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For a point $P$in the plane, let $d_{1}(P)andd_{2}(P)$be the distances of the point $P$from the lines $x−y=0andx+y=0$respectively. The area of the region $R$consisting of all points $P$lying in the first quadrant of the plane and satisfying $2≤d_{1}(P)+d_{2}(P)≤4,$is

Let $PR=3i^+j^ −2k^andSQ=i^−3j^ −4k^$determine diagonals of a parallelogram $PQRS,andPT=i^+2j^ +3k^$be another vector. Then the volume of the parallelepiped determine by the vectors $PT$, $PQ$and $PS$is$5$b. $20$c. $10$d. $30$

Q. The value of is equal $k=1∑13 (sin(4π +(k−1)6π )sin(4π +k6π )1 $ is equal

In R', consider the planes $P_{1},y=0$ and $P_{2}:x+z=1$. Let $P_{3}$, be a plane, different from $P_{1}$, and $P_{2}$, which passes through the intersection of $P_{1}$, and $P_{2}$. If the distance of the point $(0,1,0)$ from $P_{3}$, is $1$ and the distance of a point $(α,β,γ)$ from $P_{3}$ is $2$, then which of the following relation is (are) true ?

Let $f:R→(0,∞)$ and $g:R→R$ be twice differentiable functions such that f" and g" are continuous functions on R. suppose $f_{prime}(2)=g(2)=0,f(2)=0$ and $g_{′}(2)=0$, If $x→2lim f_{′}(x)g_{′}(x)f(x)g(x) =1$ then

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

A circle C of radius 1 is inscribed in an equilateral triangle PQR. The points of contact of C with the sides PQ, QR, RP are D, E, F, respectively. The line PQ is given by the equation $3 x+y−6=0$ and the point D is (3 sqrt3/2, 3/2). Further, it is given that the origin and the centre of C are on the same side of the line PQ. (1)The equation of circle C is (2)Points E and F are given by (3)Equation of the sides QR, RP are

Let $f(x)=7tan_{8}x+7tan_{6}x−3tan_{4}x−3tan_{2}x$for all $x∈(−2π ,2π )$ . Then the correct expression (s) is (are) (a) $∫_{0}xf(x)dx=121 $ (b)$∫_{0}f(x)dx=0$(c)$∫_{0}xf(x)=61 $ (d) $∫_{0}f(x)dx=121 $