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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

Box 1 contains three cards bearing numbers 1, 2, 3; box 2 contains five cards bearing numbers 1, 2, 3,4, 5; and box 3 contains seven cards bearing numbers 1, 2, 3, 4, 5, 6, 7. A card is drawn from each of the boxes. Let $x_{i}$ be the number on the card drawn from the ith box, i = 1, 2, 3.The probability that $x_{1}+x_{2}+x_{3}$ is odd isThe probability that $x_{1},x_{2},x_{3}$ are in an aritmetic progression is

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)

Suppose that the foci of the ellipse $9x_{2} +5y_{2} =1$are $(f_{1},0)and(f_{2},0)$where $f_{1}>0andf_{2}<0.$Let $P_{1}andP_{2}$be two parabolas with a common vertex at (0, 0) and with foci at $(f_{1}.0)and$(2f_2 , 0), respectively. Let$T_{1}$be a tangent to $P_{1}$which passes through $(2f_{2},0)$and $T_{2}$be a tangents to $P_{2}$which passes through $(f_{1},0)$. If $m_{1}$is the slope of $T_{1}$and $m_{2}$is the slope of $T_{2},$then the value of $(m121 +m22)$is

Let $y(x)$ be a solution of the differential equation $(1+e_{x})y_{prime}+ye_{x}=1.$ If $y(0)=2$ , then which of the following statements is (are) true? (a)$y(−4)=0$ (b)$y(−2)=0$ (c)$y(x)$ has a critical point in the interval $(−1,0)$ (d)$y(x)$ has no critical point in the interval$(−1,0)$

If $y=y(x)$ satisfies the differential equation $8x (9+x )dy=(4+9+x )_{−1}dx,x>0$ and $y(0)=7, $ then $y(256)=$ (A) 16 (B) 80 (C) 3 (D) 9

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,

Â·If the normals of the parabola $y_{2}=4x$ drawn at the end points of its latus rectum are tangents to the circle $(x−3)_{2}(y+2)_{2}=r_{2}$ , then the value of $r_{2}$ is