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Area bounded the point (x,y) in certesian plane satesfying $xy≤8$ and $1≤y≤x_{2}$ wll be

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In a triangle PQR, P is the largest angle and $cosP=31 $. Further the incircle of the triangle touches the sides PQ, QR and RP at N, L and M respectively, such that the lengths of PN, QL and RM are consecutive even integers. Then possible length(s) of the side(s) of the triangle is (are)

If $w=α+iβ,$where $β=0$and $z=1$, satisfies the condition that $(1−zw−wz )$is a purely real, then the set of values of $z$is $∣z∣=1,z=2$ (b) $∣z∣=1andz=1$$z=z$ (d) None of these

Let w = ($3 +2ι )$ and $P={w_{n}:n=1,2,3,…..},$ Further $H_{1}={z∈C:Re(z)>21 }andH_{2}={z∈c:Re(z)<−21 }$ Where C is set of all complex numbers. If $z_{1}∈P∩H_{1},z_{2}∈P∩H_{2}$ and O represent the origin, then $∠Z_{1}OZ_{2}$ =

Let $n≥2$be integer. Take $n$distinct points on a circle and join each pair of points by a line segment. Color the line segment joining every pair of adjacent points by blue and the rest by red. If the number of red and blue line segments are equal, then the value of $n$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), $(a_{2}m_{2}+1 −a_{2}m ,a_{2}m_{2}+1 −1 )$If a tangent to a suitable conic (Column 1) is found to be $y=x+8$and its point of contact is (8,16), then which of the followingoptions is the only CORRECT combination?(III) (ii) (Q) (b) (II) (iv) (R)(I) (ii) (Q) (d) (III) (i) (P)

If $g(x)=∫_{sinx}sin_{−1}(t)dt,then:$ (a)$g_{prime}(2π )=−2π$ (b) $g_{prime}(−2π )=−2π$ (c)$g_{prime}(−2π )=2π$ (d) $g_{prime}(2π )=2π$

Let $M$be a $2×2$symmetric matrix with integer entries. Then $M$is invertible ifThe first column of $M$is the transpose of the second row of $M$The second row of $M$is the transpose of the first column of $M$$M$is a diagonal matrix with non-zero entries in the main diagonalThe product of entries in the main diagonal of $M$is not the square of an integer

A rectangular sheet of fixed perimeter with sides having their lengths in the ratio $8:15$is converted into anopen rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed squares is 100, the resulting box has maximum volume. Then the length of the sides of the rectangular sheet are24 (b) 32 (c) 45 (d) 60