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Equation of three lines $r=λi^,r=μ(i^+j^ ),r=γ(i^+j^ +k^)$ and a plane $x+y+z=1$ are given then area of triangle formed by point of intersectioin of line and plane is $Δ$then $(6Δ)_{2}$equals

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let L be a straight line passing through the origin. Suppose that all the points on L are at a constant distance from the two planes $P_{1}:x+2y−z+1=0$ and $P_{2}:2x−y+z−1=0$, Let M be the locus of the feet of the perpendiculars drawn from the points on L to the plane $P_{1}$. Which of the following points lie(s) on M?

The function $y=f(x)$ is the solution of the differential equation $dxdy +x_{2}−1xy =1−x_{2} x_{4}+2x $ in $(−1,1)$ satisfying $f(0)=0.$ Then $∫_{23}f(x)dx$ is

Four person independently solve a certain problem correctly with probabilities $21 ,43 ,41 ,81 ˙$Then the probability that he problem is solve correctly by at least one of them is$256235 $b. $25621 $c. $2563 $d. $256253 $

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,

Consider the hyperbola $H:x_{2}−y_{2}=1$ and a circle S with centre $N(x_{2},0)$ Suppose that H and S touch each other at a point $(P(x_{1},y_{1})$ with $x_{1}>1andy_{1}>0$ The common tangent to H and S at P intersects the x-axis at point M. If (l,m) is the centroid of the triangle $ΔPMN$ then the correct expression is (A) $dx_{1}dl =1−3x_{1}1 $ for $x_{1}>1$ (B) $dx_{1}dm =3(x _{1}−1)x_{!} )forx_{1}>1$ (C) $dx_{1}dl =1+3x_{1}1 forx_{1}>1$ (D) $dy_{1}dm =31 fory_{1}>0$

If $f:RR$ is a twice differentiable function such that $f(x)>0forallxR,and$ $f(21 )=21 ,f(1)=1$ then: $f_{prime}(1)>1$ (b) $f_{prime}(1)≤0$(c)$21 <f_{prime}(1)<1$ (d)="" $0<f_{prime}(1)≤21 $

Let $f:[21 ,1]→R$ (the set of all real numbers) be a positive, non-constant, and differentiable function such that $f_{prime}(x)<2f(x)andf(21 )=1$ . Then the value of $∫_{21}f(x)dx$ lies in the interval (a)$(2e−1,2e)$ (b) $(3−1,2e−1)$(c)$(2e−1 ,e−1)$ (d) $(0,2e−1 )$

The minimum number of times a fair coin needs to be tossed, so that the probability of getting at least two heads is at least 0.96 is :