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if a matrix M is given by $⎣⎡ 013 121 231 ⎦⎤ $ and if $M⎣⎡ αβγ ⎦⎤ =⎣⎡ 123 ⎦⎤ $ then

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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) $x_{2}+y_{2}=x_{2}y_{2}$

The number of 5 digit numbers which are divisible by 4, with digits from the set ${1,2,3,4,5}$and the repetition of digits is allowed, is ________.

Perpendiculars are drawn from points on the line $2x+2 =−1y+1 =3z $ to the plane $x+y+z=3$ The feet of perpendiculars lie on the line (a) $5x =8y−1 =−13z−2 $ (b) $2x =3y−1 =−5z−2 $ (c) $4x =3y−1 =−7z−2 $ (d) $2x =−7y−1 =5z−2 $

Let $O$be the origin and let PQR be an arbitrary triangle. The point S is such that$OPO˙Q+ORO˙S=ORO˙P+OQO˙S=OQ$.$OR+OPO˙S$Then the triangle PQ has S as its:circumcentre (b) orthocentre (c) incentre (d) centroid

Let n be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that all the girls stand consecutively in the queue. Let m be the number in which 5 boys and 5 girls stand in such a way that exactly four girls stand consecutively in the queue. Then the value of $nm $ is ____

There are five students $S_{1},S_{2},S_{3},S_{4}$and $S_{5}$in a music class and for them there are five seats $R_{1},R_{2},R_{3},R_{4}$and $R_{5}$arranged in a row, where initially the seat $R_{i}$is allotted to the student $S_{i},i=1,2,3,4,5$. But, on the examination day, the five students are randomly allotted five seats.The probability that, on the examination day, the student $S_{1}$gets the previously allotted seat $R_{1}$, and NONE of the remaining students gets the seat previously allotted to him/her is$403 $(b) $81 $(c) $407 $(d) $51 $

If $Ik=1∑98 ∫_{k}x(x+1)k+1 dx,then:$$I<5049 $ (b) $I>(g)_{e}99$$I>5049 $ (d) $I<(g)_{e}99$

A cylindrica container is to be made from certain solid material with the following constraints: It has a fixed inner volume of $Vm_{3}$, has a 2 mm thick solid wall and is open at the top. The bottom of the container is a solid circular disc of thickness 2mm and is of radius equal to the outer radius of the container. If the volume the material used to make the container is minimum when the inner radius of the container is $10mm$. then the value of $250πV $ is