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JEE Advanced

When disc B is brought in contact with disc A, they acquire a common angular velocity in time t. The advantate frictional torque on one disc by the other during this period is

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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

Let $b_{1}>1$ for $i=1,2,……,101.$ Suppose $g_{e}b_{1},g_{e}b_{10}$ are in Arithmetic progression $(A.P.)$ with the common difference $g_{e}2.$ suppose $a_{1},a_{2}……….a_{101}$ are in A.P. such $a_{1}=b_{1}anda_{51}=b_{51}.$ If $t=b_{1}+b_{2}+……+b_{51}ands=a_{1}+a_{2}+……+a_{51}$ then

Let $s,t,r$be non-zero complex numbers and $L$be the set of solutions $z=x+iy(x,y∈R,i=−1 )$of the equation $sz+tz+r=0$, where $z=x−iy$. Then, which of the following statement(s) is (are) TRUE?If $L$has exactly one element, then $∣s∣=∣t∣$(b) If $∣s∣=∣t∣$, then $L$has infinitely many elements(c) The number of elements in \displaystyle{\Ln{{n}}}{\left\lbrace{z}\right|}{z}-{1}+{i}{\mid}={5}{\rbrace}is at most 2(d) If $L$has more than one element, then $L$has infinitely many elements

For any positive integer $n$, define $f_{n}:(0,∞)→R$as $f_{n}(x)=j=1∑n tan_{−1}(1+(x+j)(x+j−1)1 )$for all $x∈(0,∞)$.Here, the inverse trigonometric function $tan_{−1}x$assumes values in $(−2π ,2π )˙$Then, which of the following statement(s) is (are) TRUE?$j=1∑5 tan_{2}(f_{j}(0))=55$(b) $j=1∑10 (1+fj_{′}(0))sec_{2}(f_{j}(0))=10$(c) For any fixed positive integer $n$, $(lim)_{x→∞}tan(f_{n}(x))=n1 $(d) For any fixed positive integer $n$, $(lim)_{x→∞}sec_{2}(f_{n}(x))=1$

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 $∣(p ×q )×(r×(t)∣is$

If a chord, which is not a tangent of the parabola $y_{2}=16x$has the equation $2x+y=p,$and midpoint $(h,k),$then which of the following is(are) possible values (s) of $p,handk?$$p=−1,h=1,k=−3$ $p=2,h=3,k=−4$ $p=−2,h=2,k=−4$ $p=5,h=4,k=−3$

Let $S$be the set of all non-zero real numbers such that the quadratic equation $αx_{2}−x+α=0$has two distinct real roots $x_{1}andx_{2}$satisfying the inequality $∣x_{1}−x_{2}∣<1.$Which of the following intervals is (are) a subset (s) of $S?$$(21 ,5 1 )$b. $(5 1 ,0)$c. $(0,5 1 )$d. $(5 1 ,21 )$

The function $f(x)=2∣x∣+∣x+2∣=∣∣x∣2∣−2∣x∣∣$has a local minimum or a local maximum at $x=$$−2$ (b) $−32 $ (c) 2 (d) $32 $