_{2}=2xy$(d) $x_{2}+y_{2}=x_{2}y_{2}$